ARENBERG DOCTORAL SCHOOL Faculty of Science...sequence to the white-dwarf stage will be illustrated....

ARENBERG DOCTORAL SCHOOLFaculty of Science

Exploring the evolution of widehot-subdwarf binaries

Joris Vos

Dissertation presented in partialfulfillment of the requirements for the

degree of Doctor in Science

January 2015

Supervisor:Prof. Dr. Hans van WinckelDr. Roy Østensen, co-supervisor

Exploring the evolution of wide hot-subdwarf binaries

Joris VOS

Examination committee:Prof. Dr. Christoffel Waelkens, chairProf. Dr. Hans van Winckel, supervisorDr. Roy Østensen, co-supervisorProf. Dr. Conny AertsProf. Dr. Rony KeppensProf. Dr. Gijs Nelemans(Radboud University, The Netherlands)

Prof. Dr. Ulrich Heber(University of Bamberg, Germany)

Dr. Robert Izzard(Univeristy of Cambridge, UK)

Dissertation presented in partialfulfillment of the requirements forthe degree of Doctorin Science

January 2015

Cover imageThe authors impression of a red giant + main sequence system during Roche-lobe overflow above the Mercatortelescope (Photo of Mercator by Péter I. Pápic). These RG + MS systems undergoing stable RLOF can be theprogenitors of the wide sdB + MS binaries studied in this thesis (see chapter 6).

AcknowledgmentsThe research presented in this PhD thesis was based on funding from the European Research Council under theEuropean Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No 227224(prosperity), as well as from the Research Council of KU Leuven grant agreements GOA/2008/04 andGOA/2013/012.

© 2015 KU Leuven – Faculty of ScienceUitgegeven in eigen beheer, Joris Vos, Celestijnenlaan 200D box 2402, B-3001 Heverlee (Belgium)

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Thanks, Danku, Ďakujem

A thesis is not the work of just one person, thus here I take the opportunity toexpress my thanks to everyone who in one way or another was involved in thecompletion of this manuscript.

Roy thank you for raising my interest in binary stars (and keeping it there) overthe past six years, for the scientific support and guidance, for the occasionalbeer/whisky, the conferences, the trip in Arizona, and much more.

Thanks to Hans for the scientific guidance and the opportunities to go observing.

Thanks to Elizabeth, Jan, Kenneth, Maja, Pablo M., Peter N., Pieter and Ulifor the scientific collaboration. Thanks to Florian, Gert, Hans, Jesus, Saskiaand Wim P. for the support at Mercator. Thanks to everyone who observedthe wide sdB binaries at Mercator or elsewhere. Thanks to Bram V., Rik, Saraand Wim D. to keep the system running (at least most of the time). Thanks toBill and all other developers of MESA. Thanks to the jury.

Furthermore, for reasons including but not limited to: making the IvS such anice place to work, being such good office mates, the occasional beer/whisky inLeuven or elsewhere, going climbing, the serious and not so serious discussions onscience/work/life/future plans, the holidays, the conferences, ... many thanks to:Alejandra, Andrew, Anne, Bart V., Bram B., Christoffel, Cole, Conny, Ehsan,Hamed, Jeroen, Jonas, Jonathan, Katrien, Katrijn, Kenneth, Konstanze, Laura,Marie, Michel, Pablo A., Paul, Peter N., Péter P., Pieter D., Piotr, Rajeev,Robin, Santiago, Sigrid, Steven, Tähti, Timothy, Valentina, Ward and Wesley

Maja, thanks for giving me the opportunity to continue my research in Chile.

Ma, Pa, Dieter, danku om me de mogelijkheid te geven om te studeren, voor desteun, om me mijn eigen weg te laten zoeken ook als die mij nogal vaak naarhet buitenland bracht, en nog zoveel meer. Aan mijn grootouders, danku voorde steun, de taart, de gesprekken over vroeger en nu, de kaarskes in de examenperiodes, ...

Ďakujem Martininým rodicom Ľubici, Petrovi a jej bratovi Petrovi za miléprivítanie v ich dome v Bratislave, za ich snahu so mnou komunikovať napriekjazykovej bariére, za dovolenky, ktoré som s nimi strávil a samozrejme ďakujemza uvedenie do slovenskej kultúry.

Last but not least, many thanks to my fiancée Martina for always being there.

Joris,August 2015

Abstract

This thesis covers the evolution of wide hot-subdwarf B (sdB) binaries. Thesehot subdwarfs are core-helium-burning stars with a very thin hydrogen envelope(MH < 0.02 M). The formation of sdB stars has been controversial for a longtime: how can a star lose almost all of its envelope at the tip of the red-giantbranch (RGB). Currently it is widely accepted that sdB stars can only beformed by binary interaction, making them an ideal testing ground for binaryinteraction processes. There are three main formation channels proposed toform sdB stars: 1) common-envelope ejection creating a short-period binarycontaining an sdB and a white dwarf (WD) or main-sequence (MS) component,2) stable Roche-lobe overflow (RLOF) at the tip of the RGB creating long-periodsdB + MS binaries, and 3) merger channels creating a single sdB star. Manyshort-period sdB binaries are known, and the observed population correspondswell with the theoretical models. However, there are only few long-period sdBbinaries known, and only one had been studied in detail when this researchbegan. In this thesis we focus on this formally unsampled population of widesdB binaries. The text is divided in two parts, the first covering the observationsand analysis of a wide binary sdB sample while the second part focuses onspecific binary evolution models which produce the observed wide sdB systems.

In the first part, the analysis of a wide-sdB-binary sample containing eightsystems is discussed. These systems have orbital periods varying from 2 to over3 years. These binaries have been observed with the HERMES spectrographattached to the 1.2m Mercator telescope on La Palma since 2009. The highresolution and efficiency of HERMES, and the accessibility of the 1.2m Mercator

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iv ABSTRACT

telescope for long-term projects like ours, allows to obtain the long orbits withappropriate sampling to determine accurate radial velocities of both the coolcompanion and the sdB component. Based on the RV curves, the orbitalproperties of the systems could be determined. Because of the large differencein surface gravity between the two components, we were able to observe thegravitational redshift of the sdB component, and use it to estimate its surfacegravity. This was a first for sdB binaries. To determine the effective temperatureand surface gravity of both components, we developed algorithms that couldfit binary model atmospheres to the spectral-energy distribution (SED) fromphotometry available in the literature.

The most important discovery is that all wide sdB binaries in our sample havesignificantly eccentric orbits. In combination with three other systems publishedin the literature, there are ten eccentric systems and only one possibly circularsystem. This is unexpected as tidal forces are predicted to circularize thebinaries when the sdB progenitor is ascending the RGB. When analysing theperiod-eccentricity distribution of the sample, there is a clear trend visible thathigher eccentricities are detected in systems with longer orbital periods. Theeccentricity varies from e = 0.04 at P ≈ 750 days, to e = 0.17 at P ≈ 1250days. The circular system has the shortest orbital period at ∼730 days.

In the second part we tested three possible eccentricity-pumping mechanismsdescribed in the literature. These are: 1) phase-dependent tidally-enhanced-windmass loss, 2) phase-dependent mass loss during RLOF and 3) the interactionbetween a circumbinary (CB) disk with the binary in combination with phase-dependent RLOF. To test these processes we used the evolution code Modulesfor Experiments in Stellar Astrophysics (MESA, Paxton et al. 2011, 2013),which, next to the stellar evolution part, also contains a binary module. Weadapted this binary module to include the physical processes necessary to studythe evolution of eccentric binary systems.

The tidally-enhanced-wind-mass-loss models were not able to create eccentricsdB binaries in the covered parameter space. To remain on an eccentric orbit, thesdB progenitor needs to lose too much mass to still be capable of igniting helium.All formed eccentric binaries consist of a MS and He-WD component, while theformed sdB + MS binaries are all circular. Models including phase-dependentRLOF or a CB disk are capable of creating eccentric sdB + MS binaries.Using only phase-dependent RLOF, the systems with low and intermediateeccentricities on orbital periods up to ∼1100 days can be explained. When alsoa CB disk is added, also the higher eccentricities and orbital periods up to 1600days can be reached. Circular systems on short orbital periods (.1000 days)can not be formed when the eccentricity-pumping mechanisms are active. Eventhough the models allow all observed eccentric systems, they do not predictthe observed trend of higher eccentricities at higher orbital periods. Both

ABSTRACT v

eccentricity-pumping mechanisms are more effective at shorter orbital periods.

We conclude that our observed sample of sdB binaries can contribute significantlyto theoretical studies of binary-interaction mechanisms. Two of the testedeccentricity-pumping mechanisms allow for the existence of the eccentric wide-sdB-binary population, even though they can not predict the observed trend inthe period-eccentricity distribution. Further theoretical research is necessary toimprove the eccentricity-pumping models. An improved understanding of thecool companions in wide sdB binaries can help to constrain the free parametersin the binary-interaction mechanisms. Our findings have wider implications asalso in the more evolved post-AGB binaries, wide eccentric systems are observedbut not understood.

Beknopte samenvatting

Dit proefschrift behandelt de evolutie van hete subdwerg sterren in langperiodieke binaire systemen. De zogenaamde hete subdwergen zijn geëvolueerdesterren die helium verbranden in hun kern. Het grote verschil met standaardster evolutie is dat ze het grootste deel van hun waterstof enveloppe verlorenhebben voordat de heliumverbranding start. Een belangrijk vraagstuk voor dezesterren is hoe een rode reus bijna zijn volledige enveloppe verliest juist voordatheliumverbranding start. Het is momenteel algemeen aangenomen dat dezesterren enkel gevormd kunnen worden door binaire interactieprocessen. Hierdoorzijn ze belangrijk om deze interactieprocessen beter te kunnen begrijpen, ookin andere soorten binaire systemen. Er zijn drie vormings scenarios die hetesubdwergen kunnen vormen: 1) Onstabiel massaverlies resulterend in eengemeenschappelijke enveloppe die uitgestoten wordt als genoeg energie vande baan naar de enveloppe getransfereerd wordt. Dit resulteert in een hetesubdwerg met een hoofdreeks ster of een witte dwerg als begeleider op een kortperiodieke baan. 2) Stabiel massaverlies op de rode reuzen tak resulterendin een hete subdwerg en een hoofdreeks ster op een lang periodieke baan. 3)De fusie van twee helium witte dwergen resulterend in een alleenstaande hetesubdwerg. Er zijn veel kort periodieke hete subdwerg systemen bekend, enhun geobserveerde eigenschappen komen overeen met de voorspellingen vande theoretische modellen. In tegenstelling tot de kort periodieke systemen,zijn er slechts enkele lang periodieke systemen bekend, en slechts één systeemwas in detail bestudeerd voor we aan dit thesisonderzoek begonnen. Hetonderwerp van dit proefschrift is dan ook lang periodieke binaire systemenmet een hete subdwerg component. Het proefschrift is onderverdeeld in twee

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viii BEKNOPTE SAMENVATTING

delen, het eerste behandelt de analyse van waarnemingen (tijdreeksen vanspectra en fotometrische metingen over een groot golflengtebereik) van dezehete subdwergen terwijl het tweede deel theoretisch van aard is.

In het eerste deel worden onze waarnemingen van acht langere periode binairesystemen met een hete subdwerg component besproken. Deze systemen hebbenorbitale periodes variërend van twee tot meer dan drie jaar. Ze zijn geobserveerdsinds 2009 met de HERMES spectrograaf aangesloten aan de 1.2m Mercatortelescoop op La Palma. Door de hoge resolutie en efficiëntie van de HERMESspectrograaf is het mogelijk om de snelheid van beide componenten te meten,en kunnen we zo een accurate beschrijving van de baan geven, als ook demassaverhouding van de componenten berekenen. Omdat de hete subdwergeen grotere graviteit heeft dan de koudere component, is er een gravitationeleroodverschuiving zichtbaar in de spectraal lijnen van de hete subdwerg. Metdeze roodverschuiving hebben we de graviteit van de hete subdwergen kunnenberekenen. Dit is de eerste keer dat dit werd toegepast voor dit type sterren.Verder hebben we een programma ontwikkeld dat door middel van atmosfeermodellen en fotometrische metingen uit de literatuur de spectrale eigenschappenvan beide componenten bepaalt.

De belangrijkste ontdekking is dat alle door ons geanalyseerde systemen eenduidelijk excentrische baan hebben. In combinatie met nog drie andere systemendie door andere teams zijn gepubliceerd, zijn er in totaal 10 excentrische langperiodieke binaire systemen met een hete subdwerg, en één systeem dat circulairis. Deze excentrische banen zijn onverwacht omdat de voorloper van de hetesubdwerg tijdens zijn evolutie een rode reus wordt, en in deze fase zijn degetijdekrachten in het systeem zo groot dat de baan cirkelvormig zou moetenworden volgens de theoretische voorspellingen. Een tweede vaststelling is dat deexcentriciteit van de systemen groter wordt naarmate de periode langer wordt.De excentriciteit variëert van e = 0.04 bij P ≈ 750 dagen, tot e = 0.17 bijP ≈ 1250 dagen. Het circulaire systeem heeft de kortste periode met ∼730dagen.

In het tweede deel worden mogelijke evolutionaire modellen die deze systemenkunnen verklaren voorgesteld. In de wetenschappelijke literatuur hebben we drieprocessen gevonden die de excentriciteit van binaire systemen kunnen behoudenof verhogen: 1) fase afhankelijk massaverlies in een sterrenwind versterkt doorde getijdenwerking tussen de twee componenten in een binair systeem, 2) faseafhankelijk massaverlies tijdens de massatransfer van de rode reus donor naarzijn begeleider, en 3) de interactie tussen een stofschijf rond het systeem enhet binair systeem zelf in combinatie met fase afhankelijk massaverlies. Omdeze drie processen te testen hebben we gebruik gemaakt van de evolutie codeMESA, die naast een stellaire evolutie deel ook modules voor de evolutie vanbinaire systemen bevat. We hebben deze binaire modules uitgebreid met alle

BEKNOPTE SAMENVATTING ix

fysische processen noodzakelijk om de evolutie van excentrische systemen tebestuderen.

We kunnen besluiten dat massaverlies in een wind versterkt door getijdenwerkinggeen excentrische systemen met een hete subdwerg kan vormen. De rodereus moet zoveel massa verliezen dat ze uiteindelijk te licht is om helium teontbranden, en een helium witte dwerg wordt. Als dit proces toch een hetesubdwerg vormt, dan is dat enkel in een cirkelvormige baan. De andere tweeprocessen, fase afhankelijk massaverlies tijdens massa transfer en het effect vaneen stofschijf, kunnen excentrische binaire systemen met een hete subdwergcomponent vormen. Als enkel fase afhankelijk massaverlies actief is, kunnende systemen met lage tot gemiddelde excentriciteit op banen met periodes tot1100 dagen verklaard worden. Als ook het effect van een stofschijf in rekeninggebracht wordt kunnen alle geobserveerde excentrische systemen verklaardworden. Het waargenomen cirkelvormige systeem op een kort periodieke baankan echter niet gevormd worden als de excentriciteitsverhogende mechanismenactief zijn. Een tweede belangrijke opmerking is dat alhoewel deze mechanismende excentrische systemen toelaten, ze niet de geobserveerde trend van een hogereexcentriciteit bij langere periode voorspellen. Beide mechanismen zijn effectieverbij korte periodes dan bij lange periodes.

We besluiten dat de systemen die we geanalyseerd hebben een belangrijkebijdrage kunnen leveren om de interacties tussen twee sterren in een binairsysteem te kunnen bestuderen. Twee van de processen die we hebben getestkunnen inderdaad de excentriciteit en periode distributie van de geobserveerdesystemen verklaren, maar niet de geobserveerde trend. Verder theoretisch werkis nodig om deze modellen te verbeteren. Ook observationeel kan een bijdragegeleverd worden door de begeleiders van de hete subdwergen te analyseren.Verder moet het bestaan, de massadistributie en evolutietijdschaal van eventuelestofschijven rond voorlopers van sdB sterren worden onderzocht. Deze kunnenimmers helpen om limieten te stellen op de parameters van de interactiemechanismen.

Contents

Abstract iii

Contents xi

1 Introduction 1

1.1 Stellar evolution in a nutshell . . . . . . . . . . . . . . . . . . . 1

1.1.1 Single stars . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Binary stars . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Hot subdwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.2 The life of an sdB star . . . . . . . . . . . . . . . . . . . 17

1.2.3 Formation channels . . . . . . . . . . . . . . . . . . . . . 22

1.2.4 Why study sdBs? . . . . . . . . . . . . . . . . . . . . . . 25

1.3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 25

xi

xii LIST OF SYMBOLS

I The orbits of sdB + MS binaries 27

2 The sdB+G0 system PG1104+243 29

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.1 Radial Velocities . . . . . . . . . . . . . . . . . . . . . . 34

2.2.2 Orbital Parameters . . . . . . . . . . . . . . . . . . . . . 38

2.3 Spectral energy distribution . . . . . . . . . . . . . . . . . . . . 41

2.3.1 Photometry . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.2 SED fitting . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . 51

2.5 Absolute dimensions . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6 Atmospheric parameters from Fe i-ii lines . . . . . . . . . . . . 53

2.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . 56

3 Three eccentric sdB binaries 59

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.1 Radial Velocities . . . . . . . . . . . . . . . . . . . . . . 64

3.2.2 Orbital parameters . . . . . . . . . . . . . . . . . . . . . 68

3.3 Spectral Energy Distribution . . . . . . . . . . . . . . . . . . . 70

3.3.1 Photometry . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4.1 Disentangling . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4.2 Atmospheric parameters and abundances . . . . . . . . 80

3.4.3 Spectral fitting . . . . . . . . . . . . . . . . . . . . . . . 82

LIST OF SYMBOLS xiii

3.5 Gravitational redshift . . . . . . . . . . . . . . . . . . . . . . . 83

3.6 Absolute parameters . . . . . . . . . . . . . . . . . . . . . . . . 85

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 The observed period-eccentricity distribution of wide sdB binaries 89

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Balloon 82800003 and BD−7o5977 . . . . . . . . . . . . . . . . 89

4.3 PG1018−047 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 EC11031−1348 and TYC2084−448−1 . . . . . . . . . . . . . . 96

4.5 PG1449+653 and PG1701+359 . . . . . . . . . . . . . . . . . 99

4.6 Period-eccentricity distribution . . . . . . . . . . . . . . . . . . 102

II Testing eccentricity pumping processes 105

5 Modules for Experiments in Stellar Astrophysics 107

5.1 MESA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.1.1 Microphysics . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.2 Macrophysics . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Binary physics in MESA . . . . . . . . . . . . . . . . . . . . . . 109

5.2.1 Evolution of the orbital parameters . . . . . . . . . . . . 110

5.2.2 Roche-lobe overflow . . . . . . . . . . . . . . . . . . . . 112

5.2.3 Wind mass loss . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.4 Phase-dependent mass loss . . . . . . . . . . . . . . . . 117

5.2.5 Tidal forces . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.6 Circumbinary disks . . . . . . . . . . . . . . . . . . . . . 120

5.2.7 Gravitational wave radiation and magnetic braking . . . 125

6 Testing eccentricity pumping processes 127

xiv LIST OF SYMBOLS

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2 MESA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2.1 Stellar input parameters . . . . . . . . . . . . . . . . . . 130

6.3 Modelling methodology . . . . . . . . . . . . . . . . . . . . . . 132

6.4 Tidally enhanced wind mass loss . . . . . . . . . . . . . . . . . 133

6.5 Phase dependent RLOF . . . . . . . . . . . . . . . . . . . . . . 135

6.5.1 Model and input parameters . . . . . . . . . . . . . . . 136

6.5.2 Parameter study . . . . . . . . . . . . . . . . . . . . . . 137

6.6 Circumbinary Disks . . . . . . . . . . . . . . . . . . . . . . . . 146

6.6.1 Model and input parameters . . . . . . . . . . . . . . . 147

6.6.2 Parameter study . . . . . . . . . . . . . . . . . . . . . . 148

6.7 Period-eccentricity distribution . . . . . . . . . . . . . . . . . . 156

6.8 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 159

7 Conclusions and future prospects 163

7.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 163

7.2 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Appendix A Mesa inlists 169

A.1 Single stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A.2 Binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Bibliography 173

Curriculum Vitae 183

1Introduction

The big picture in stellar evolution is well known, but there are still many detailsthat remain unclear or even unknown at all. If the evolution is complicated byinteraction with a binary companion, the level of complexity increases further.Many binary-interaction mechanisms are known to impact stellar evolution,but their exact workings remain elusive. This thesis aims to uncover some ofthe interactions in wide binaries by studying long-period hot-subdwarf binaries.In this introduction the basics of stellar and binary evolution are described,followed by a section on hot subdwarfs.

1.1 Stellar evolution in a nutshell

In this section I will briefly describe the evolution of single stars, after which theimpact of a binary companion on stellar evolution will be discussed. As this thesisconcerns hot subdwarf stars, the main focus is on low mass stars which ignitehelium under degenerate conditions. The evolution of an intermediate-massstar with helium ignition under non-degenerate conditions is briefly summarizedas a comparison. All figures describing the evolution of single stars are createdwith the stellar evolution code Modules for Experiments in Stellar Astrophysics(MESA, Paxton et al. 2011, 2013), based on my own calculations. Exampleinlists for MESA are given in AppendixA. The sections on single star evolutionare based on Iben & Renzini (1983); Kippenhahn et al. (2012) and Hansenet al. (2004), while those on binary evolution are based on Iben & Livio (1993);Hilditch (2001); Tauris & van den Heuvel (2006) and Podsiadlowski (2008).

1

2 INTRODUCTION

1.1.1 Single stars

In this section the evolution of low- and intermediate-mass stars from the mainsequence to the white-dwarf stage will be illustrated. As an example of alow-mass star we use a 1 M model with a quasi-solar composition (X = 0.7,Z = 0.02). The evolution of the solar-mass star in the Hertzsprung-Russel(HR) diagram is plotted in Fig. 1.1 and in Fig. 1.2 a so-called “Kippenhahn”diagram is shown together with the evolution of the luminosity, radius andcentral abundances. In the Kippenhahn diagram the hydrogen-burning zonesare shown in red, while the helium-burning zone is plotted in blue. Convectivezones are shown in green. The letters on the HR and Kippenhahn diagramsindicate stages in the evolution and are referred to in the text.

Main sequence evolution

Stars spend most of their life on the main sequence (A-B), slowly fusing hydrogeninto helium in their core. Evolution on the main sequence happens slowly, anddepending on their mass, stars spend between 106 to 109 years in this phase.During the main sequence, the star evolves away from the zero-age main sequence(ZAMS) towards higher luminosities and larger radii. Low-mass stars evolvetowards higher effective temperature (Teff) while higher-mass stars evolve tolower Teff , but increase more strongly in radius.

The star is in hydrostatic and thermal equilibrium, as the gass pressure fromthe energy produced in the hydrogen fusion balances the gravitational pull.The high-temperature sensitivity of the nuclear-energy-generation rates (ε)make the nuclear reactions act like a thermostat in the central regions. If thecentral temperature drops, the energy generation diminishes, which decreasesthe outwards pressure. The outer layers of the star will fall inwards, increasingthe pressure in the core, and thus increasing the temperature again. If thecentral temperature increases, nuclear burning will increase, increasing theoutwards pressure. The layers outside the core will produce less pressure on thecore, and the temperature will decrease again, resulting in an equilibrium inthe center.

Hydrogen fusion into helium happens through two cycles: the proton-proton(p-p) chain, and the carbon-nitrogen-oxygen (CNO) cycle. In low-mass stars(. 1.3 M), the central temperature is too low to ignite the CNO cycle. Theirmain energy production originates in the p-p chain, where four protons are, inseveral steps, fused into a helium ion. In higher-mass stars with a higher centraltemperature, the CNO cycle dominates the energy production. Due to the high

STELLAR EVOLUTION IN A NUTSHELL 3

temperature sensitivity of this process (εCNO ∝ T 18 while εpp ∝ T 4), the moremassive stars expand considerably during the MS evolution.

Hydrogen shell burning

When low-mass stars leave the main sequence, they have dense cores that areclose to being degenerate. At point B core hydrogen is practically exhausted(Hc < 0.001). Hydrogen fusion moves slowly to a thick shell surrounding thehelium core. During phase B-C (subgiant branch), the core slowly grows inmass and contracts further, while the envelope expands, and the H-burningshell gradually becomes thinner. The outer envelope expands, cools down, theopacity rises, and, starting at the surface, becomes convective. At point C, thehelium core becomes electron degenerate, and the star is now at the base of thered giant branch (RGB). At point D, the convective envelope is at its deepestpoint in mass and reaches into layers that were processed by H burning duringthe main sequence (first dredge-up). As the star continues to climb the RGB,the envelope becomes more loosely bound, and stellar-wind-mass loss increases.Between point D and the tip of the RGB (point E), the example star of 1 Mloses roughly 0.15 M in a stellar wind. The amount of mass lost in a stellarwind depends on the used prescription, which is not well known. In this case aReimers wind with ηreimers = 0.5 (Reimers 1975) was used.

He core flash

Helium burning in low-mass stars differs from intermediate mass stars in twomain ways. First, in low mass stars, the helium core ignites under electron-degenerate circumstances, giving rise to a helium flash. Second, the mass of theHe core at ignition is almost the same for all low-mass stars: MC ≈ 0.47M.The luminosity of low-mass helium-core-burning stars is therefore independentof their mass, a property that results in the horizontal branch and the red clump(point F).

He burning in a degenerate core is unstable. The 3α reaction causes an increasein temperature. Because in a degenerate core the temperature is decoupled fromthe pressure, the latter does not increase, and thus the He ignition initiates athermonuclear runaway. This thermonuclear runaway leads to an overproductionof energy. At its maximum, the luminosity in the core can be up to 1010Lfor a few seconds, comparable with the luminosity of a small galaxy. All thisnuclear energy is absorbed by expanding the outer layers of the core and thenon-degenerate layers around the core, hiding most of the energy from an outsideobserver. This is called the helium flash (point E in Figs. 1.1, 1.2 and 1.11).

4 INTRODUCTION

3.453.503.553.603.653.703.753.80

log Teff/K

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Figure 1.1: Hertzsprung-Russell diagram of the evolution of a 1 M star fromthe ZAMS to the end of shell helium burning. The letters on the plot indicateevents discussed in the text.


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Figure 1.2: Evolution of a 1 M star from the ZAMS to the end of shellhelium burning. The letters on the plot indicate events discussed in the text.Top panel: Kippenhahn diagram. red colors indicate hydrogen burning, blueindicates helium burning and convective zones are plotted in green. The radiusevolution is plotted on the right axes. The x-axes is subdivided in three partswith different time scales to improve clarity. Middle panel: Evolution of theluminocity. Bottom panel: Evolution of the central abundances.

6 INTRODUCTION

3.53.63.73.83.94.04.14.2

log Teff/K

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Figure 1.3: Hertzsprung-Russell diagram of the evolution of a 5 M star fromthe main sequence to the end of shell helium burning. The letters on the plotindicate events discussed in the text.


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Figure 1.4: Evolution of a 5 M star from the main sequence to the end ofshell helium burning. The letters on the plot indicate events discussed in thetext. Top panel: Kippenhahn diagram. red colors indicate hydrogen burning,blue indicates helium burning and convective zones are plotted in green. Theradius evolution is plotted on the right axes. Middle panel: Evolution of theluminocity. Bottom panel: Evolution of the central abundances.

8 INTRODUCTION

In practice, the He flash does not take place in the core. In Fig. 1.11 a close upof the He burning phases of a 1 M star is shown. From this figure, it is clearthat the main He flash takes place in a shell around mass coordinate 0.20 M.The reason for this is that the core itself cools down due to energy loss byneutrinos in the preceding red-giant phase. These neutrinos are created bythe plasmon-neutrino process (Yakovlev et al. 2001; Odrzywołek 2007), whichincreases in efficiency with increasing density. The most dense part of thecore will thus be cooler than the surrounding layers. This main helium flashis followed by a series of smaller flashes that reach closer and closer to thestellar center. As the temperature of the core keeps increasing, the degeneracyis eventually lifted, re-coupling temperature to pressure, and further nuclearburning is stable. The number of flashes before the core starts stable He burningdepends on the total mass of the star.

The core now startes stable He burning, and the star emerges on the horizontalbranch (point F). The luminosity and radius of the star have decreased by morethan a magnitude compared to before the He flash. The core has expanded, andthe outer shell above the H-burning shell has contracted. The life time duringstable He burning is roughly 108 yr (phase F-G), and is rather independentof stellar mass. The luminosity during stable He burning, roughly 50 L, ismainly determined by the core mass, which is ≈ 0.47 M for all low-mass stars.

In a population of stars with a given composition, for example in a cluster, onlythe envelope mass of the He-burning stars will vary. At solar metallicity, all suchstars cluster together in the HR diagram, giving rise to the red clump, whichis observed in low-mass populations. The radius and effective temperature ofstars on the He main sequence depend on envelope mass. Stars with a lowerenvelope mass can be substantially hotter than those with higher envelopemasses. He-burning stars with different envelope masses will form the horizontalbranch observed in old stellar populations. He-burning stars with no hydrogenenvelope are found at the left end of the horizontal branch.

After central He is exhausted, the carbon-oxygen (CO) core contracts, He shellburning starts, and the star enters the asymptotic-giant branch (AGB). Thestar consists of a CO core, a He-burning shell, a H-burning shell and a large Henvelope. The star increases in radius and moves up in the HR diagram alongthe AGB (phase G-H). The ashes of the H-burning shell cause the He-burningshell to build up. This can, in time, cause a thermal pulse. As the mass of theCO core increases, the star will increase in radius and luminosity.

When the star climbs the AGB, it develops a stellar wind outside its envelope,which will blow the outer layers into space. In combination with strong stellarpulsations, this wind can cause mass loss up to 10−4 Myr−1. In roughly 10 000yr, the envelope is removed, leaving only a hot degenerate CO core, a CO white


dwarf, surrounded by a planetary nebula. This WD radiates energy and coolson the WD cooling track.

Evolution of intermediate mass stars

To illustrate the evolution of intermediate-mass stars, we use a 5 M modelwith quasi-solar composition (X = 0.7, Z = 0.02). The evolution of the 5 Mstar in the Hertzsprung-Russel (HR) diagram is plotted in Fig. 1.3 and in Fig. 1.4a Kippenhahn diagram is shown together with the evolution of the luminosity,radius and central abundances. In the Kippenhahn diagram the H-burningzones are shown in red, while He-burning is plotted in blue. Convective zonesare shown in green.

The main-sequence evolution of an intermediate-mass star is similar to that ofa low-mass star, with the difference that the CNO cycle is the most importantsource of H fusion. Intermediate-mass stars will thus expand more during theirevolution along the MS.

Once central hydrogen is getting depleted, hydrogen fusion in the core diminishes(Hc = 0.03 at point B). As the outward pressure of hydrogen fusion is declining,the stellar core contracts. At point C, central hydrogen is completely exhausted,and H fusion in the core ceases. In intermediate-mass stars the shell burningcan be divided into two phases, thick- and thin-shell burning. After the corecontraction (point C), there is a quick transition from core H burning to shellH burning. Directly after the core contraction the temperature and densitygradients between the core and envelope are small (phase C-D). The burning shellthen occupies a large mass region, hence the name thick-shell burning, and theburning is relatively slow. During this H-shell burning the core increases in massuntil the Schönberg-Chandrasekhar limit is reached and the core contractionspeeds up. The envelope of the star will expand due to the increased radiationpressure. This leads to stronger temperature and density gradients, thus theH-burning shell will become thinner as the star reaches point D. As point Dis approached, the stellar envelope cools down, and its opacity rises. Startingfrom the surface the envelope becomes convective. During phase D-E, the staris an expanding red giant with a deep convective envelope.

If the He core reaches a temperature of 108 K, the star starts helium burningin the 3α → C reaction. In intermediate- and high-mass stars with a non-degenerate core, the nuclear burning is thermally stable and the ignition proceedsquietly. The He-burning region is highly concentrated towards the center dueto the strong temperature sensitivity of the 3α process. Surrounding the He-burning core are a H-burning shell, and a very extended outer envelope. Theluminosity of the star slightly increases, even though the energy yield of He

10 INTRODUCTION

Figure 1.5: Binary statistics by spectral type. The thick solid and dashed linesare binary statistics calculated by Raghavan et al. (2010), while thin solid linesshow results from other studies summaried by them. Figure 12 from Raghavanet al. (2010).

burning is only 10% of that of H burning. The reason why the star can maintainits luminosity is because the majority of the energy is produced by H-shellburning. This is shown clearly in the middle panel of Fig. 1.4.

Intermediate-mass stars evolve through a loop in the HR diagram when burningHe in their core (E-H), also called the blue loop. The evolution from the Heshell burning through the AGB, post-AGB and planetary-nebula phase to theWD cooling track is similar as with the low-mass stars described before.

1.1.2 Binary stars

In the previous sections only single-star evolution was considered. However,a large part of the stellar population resides in binary or multiple systems.Raghavan et al. (2010) shows that depending on the spectral type 20 to 70 %of all stars reside in binaries (see Fig. 1.5). Binary interaction can lead to manyinteresting types of systems, including but certainly not limited to: Cataclysmicvariables, type Ia supernovae, pulsars, hot subdwarfs.


Roche lobes

The interaction between two stars in a binary system is described using theirgravitational wells, also referred to as their Roche lobes1. A Roche lobe isthe equipotential surface within which matter is gravitationally bound to onestar in the binary system. The Roche lobes of the two components of a binarysystem meet at the inner Lagrange point (L1), a saddle point where mass can betransferred from one star to its companion (see Fig. 1.6). The classical way toderive the Roche lobes is using the assumtion of point masses on synchronizedcircular orbits.

Binary evolution can be described based on the radii of the components withrespect to their Roche-lobe radii. As long as both stars are well within theirRoche lobes, they evolve as if they are single stars. These detached binariescan be excellent sources for probing stellar parameters, especially when theyare eclipsing. However, when one of the stars evolves onto the RGB or further,its radius increases significantly. Depending on the ratio between the stellarradii and the orbit we differentiate between different channels (see also Fig. 1.6).1) If the binary is very wide, the primary stays contained in its Roche lobeand further evolution continues as if it were a single star. 2) The primarystarts to fill its Roche lobe, and it starts to lose mass, which can be accretedby its companion star or lost entirely from the system. The binary becomesa semi-detached system. 3) If the primary increases so much in size that itengulfs its companion, a contact binary is formed. The companion star thenorbits within the atmosphere of the primary, and spirals in. This results in acommon-envelope ejection or, if not enough energy is available, a merger.

Roche-lobe overflow

In the case of stable Roche lobe overflow (RLOF), the donor star (or primary)has to continue filling its Roche lobe. We discriminate between two ways. Thedonor continues expanding because of its own internal evolution, or the systemloses angular momentum, causing the orbit and hence the Roche lobes to shrink.Mass loss can continue as long as there is sufficient mass left in the envelope ofthe donor star. At the end of RLOF, the donor star is left with only a smallenvelope.

Depending on the evolutionary phase of the donor star, three different types ofmass loss are distinguished.

1Named after the French astronomer Edouard Roche

12 INTRODUCTION

Figure 1.6: Top: sections in the orbital plane of Roche equipotentials for asystem with q = 0.5. The Roche lobes are shown in red, the Lagrange pointsin blue and the location of both stars as black dots. Bottom: Possible binaryconfigurations; detached, semi-detached with RLOF and common envelope. Foreach a schematic representation of the equipotential well and the filling of theRoche lobes is shown. Figure adapted from Jorissen (2003).


Type-A Mass transfer starts when the donor star is on the MS. This masstransfer will occur on the very long nuclear time scale. As this masstransfer is very slow, the donor can adapt to the changes in its Roche-loberadius without deviating from thermal equilibrium.

Type-B Mass loss starts when the donor is in its H-shell-burning phase, andis ascending on the RGB. This mass loss occurs on a shorter time scale.Whether the donor can adapt to the changing Roche lobe depends on thesystem.

Type-C In this case the donor star has already ignited He, for example on theAGB. The mass transfer is so extreme that the donor cannot stay withinits Roche lobe.

Effect of mass transfer on stellar structure

When mass loss has been initiated, and the mass loss is conservative, the Rochelobe of the donor star starts to shrink. As the stellar radius cannot be muchlarger than the Roche-lobe radius, the donor itself also has to shrink. Theevolution of the donor then depends on whether it can let its radius evolvewith the changing Roche lobe, and still remain in equilibrium. The hydrostaticequilibrium of the donor star is not disrupted by mass loss, as the time scalefor restoring the equilibrium is very short compared to the mass-loss time scale.The thermal equilibrium on the other hand can be disturbed by mass loss.

If the donor star has a radiative envelope during mass loss, the envelope willshrink while the core expands. Due to the expanding core the nuclear reactionrate decreases and the donor will remain in thermal equilibrium. Donor starswith a convective envelope will expand when losing mass, and will therefore notbe able to remain in thermal equilibrium. Depending on how the orbit evolves,the mass loss can occur on a short dynamical time scale, and can lead to a CE.If the orbit expands, the mass loss can still be stable.

The evolution of the accreting star is more extreme than for the donor. Themass stream of the donor will form an accretion disc from which the mass isaccreted onto the companion. The accretor will not only accrete mass, but alsogain momentum, causing it to spin up. In certain cases it is possible that thecompanion reaches its break-up velocity.

If mass transfer is close to conservative, the mass gain of the secondary issignificant. This has two main effects on its further evolution. Firstly, theconvective core will grow, mixing in new fuel from the surrounding layers. Thisprocess, also known as rejuvenation, causes the lifetime on the MS to extend.A second effect is that due to the increased mass, nuclear fusion speeds up and

14 INTRODUCTION

the lifetime on the MS decreases. If the second process is stronger than thefirst, the secondary can overtake the evolution of the primary.

Common-envelope evolution

When the donor star is losing mass from a convective envelope, this envelopemight respond to the mass loss by increasing even further in radius. This resultsin runaway mass loss, and a common-envelope situation. There are currentlytwo different mechanisms to model a common-envelope ejection.

α formalism The classical look at CE evolution is that of a companionorbiting inside the envelope of a giant (Paczynski 1976). The companionexperiences a drag force due to the friction when moving inside the CE.Due to this drag, it will spiral in towards the core of the giant. In thisprocess, orbital energy is transferred to heat and motion of the gas in theCE, and eventually into kinetic energy that causes the CE to expel. Thisenvelope ejection effectively ends the spiral-in phase, and the system isnow a much closer binary than before the CE phase. This CE evolutionis considered the main mechanism to convert wide binaries into veryclose binaries. General consensus is that the companion will not accretemuch matter during the very short CE phase, and will thus not changesignificantly.Current models of the α formalism are based on the energy balance of thesystem, assuming angular-momentum conservation. The orbital energy ofthe binary, assuming some unknown efficiency factor is used to expel thecommon envelope. However, it is found from some observations that theefficiency of this mechanism has to be higher than unity, indicating thatthere is another energy source at play.

γ formalism Nelemans et al. (2000, 2001) proposed an alternative CE-ejectionchannel. This γ formalism assumes that a CE caused by runaway massloss will be in co-rotation with the orbit. There is thus very little drag onthe components, and no or very little spiral-in. The energy necessary toexpel the envelope can be supplied by the luminosity of the giant or tidalheating.The energy balance in the γ formalism is created by linking the change inangular momentum to the total amount of mass that is lost when the CEis ejected.

A more extreme version of the CE evolution occurs when the energy releasedduring the CE phase is not sufficient to eject the common envelope. The

HOT SUBDWARFS 15

companion will then spiral-in until eventually the core of the giant will mergewith the core of the companion. The end product is a rapidly-rotating singlestar. Binary mergers are thought to be responsible for many observed eruptiveevents and other peculiar stars.

An overview of the current state on common-envelope evolution is given in thereview paper by Ivanova et al. (2013).

1.2 Hot subdwarfs

1.2.1 A brief history

The discovery of hot subdwarfs dates back to 1947, when Humason & Zwicky(1947) discovered a few subluminous blue stars at high Galactic latitudes in aphotometric survey of the North Galactic pole. The definition of hot subdwarf-B stars originates from Sargent & Searle (1968), who defined them as starswith colours similar to main-sequence-B stars, but much broader Balmer lines,indicating a high surface gravity and corresponding low luminosity. Temperatureand surface gravities of sdB and sdO stars were first determined by Greenstein& Sargent (1974). They placed them in the HR diagram, where they residebetween the MS and the WD cooling track (see Fig. 1.7). It took until thePalomar-Green survey (Green et al. 1986) to uncover a large population of hotsubdwarfs. In this survey, the sdB stars outnumbered all other types of faintblue objects, and were found in all galactic populations.

With the improvements in model atmospheres calculated in non-local-thermodynamic-equilibrium (NLTE) conditions, and improved optical spec-troscopy, it was found that the hot subdwarfs are extreme-horizontal-branch(EHB) stars, and are thus core-He-burning stars (Heber et al. 1984; Heber1986). The difference from normal horizontal-branch stars is that they lack aH-burning-shell (MH . 0.02M) and have masses close to the core-helium-flashmass ∼ 0.47 M. Their effective temperatures range from 20 000 to 40 000 K,while their surface gravities vary between log(g) = 5.0 and 6.0. More recentlyit was discovered that they are the main source for the UV-upturn in early-type galaxies (Greggio & Renzini 1990; Brown et al. 1997), and that theirphotospheric chemical composition is governed by diffusion processes causingstrong He-depletion, up to a factor 100 relative to the solar He abundance, andother chemical peculiarities (Heber 1998).

The origin of the hot subdwarfs was unknown for a long time. The problembeing: how can almost all envelope mass be removed at the same time as the He

16 INTRODUCTION

Figure 1.7: Hertzsprung-Russell diagram indicating the location of the hotsubdwarf stars (sdB and sdO) at the extreme end (EHB) of the blue-horizontalbranch (BHB). The location of the cool subdwarfs under the main sequence isindicated as well. Figure taken from Heber (2009).

core reached the minimum mass for He ignition. An overview of hot subdwarfstars is given in the review paper of Heber (2009).

HOT SUBDWARFS 17

1.2.2 The life of an sdB star

The evolution of an sdB star in the HR diagram is plotted in Fig. 1.8, for a 1.2M star with solar-like composition at the start of its evolution, and endingwith 0.47 M during its sdB phase. The early evolution of an sdB star onthe main sequence (A-B) and red-giant branch (B-E) with the first dredge-upat point D, is similar to that of a regular star as explained in section 1.1.1.However, the evolution differs when the sdB progenitor reaches the tip of theRGB. For an sdB star to appear on the far left of the EHB, the sdB progenitorneeds to lose almost all of its hydrogen envelope, at the time when the He corereached the minimum mass required to ignite (point E). Currently is it widelyaccepted that binary processes are responsible for this mass loss (see section1.2.3).

Low-mass stars will undergo the He flash at the tip of the RGB. In the case of ansdB star, the mass loss will cause the star to depart from the RGB and undergothe first He flash while it is descending on the WD cooling track. This is alsoknown as a hot flasher. These hot flashers can be subdivided in two groupsdepending on when the actual He ignition takes place. If the He flash occursshortly after the departure from the RGB, during the evolution at constantluminosity, it is called an early hot flasher. The resulting sdB star has anenvelope with a standard H/He composition. If the He ignition takes place onthe WD cooling track, there will be mixing in the envelope. If the flash occursearly on the cooling track, the mixing will be shallow and the atmosphere of thesdB star will be somewhat enriched in He. This is a late hot flasher. Modelsshow that the moment of ignition is determined by the remaining mass of thesdB progenitor after mass loss ends. The more massive the sdB progenitor, theearlier the He flash. This is shown in Fig. 1.9 for a 0.46 M late hot flasher, a0.47 M early hot flasher, and a star igniting He on the RGB.

The empirical mass distribution of sdB stars has been determined by Fontaineet al. (2012) and Van Grootel et al. (2014) based on masses derived fromasteroseismology and eclipsing binary systems. They find that based on asample of 22 stars, the mean sdB mass is 0.470 M and the mass rangecontaining 68.3% of the sdBs is 0.439−0.501M. From this they concludethat the observed distribution compares very well with the expectations fromstellar evolution theory. The histogram of the sdB mass distribution is shown inFig. 1.10. Even though this mass range is small, sdBs can form in large enoughnumbers to be observed frequently. The reason for this is that the radius of thesdB progenitor increases very strongly when it reaches the tip of the RGB (seeFig. 1.2). A small mass range will thus result in a large range in radii, necessaryfor binary interaction to take place near the tip of the RGB.

18 INTRODUCTION

3.43.63.84.04.24.44.64.85.0

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Figure 1.8: The evolution of an sdB star (early hot flasher) in the HR diagram.Starting mass is 1.2 M, mass on the sdB track in 0.471 M. The pre-main-sequence evolution is shown with a dotted black line. Evolutionary stages whereH fusion is the most important luminosity source are plotted in red: the MS infull red line and the RGB until He ignition with a dashed red line. He burningis shown in blue: the He flashes with a blue dotted line, He-core burning witha blue thick solid line and He-shell burning with a blue dashed line. Finallythe WD cooling track is plotted in black. Evolutionary track calculated withMESA.

HOT SUBDWARFS 19

3.64.04.44.8

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Figure 1.9: HR diagram of a late hot flasher (left panel), early hot flasher(middle panel) and He ignition on the RGB (right panel). The exact point ofthe He flash is indicated with a red dot. The mass at He ignition is given foreach case. Evolutionary tracks calculated with MESA.

The sdB starts He burning under electron-degenerate circumstances, in a similarHe-flash process as explained in section 1.1.1. The He-burning evolution ofan sdB star is compared to the canonical He-burning evolution of a 1 Mstar with solar composition in Fig. 1.11. This figure is split in three parts: Heflashes (E-F), He-core burning (F-G) and He-shell burning (G-H). The timescale is linear in each part, but differs between the parts. The core-He-burningphase in sdB stars lasts roughly 80 Myr, while in the 1 M AGB star it issomewhat shorter. These models are calculated without core overshooting.When adding overshooting to the core, the He-core-burning phase will lastsignificantly longer (∼ 120 Myr, Schindler et al. 2014), and the He-shell-burningphase is correspondingly shortened.

As the envelope of sdB stars is too thin to sustain nuclear burning, these starswill not go to the asymptotic giant branch and planetary-nebula phase. Whenthe core helium of an sdB is exhausted, they will start a phase of He-shell burning(G-H), during which they expand and heat up. If there is some hydrogen left inthe shell, the sdB might undergo a very short H-shell burning phase when theHe-burning shell reaches the outer layers of the core. This is the small bumpin luminosity at the end of the He-shell burning. As the core does not get hotenough for carbon burning, the sdB star ends its life as a cooling CO-WD (pointH - end).

20 INTRODUCTION

Figure 1.10: Histogram of the observed mass distribution of sdB stars based onmasses derived from asteroseismology and binary systems. The histogram forthe pulsating systems alone is plotted in red hatched bars, while the histogramcontaining all systems is plotted in blue hatched bars. The histograms are madebased on observed masses while taking into account their errors. Figure takenfrom Fontaine et al. (2012).

HOT SUBDWARFS 21

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Time

Figure 1.11: Comparison of the He burning phases of a regular 1.0 M star(upper two panels), and an sdB star with a final mass of 0.471 M duringHe burning (lower two panels). For each star, the He burning is split in threephases, He flashes (E-F), He-core burning (F-G) and He-shell burning (G-H).On the Kippenhahn diagram the He-burning regions are plotted in blue and theH-burning regions in red. Convective zones are shown in green. The radius isplotted in black on the right axes. The luminosity due to H-burning is plottedin red while luminosity due He-burning is plotted in blue.

22 INTRODUCTION

1.2.3 Formation channels

Different possible evolution scenarios for sdBs have been proposed in theliterature. The main difficulty for the evolution scenarios is the large amountof mass loss on the RGB necessary to remove the H envelope just beforeHe ignition. Several single-star evolution scenarios have been proposed. Forexample: enhanced stellar-wind mass loss on the RGB (D’Cruz et al. 1996),increased mass loss due to rotationally-driven helium mixing on the RGB(Sweigart 1997) and envelope-stripping processes in dense clusters (Mariettaet al. 2000). All these scenarios require exact fine tuning of the parameters,or extreme environmental conditions, making them unlikely to occur for themajority of the observed sdB stars.

Binary-interaction processes have been proposed as possible formationmechanisms as early as Mengel et al. (1975). The finding that many sdBsseemed to reside in binary systems (Maxted et al. 2001) caused this idea toresurface. By now the general consensus is that sdBs are formed by binary-interaction processes alone, and several binary-evolution channels have beendeveloped. In each of them binary-interaction physics plays a major role.Currently there are three main formation channels that are thought to producesdB stars: common-envelope evolution (Paczynski 1976), RLOF evolution (Hanet al. 2000, 2002), and a white-dwarf merger (Webbink 1984). Han et al. (2002,2003) addressed these three binary formation channels, and performed a binary–population-synthesis study for two CE channels, two RLOF channels and theWD-merger channel. Three of the possible formation channels are shown inFig. 1.12.

CE channels In the CE channel based on the α formalism, the sdB progenitorfills its Roche lobe near the tip of the RGB, and triggers dynamical masstransfer. The companion star can not keep up with the high mass-transferrates, and will eventually overfill its Roche lobe. A CE is formed. Thetwo components spiral in until enough orbital energy is released to ejectthe CE. The remaining core of the red giant, becomes the sdB star, whilethe companion does not change significantly during the fast CE phaseand stays on the MS. The end product of the CE is then an sdB + MSbinary with a period between 0.1 and 10 days.When the evolution of this short-period system continues, the MS starwill start to ascend the RGB. If it then starts to fill its Roche lobe, asecond CE can be formed. If enough energy is present in the orbit to ejectthis second CE, a short-period binary containing a WD and an sdB starcan be formed.

HOT SUBDWARFS 23

Figure 1.12: Three possible binary sdB formation channels. The evolution of thesystem proceeds from top to bottom. Left: stable RLOF followed by a CE phase,which results in a short period sdB + WD binary. Middle: unstable RLOFcreating a CE phase which results in a short period sdB + MS binary. Right:stable RLOF occurring near the tip of the RGB phase of the sdB progenitor,resulting in a wide sdB + MS binary. Figure taken from Heber (2009).

24 INTRODUCTION

Soker & Regev (1998) suggested that sub-stellar objects may also beswallowed by their host star. If such objects are sufficiently heavy(& 10 MJ), they can survive a CE-ejection episode, and the resulting systemwill consist of an sdB with a close sub-stellar companion. Observationalevidence seems to indicate that this channel might indeed exist (Geieret al. 2011b; Geier 2015).When the γ formalism for CE ejection without spiral in of the componentsis applied to sdBs, sdB + MS binaries can be created with orbital periodsaround 1-2 years (Nelemans 2010).

RLOF channels In the case RLOF is dynamically stable, no CE forms. Themass lost by the red giant can be slowly accreted by the companion, orlost entirely from the system. If the red giant is able to lose its entireenvelope during RLOF, it can become an sdB star in a long-period binarywith a MS component. A second phase of stable RLOF is not predictedto contribute to the sdB population. However, the stable RLOF can befollowed by a CE channel if the companion ascends the RGB, and leadagain to a short period (0.1 - 10 d) sdB + WD binary.The expected orbital periods of the sdB binaries formed by RLOF arepredicted to range from 10 to 500 days. Chen et al. (2013) revisited themodels of Han et al. (2003) with a more sophisticated treatment of angular-momentum loss, and the inclusion of atmospheric RLOF. Their revisedmodels show mass-orbital-period relations that increase as a function ofcomposition, with solar-metallicity models reaching periods up to 1100days. sdB models including atmospheric RLOF can reach periods up to1600 days.

Merger channels He-WD mergers could explain the population of singlesdB stars. Short period He-WD binaries can lose orbital energy throughgravitational waves, and spiral in. When the orbit shrinks, the less massiveWD will be disrupted and accreted onto the primary WD, leading to Heignition. This merger product can result in a fast rotating single sdBstar. sdB stars formed through this channel can reach higher masses, upto 0.65 M. However, the observed apparent single sdBs rotate slowly(Geier & Heber 2012), and the masses obtained through asteroseismologyare much narrower than predicted by this channel (Fontaine et al. 2012).The He-WD merger is thus an unlikely formation channel for single sdBs.More recently, the merger with a sub-stellar companion was proposed bySoker (2014), but not yet modelled. This merger channel could producesingle sdB binaries with a much more narrow mass distribution centeredaround the canonical sdB mass, compared to the WD-merger channel, andresults in a slow rotating sdB star. An alternative is that the sub-stellar

OUTLINE OF THIS THESIS 25

companion has a low mass (for example a small planet < 10 MJ), andevaporates when entering a CE phase (Geier 2015).

1.2.4 Why study sdBs?

The formation of sdBs has been a challenge for stellar- and binary-evolutionmodels, and continues to be so today. sdBs are believed to be remnants ofbinary-interaction processes, but our current fundamental understanding ofmany of these is still very limited. This shows for example in the necessaryparametrization of many phenomena. Concrete examples are the efficiencies ofcommon-envelope ejection both in the α and γ formalism, the amount of massand angular momentum lost during stable mass transfer, the amount of massaccreted by the companion or lost to infinity, criteria for stable mass transfer,and many more.

To improve our understanding of these processes there are two necessaryelements: theoretical binary models, and an observed population to comparethem with. The hot-subdwarf stars form a very interesting population for thesetests. While there are other evolved stars that might undergo similar evolutionprocesses (for example white dwarfs), they are usually the exception of theirpopulation. The advantage of sdBs is that all of them are formed throughnon-standard channels involving binary interaction. This makes each observedsdB star a potentially important marker for binary-evolution models. Thus, bycomparing the observed sdB population with for example binary-population-synthesis studies, one can try to constrain currently unknown parameters inbinary-interaction processes.

Apart from binary-interaction processes, the hot subdwarfs themselves exhibita number of interesting mechanisms such as gravitational settling, thermaldiffusion, radiative levitation and turbulence, which all conspire to producetheir chemically-peculiar envelopes. Furthermore, sdBs have been shown to beinteresting testbeds for asteroseismology, with four distinct known families ofhot-subdwarf pulsators. This technique has yielded reliable determination oftheir internal properties, including constraints on the core-He burning.

1.3 Outline of this thesis

Short-period binary systems containing an sdB component have been extensivelystudied, see for example: Koen et al. (1998); Maxted et al. (2000, 2001); Heberet al. (2002); Morales-Rueda et al. (2003); Napiwotzki et al. (2004); Copperwheat

26 INTRODUCTION

et al. (2011); Geier et al. (2011a). Currently over 142 of these systems areknown (Kupfer et al. 2015), and the observed systems correspond very wellwith the theoretical results from binary-population-synthesis studies. For thewide sdB binaries, the situation is completely different. At the start of thisthesis, only a few long period systems were known (Green et al. 2001; Østensen& Van Winckel 2011, 2012), and only one system, PG1018−047, was studied indetail by Deca et al. (2012), and all those only with very preliminary results.

The first part of this thesis will focus on observations of long-period sdBbinaries and their analysis. These observations have all been performed withthe HERMES spectrograph (Raskin et al. 2011), attached to the Mercatortelescope at the Roque de los Muchachos Observatory in La Palma. This part issubdivided in three chapters. In Chapter 2 the analysis methods are describedafter which they are applied to the long-period sdB binary PG1104+243.In Chapter 3, three more long-period sdB systems are discussed in detail:BD+29o3070, BD+34o1543 and Feige 87. All of these systems have eccentricorbits. The first part is concluded by Chapter 4. In this chapter, all observationalresults presented in this thesis, or published in the literature are summarized.Furthermore, the orbits of two extra systems that are not completely coveredyet are presented. Lastly, the observed period-eccentricity distribution of allknown long-period sdB binaries will be discussed.

The second part of this thesis is more theoretically oriented. The observationspresented in the first part have shown that the majority of the known long-periodsdB binaries have significantly eccentric orbits. However, current theoreticalmodels predict that binaries of which one component is ascending the red giantbranch, will completely circularise their orbits even before the onset of RLOF.In this part, we will test three different eccentricity-pumping processes describedin the literature, and check if any can recreate the observed period-eccentricitydistribution of the wide sdB binaries. To this end the stellar evolution codeMESA (Modules for Experiments in Stellar Astrophysics), extended with abinary module, is used. In Chapter 5 the MESA code is introduced, and anoverview of the binary physics implemented in it, is given. In Chapter 6, thethree different eccentricity pumping processes are presented and analysed. Theobserved period-eccentricity diagram is then compared to that of the models.We evaluated our model calculations and confront them with the observationsof previous chapters.

In Chapter 7 all results are summarized, and necessary future work is discussed.

IThe orbits of subdwarf B +main sequence binaries

27

2The sdB+G0 system PG1104+243

In this chapter the methods used to analyse the observed long period sdBbinaries are described and applied to PG1104+243. These results have beenpublished, but here we present an updated version of the analysis using allavailable HERMES spectra, and an improved SED fitting method. The mostimportant change with respect to the original article is that with more coveredorbits, we find a small but likely non-zero eccentricity.

The original main body of this chapter was published as:A&A 548, A6 (2012)DOI: 10.1051/0004-6361/201219723© ESO 2012

The orbits of subdwarf-B + main sequence binariesI. The sdB+G0 system PG1104+243

J. Vos, R.H. Østensen, P. Degroote, K. De Smedt, E.M. Green, U. Heber,H. Van Winckel, B. Acke, S. Bloemen, P. De Cat, K. Exter, P. Lampens, R. Lombaert,T. Masseron, J. Menu, P. Neyskens, G. Raskin, E. Ringat, T. Rauch, K. Smolders,

A. Tkachenko

Author contributionsJ. Vos obtained part of the observations with Mercator, analysed the spectra and derived theorbital parameters. Furtheremore, he preformed the SED fitting, and the determination ofthe sdB surface gravity from the gravitational redshift. R.H. Østensen and H. Van Winckelstarted the long-term observing project of wide sdB binaries, and contributed to the discussion

29

30 THE SDB+G0 SYSTEM PG1104+243

of the results. P. Degroote contributed to the development of the SED fitting algorithms.K. De Smedt derived atmospheric parameters of the MS component using MOOG. E.M. Greenprovided and reduced the intermediate-resolution spectra obtained with the Multi MirrorTelescope, and derived radial velocities for the MS component from those spectra. U. Heberprovided a grid of LTE high-resolution sdB spectra used to determine the radial velocities ofthe sdB component, and contributed to the discussion of the results. B. Acke, S. Bloemen,P. De Cat, K. Exter, P. Lampens, R. Lombaert, T. Masseron, J. Menu, P. Neyskens, G. Raskin,K. Smolders and A. Tkachenko observed PG1104+243 with the Mercator telescope. E. Ringatand T. Rauch are included as co-authors as they provided a copy of the Tübingen NLTEModel-Atmosphere Package to our institute, and this was the first published article usingthose models.

ABSTRACT

Context. The predicted orbital period histogram of a subdwarf B (sdB)population is bimodal with a peak at short ( < 10 days) and long ( > 250 days)periods. Observationally, however, there are many short-period sdB systemsknown, but only very few long-period sdB binaries are identified. As thesepredictions are based on poorly understood binary interaction processes, it is ofprime importance to confront the predictions to well constrained observationaldata. We therefore initiated a monitoring program to find and characterizelong-period sdB stars.Aims. In this contribution we aim to determine the absolute dimensions ofthe long-period binary system PG1104+243 consisting of an sdB and a main-sequence (MS) component, and determine its evolution history.Methods. High-resolution spectroscopy time-series were obtained with HER-MES at the Mercator telescope at La Palma, and analyzed to determine theradial velocities of both the sdB and MS components. Photometry from theliterature was used to construct the spectral energy distribution (SED) ofthe binary. Atmosphere models were used to fit this SED and determinethe surface gravity and temperature of both components. The gravitationalredshift provided an independent confirmation of the surface gravity of the sdBcomponent.Results. An orbital period of 755 ± 3 d and a mass ratio of q = 0.70 ± 0.02 werefound for PG1104+243 from the radial velocity curves. The sdB componenthas an effective temperature of Teff = 33800 ± 2500 K and a surface gravity oflog g = 5.94 ± 0.20 dex, while the cool companion is found to be a G-type starwith Teff = 5970 ± 200 K and log g = 4.38 ± 0.11 dex. When a canonical massof MsdB = 0.47 M is assumed, the MS component has a mass of MMS = 0.67± 0.07 M, and its temperature corresponds to what is expected for a terminalage main-sequence star with sub-solar metallicity.Conclusions. PG1104+243 is the first long-period sdB binary in which

INTRODUCTION 31

accurate and consistent physical parameters of both components could bedetermined, and the first sdB binary in which the gravitational redshift ismeasured. Furthermore, PG1104+243 is the first sdB+MS system that showsconsistent evidence for being formed through stable Roche-lobe overflow. Ananalysis of a larger sample of long-period sdB binaries will allow for therefinement of several essential parameters in the current formation channels.

2.1 Introduction

Currently there is a consensus that sdB binaries are formed through binaryinteraction only. Short period sdB binaries can be formed by common-envelope(CE) ejection, or a merger of two He-WD stars. Long-period binaries are formedthrough stable RLOF at the tip of the RGB. There are many short-period sdBbinaries observed, and their parameters correspond with the theoretical models.However, only very few wide sdB binaries are known.

Clausen et al. (2012) concluded that, although the currently known populationof sdB binaries with a white dwarf (WD) or M dwarf companion can constrainsome parameters used in the binary evolution codes, many parameters remainopen. Furthermore they concluded that sdB binaries with a main-sequence(MS) component will be able to provide a better understanding of the limitingmass ratio for dynamically stable mass transfer during the RLOF phase, theway that mass is lost to space, and the transfer of orbital energy to the envelopeduring a common envelope phase. The period distribution of sdB + MS binariescan support the existence of stable mass transfer on the RGB. If long-periodsdB + MS binaries are found, they are thought to be formed during a stablemass transfer phase in the RLOF evolutionary channel. If no long-period sdB+ MS binaries can be found, this would indicate that mass transfer on the RGBis unstable.

The goal of this chapter is to describe the methods we used to determine radialvelocities of both the main-sequence and subdwarf B component in sdB +MS systems (Sect. 2.2), and the use of spectral energy distributions (SEDs) todetermine the spectral type of both components (Sect. 2.3). Furthermore, if themass ratio is known from spectroscopy, the SEDs from high-precision broad-band photometry alone can be used to determine their surface gravities with anaccuracy up to ∆ log g = 0.05 dex (cgs) and the temperatures with an accuracyas high as 5%. The resulting surface gravities are independently confirmed bythe surface gravities derived from the gravitational redshift (Sect. 2.4). Afterwhich all obtained parameters are summarized (Sect. 2.5). The equivalent


widths of iron lines in the spectra can be used to independently confirm theatmospheric parameters of the cool companion (Sect. 2.6). These methods wereapplied to PG1104+243. The cool companion of PG1104+243 is presumed tobe on the main-sequence, and we will refer to it as the MS component in thischapter. PG1104+243 is part of a long-term spectroscopic observing program,and preliminary results of this and seven more systems in this program werepresented by Østensen & Van Winckel (2011, 2012).

2.2 Spectroscopy

The high-resolution spectroscopy of PG1104+243 was obtained with theHERMES spectrograph (High Efficiency and Resolution Mercator EchelleSpectrograph, R = 85000, 55 orders, 3770-9000 Å, Raskin et al. 2011) attachedto the 1.2–m Mercator telescope at the Roque de los Muchachos Observatory,La Palma. HERMES is connected to the Mercator telescope by an opticalfiber, and is located in a temperature controlled enclosure to ensure optimalwavelength stability. In total 99 spectra of PG1104+243 were obtained betweenJanuary 2010 and March 2015. The date and exposure time of each spectrum isshown in Table 2.1. HERMES was used in high-resolution mode, and Th-Ar-Neexposures were made at the beginning and end of the night, with the exposuretaken closest in time used to calibrate the wavelength scale. The exposure timeof the observations was adapted to reach a signal-to-noise ratio (S/N) of 25 inthe V –band. The HERMES pipeline v6.0 was used for the basic reduction ofthe spectra, and includes barycentric correction. Part of a sample spectrumtaken with HERMES is shown in Fig. 2.1.

Furthermore, five intermediate-resolution spectra (R = 4100) were obtained withthe Blue Spectrograph at the Multi Mirror Telescope (MMT) in 1996-1997. TheBlue Spectrograph was used with the 832/mm grating in 2nd order, coveringthe wavelength region 4000-4950 Å. The data were reduced using standardIRAF tasks. An overview of these spectra is given in Table 2.4.

To check the wavelength stability of HERMES, 38 different radial velocitystandard stars of the IAU were observed over a period of 1481 days, onestandard star per night, coinciding with the observing period of PG1104+243.These spectra are cross-correlated with a line mask corresponding to the spectraltype of each star to determine the radial velocity. For this cross-correlation(CC), the hermesVR method of the HERMES pipeline, which is designed todetermine the radial velocity of single stars, is used. This method handles eachline separately, starting from a given line-mask. The method uses the extractedspectra, after the cosmic clipping was performed but prior to normalization.

SPECTROSCOPY 33Ta

ble2.1:

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BJD

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BJD

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Obs.

BJD

Exp.

Obs.

BJD

Exp.

Obs.

–2450000

s–2450000

s–2450000

s–2450000

s5204.70741

1400

K.S.

5685.44968

1200

R.L.

6049.45661

1200

R.O

.6334.65722

1200

S.S.

5204.72527

1400

K.S.

5718.44565

1600

A.T.

6049.47121

1200

R.O

.6338.64495

1800

A.M

.5217.67376

1500

C.W

.5886.76920

1253

K.S.

6054.41725

1500

R.O

.6346.58153

1600

A.M

.5217.69173

1500

C.W

.5909.77232

1300

P.L.

6056.39538

1200

R.O

.6348.55338

1500

A.M

.5234.60506

900

K.E.

5914.76440

1500

P.L.

6056.40985

1200

R.O

.6381.42379

1500

P.A.

5234.61608

900

K.E.

5937.71439

1800

C.W

.6056.42432

1200

R.O

.6381.48281

1200

P.A.

5234.62710

900

K.E.

5943.63769

1800

N.G

.6060.43931

1800

R.O

.6384.44310

1500

J.V.

5264.54423

2700

T.M

.5953.76375

1200

J.M.

6060.46108

1800

R.O

.6388.57609

1200

J.V.

5340.42821

1200

R.L.

5957.72053

1200

J.M.

6064.48563

2400

S.B.

6390.55968

1500

J.V.

5351.37760

1500

S.B.

5959.59752

1200

J.M.

6065.39677

1800

S.B.

6415.54644

1300

P.N.

5553.70059

1200

K.E.

5963.64527

3600

J.V.

6066.40727

1800

S.B.

6437.47882

1300

L.S.

5569.69768

1200

S.B.

5964.53129

3500

J.V.

6068.41460

1800

S.B.

6454.40407

1670

W.H

.5579.56553

1500

P.D.

5964.65640

2400

J.V.

6082.39288

1200

M.M

.6456.39198

1700

W.H

.5579.58348

1500

P.D.

5966.69797

2400

J.V.

6088.40329

1800

H.W

.6619.72286

1500

M.H

.5589.75559

1200

P.L.

5968.61334

1800

J.V.

6255.69802

1200

K.D

.6663.67471

1800

C.W

.5611.64081

1200

P.C.

5980.56900

1400

M.A

.6273.73978

1200

J.M.

6664.71928

1800

C.W

.5622.61268

1200

N.C.

5980.58579

1400

M.A

.6298.71807

1200

C.W

.6677.68739

1200

P.D.

5639.58778

1500

N.G

.5983.60009

800

K.E.

6306.72126

1200

C.W

.6707.61905

1840

S.T.

5650.50737

1500

B.A.

5983.61109

1000

K.E.

6308.63290

1400

A.S.

6762.58739

1200

E.C.

5652.55550

1300

P.N.

5983.62325

1000

K.E.

6318.62464

1500

H.W

.6789.44118

1200

D.K

.5655.60537

1396

P.N.

5994.49229

1800

P.C.

6321.63194

1200

H.W

.6798.42188

1200

H.W

.5658.47913

1426

P.N.

6012.54878

1500

P.R.

6322.66516

1200

H.W

.6809.41668

1200

R.N

.5666.43280

1200

S.B.

6017.54648

1200

J.P.

6323.72192

1200

H.W

.6999.74350

1200

T.M

.5666.45153

1200

S.B.

6028.45960

1600

S.B.

6332.69252

1500

S.S.

7082.64782

1200

R.M

.5672.53688

1500

S.B.

6035.39508

1800

S.B.

6333.73084

1800

S.S.


5825 5850 5875 5900 5925

Wavelength (A)

0.6

0.7

0.8

0.9

1.0

Flu

x

He I

Na I - D

Figure 2.1: A sample normalized spectrum of PG1104+243 (black), showing the5875 Å He I line of the sdB component, and several lines as e.g. the Na doubletat 5890-5896 Å of the MS component. Furthermore, a synthetic templatespectrum of the MS component (green dashed line) and sdB component (redfull line) are shown.

It performs a cross-correlation for each line in the mask, and sums the CCfunctions in each order. The radial velocity is derived by fitting a Gaussianfunction to the CC function (Raskin et al. 2011). To derive the final radialvelocity only orders 55→74 (5966 – 8920 Å) are used, as they give the bestcompromise between maximum S/N for G-K type stars and, after masking thetelluric bands, absence of telluric influence (Raskin et al. 2011). The resultingstandard deviation of these radial velocity measurements is 80 m/s, and theyare in agreement with the IAU radial velocity standard scale.

2.2.1 Radial Velocities

The determination of the radial velocities of PG1104+243 from the HERMESspectra was performed in several steps. First, there were several spectra taken ina short period of time. As the orbital period of PG1104+243 was determined at752 ± 14 d by Østensen & Van Winckel (2012), spectra that were taken on thesame night, or with only a couple of nights in between are summed to increasethe S/N. We experimented with several different intervals, and determined thatwhen spectra taken within a five-day interval (corresponding to 0.7% of theorbital period, or a maximum radial velocity shift of 0.05 km/s) were merged,there was no significant smearing or broadening of the spectral lines. After thismerging, 81 spectra with a S/N ranging from 25 to 50 remained.

SPECTROSCOPY 35

The MS component has many spectral lines in every echelle order, most ofwhich are not disturbed by the He lines of the sdB component. Because of this,the CC-method of the HERMES pipeline (hermesVR) can be used to determinethe radial velocities of the MS component without any loss of accuracy. For theMS component of PG1104+243, a G2-type mask was used. The final errorson the radial velocities take into account the formal errors on the Gaussian fitto the normalized cross-correlation function, the error due to the stability ofthe wavelength calibration and the error arising due to the used mask. Theresulting velocities and their errors are shown in Table 2.2.

The determination of the radial velocities of the sdB component is morecomplicated. Except for the Balmer lines, there are only a few broad Helines visible in the spectrum. After comparison of a synthetic spectrum of aG2-type star with the spectra, it was found that only the 5875.61 Å He I blendis not contaminated by lines of the MS component. For this single blend, wehad to deploy another method to obtain an accurate radial velocity. As onlyone line is used, anomalies in the spectra can cause significant errors in theradial velocity determination. To prevent this the region around the He I line iscleaned from any remaining cosmic rays by hand. The normalization process isdesigned to run automatically, making it possible to quickly process multiplelines is many spectra, and consists of two steps. In the first, a spline function isfitted through a 100 Å region centered at the He I line to remove the responsecurve of HERMES, and the main features of the continuum. In the second step,a small 10 Å region around the He I line is further normalized by fitting andre-fitting a low degree polynomial through the spectrum. After each iteration,the flux points lower than the polynomial are discarded. This is repeated untilthe polynomial fits the middle of the continuum. An example of a spectrumnormalized using this method is shown in Fig. 2.1.

These cleaned and normalized spectra can then be cross-correlated to a syntheticsdB spectrum. For this purpose a high-resolution LTE spectrum of Teff = 35000K and log g = 5.50 from the grids of Heber et al. (2000) was used. The resolutionof this synthetic spectrum is matched to the resolution of the HERMES spectra.Synthetic spectra with different temperatures and surface gravities were tested,to check the temperature and log g dependence of the radial velocities. Changingeither of those parameters causes a systemic shift of maximum 0.2 km/s in theresulting radial velocities. The cross-correlation is performed in wavelengthspace by starting at the expected radial velocity, calculated from the radialvelocities of the MS component. To determine the radial velocity, a Gaussianis fitted to the cross-correlation function. The error on the radial velocitiesis obtained from a Monte-Carlo simulation. This is done by adding Gaussiannoise to the spectrum and repeating the cross-correlation with the syntheticspectrum. The level of the Gaussian noise is determined from the noise level in


the continuum near the He I line. The final error is determined by the standarddeviation from the radial velocity results of 1000 simulations, the wavelengthstability of HERMES and the dependence on the template. The resulting radialvelocities and their errors are shown in Table 2.2.

To derive the radial velocities for the MS component from the MMT spectra,the spectra were cross-correlated against two high S/N G0V spectra (HD13974and HD39587) using the IRAF task FXCOR, where the cross-correlation wasdone only in the wavelength ranges containing absorption lines from the MScompanion, i.e. avoiding the Balmer lines, Heλ 4026, Heλ 4387, Heλ 4471,Heλ 4686, Heλ 4712, and Heλ 4921. The velocities derived from the two G0Vstars were averaged to get the velocity of the MS companion for each of thefive PG1104+243 spectra. The proper scaling factor for the velocity errorsoutput by FXCOR was determined using the standard deviation of the velocitiesobtained from each template relative to the average velocity for each epoch.The resulting radial velocities and errors are given in Table 2.4.

Table 2.2: The radial velocities of both components of PG1104+243.

MS component sdB componentBJD RV Error RV Error−2450000 km s−1 km s−1 km s−1 km s−1

5204.7163 -11.865 0.051 -18.92 0.825217.6827 -11.997 0.074 -20.17 0.895234.6161 -11.311 0.073 -20.85 1.065264.5442 -10.252 0.081 -19.79 0.655340.4282 -11.948 0.089 -20.22 0.555351.3776 -11.985 0.115 -18.82 0.655553.7006 -17.683 0.072 -10.49 0.775569.6977 -18.415 0.070 -9.14 0.775579.5745 -18.644 0.079 -9.20 0.905589.7556 -19.089 0.093 -7.38 1.215611.6408 -19.685 0.073 -7.12 0.655622.6127 -19.671 0.105 -7.82 1.355639.5878 -18.989 0.107 -6.88 0.825650.5074 -19.466 0.081 -7.55 0.825652.5555 -19.415 0.067 -7.34 0.895655.6054 -19.292 0.092 -6.60 1.255658.4791 -19.803 0.065 -7.26 1.065666.4422 -20.151 0.036 -7.71 0.855672.5369 -20.223 0.092 -6.79 1.135685.4497 -20.315 0.094 -7.69 0.85

Continued on next page

SPECTROSCOPY 37

Table 2.2 – continued from previous pageMS component sdB component

BJD RV Error RV Error−2450000 km s−1 km s−1 km s−1 km s−1

5718.4457 -20.120 0.116 -9.24 0.765886.7692 -13.511 0.071 -17.57 0.675909.7723 -13.027 0.066 -16.99 0.655914.7644 -13.066 0.041 -18.43 0.765937.7144 -12.573 0.053 -19.72 0.565943.6377 -12.273 0.076 -19.96 0.675953.7638 -12.260 0.087 -20.47 0.685957.7205 -12.087 0.090 -19.31 0.585959.5975 -12.145 0.057 -19.61 0.725964.0883 -11.863 0.698 -18.14 1.205964.6564 -12.046 0.070 -17.18 0.725966.6980 -11.560 0.096 -21.25 0.755968.6133 -12.028 0.068 -17.69 0.665980.5774 -11.131 0.121 -19.17 0.515983.6115 -11.373 0.096 -19.27 0.795994.4923 -10.424 0.111 -21.49 0.676012.5488 -11.447 0.039 -21.66 0.546017.5465 -11.573 0.083 -20.49 0.686028.4596 -11.655 0.072 -22.02 0.816035.3951 -11.648 0.103 -21.15 0.706049.4639 -11.695 0.075 -20.02 0.586054.4173 -11.592 0.096 -19.91 0.616056.4098 -11.474 0.108 -21.22 0.736060.4502 -11.576 0.157 -19.80 0.796064.9412 -11.688 0.095 -17.89 0.666066.4073 -11.559 0.093 -23.35 0.846068.4146 -11.562 0.073 -21.91 0.536082.3929 -11.584 0.104 -21.38 0.506088.4033 -11.399 0.113 -21.15 0.646255.6980 -16.151 0.070 -14.18 1.136273.7398 -16.693 0.051 -12.21 0.616298.7181 -17.748 0.062 -11.88 0.566306.7213 -17.830 0.039 -10.20 0.676308.6329 -18.088 0.067 -9.48 0.596318.6246 -18.544 0.045 -10.03 0.596321.6319 -18.794 0.105 -9.28 0.526322.6652 -18.725 0.064 -11.38 0.49



Table 2.2 – continued from previous pageMS component sdB component

BJD RV Error RV Error−2450000 km s−1 km s−1 km s−1 km s−1

6323.7219 -18.672 0.090 -10.58 0.756332.6925 -18.986 0.093 -10.41 0.746334.1940 -18.967 0.069 -11.32 0.636338.6450 -19.239 0.064 -6.95 0.586346.5815 -19.200 0.061 -10.76 0.696348.5534 -19.071 0.095 -11.63 0.596381.4533 -19.209 0.081 -10.36 0.626384.4431 -19.37 0.072 -7.08 0.866388.5761 -19.294 0.082 -5.67 0.906390.5597 -19.525 0.058 -7.80 0.836415.5464 -20.538 0.102 -8.43 0.746437.4788 -20.999 0.123 -8.58 0.716454.4041 -20.260 0.080 -9.77 0.526456.3920 -20.483 0.097 -8.97 0.596619.7229 -14.580 0.093 -14.41 0.636663.6747 -13.402 0.031 -16.33 0.556664.7193 -13.296 0.060 -20.14 0.546677.6874 -13.303 0.068 -16.78 1.296707.6190 -11.881 0.111 -18.39 0.536762.5874 -12.588 0.150 -18.74 2.406789.4412 -11.537 0.073 -23.04 1.086798.4219 -11.355 0.090 -20.40 0.806809.4167 -11.222 0.094 -18.63 0.926999.7435 -15.575 0.088 -12.30 1.687082.6478 -18.499 0.149 -10.56 1.12

2.2.2 Orbital Parameters

The orbital parameters were calculated by fitting a Keplerian orbit through theradial-velocity measurements, while adjusting the period (P ), time of periastron(T0), eccentricity (e), angle of periastron (ω), amplitude (K) and systemicvelocity (γ), using the orbital period determined by Østensen & Van Winckel(2012) as a first guess for the period. The MS and sdB component were treatedseparately in this procedure. The radial velocities of both components wereweighted according to their errors as w = 1/σ. There is no difference in the

SPECTROSCOPY 39

Table 2.3: The phase binned radial velocities of both components ofPG1104+243.

MS component sdB componentPhase RV Error RV Error

km s−1 km s−1 km s−1 km s−1

0.129 -14.99 0.042 -14.41 0.630.158 -13.963 0.046 -17.57 0.670.193 -13.078 0.047 -17.86 0.760.235 -12.403 0.027 -19.30 0.590.258 -11.970 0.026 -19.11 0.760.280 -11.568 0.026 -19.46 0.730.298 -10.925 0.036 -21.24 0.860.327 -11.423 0.027 -20.56 1.060.355 -11.243 0.025 -21.54 0.850.381 -11.234 0.022 -19.98 0.730.396 -11.323 0.030 -20.95 0.680.426 -11.563 0.023 -20.92 0.560.449 -11.699 0.056 -18.82 0.650.639 -16.109 0.054 -13.43 1.400.671 -16.988 0.047 -12.21 0.610.713 -18.018 0.024 -10.55 0.650.738 -18.570 0.021 -10.21 0.710.760 -18.967 0.024 -9.72 0.770.805 -19.681 0.027 -8.59 0.870.824 -19.774 0.025 -6.88 0.850.852 -19.966 0.025 -7.53 0.950.871 -20.089 0.037 -7.32 0.990.890 -20.085 0.021 -8.17 0.780.911 -19.735 0.022 -9.40 0.550.935 -19.486 0.061 -9.24 0.76


Table 2.4: The observing date (mid-time of exposure), exposure time and radialvelocities of the MS component of PG1104+243, determined from the spectraobserved with the Blue Spectrograph at the MMT.

BJD Exp. RV Error-2450000 s km s−1 km s−1

435.0168 300 -20.21 0.60436.0515 300 -19.21 0.63476.9221 400 -18.24 0.66510.8811 120 -18.94 0.61836.0087 200 -11.94 0.58

resulting orbital parameters when including or excluding the radial velocitiesdetermined from the MMT spectra.

The residuals of the orbital fit to the MS component are larger than the errorson the radial velocities, and there seems to be a sine wave in the residuals.However, no period for this wave could be determined. The most likely causefor these variations are stellar spots, as the spectroscopic parameters of the MScomponent indicate that it is similar to the Sun in terms of effective temperatureand log g, and it is not in any known instability strip in the HR diagram. Asthese spots are not periodic, their signal cannot be filtered out by fitting a sinewave. To increase the reliability and accuracy of the radial velocities, we firstdetermined the orbital period and T0. And based on this P and T0, we mergedspectra that are close in phase to average out the effects of spots. We use phasebins of M phase = 0.02. We rederived the radial velocities from the binnedspectra, and determined e, ω, the amplitudes and system velocities from thesenew radial velocities. The radial velocities obtained from the phase binnedspectra are given in Table 2.3.

Fitting the orbit resulted in a low eccentricity (e = 0.04 ± 0.02). To check ifthe eccentricity is real or if the orbit of the sdB+MS binary is circularized, theLucy & Sweeney (1971) test for circularization was applied:

P =(∑

(o− c)2ecc∑

(o− c)2circ

)(n−m)/2

, (2.1)

where ecc indicates the residuals of an eccentric fit, and circ the residuals ofa circular fit, n the total number of observations, and m = 6, the number offree parameters in an eccentric fit. P is the probability of falsely rejecting thecircular orbit, under the assumption that the radial velocity measurementsare homogeneously distributed in phase space. For a low P value it would beunreasonable to assume a circular orbit, while for a large value it is unreasonable

SPECTRAL ENERGY DISTRIBUTION 41

Table 2.5: Spectroscopic orbital solution for both the main-sequence (MS) andsubdwarf B (sdB) component of PG1104+243.

Parameter MS sdBP (d) 755 ± 3T0 2450480 ± 8e 0.04 ± 0.02q 0.70±0.02γ (km s−1) −15.59 ± 0.06 −13.8 ± 0.2K (km s−1) 4.42 ± 0.04 6.34 ± 0.2a sin i (R) 66 ± 1 95 ± 2M sin3 i (M) 0.057 ± 0.002 0.039 ± 0.004σ 0.08 0.53

a denotes the semi-major-axis of the orbit. The quoted errors are the standard deviationfrom the results of 5000 iterations in a Monte Carlo simulation.

to assume an eccentric orbit. Lucy & Sweeney (1971) proposes a 5% significancelevel. Meaning that when P < 0.05, the orbit is significantly eccentric, ifP ≥ 0.05, the orbit is effectively circular. In the case of PG1104+243 we findP = 0.01, indicating a likely small but non-zero eccentricity.

To derive the final orbital parameters, both components were first treatedseparately to measure the difference in systemic velocity (see Sect. 2.4). Thenthat difference was subtracted from the radial velocities of the sdB component,and both components are solved together to obtain their spectroscopicparameters. The uncertainties on the orbital parameters were determinedby using Monte Carlo-simulations. The radial velocities were perturbed basedon their errors, and the errors on the parameters were determined by theirstandard deviation after 5000 iterations. The spectroscopic parameters ofPG1104+243 are shown in Table 2.5. The radial-velocity curves and the bestfit solution are plotted in Fig 2.2 both for the HJD-binned and phase-binnedspectra.

2.3 Spectral energy distribution

To derive the spectral type of the main-sequence and subdwarf component, wefitted the photometric spectral energy distribution (SED) of PG1104+243 withmodel SEDs. With this method we can determine the effective temperatureand surface gravity of both components with good accuracy.


−20

−15

−10

−5

−1

0

1

5500 6000 6500 7000

HJD - 2450000

−4

−2

0

2

4

−22

−20

−18

−16

−14

−12

−10

−8

−6

−0.2

0.0

0.2

0.4

0.0 0.2 0.4 0.6 0.8 1.0

Phase

−1

0

1

2

Figure 2.2: Radial velocity curves of PG1104+243. Left panel: RV as afunction of HJD. Right panel: binned RV measurements as a function of phase.Top: spectroscopic orbital solution (solid line: MS, dashed line: sdB), and theobserved radial velocities (blue filled circles: MS component, green open circles:sdB component). The measured system velocity of both components is shownby a dotted line. Middle: residuals of the MS component. Bottom: residuals ofthe sdB component.

2.3.1 Photometry

Photometry of PG1104+243 was collected using the subdwarf database1(Østensen 2006), which contains a compilation of data on hot subdwarf starscollected from the literature. In total, fifteen photometric observations werefound in four different systems: Johnson B and V , Cousins R and I, Strömgrenuvby, and 2MASS J , H and Ks. Four observations have uncertainties largeror equal than 0.1 mag and are discarded. The observations used in the SEDfitting process are shown in Table 2.6. Accurate photometry at both short and

1http://catserver.ing.iac.es/sddb/


Table 2.6: Photometry of PG1104+243 collected from the literature.

Band Wavelength Width Magnitude ErrorÅ Å mag mag

Johnson B 4450 940 11.368 0.010Johnson V 5500 880 11.295 0.013Cousins R 6500 1380 11.161 0.010Cousins I 7880 1490 11.001 0.010Strömgren u 3460 300 11.513 0.045Strömgren v 4100 190 11.485 0.045Strömgren b 4670 180 11.334 0.045Strömgren y 5480 230 11.256 0.0062MASS J 12410 1500 10.768 0.0262MASS H 16500 2400 10.520 0.0272MASS Ks 21910 2500 10.510 0.023

Johnson and Cousins photometry obtained from Allard et al. (1994), Strömgren photometryobtained from Wesemael et al. (1992) and 2MASS photometry obtained from Skrutskie et al.(2006)

long wavelengths are used to establish the contribution of both the hot sdBcomponent and the cooler MS component.

2.3.2 SED fitting

In the SED of PG1104+243 in Fig. 2.3, we see both the steep rise in fluxtowards the shorter wavelengths of the sdB component and the bulk in flux inthe red part of the SED of the MS component. To fit a synthetic SED to theobserved photometry, Kurucz atmosphere models (Kurucz 1993) are used forthe MS component, and TMAP (Tübingen NLTE Model-Atmosphere Package,Werner et al. 2003) atmosphere models for the sdB component. The Kuruczmodels used in the SED fit have a temperature range from 4000 to 9000 K, and asurface gravity range of log g=3.0 dex (cgs) to 5.0 dex (cgs). The TMAP modelswe have used cover a temperature range from 20000 K to 50000 K, and log g from4.5 dex (cgs) to 6.5 dex (cgs). They are calculated using an atmospheric mixtureof 97 % hydrogen and 3 % helium in mass due to the expected He-depletion inthe atmosphere of sdB stars. The original TMAP atmosphere models cover awavelength range of 2500–15000 Å. To include the 2MASS photometry as well,they are extended with a black body of corresponding temperature to 24000 Å.

The SEDs are fitted in two steps. Firstly, a grid-based approach is used to scanthe entire parameter space. Secondly, the grid point with the lowest χ2 is used


as starting point for a least-squares minimizer which will then determine thefinal result.

The grid based approach used, is the procedure described in Degroote et al.(2011) extended to include constraints from binarity. In the binary scenario,there are eight free parameters to consider; the effective temperatures (Teff,MSand Teff,sdB), surface gravities (gMS and gsdB) and radii (RMS and RsdB) ofboth components. The interstellar reddening E(B − V ) is naturally assumed tobe equal for both components, and is incorporated using the reddening law ofFitzpatrick (2004) with RV = 3.1. To increase the accuracy, the models are firstcorrected for interstellar reddening, and then integrated over the photometricpassbands. The distance (d) to the system, acts as a global scaling factor.

The radii of both components are necessary as scale factors for the individualfluxes when combining the atmosphere models of both components to a singleSED. Including the distance to the system, the flux of the combined SED isgiven by:

Ftot(λ) = 1d2

(R2

MSFMS(λ) +R2sdBFsdB(λ)

), (2.2)

where FMS is the flux of the MS model, and FsdB is the flux of the sdB model, in aparticular passband. However, if the masses of both components are known, theradii can be derived and removed as free parameters. Assuming that the mass ofthe sdB component is MsdB = 0.47 M as predicted by stellar evolution models(see Sect. 2.1), the mass of the MS component can be calculated using the massratio derived from the radial velocity curves (see Sect. 2.2.2): MMS = 1/q MsdB.For each model in the grid, the radii of both components are derived from theirsurface gravities and masses, according to:

Ri =

√GMi

gi, i = sdB,MS, (2.3)

in which G is the gravitational constant. This relation can be used to eliminatethe radius dependence in Eq. 2.2:

Ftot(λ) = G MMS

d2 gMS

(FMS(λ) + q

gMS

gsdBFsdB(λ)

). (2.4)

The distance to the system is computed by shifting the combined syntheticmodel flux (Ftot) to the photometric observations. As the effect of the massof the main-sequence component (MMS) in this equation can be adjusted bythe distance, the resulting flux is only dependent on the mass ratio of thecomponents, and not the presumed mass of the sdB component. By using thismass ratio as a limiting factor on the radii of both components, the number offree parameters in the SED fitting process is reduced from eight to six.


To select the best model, the χ2 value of each fit is calculated using the sum ofthe squared errors weighted by the uncertainties (ξi) on the observations:

χ2 =∑i

(Oi − Ci)2

ξ2i

, (2.5)

with Oi the observed photometry and Ci the calculated model photometry. Theexpectation value of this distribution is k = Nobs −Nfree, with Nobs the numberof observations and Nfree the number of free parameters in the fit. In our case,k = 11−6 = 5. Based on this χ2 statistics the error bars on the photometry canbe checked. If the obtained χ2 is much higher than the expectation value, theerrors on the photometry are underestimated. If on the other hand, the obtainedχ2 is much lower than the expectation value, the errors on the photometry areoverestimated or we are over fitting the photometry.

The uncertainties on the final parameters are determined by calculating two-dimensional confidence intervals (CI) for all parameter pairs. This is done bycreating a 2D-grid for each parameter pair. For each point in this grid, the twoparameters for which the CI is calculated are kept fixed on the grid-point value,while the least-squares minimizer is used to find the best-fitting values for allother parameters. The resulting χ2 for each point in this grid is stored. Allthese χ2s are then rescaled so that the χ2 of the best fit has the expected valuek = Nobs −Nfree, with Nobs the number of observations and Nfree the numberof free parameters in the fit. The cumulative density function (CDF) is used tocalculate the probability of a model to obtain a certain χ2 value as:

F (χ2, k) = P

(k

2 ,χ2

2

), (2.6)

where P is the regularized Γ-function. Based on the obtained probabilitydistribution, the uncertainties on the parameters can be derived.

To fit the SED, first a grid of composite binary spectra is calculated for∼2 000 000points randomly distributed in the Teff , log g and E(B-V) intervals of bothcomponents. Then the 50 best-fitting grid points are used as starting valuesfor the least-squares minimizer. For this best fit the 95% probability intervalsare estimated, and used to limit the original ranges on the parameters, afterwhich the fitting process (grid search and least-squares minimizer) is repeated.To determine the CIs of the parameters, for each two parameters a grid witha resolution of 50 × 50 points was used, with the limits adjusted based onthe resulting confidence intervals. When the model atmospheres allowed, thelimits of the grid are set to include the 95% CI, when this was not possible,the limits of the model atmosphere grid were used. The final uncertaintieson the parameters are an average of the 95% CIs for that parameter in alltwo-dimensional CIs that contain that parameter.


2.3.3 Results

For PG1104+243, the best fit has a reduced χ2 of 5.4, and with five degreesof freedom, the χ2 values in the grid were rescaled by a factor 1.10 which isequivalent to slightly increasing the photometric uncertainties. The SED fitresulted in an effective temperature of 5970 ± 200 K and 33800 ± 2500 Kfor the MS and sdB components respectively, while the surface gravity wasdetermined at respectively 4.38 ± 0.11 dex and 5.94 ± 0.20 dex for the MS andsdB component. The reddening of the system was found to be 0.002+0.017

−0.002 mag.This reddening can be compared to the dust map of Schlegel et al. (1998), whichgives an upper bound of E(B-V) = 0.018 for the reddening in the directionof PG1104+243. The reddening of E(B-V) = 0.002+0.017

−0.002 mag indicates thatPG1104+243 is located in front of the interstellar molecular clouds. This isconfirmed spectroscopically, as there are no sharp interstellar absorption lines(e.g. Ca ii K-H, K i) visible in the spectra. These parameters indicate that thecool companion is a G-type star.

The results of the fit together with the probability intervals are shown in Table 2.7.The optimal SED fit is plotted in Fig. 2.3, while the confidence intervals forseveral parameter pairs are plotted in Fig. 2.4. The probability distributionfor the surface gravity and effective temperature of the MS component followsan elongated Gaussian pattern, while the probability distribution for the sameparameters of the sdB component is stretched out in Teff . This latter effect isrelated to a moderate correlation between log g and Teff . Thus, the effect onthe models caused by an increase in Teff , can be diminished by increasing log gas well. This correlation is stronger towards shorter wavelengths, so that themain-sequence component, which is mainly visible in the red part of the SED,is less affected. The surface gravities of both components (log gMS and log gsdB)are strongly correlated as well. This is expected as the distance, which acts likea global scaling factor, is a free parameter. Thus when increasing/decreasingthe log g of either component, this can be counteracted by increasing/decreasingthe log g of the other component, and decreasing/increasing the distance.

As a check of the results, a SED fit of PG1104+243 without any assumptionson the mass of both components is performed as well. In this case, the radiiof the components are randomly varied between 0.5 R and 2.0 R for theMS component and between 0.05 R and 0.5 R for the sdB component. Theresults of this fit are given in the lower half of Table 2.7. The resulting best fitparameters are very close to the results from the mass ratio-constrained fit, butapart from the uncertainty on the MS effective temperature, the uncertaintieson the parameters are much larger. For both the MS and sdB surface gravity,the 95% probability interval are larger than the range of the models. Theeffective temperature and surface gravity of both components of PG1104+243


0.2

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0.6

0.8

1.0

λFλ

·109

(erg

/s/c

m2)

−0.03

0.00

0.03

0.06

(O−

C)

3 · 103 1 · 104

Wavelength (A)

Figure 2.3: Top: The spectral energy distribution of PG1104+243. Themeasurements are given in blue, the integrated synthetic models are shown inblack, where a horizontal error bar indicates the width of the passband. Thebest fitting model is plotted in red. Bottom: The (O-C) value for each syntheticflux point.


5.8 5.9 6.0 6.1 6.2

Teff MS (1000 K)

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

logg

MS

(dex

)

30 35 40 45

Teff sdB (1000 K)

5.6

5.7

5.8

5.9

6.0

6.1

logg

sdB

(dex

)

5.7 5.8 5.9 6.0 6.1 6.2

Teff MS (1000 K)

25

30

35

40

45

Teff

sdB

(100

0K

)

4.2 4.3 4.4 4.5 4.6

log g MS (dex)

5.7

5.8

5.9

6.0

6.1

logg

sdB

(dex

)

58 66 75 83 92 100

Figure 2.4: The confidence intervals of the SED fit of PG1104+243. Thedifferent colors show the cumulative density probability connected to the χ2

statistics given by Equation 2.6, with the χ2 values of the grids rescaled by afactor of 1.26. The best fitting values and the errors are plotted in black.


Table 2.7: The results of the SED fit, together with the 95% probability intervals.In the left part (fixed radius) the results when the radii of both components aredetermined from the log g and mass, are shown. In the right part (free radius)the results with unconstrained radii are shown. For some parameters the 95%probability intervals could not be determined as they are larger than the rangeof the models.

Fixed radius Free radiusParameter Best fit 95% Best fit 95%Main sequence componentTeff (K) 5970 5800–6180 6020 5630–6290log g (dex) 4.38 4.34–4.57 4.35 /E(B-V) (mag) 0.002 0–0.013 0.001 0–0.1

Subdwarf B componentTeff (K) 33800 31000–37000 33500 27500–45100log g (dex) 5.94 5.70–6.05 5.98 4.7– /E(B-V) (mag) 0.002 0–0.013 0.001 0–0.1

correspond with the ionization balance of the iron lines seen in the spectrum ofMS component, and the Balmer lines of the sdB component.

The reason for the high accuracy of the mass ratio-constrained fit comparedto a fit in which the radii are unconstrained is shown in Fig. 2.5. Here fivemodel SEDs are shown, calculated using the best fit parameters from Table 2.7,but with varying surface gravity, and accordingly adjusted radii, for the sdBcomponent. The best fit model with log g = 5.94 dex (cgs) is shown in blue,while models with a log g at the edge of the confidence interval (log g = 5.70and 6.05 dex) are plotted in black and green. When varying log g, the radius ofthe sdB component changes, while the radius of the MS component remainsconstant. It is this radius change that causes the large deviation in the models,and allows for an accurate determination of the surface gravity. When onlythe gravity is changed, and the radii of both components is kept constant, thedifference in absolute magnitude in the Strömgren-u band is of the order of10−4 mag, while when the radius is adapted using the mass, the difference is onthe order of 10−2 mag. Furthermore, because the absolute flux depends on theradius, this method has the advantage that all photometric observations areused to constrain the log g of each component, instead of only the observationsin the blue or the red part. This analysis of PG1104+243 clearly shows thepower of binary SED fitting when the mass ratio of the components is known.


3600 4000 4400 4800

Wavelength (A)

0.6

0.8

1.0

1.2

1.4

λFλ

∗109

(erg

/s/c

m2)

Figure 2.5: SED models calculated for the best fit parameters of PG1104+243as given in Table 2.7, but with varying surface gravity of the sdB component.From bottom to top, the models have a log g of respectively 6.00, 5.85, 5.81,5.77 and 5.00 dex (cgs). The crosses show the integrated model photometry.The photometric observations with their error bars are given in red.

GRAVITATIONAL REDSHIFT 51

2.4 Gravitational Redshift

In an sdB + MS binary the difference in surface gravity between the twocomponents is substantial. The surface gravity of a star gives rise to a frequencyshift in the emitted radiation, which is known as the gravitational redshift.General relativity shows that the gravitational redshift as a function of the massand surface gravity of the star is given by (Einstein 1916):

zg = 1c2

√GMg, (2.7)

where zg is the gravitational redshift, c the speed of light, G the gravitationalconstant, M the mass, and g the surface gravity. This zg will effectivelychange the apparent systemic velocity for the star. In a binary system thedifference in surface gravity for both components will be visible as a differencein systemic velocity between both components. As the zg is proportional to thesquare root of the surface gravity, this effect is only substantial when there is alarge difference in log g between both components, as is the case for compactsubdwarfs and main-sequence stars.

The measured difference in systemic velocity can be used to derive an estimateof the surface gravity of the sdB component. Using the mass ratio from thespectroscopic orbit, and the canonical mass of the sdB component, the mass ofthe MS component can be calculated. Combined with the surface gravity of theMS component from the SED fit, the zg of the MS component can be calculated.The zg of the sdB component can be obtained by combining the zg of the MScomponent with the measured difference in systemic velocity, and can then beconverted to an estimated surface gravity of the sdB component. A caveat hereis that the wavelength of the He I λ 5875 multiplet is only known with a presisionof ∼ 0.01 Å, corresponding to a systematic radial velocity uncertainty of up to∼ 0.5 km s−1. The measured shift in systemic velocity could be partly due tothis uncertainty, but since the observations are consistent with the predictedgravitational redshift we believe this to be the main contributor.

Using a canonical value of 0.47 M for the sdB component, and the mass ratioderived in Sect. 2.2.2, we find a mass of 0.74 ± 0.07 M for the MS component.The SED fit resulted in a log gMS of 4.38 ± 0.11 dex. Using equation 2.7, thegravitational redshift of the MS component is czg = 0.49 ± 0.10 km s−1. Themeasured difference in systemic velocity is 1.78 ± 0.21 km s−1, resulting in atotal gravitational redshift of czg = 2.28 ± 0.20 km s−1 for the sdB component.This is equivalent to a surface gravity of log gsdB = 5.87 ± 0.09 dex, which isvery close to the results of the SED fit.


2.5 Absolute dimensions

Combining the results from the SED fit with the orbital parameters derivedfrom the radial velocities, the absolute dimensions of PG1104+243 can bedetermined. With an assumed mass for the sdB component, the inclination ofthe system can be derived from the reduced mass determined in Sect. 2.2.2.Using this inclination, the semi-major axis of the system can be calculated. Thesurface gravity of the sdB component as determined by the SED fit and thegravitational redshift correspond within errors. For the final surface gravity,the average of both values is taken. The radius of both components is derivedfrom their mass and surface gravity. The absolute dimensions of PG1104+243are summarized in Table 2.8

For an sdB star of the canonical mass and a surface gravity of log g = 5.94dex a temperature around 33800 K is expected, consistent with the result ofthe SED fit and the strength of the He i/He ii lines. This high surface gravityand temperature is consistent with the binary population models of Brown(2008) that find that low metallicity produces sdB stars located at the highlog g end of the extreme horizontal branch. The canonical sdB mass assumptiontogether with the derived mass ratio implies that the MS component must havea sub-solar metallicity. At solar metallicity it should have a radius of 0.8 Rand a temperature of ∼4800K, which is clearly ruled out by the SED fit. If wedrop the canonical sdB mass assumption, and force the MS star to have normalmetallicity, both the MS and sdB star must be more massive in order to beconsistent with the temperature of the MS component and the spectroscopicmass ratio. From the He-MS models of Paczyński (1971), one finds that acore-He burning model with M > 0.6M would have Teff > 40000K, which isruled out by both SED fit and spectroscopy.

Recent evolutionary models calculated with MESA (Modules for Experiments inStellar Evolution, Paxton et al. 2011, 2013)2 for main sequence stars with variousmetalicities, are perfectly consistent with the parameters from the SED fit foran 0.86 M model, provided that the metallicity is assumed to be substantiallysub-solar, at around Z = 0.005. Taking the error on Teff and log g into account,the cool companion can be fitted by models with a mass varying from 0.78 Mto 0.95 M. Such tracks are shown in Fig. 2.6, where one can see that an 0.86M star has an expected temperature of around 6000 K, and a surface gravityof log g = 4.38 near the end of the main sequence, corresponding with an age of8.5 ± 2.5 Gyr for the MS component.

The luminosity of both components can be calculated using L = 4πσR2T 4,2Example inlists for MESA are given in AppendixA.

ATMOSPHERIC PARAMETERS FROM FE I-II LINES 53

resulting in LMS = 0.88 ± 0.26 L and LsdB = 19.43 ± 4.5 L. The apparentV magnitudes of both components are obtained directly from the SED fittingprocedure and are given by: V0,MS = 12.20 ± 0.10 mag and V0,sdB = 12.05 ±0.15 mag. The absolute magnitude can be obtained by integrating the bestfit model SEDs over the Johnson V band, and scaling the resulting flux toa distance of 10 pc, resulting in MV,MS = 4.94 ± 0.10 mag and MV,sdB =4.80 ± 0.15 mag. The distance to the system can then be calculated fromlog d = (mV −MV + 5)/5, which places the system at a distance of d = 282 ±15 pc. Obviously, the distance for both components comes out as exactly thesame, as this is a fixed condition in the SED fitting process.

The proper motion of PG1104+243 as measured by Høg et al. (2000) is:

(µα, µδ) = (−63.4,−22.1)± (1.8, 1.9) mas yr−1. (2.8)

Using the method of Johnson & Soderblom (1987), these numbers together withthe measured value of γ, can be used to compute the galactic space velocityvector

(U, V,W ) = (−56.8,−50.9,−49.6)± (2.8, 2.8, 1.4) km s−1, (2.9)

where U is defined as positive towards the galactic center. Using the values forthe local standard of rest from Dehnen & Binney (1998), we get

(U, V,W )LSR = (−46.8,−45.7,−42.4)± (2.9, 2.7, 1.4) km s−1. (2.10)

Following the selection criteria of Reddy et al. (2006), PG1104+243 is boundto the galaxy, and belongs to the thick disk population.

2.6 Atmospheric parameters from Fe i-ii lines

In this section the equivalent widths (EWs) of iron lines are used to derive theatmospheric parameters and metallicity of the MS component. This methodis independent of the sdB mass assumption used in the SED fit, and is usedto confirm the effective temperature and surface gravity obtained with thatmethod. Furthermore, the mass of the sdB component can be derived by usingevolutionary tracks to fit the mass of the MS component, and the mass ratiofrom the radial velocity curves.

Before the EWs can be measured, the continuum contribution of the sdBcomponent must be subtracted from the HERMES spectra, which are thenshifted to the zero velocity based on the MS radial velocities and summed.The LTE abundance calculation routine MOOG (Sneden 1973) was used to


Table 2.8: Fundamental properties for both the main-sequence (MS) andsubdwarf (sdB) component of PG1104+243.

Systemic parametersP (d) 755 ± 3T0 (HJD) 2450480 ± 8e 0.04 ± 0.02ω 3.83 ± 0.02γ (km s−1) −15.17 ± 0.07q 0.70 ± 0.02a (R) 322 ± 12i (o) 32 ± 3E(B − V ) 0.002 ± 0.017d (pc) 282 ± 15Component parameters

MS sdBK (km s−1) 4.42 ± 0.04 6.3 ± 0.2M (M) 0.67 ± 0.07 0.47 ± 0.05log g (cgs) 4.38 ± 0.11 5.94 ± 0.20R (R) 0.88 ± 0.12 0.13 ± 0.02Teff (K) 5970 ± 200 33800 ± 2500L (L) 0.88 ± 0.26 19.43 ± 4.5V0 (mag) 12.20 ± 0.10 12.05 ± 0.15MV (mag) 4.94 ± 0.10 4.80 ± 0.15

Table 2.9: Atmospheric parameters obtained based on the EW of iron lines,and derived properties. See Sect. 2.6 for further explanation.

MS sdBM (M) 0.86 ± 0.15 0.63 ± 0.15log g (cgs) 4.50 ± 0.50 5.74 ± 0.11R (R) 0.88 ± 0.15 0.18 ± 0.05Teff (K) 6000 ± 250 /L (L) 0.91 ± 0.20 /V0 (mag) 12.59 ± 0.15 /MV (mag) 4.90 ± 0.15 /d (pc) 345 ± 60 /

ATMOSPHERIC PARAMETERS FROM FE I-II LINES 55

5000520054005600580060006200

Teff (K)

3.0

3.5

4.0

4.5

log

g(d

ex)

0.95 M⊙ 0.86 M⊙ 0.78 M⊙

0 1500 3000 4500 6000 7500 9000 10500 12000

Age (Myr)

Figure 2.6: Evolutionary tracks calculated with MESA with a metallicity of Z= 0.005, for three different masses: 0.78 M, 0.86 M and 0.95 M. The ageof the models is shown by the color bar. The resulting Teff and log g obtainedfor the cool companion of PG1104+243 is plotted in red. See also Sect. 2.5

determine the atmospheric parameters of the MS component using the EWsof 45 Fe i and 10 Fe ii lines in the wavelength range 4400–6850 Å, excludingblended lines, and wavelength ranges contaminated by Balmer lines, He i orHe ii lines from the sdB component. The atmospheric parameter determinationalso provides the metallicity of the MS component. The atomic data for theiron lines was taken from the VALD line lists (Kupka et al. 1999), and theused oscillator strengths were laboratory values. The EWs are measured viaintegration and abundances are computed by an iterative process in MOOG inwhich for a given abundance, the theoretical EWs of single lines are computedand matched to the observed EWs. The effective temperature is derived by theassumption that the abundance of individual Fe I lines is independent of lowerexcitation potential, only Fe I lines are used because they cover a large range inlower excitation potential, resulting in Teff = 6000 ± 250 K. A surface gravityof log g = 4.5 ± 0.5 dex is derived by assuming ionization balance between theiron abundance of individual Fe i and Fe ii lines. Assuming an independence


between the iron abundance and the reduced EW, a microturbulence velocityof 2.0 ± 0.5 km/s can be derived. The final metallicity is based on the averageabundance of all used lines, and result in a metallicity of [Fe/H] = -0.58 ±0.11 dex, corresponding to Z = 0.005 ± 0.001 using the conversion of Bertelliet al. (1994). The error on the metallicity is the line-to-line scatter for the Feabundances for the derived atmospheric parameters. The used Fe i lines arefitted by creating synthetic spectra in MOOG using the derived atmosphericparameters, which confirm the obtained metallicity. A detailed description ofthe method used to determine the atmospheric parameters can be found in DeSmedt et al. (2012, Sect. 3). These atmospheric parameters correspond verywell with the effective temperature and surface gravity derived from the SEDfitting process.

The atmospheric parameters derived from the spectral analysis can be used toget a mass estimate for both the MS and sdB component independently fromthe SED fitting process, by using evolutionary models. Solar scaled evolutionarytracks calculated with MESA with a metallicity of Z = 0.005 are compared tothe derived surface gravity (log g = 4.5 ± 0.5 dex) and effective temperature(Teff = 6000 ± 250 K). The best fitting model has a mass of MMS = 0.90 ±0.15 M, with as corresponding age 5.0 ± 5 Gyr. Using the mass ratio fromthe radial velocity curves, a mass of MsdB = 0.63 ± 0.15 M is obtained forthe sdB component. This value is higher than the canonical value of MsdB =0.47 ± 0.05 M. One has to take into account that the cool companion couldhave accreted mass from the sdB progenitor, an effect that is not taken intoaccount in the evolution models.

Using the newly obtained mass of the sdB component to derive its surface gravityfrom the gravitational redshift we find log gsdB = 5.74 ± 0.11 dex. Furthermore,the radius, luminosity and magnitudes in the V band can be derived using themethods described in Sect. 2.5, the results are shown in Table 2.9. Most of thederived parameters are compatible with those derived in Sect. 2.5.

2.7 Discussion and Conclusions

From an analysis of literature photometry and observed spectra, detailedastrophysical parameters of PG1104+243 have been established. The surfacegravity determined from the binary SED fitting method agrees very wellwith the surface gravity determined from the observed gravitational redshift.Furthermore, the long time-base observations made it possible to accuratelyestablish the orbital period at 755 ± 3 d. The sdB component is consistent witha canonical post-core-helium-flash model with a mass of ∼0.47 M, formed

DISCUSSION AND CONCLUSIONS 57

through stable Roche lobe overflow.

When comparing our results with the distribution predicted by the binary-population-synthesis (BPS) models, we find that the orbital period ofPG1104+243 is larger than the most likely outcome at just over 100 daysresulting from Han et al. (2003, Fig. 21). However, the period is in agreementwith the observed estimate of three to four years noted by Green et al. (2001).Furthermore, the period is also consistent with BPS models computed withinthe γ-formalism by (Nelemans 2010), and with the revised model from Chenet al. (2013). In the latter, the original BPS studies of Han et al. (2003) areimproved by adding atmospheric RLOF. Our results therefore provide strongobservational constraints for which BPS models are viable.

Clausen et al. (2012) published several limitations on the limiting mass ratiofor dynamically stable mass transfer during the RLOF phase, the way thatmass is lost to space and the transfer of orbital energy to the envelope during acommon envelope phase, based on the observed period distribution of sdB +MS binaries. They state that if long-period (Porb > 75 d) sdB+MS binariesexist, mass transfer on the red giant branch (RGB) is stable, and if sdB + MSbinaries with periods over 250 d exist, mass that leaves the system carries awaya specific angular momentum proportional to that of the donor. In both casesthe efficiency at which orbital energy is transferred to the envelope during thecommon envelope phase is below 75 %. Furthermore, in several of their BPSmodels the resulting distributions for sdB + MS binaries would include periodsup to 500 days (Clausen et al. 2012, Fig. 16). However, this formation channelcannot explain the low mass of the cool companion. In this formation channel,the cool companion would be expected to have a mass ∼ 1M.

The low mass of the MS component (0.67 ± 0.07 M) indicates that duringthe stable Roche–Lobe overflow phase only a small amount of the mass wasaccreted, and the major part must have been lost to infinity. The evolvedstate of the MS component is further evidence of this. If the companion hadaccreted a significant fraction of the original envelope of the sdB component,its progenitor should have had a much lower mass as the current 0.67 ± 0.07M. In this case, the MS component should be almost un-evolved. As the timesince the mass transfer ended cannot be longer than the core-helium burninglifetime of only 80-120 Myr, no significant evolution of the MS component canhave taken place since then.

Based on the Teff and log g determined from the Iron lines for the MS component,evolution models indicate a MS mass of 0.90 ± 0.15 M. Compared to 0.67± 0.07 M derived from the RV curves when assuming am sdB mass of 0.47M. Knowing the distance to the system could help in solving this apparentdiscrepancy, as it will significantly improve the derived parameters in for example


the SED fit. The GAIA mission will provided distances, but to obtain accuratedistances for binary systems from the GAIA data will prove a challenge as thecenter of light will change with the binary motion.

Deca et al. (2012) published a study of the long-period sdB+K system PG1018–047, which shares many similar properties with PG1104+243. It has a periodof 760 ± 6 d, a mass ratio of MsdB/MK = 0.63 ± 0.11, and they find aneffective temperature of 30500 ± 200 K and log g of 5.50 ± 0.02 dex for the sdBcomponent. However, while the cool companion in PG1104+243 contributes∼52% of the light in the V -band, in PG1018–047 it only contributes ∼6%, andis photometrically and spectroscopically consistent with a mid-K type star. Decaet al. (2012) speculate that the orbit of PG1018-047 may be quite eccentric,and that it can have formed through the merger scenario of Clausen & Wade(2011), meaning that the K-star was not involved in the evolution of the sdBstar. The low eccentricity of PG1104+243, makes such a scenario unlikely, andwe conclude that it is the first sdB+MS system that shows consistent evidencefor being formed through post-Roche-lobe overflow.

Thus, we have used high-resolution spectroscopy to solve the orbits of bothcomponents in an sdB+MS binary system, and have for the first time derivedaccurate and consistent physical parameters for both components. Furthermore,the accurate radial velocities allowed the first measurement of gravitationalredshift in an sdB binary. PG1104+243 is part of an ongoing long-termobserving program of sdB+MS binaries with HERMES at Mercator, and ananalysis of the complete sample will make it possible to refine several essentialparameters in the current formation channels. In the next chapter we presentadditional results of our ongoing effort to characterise sdB stars in wide systems.

3Three eccentric sdB binaries:

BD+293070, BD+341543 and Feige 87

In this chapter the analysis of three eccentric sdB binaries is presented using themethods explained in Chapter 2. These results have been published, but here wepresent an updated version of the analysis using all available HERMES spectra.These three binaries were the first confirmed eccentric wide sdB binaries.


The orbits of subdwarf-B + main sequence binariesII. Three eccentric systems; BD+293070, BD+341543 and

Feige 87

J. Vos, R.H. Østensen, P. Németh, E.M. Green, U. Heber, H. Van Winckel

Author contributionsJ. Vos obtained part of the observations with Mercator, analysed the spectra and derivedthe orbital parameters. Furthermore, he performed the SED fitting, spectral disentangling,spectral analysis of the disentangled MS spectra using VWA, and the determination of thesdB surface gravity from the gravitational redshift. R.H. Østensen and H. Van Winckelstarted the long term observing project of wide sdB binaries, and contributed to the discussionof the results. P. Németh analysed the flux-calibrated spectra observed with the B&Cspectrograph, which were provided by E.M. Green. U. Heber provided the high-resolution,

59

60 THREE ECCENTRIC SDB BINARIES

high signal-to-noise spectra of BD+341543 observed with the FOCES spectrograph.

ABSTRACT

Context. The predicted orbital-period distribution of the subdwarf-B (sdB)population is bi-modal with a peak at short ( < 10 days) and long ( > 250 days)periods. Observationally, many short-period sdB systems are known, but thepredicted long period peak is missing as orbits have only been determined for afew long-period systems. As these predictions are based on poorly understoodbinary-interaction processes, it is of prime importance to confront the predictionswith reliable observational data. We therefore initiated a monitoring programto find and characterize long-period sdB stars.Aims. In this paper we aim to determine the orbital parameters of the three long-period sdB+MS binaries BD+293070, BD+341543 and Feige 87, to constraintheir absolute dimensions and the physical parameters of the components.Methods. High-resolution spectroscopic time series were obtained withHERMES at the Mercator telescope on La Palma, and analyzed to determinethe radial velocities of both the sdB and MS components. Photometry from theliterature was used to construct the spectral-energy distribution (SED) of thebinaries. Atmosphere models were used to fit these SEDs and to determine thesurface gravities and temperatures of both components of all systems. Spectralanalysis was used to check the results of the SEDs.Results. An orbital period of 1254 ± 5 d, a mass ratio of q = 0.37 ± 0.01 and asignificant non-zero eccentricity of e = 0.15 ± 0.01 were found for BD+293070.For BD+341543 we determined Porb = 972 ± 2 d, q = 0.57 ± 0.01 and againa clear non-zero eccentricity of e = 0.16 ± 0.01. Last, for Feige 87 we foundPorb = 938 ± 2 d, q = 0.55 ± 0.01 and e = 0.11 ± 0.01.Conclusions. BD+293070, BD+341543 and Feige 87 are long period sdB +MS binaries on clearly eccentric orbits. These results are in conflict with thepredictions of stable Roche-lobe overflow models.

3.1 Introduction

Hot subdwarf-B (sdB) stars are core helium burning stars with a very thinhydrogen envelope (MH < 0.02 M), and a mass close to the core heliumflash mass ∼ 0.47 M (Saffer et al. 1994; Brassard et al. 2001). Currently,there is a consensus that sdB stars are formed by binary evolution only, andseveral evolutionary channels have been proposed, where binary-interactionphysics plays a major role. Close binary systems can be formed in a common

SPECTROSCOPY 61

envelope (CE) ejection channel (Paczynski 1976), while stable Roche-lobeoverflow (RLOF) can produce wide sdB binaries (Han et al. 2000, 2002). Analternative formation channel forming a single sdB star is the double whitedwarf (WD) merger, where a pair of white dwarfs spiral in to form a single sdBstar (Webbink 1984). The focus of this paper is on the long period sdB binaries.

Until now only a few long period sdB binaries are known (Green et al. 2001;Østensen & Van Winckel 2011, 2012; Deca et al. 2012; Barlow et al. 2012; Voset al. 2012), and the current studies show that there are still large discrepanciesbetween theory and observations (Geier 2013). In a recent response to thesediscoveries Chen et al. (2013) have revisited the RLOF models of Han et al.(2003) with more sophisticated treatment of angular momentum loss. Theirrevised models show mass – orbital period relations that increase substantiallyas a function of composition, with solar metallicity models reaching periods upto 1100 d. They also note that by allowing the transfer of material extendingbeyond the classic Roche lobe (atmospheric RLOF) they can reach periods aslong as ∼1600 d.

In this paper we present the orbital and atmospheric parameters of the threelong-period sdB + MS binaries BD+293070, BD+341543 and Feige 87, usingthe methods described in Vos et al. (2012). In Sect. 3.2 the radial velocities aredetermined for both components after which the orbital parameters are derived.Using the obtained mass ratio, the atmospheric parameters are derived fromthe spectral-energy distribution (Sect. 3.3), and spectral analysis (Sect. 3.4).Furthermore, the surface gravity of the sdB component is estimated based onthe gravitational redshift in Sect. 3.5. Finally in Sect. 3.6 and 3.7 all results aresummarized. BD+293070, BD+341543 and Feige 87 are part of a long-termspectroscopic monitoring program, and preliminary results of these and fivemore systems in this program were presented in Østensen & Van Winckel (2011,2012).

3.2 Spectroscopy

High-resolution spectroscopic observations of BD+293070, BD+341543 andFeige 87 were obtained with the HERMES spectrograph between Januari 2009and March 2015. The observing mode and wavelength stability of HERMES isdiscussed in the Sect. 2.2. In total there were 42 spectra taken of BD+293070,54 of BD+341543 and 45 of Feige 87. These observations are summarized inTable 3.1. The HERMES pipeline v6.0 was used for the basic reduction of thespectra, including barycentric correction.

For BD+341543 there is one more high-resolution spectrum available, taken


with the FOCES spectrograph (Fiber-Optics Cassegrain Echelle Spectograph, R= 30 000, 3600-6900 Å) attached to the 2.2-m telescope at Calar Alto observatory,Spain. This spectrum was observed in February 2000 (HJD = 2451576.5166),and is reduced as described in Pfeiffer et al. (1998) using the IDL macrosdeveloped by the Munich Group.

Flux-calibrated spectra of BD+293070, BD+341543, and Feige 87 were takenwith the Boller and Chivens (B&C) spectrograph attached to the Universityof Arizona’s 2.3m Bok telescope located on Kitt Peak. All three stars wereobserved using a 2.5” slit and 1st order 400/mm grating blazed at 4889 Å,with a UV-36 filter to block 2nd order light. These parameters provided a 9 Åresolution over the wavelength range 3600-6900 Å. BD+293070 was observed on25-06-2000 with an exposure time of 30 s, resulting in an overall S/N of 235 perresolution element (134 per pixel, and slightly higher in the range 3600-5000 Å).BD+341543 was observed once on 17-09-1998 and five additional times between2005 and 2007, for a total exposure time of 260 s and a (formal) S/N ∼ 750 (435per pixel). Feige 87 was observed twice, on 10-03-1999 and 06-06-1999, for a totalexposure time of 210 s and S/N of 345 (195 per pixel). The spectra were bias-subtracted, flat-fielded, optimally extracted, and wavelength calibrated usingstandard IRAF1 tasks. They were flux calibrated using either BD+284211 orFeige 34 as flux standards. The individual spectra for BD+341543 and Feige 87were combined by determining the cross-correlation velocities using only theBalmer and helium lines, and shifting each spectrum to the mean sdB velocitybefore combining (although the velocity shifts were always small, less than 1/3of a pixel, compared to the spectral resolution 3.15 pixels). While these spectrahave a too low resolution to obtain radial velocities, they are used to determinespectroscopic parameters of both components in Section 3.4.3.

Table 3.1: The observing dates (mid-time of exposure) and exposure times ofthe spectra of BD+293070, BD+341543 and Feige 87

BJD Exp. BJD Exp. BJD Exp.–2450000 s –2450000 s –2450000 s

BD+2930705029.46427 1800 5701.55126 1750 6139.42921 14005040.45677 1800 5718.47286 2000 6145.43519 10005055.46749 1200 5745.42605 1950 6176.47063 9005060.39796 1210 5761.56846 1800 6310.78677 7205321.61365 1500 5772.58405 1950 6374.67825 1500


1IRAF is distributed by the National Optical Astronomy Observatory; seehttp://iraf.noao.edu/

SPECTROSCOPY 63

Table 3.1 – continued from previous pageBJD Exp. BJD Exp. BJD Exp.–2450000 s –2450000 s –2450000 s5351.47390 1500 5795.38941 1500 6393.53333 15005425.40235 1600 5808.37209 1500 6439.54435 15005619.71851 1500 5829.36956 1500 6452.46763 8505638.66109 1700 5999.76186 900 6711.73992 10205648.67319 1500 6027.69896 1500 6771.53161 15005658.68162 1600 6049.55591 1500 6809.45787 13505670.52248 1500 6082.54844 1150 6833.56340 15005678.54949 1940 6135.43246 800 6835.63120 10105686.53274 1800 6135.44229 800

BD+3415435166.52522 700 5942.60393 1200 6358.38785 9505218.57295 600 5949.51901 900 6374.43191 9005238.55047 900 5954.49672 1000 6389.43526 14005238.56638 900 5965.42615 1100 6416.38608 7205238.57745 900 5968.63459 900 6535.74589 9005240.53158 1800 5978.57215 1050 6557.68022 9005475.69181 900 6007.50119 820 6609.57142 9005502.72812 900 6043.43008 900 6615.71341 6905507.69553 900 6058.39694 1500 6617.56077 9005553.57671 900 6066.37564 1800 6664.67231 9005614.56681 900 6068.38939 1250 6708.61257 10205621.55596 800 6069.36684 900 6771.43562 8005658.34764 594 6175.67953 900 6790.39303 9005807.73260 600 6193.72815 1000 6791.40188 9005844.67076 900 6254.74395 900 6904.73029 4805861.71311 900 6312.67995 900 6996.69116 8005887.68071 900 6321.59109 900 6996.73055 7005932.58690 900 6322.60697 900 7082.53759 900

Feige 875028.44883 1800 5663.54111 1200 6141.40159 18005028.47026 1800 5666.53453 1800 6196.34153 18005028.49174 1800 5937.79667 1800 6301.76786 18005251.71445 890 5964.69002 3400 6318.75464 18005251.72468 390 5965.58427 2400 6320.70017 18005341.57223 1603 5968.65170 1800 6392.51615 24005351.41840 2700 5990.72249 1800 6444.39039 18005371.41748 2700 6017.61446 1500 6621.76365 10805566.78096 2700 6042.47734 1800 6707.72323 1100



Table 3.1 – continued from previous pageBJD Exp. BJD Exp. BJD Exp.–2450000 s –2450000 s –2450000 s5616.64021 2100 6060.41466 1800 6716.58439 18005622.70741 720 6064.56735 1800 6753.59569 18005622.73169 2000 6066.46515 1800 6809.43298 12005656.60318 911 6068.45739 1800 6834.43462 17005658.58702 1210 6082.45036 1800 7009.78027 18005660.61069 704 6138.43744 1800 7083.64505 1800

3.2.1 Radial Velocities

Østensen & Van Winckel (2011) determined preliminary orbital periods ofBD+293070, BD+341543 and Feige 87 based on the radial velocities of thecool companion and assuming circular orbits, resulting in respectively 1160 ±67 days, 818 ± 21 days and 915 ± 16 days. These long periods allow us tosum spectra that are taken within a five-day interval to increase the signal tonoise (S/N), without significantly smearing or broadening the spectral lines.This five-day interval corresponds to about 0.5 % of the orbital period, and amaximum radial velocity shift of 0.06 km s−1. After this merging, 38 spectraof BD+293070, 42 spectra of BD+341543 and 32 spectra of Feige 87 remain,with a S/N varying from 25 to 50. These spectra with the averaged BJD incase of the merged spectra are displayed in Table 3.2, 3.3 and 3.4.

The determination of the radial velocites of both the MS and sdB component isperformed as explained in Sect. 2.2.1. For the MS components of BD+341543and Feige 87, a G2-type mask was used, while for BD+293070 and F0-typemask was used with the hermesVR cross-correlation tool v6.0. For the sdBcomponent the He i 5875.61 Å line is cross correlated with a high-resolutionsynthetic sdB spectrum from the LTE grids of Heber et al. (2000). The finalradial velocities of both the MS and sdB component of BD+293070 togetherwith their errors are given in Table 3.2, while those of BD+341543 and Feige 87can be found in Table 3.3 and 3.4 respectively.

The FOCES spectrum of BD+341543 is analyzed in exactly the same way asthe HERMES spectra to determine the radial velocities of both components.The results are given in Table 3.3 together with the HERMES results.

SPECTROSCOPY 65

Table 3.2: The radial velocities of both components of BD+293070.

MS component sdB componentBJD RV Error RV Error-2450000 km s−1 km s−1 km s−1 km s−1

5029.4643 -52.94 0.23 -68.89 0.725040.4568 -52.65 0.24 -66.79 0.575057.9327 -53.12 0.19 -67.00 0.515321.6137 -62.66 0.25 -39.14 0.535351.4739 -62.66 0.22 -41.02 0.575425.4024 -63.47 0.28 -37.21 0.915619.7185 -62.06 0.26 -47.97 0.455638.6611 -61.42 0.27 -46.00 0.625648.6732 -60.68 0.22 -49.70 0.435658.6816 -61.02 0.22 -49.85 0.505670.5225 -60.40 0.32 -49.67 0.665678.5495 -60.55 0.26 -49.98 0.475686.5327 -59.74 0.25 -50.57 1.615701.5513 -60.23 0.20 -50.18 0.455718.4729 -59.44 0.26 -51.09 0.545745.4260 -58.83 0.28 -53.07 0.625761.5685 -58.14 0.27 -54.30 0.535772.5840 -58.72 0.39 -54.23 1.125778.4452 -58.00 0.24 -56.57 0.535795.3894 -57.77 0.24 -55.21 0.535808.3721 -57.33 0.24 -55.38 0.695829.3696 -56.26 0.24 -60.02 0.505999.7619 -53.28 0.23 -68.55 0.636027.6990 -52.72 0.21 -69.76 0.556049.5559 -52.41 0.26 -70.67 0.536082.5484 -52.19 0.24 -72.15 0.606136.7680 -51.15 0.23 -72.23 0.536145.4352 -51.58 0.26 -73.50 0.606176.4706 -51.48 0.26 -70.21 0.666310.7868 -54.53 0.28 -64.58 0.796374.6782 -55.42 0.27 -60.60 0.496393.5333 -56.98 0.26 -59.76 0.646439.5444 -58.07 0.29 -55.67 0.636452.4676 -59.02 0.21 -53.75 0.596711.7399 -63.68 0.26 -40.05 0.496771.5316 -63.61 0.24 -40.47 0.706809.4579 -62.88 0.25 -44.35 0.646834.5973 -61.79 0.22 -42.98 0.41


Table 3.3: The radial velocities of both components of BD+341543.


1576.5166a 29.23 0.25 38.66 1.025166.5252 38.09 0.09 23.03 1.285218.5730 38.07 0.04 23.43 0.715239.0565 38.33 0.03 21.81 0.415475.6918 28.83 0.07 38.96 0.695505.2118 28.09 0.04 39.47 0.495553.5767 26.43 0.04 41.83 0.475621.5560 26.07 0.04 42.88 0.625658.3476 26.57 0.07 41.32 0.815807.7326 29.51 0.04 38.22 0.505844.6708 30.73 0.04 36.82 0.545861.7131 31.04 0.03 35.74 0.485887.6807 31.87 0.04 33.10 0.435932.5869 33.08 0.04 30.10 0.445942.6039 33.57 0.03 30.98 0.405952.0079 34.45 0.02 30.51 0.385967.0304 34.77 0.03 29.09 0.435978.5721 34.39 0.04 27.55 0.676007.5012 35.18 0.03 26.34 0.546043.4301 36.19 0.04 25.47 0.596058.3969 36.27 0.08 23.77 1.296068.0440 35.78 0.04 24.79 0.426254.7439 37.77 0.02 24.03 0.616312.6799 35.70 0.03 27.09 0.426322.0990 35.31 0.02 26.85 0.406358.3879 33.24 0.04 31.66 0.446374.4319 32.56 0.04 33.10 0.486389.4353 31.72 0.02 33.12 0.366416.3861 30.20 0.04 35.39 0.626535.7459 26.07 0.06 43.61 0.716557.6802 26.30 0.10 43.80 0.436609.5714 26.24 0.04 41.41 0.876616.6371 26.10 0.06 42.35 0.756664.6723 27.24 0.03 39.52 0.576708.6126 27.56 0.04 40.59 0.616771.4356 29.36 0.04 37.34 0.616904.7303 33.02 0.07 30.98 0.686996.7109 35.72 0.02 28.05 0.377082.5376 37.36 0.04 23.82 0.96

(a) FOCES spectrum

SPECTROSCOPY 67

−70

−60

−50

−40

−101 0

.00.2

0.4

0.6

0.8

1.0

−303

20

25

30

35

40

45

−101 0

.00.2

0.4

0.6

0.8

1.0

−303

20

25

30

35

40

45

50

−101 0

.00.2

0.4

0.6

0.8

1.0

−202

Radialvelocity(km/s)

Pha

se

BD

+29

307

0B

D+

34 1

543

Feig

e87

Figu

re3.1:

The

radial-velocity

curves

forBD+29 3070(le

ft),BD+34 1543(center)

and

Feige8

7(right).

Top:

spectroscopicorbitals

olution(solid

line:

MS,

dashed

line:

sdB),

andtheob

served

radial

velocitie

s(H

ERMES

MS

comp.:blue

circles,

HER

MES

sdB

comp.:greenop

encircles,

FOCES

MScomp.:magenta

triang

le,F

OCES

sdB

comp.:m

agenta

open

squa

re).

Themeasuredsystem

velocit

iesof

both

compo

nentsa

reshow

nby

ado

tted

line.

Middle:

resid

uals

oftheMScompo

nent.Bo

ttom

:resid

uals

ofthesdB

compo

nent.


Table 3.4: The radial velocities of both components of Feige 87.


5028.4703 27.84 0.02 42.68 0.345341.5722 31.03 0.05 40.81 0.475351.4184 31.73 0.16 36.85 0.615371.4175 32.33 0.10 35.76 0.285616.6402 40.54 0.13 22.91 0.455622.7196 40.04 0.10 20.98 0.325658.6003 40.33 0.08 22.48 0.605665.0378 40.17 0.08 21.12 0.395937.7967 29.72 0.06 42.02 0.395966.3087 28.28 0.11 43.66 0.285990.7225 26.68 0.07 47.41 0.476017.6145 24.14 0.12 47.51 0.796042.4773 24.42 0.15 50.36 0.596062.4910 24.40 0.14 51.33 0.696082.4504 23.80 0.11 51.54 0.436139.9195 23.68 0.06 51.56 0.366301.7679 32.12 0.12 36.58 0.386319.7274 33.35 0.05 34.73 0.276707.7232 37.80 0.22 26.96 0.456716.5844 37.68 0.11 25.48 0.426753.5957 36.51 0.12 29.62 0.376834.4346 31.60 0.04 33.22 0.507009.7803 23.89 0.03 51.39 0.697083.6450 24.60 0.11 49.37 0.51

3.2.2 Orbital parameters

Similar as outlined in Sect. 2.2.2, the orbital parameters of the sdB and MScomponents are calculated by fitting a Keplerian orbit to the radial velocitymeasurements, while adjusting P , T0, e, ω, KMS, KsdB, γMS and γsdB. As afirst guess, the results of Østensen & Van Winckel (2011) were used. The radialvelocity measurements were weighted according to their errors as w = 1/σ.For each system, the Lucy & Sweeney (1971) test was used to check if theorbit is significantly eccentric. In the fitting process, the system velocities ofboth components are allowed to vary independently of each other, to allow forgravitational redshift effects in the sdB component (see Sects. 2.4 and 3.5). The

SPECTROSCOPY 69

Table3.5:

Spectroscopicorbitals

olutions

forbo

ththemain-sequ

ence

(MS)

andsubd

warf-B

(sdB

)compo

nent

ofBD+29 3070,B

D+34 1543an

dFe

ige8

7.ade

notesthesemi-m

ajor-axisof

theorbit.

The

quoted

errors

arethe

stan

dard

deviationfro

mtheresults

of5000

iteratio

nsin

aMon

teCarlo

simulation.

BD+29 3070

BD+34 1543

Feige8

7Pa

rameter

MS

sdB

MS

sdB

MS

sdB

P(d)

1254±

5972±

2938±

2T

02453966±

102451519±

112454232±

20e

0.15±

0.01

0.16±

0.01

0.11±

0.01

ω1.88±

0.09

1.58±

0.07

0.08±

0.15

q0.37±

0.02

0.57±

0.01

0.55±

0.01

γ(km

s−1 )

−57.79±

0.15

−56.2±

0.5

32.10±

0.06

33.12±

0.15

32.90±

0.08

34.95±

0.17

K(km

s−1 )

6.22±

0.25

16.7±

0.5

5.91±

0.07

10.31±

0.22

8.40±

0.10

15.15±

0.25

asin

i(R)

153±

15407±

35112±

2196±

4155±

2278±

5M

sin3i(M)

1.11±

0.15

0.41±

0.05

0.27±

0.01

0.15±

0.01

0.80±

0.03

0.45±

0.02


uncertainties on the final parameters are obtained using 5000 iterations in aMonte-Carlo simulation where the radial velocities are perturbed based on theirerrors. The spectroscopic parameters of BD+293070 and BD+341543 areshown in Table 3.5. The radial-velocity curves and the best fits are plotted inFig. 3.1.

Feige 87 (= PG1338+611) has been studied by Barlow et al. (2012) as part ofa long term observing program with the Hobby-Eberly telescope lasting fromJanuary 2005 till March 2008. They published radial velocities for both the MSand sdB component. However, Barlow et al. (2012) find a difference in systemicvelocity for the MS and sdB component of γsdB − γMS = −2.1 ± 1.0 km s−1

(compared to γsdB−γMS = 2.05±0.18 km s−1 for the HERMES spectra), whichthey attribute to gravitational redshift. If this was caused by gravitationalredshift, this shift would mean that the MS component has a higher surfacegravity than the sdB component, a highly unlikely situation (see also Sect.3.5). A more plausible cause can be found in the lines used to derive the radialvelocities of the sdB component. Barlow et al. (2012) used both the He i λ4472 and He i λ 5876 lines in their cross correlation, but when comparing theHe i λ 4472 line with a synthetic G2-type spectrum, it is clear that this lineis significantly contaminated by spectral features of the cool companion. Inthe analysis of the radial-velocity curves of Feige 87, we used the results ofBarlow et al. (2012) of the MS component, but discarded the results of the sdBcomponent. The phase-folded radial velocity curve of the HERMES data isshown in Fig. 3.1, while the radial velocity curve of both HERMES and Barlowet al. (2012) is shown in Fig. 3.2. The spectroscopic parameters of Feige 87 aregiven in Table 3.5.

3.3 Spectral Energy Distribution

The spectral-energy distribution (SED) of the systems can be used to determinethe spectral type of the MS and sdB component. We used photometric SEDswhich were fitted with model SEDs to determine both the effective temperatureand surface gravity of both components. The method used to fit binary modelspectra to the observed SEDs is described in Sect. 2.3.2.


3500

4000

4500

5000

5500

6000

20

30

40

50


BJD

-245

0000

Figu

re3.2:

The

radial

velocity

curveof

Feige8

7,show

ingthespectroscopicorbitals

olution(solid

line:

MS,

dashed

line:

sdB),

andtheob

served

radial

velocitie

s(H

ERMES

:bluecircles,

Barlow

etal.:

greensqua

res,

filledsymbo

ls:MScompo

nent,o

pensymbo

ls:sdB

compo

nent).

The

measuredsystem

velocitie

sof

both

compo

nentsba

sedon

the

HER

MES

data

andtheMSradial

velocitie

sof

Barlo

wet

al.areshow

nby

ado

tted

redlin

e,while

thesystem

velocity

ofthesdB

compo

nent

basedon

theda

taof

Barlo

wet

al.is

plottedin

agreendo

tted

line.


3.3.1 Photometry

To collect the photometry of both systems the subdwarf database2 (Østensen2006), which contains a compilation of data on hot subdwarf stars collectedfrom the literature, is used. These photometric measurements are supplementedwith photometry obtained from several other catalogs as listed in Table 3.6.In total we obtained 17 photometric measurements for BD+293070, 14 forBD+341543 and 11 for Feige 87. Both accurate photometric measurements atshort and long wavelengths are used to establish the contribution of the hotsdB component and the cool MS component.

Table 3.6: Photometry of BD+293070, BD+341543 and Feige 87 collectedfrom the literature.

Band Wavelength Width Magnitude ErrorÅ Å mag mag

BD+293070Johnson Ua 3640 550 10.030 0.020Johnson Ba 4450 940 10.600 0.020Johnson V a 5500 880 10.420 0.020Johnson Bb 4450 940 10.590 0.013Johnson V b 5500 880 10.416 0.013Johnson V c 5500 880 10.376 0.052Cousins Ic 7880 1490 10.026 0.053Geneva Ua 3460 170 10.264 0.010Geneva Ba 4250 283 9.638 0.010Geneva V a 5500 298 10.359 0.010Geneva B1a 4020 171 10.523 0.011Geneva B2a 4480 164 11.103 0.011Geneva V 1a 5400 202 11.075 0.011Geneva Ga 5810 206 11.476 0.0112MASS Jd 12410 1500 9.773 0.0202MASS Hd 16500 2400 9.621 0.0232MASS Ks

d 21910 2500 9.546 0.022BD+341543

Johnson Bb 4450 940 10.293 0.014Johnson V b 5500 880 10.145 0.013Johnson V c 5500 880 10.156 0.048Cousins Ic 7880 1490 9.758 0.077


2http://catserver.ing.iac.es/sddb/


Table 3.6 – continued from previous pageBand Wavelength Width Magnitude Error

Å Å mag magGeneva Ua 3460 170 9.759 0.008Geneva Ba 4250 283 9.362 0.009Geneva V a 5500 298 10.140 0.007Geneva B1a 4020 171 10.215 0.010Geneva B2a 4480 164 10.865 0.010Geneva V 1a 5400 202 10.854 0.010Geneva Ga 5810 206 11.260 0.0102MASS Jd 12410 1500 9.485 0.0232MASS Hd 16500 2400 9.326 0.0332MASS Ks

d 21910 2500 9.207 0.018Feige 87

Johnson Be 4450 940 11.598 0.050Johnson V e 5500 880 11.693 0.050Cousins Re 6470 1515 11.671 0.050Cousins Ie 7880 1490 11.623 0.050Stromgren uf 3500 300 11.800 0.045Stromgren vf 4110 190 11.747 0.045Stromgren bf 4670 180 11.698 0.045Stromgren yf 5470 230 11.730 0.0902MASS Jd 12410 1500 11.484 0.0222MASS Hd 16500 2400 11.359 0.0262MASS Ks

d 21910 2500 11.312 0.022(a)Mermilliod et al. (1997); (b)Kharchenko (2001); (c)Richmond (2007)(d)Cutri et al. (2003); (e)Allard et al. (1994); (f)Bergeron et al. (1984)

3.3.2 Results

The SEDs of all three systems are plotted in Fig. 3.3, while the confidenceintervals are plotted in Fig. 3.4. The best fit parameters together with their95% confidence intervals are given in Table 3.7.

The photometry of BD+293070 has a rather large spread, which results in largeuncertainties on the derived parameters. The best fitting effective temperaturesare Teff,MS = 6630 ± 550 K and Teff,sdB = 30600 ± 5000 K, with surface gravitiesof log gMS = 4.43 ± 0.5 dex and log gsdB = 5.82 ± 0.5 dex. The reddeningis E(B − V ) = 0.008+0.042

−0.008 mag. This value is consistent with the maximumreddening E(B − V )max = 0.052 derived from the dust maps of Schlegel et al.


1234

−303

246

−2

−101

12

−101

BD

+29

3070

BD

+34

1543

Feig

e87

3·1

031·1

043·1

031·1

043·1

031·1

04W

avel

engt

h(A

)

(O-C)·10λFλ·1013

(erg/s/cm2)

Figu

re3.3:

The

spectral

energy

distrib

utionan

d(O

-C)of

BD+29 3070(le

ft),B

D+34 1543(center)

andFe

ige8

7(right).

The

measurements

aregivenin

blue

,the

integrated

synthetic

mod

elsareshow

nin

black,

whe

reaho

rizon

tal

errorba

rindicatesthewidth

ofthepa

ss-ban

d.The

best

fittin

gmod

elis

plottedin

red.

Inthebo

ttom

pane

lsthe

resid

uals

areplottedforeach

system

.


Table3.7:

The

results

oftheSE

Dfit

forBD+29 3070,

BD+34 1543an

dFe

ige8

7,together

with

the95%

prob

ability

intervalsde

rived

from

theconfi

denc

eintervalsplottedin

Fig.

3.4.

BD+29 3070

BD+34 1543

Feige8

7Pa

rameter

Best

fit95%

Best

fit95%

Best

fit95%

Teff,M

S(K

)6630

6200

–7200

6210

6000

–6440

5840

5300

–6400

logg

MS(dex)

4.43

3.95

–4.90

4.19

4.05

–4.35

4.40

4.15

–4.60

Teff,s

dB(K

)20600

25000–35000

36700

30000–/

27400

21000–33000

logg

sdB(dex)

5.82

5.30

–6.20

5.92

5.75

–6.05

5.54

5.20

–5.80

E(B−V)

0.008

0–0.052

0.007

0–0.068

0.012

0–0.057

Forsomepa

rameterstheconfi

denceintervalscouldno

tbe

calculated

dueto

thelim

ited

rang

eof

themod

elatmosph

eres.The

uppe

rlim

itof

Teff

,sdB

=35

000K

forBD+29

30

70is

basedon

theab

senceof

Hei

ilines

initsspectrum

.


6.0 6.2 6.4 6.6 6.8 7.0 7.2

4.0

4.2

4.4

4.6

4.8

log

gM

S(d

ex)

20 25 30 35

5.4

5.6

5.8

6.0

6.2

log

gsd

B(d

ex)

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.420

25

30

35

Teff

sdB

(100

0K

)

4.2 4.4 4.6 4.8

5.4

5.6

5.8

6.0

6.2

6.4

log

gsd

B(d

ex)

0.00 0.01 0.02 0.03 0.04 0.05

6.4

6.6

6.8

7.0

7.2

Teff

MS

(100

0K

)

57 65 74 83 91 100

6.0 6.1 6.2 6.3 6.4 6.5

4.15

4.20

4.25

4.30

4.35

30 32 34 36 38 40

5.6

5.7

5.8

5.9

6.0

5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5

26

28

30

32

34

36

38

40

3.8 3.9 4.0 4.1 4.2 4.35.4

5.5

5.6

5.7

5.8

5.9

6.0

6.1

0.00 0.01 0.02 0.03 0.04 0.05 0.06

6.0

6.1

6.2

6.3

6.4

6.5

56 65 73 82 91 100

5.2 5.4 5.6 5.8 6.0 6.2 6.44.0

4.1

4.2

4.3

4.4

4.5

4.6

4.7

18 20 22 24 26 28 30 32

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.2 5.4 5.6 5.8 6.0 6.2 6.4

24

26

28

30

32

34

4.2 4.4 4.6 4.85.2

5.4

5.6

5.8

6.0

0.000 0.004 0.008 0.012

5.6

5.8

6.0

6.2

6.4

58 66 75 83 92 100

BD+293070 BD+341543

Teff MS (1000 K)

Teff sdB (1000 K)

Teff MS (1000 K)

log g MS (dex)

E(B − V ) (mag)

Feige 87

Figure 3.4: The confidence intervals of the SED fits of BD+293070,BD+341543 and Feige 87. The different colors show the cumulative densityprobability connected to the χ2 statistics given by Equation 2.6. The bestfitting values and there errors are plotted in black.


(1998). The χ2 of the best fit is 51.0 which is more than three times as highas the expected value (Nobs - Nfree = 12), indicating that the stated errors onthe photometry are too small. When calculating the confidence intervals, ascaling factor of 4.6 is used when converting the χ2 values to probabilities. Ascan be seen on the plots of the CIs in Fig. 3.4, there is no strong constraintpossible on the parameters of the MS component. From all three objects, the MSparameters of BD+293070 are the least constrained. The effective temperaturecan vary between 6400 and 7100 K and the surface gravity between 4.0 and4.8 without significantly affecting the fit. This large uncertainty on the MSparameters is likely caused by the rather large difference between the fluxesobserved in the Johnson filters and the Geneva filters. Furthemore, the surfacegravities of both components are correlated. Increasing the surface gravity ofone component can be countered by increasing the log g of the other componentas well, and thus effectively shrinking the radii of both components. For the sdBcomponent there is a strong correlation between Teff and log g visible. The effecton the atmosphere models of an increase in Teff can be diminished by increasinglog g, and thus decreasing the radius. However, it is possible to provide anupper limit on the effective temperature as the He ii lines are not visible in theHERMES spectra, indicating that the effective temperature is below 35000 K(Heber 2009). There is also no strong constraint on the reddening of the system,but this value is limited by the upper limits derived from the dust maps.

For BD+341543 fourteen photometric measurements are available, with asmaller spread than for BD+293070. The effective temperatures of thecomponents derived from the SED fit are Teff,MS = 6210 ± 250 K and Teff,sdB= 36700 ± 5000 K, while a surface gravity of log gMS = 4.19 ± 0.20 dex andlog gsdB = 5.92 ± 0.40 dex are found. The reddening is determined to beE(B−V ) = 0.007+0.061

−0.007 which is consistent with the maximum reddening foundon the dust maps of Schlegel et al. (1998), and supported by the absence of sharpinterstellar absorption lines in the spectrum. As can be seen in Fig. 3.4 theprobability distributions of the MS components parameters form a Gaussian-likepattern, and the Teff and log g of the MS component have stronger constraintsthan for BD+293070, independently of the parameters for the sdB components.The uncertainty on the effective temperature and surface gravity of the sdBcomponent is larger as both parameters are correlated in the same way as forBD+293070. However, the presence of clearly visible He i and He ii lines inthe HERMES spectra of BD+341543 indicates that the effective temperatureshould be between 35000 and 50000 K.

In the case of BD+341543 the parallax was measured by Hipparcos (vanLeeuwen 2007) to be 4.22 ± 1.72 mas. This parallax could be used to fixthe distance to the system in the SED fitting process. Since the parallax isof the same order as the projected size of the orbit for this system, one may


presume that the Hipparcos parallax is unreliable, but as both components ofBD+341543 have a very similar flux in the Hipparcos pass band, the center oflight does not change during the orbit. However, the uncertainty on the parallaxis large (∼ 40%). Using this parallax to fix the distance to the system, withoutpropagating the uncertainty on it, results in effective temperatures of Teff,MS =6230 ± 100 K and Teff,sdB = 37900 ± 3500 K, and surface gravities of log gMS= 4.04 ± 0.05 dex and log gsdB = 5.81 ± 0.15 dex. The reddening is determinedto be E(B − V ) = 0.011+0.057

−0.011. The uncertainties are much smaller as whenthe distance is included as a free parameter, especially on the surface gravity.However, due to the large uncertainty on the parallax, only the atmosphericparameters derived with the distance as a free parameter are used.

For Feige 87 there are only eleven photometric measurements found in theliterature, but there is a very good agreement between all the measurementsin the different bands. When checking the residuals of the fit, Feige 87 has thesmallest spread of all three systems, with a total χ2 of 2.9. The SED results inan effective temperature of Teff,MS = 5840±500K and Teff,sdB = 27400±5000Kfor the MS and sdB component, together with a surface gravity of log gMS =4.40 ± 0.30 dex and log gsdB = 5.50 ± 0.50 dex. The reddening of the system isfound to be E(B − V ) = 0.012+0.045

−0.012, consistent with the results from the dustmaps of Schlegel et al. (1998). On Fig. 3.4 the probability distributions showa similar pattern as for BD+293070. Although the distribution in Teff,MS –log gMS and Teff,sdB – Teff,MS have a clearer Gaussian pattern, indicating thatthey are determined more accurate as for BD+293070.

The advantage of having an estimate of the distance to the target is clear whenconsidering the uncertainties of the atmospheric parameters. Because the radiiof the components are derived from their surface gravity, limiting the distancewill also limit the radii. If the distance is accurately known, the accuracy ofthe surface gravity can be increased by a factor ten compared to when thedistance is treated as a free parameter. Determining atmospheric parametersfrom photometry will greatly benefit from the Gaia mission that will deriveaccurate distances of about a billion stars.

3.4 Spectral analysis

The atmospheric parameters determined from the SEDs can be checked using thespectra. The HERMES echelle spectra are not easy to flux calibrate accurately,and therefore not well suited to fit model atmospheres, but it is possible tosubtract the continuum contribution of the sdB component as explained in thefollowing subsection. The resulting spectra of the MS components can be used

SPECTRAL ANALYSIS 79

to derive atmospheric parameters based on the Fe i and Fe ii lines. Apart fromthe HERMES spectra, we have obtained flux calibrated long-slit spectra withthe Bok telescope (see Sect. 3.4.3). The resolution of these spectra is too lowto determine radial velocities, but they can be used to fit model atmospheresand provide an independent set of atmospheric parameters.

3.4.1 Disentangling

When the spectroscopic parameters of both components in a system are known,it is possible to extract the spectrum of the MS component from the combinedspectrum. This is done by subtracting a synthetic sdB spectrum with a surfacegravity and effective temperature determined from the SED fit. As the sdBcomponent only has a few lines and only wavelength regions that do not containBalmer or He lines are used, this is equivalent to subtracting the continuumcontribution of the sdB component.

Each HERMES spectrum is treated separately. First the HERMES responsecurve is removed from the spectrum after which its continuum is determinedby fitting a low degree polynomial to the spectrum. Then the continuumcontribution of the sdB component is subtracted following:

Ftot(λ) = FMS(λ) C(λ)1 + Frat(λ) + FsdB(λ)C(λ) · Frat(λ)

1 + Frat(λ) , (3.1)

where Ftot is the total flux in the HERMES spectrum, FMS and FsdB are thenormalized fluxes of the MS and sdB components, C is the continuum of theHERMES spectrum, and Frat is the ratio of the sdB flux to the MS flux. Asonly the regions where the sdB component does not have significant lines areused, FsdB(λ) = 1 for every λ. The normalized MS spectrum is then given by:

FMS(λ) =(Ftot(λ)− C(λ) · Frat(λ)

1 + Frat(λ)

)/

(C(λ)

1 + Frat(λ)

). (3.2)

The obtained MS spectra are shifted to zero velocity, and averaged weighted bytheir S/N. An example region of the final extracted spectra for both systems isplotted in Fig. 3.5.

This way of disentangling the spectra is dependent on the atmosphericparameters of both components, which means that the disentangling processneeds to be repeated when a new set of atmospheric parameters for the MScomponent is derived from the iron lines. This is done until a consistent resultemerges, which happened for all systems after two iterations. The parameters ofthe sdB component cannot be derived from the resulting spectra, but tests wereperformed to check their influence. As it turns out, changing the Teff or log g of


0.0 0.2 0.4 0.6 0.8 1.0

Wavelength (A)

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Flu

x

0.7

0.8

0.9

1.0

0.7

0.8

0.9

1.0

5600 5620 5640 5660 5680 5700

0.6

0.7

0.8

0.9

1.0

BD+293070

BD+341543

Feige 87

Figure 3.5: The disentangled HERMES spectra of the MS components obtainedby subtracting the sdB continuum and averaging all spectra (black full line),together with Kurucz model atmospheres based on the atmospheric parametersdetermined from the iron lines (red dotted line). Top: BD+293070, center:BD+341543, bottom: Feige 87.

the sdB component within the errors determined in the SED fitting process,does not have a significant effect on the parameters of the MS componentderived from the iron lines.

3.4.2 Atmospheric parameters and abundances

The Versatile Wavelength Analysis3 (VWA) tool developed by Bruntt et al.(2002) is used to determine the effective temperature, surface gravity,microturbulent velocity and abundances of the MS components. VWA generatessynthetic spectra using the SYNTH software of Valenti & Piskunov (1996).Atomic line data is taken from the VALD database (Kupka et al. 1999),

3https://sites.google.com/site/vikingpowersoftware/

SPECTRAL ANALYSIS 81

but the log(gf) values are adjusted so that every line measured by Wallaceet al. (1998) reproduces the atmospheric abundances by Grevesse et al. (2007).The atmosphere models are interpolated from MARCS model atmospheres(Gustafsson et al. 2008) using the solar composition of Grevesse et al. (2007).The VWA package fits abundances in a semi-automatic way. It first selects theleast blended lines in the spectra, and determines the abundances of these linesby calculating synthetic spectra for each line while iteratively changing the inputabundance until the equivalent widths of the observed and synthetic spectrummatch. The main advantage of VWA is that the synthetic spectrum includesthe contribution of neighboring lines, thus making it possible to analyze starswith a high rotational velocity. A detailed description of the VWA software canbe found in Bruntt et al. (2004, 2008); Bruntt (2009); Bruntt et al. (2010a,b).

Before the spectra are analyzed, they are carefully normalized with the RAINBOWtool of the VWA package. Then the spectra are compared to synthetic spectra,and projected rotational velocities of 52 ± 5 km s−1, 17 ± 4 km s−1 and 8 ±3 km s−1 are found for the G-star components of respectively BD+293070,BD+341543 and Feige 87. Especially BD+293070 has a high rotationalvelocity, resulting in severe line blending, which makes it difficult to derive theatmospheric parameters. These parameters are determined using only the ironlines. The effective temperature is determined by requiring the abundance ofthe Fe i lines to be independent of the excitation potential. The surface gravityis derived by requiring the same abundance for Fe i and Fe ii lines, and checkedby fitting synthetic spectra to several calcium and magnesium lines that aresensitive to changes in log g (Gray 2005). Furthermore the independence ofabundance on equivalent width gives the microturbulence velocity. When theatmospheric parameters are determined the final abundances of all measuredlines are calculated, and the overall metallicity is obtained by averaging allabundances over the measured elements weighted by the number of lines foundfor each element.

The disentangled spectrum of BD+293070 has the highest signal-to-noise ratioof the three systems (S/N ∼ 130), making it possible to derive robust parametersregardless of the high rotational blending. In total, 401 suitable lines wereselected after comparing the data with a synthetic spectrum. In the abundancedetermination process 56 lines (of which 34 Fe i and 3 Fe ii lines) were used andhad an equivalent width between 10 and 90 mÅ. The three Fe ii lines have toolarge scatter to constrain the surface gravity and are ignored for this purpose.To obtain an estimate of the surface gravity three calcium lines (Ca λ 6122, Caλ 6162, Ca λ 6439) and the magnesium triplet (Mg-1b λ 5172) are fitted withsynthetic spectra. This resulted in a surface gravity of log g = 4.3 ± 0.5 dex.Using the surface gravity determined from the Ca and Mg lines, the atmosphericparameters determined based on the Fe i lines are: Teff = 6100 ± 200 K and


vmicro = 1.50 ± 0.35 km s−1. The overall metallicity calculated using all 56lines is [M/H] = 0.09 ± 0.21. The abundances of all lines are given in Table 3.8.

Having a lower rotational velocity and still a high signal to noise (S/N ∼ 90),more useful lines (578) were found in BD+341543. After the fitting process99 lines (including 50 Fe i and 6 Fe ii lines) were fitted and had an equivalentwidth between 10 and 90 mÅ. In this case the scatter on the abundances ofthe Fe ii lines was low, we used it together with the Fe i lines to constrain thesurface gravity, resulting in log g = 4.2 ± 0.3 dex. This value is supportedby fitting synthetic spectra to the same Ca and Mg lines as for BD+293070.Based on the Fe i lines an effective temperature of Teff = 6150 ± 150 K anda microturbulence of vmicro = 1.45 ± 0.25 km s−1 were found. The averagedmetallicity is found to be slightly sub solar at [M/H] = −0.24 ± 0.12. Theabundances of all elements are shown in Table 3.8.

Feige 87 has the lowest rotational velocity, but this advantage is partly counteredby the low signal-to-noise ratio of the disentangled spectrum (S/N ∼ 35). Fromthe 428 selected lines 93 were used (including 51 Fe i lines and 3 Fe ii lines).The scatter of the Fe ii lines is high compared to the previous system, thusthey are not very reliable in constraining the surface gravity. Based on fittingsynthetic spectra to Ca and Mg lines, and on the ionization balance betweenFe i and Fe ii, we find a surface gravity of log g = 4.5 ± 0.6 dex. Based onthe Fe i lines, the effective temperature is constrained to Teff = 6175 ± 150 Kand the microturbulence results in vmicro = 1.15 ± 0.25 km s−1. The averagemetallicity based on 96 lines in total is clearly sub solar at [M/H] = −0.48 ±0.26. The abundances of all elements are shown in Table 3.8.

3.4.3 Spectral fitting

F and G type MS companions of sdB stars can be easily resolved in theoptical as they have similar brightnesses and distinct spectra. This allowsone to disentangle such composite spectra from a single observation withoutknowing the radii or fluxes of the components. Such a binary decompositionwas implemented in the XTgrid (Németh et al. 2012) spectral fitting algorithmand was used to estimate the atmospheric parameters of the components in 29composite spectra binaries. XTgrid employs the NLTE model atmosphere codeTLUSTY (Hubeny & Lanz 1995) for the subdwarf component and interpolatedMILES (Cenarro et al. 2007) template spectra for the MS companion. Thebinary spectrum is fitted with a linear combination of the two components.This method is independent from the SED and VWA analysis and was appliedfor low-resolution flux calibrated spectra obtained with the B&C spectrograph,therefore it can be used to check the consistency of the different approaches.

GRAVITATIONAL REDSHIFT 83

Table 3.8: The abundances ([El/H]) for the MS components of BD+293070,BD+341543 and Feige 87 obtained from the disentangled HERMES spectra.

BD+293070 BD+341543 Feige 87Ion [El/H] Na [El/H] Na [El/H] Na

Al i 0.26 ± 0.1 2Si i 0.24 ± 0.19 8 −0.12 ± 0.08 11 −0.38 ± 0.13 9Si ii 0.25 1 −0.21 ± 0.06 2Ca i 0.03 1 −0.14 ± 0.04 7 −0.27 ± 0.06 7Sc ii −0.09 1 −0.70 ± 0.13 3Ti i −0.05 ± 0.06 3 0.09 ± 0.25 2Ti ii −0.07 ± 0.12 2 −0.77 1Cr i −0.27 ± 0.13 4 −0.51 ± 0.04 2Cr ii 0.08 1 −0.25 1Mn i −0.07 1 −0.82 1Mg i −0.12 1Fe i 0.04 ± 0.16 34 −0.26 ± 0.07 50 −0.49 ± 0.17 51Fe ii 0.1 ± 0.18 3 −0.26 ± 0.12 6 −0.60 ± 0.4 3Ni i 0.08 ± 0.16 6 −0.34 ± 0.10 11 −0.40 ± 0.25 12Na i −0.44 1

(a) Number of lines used per ion, the stated error is the rms error computed if two or morelines are available.

Without preliminary assumptions on the spectral types XTgrid confirmed theresults of the SED and VWA analysis on BD+293070 and BD+341543 withinerror bars, but predicted a lower surface gravity of the components in Feige 87.Then, with the help of the radial velocity measurements, we could constrain thesurface gravities of the companions which helped achieving a better consistencyof the decomposition in all three cases. Our results are listed in Table 3.9 andthe disentangled binary spectra are plotted in Fig. 3.6. We note that the lowercontribution of the MS star in Feige 87 raises the uncertainties of our parameterdetermination.

3.5 Gravitational redshift

If the difference between the surface gravity of both components in a binarysystem is substantial, this can give rise to a frequency shift in the emittedradiation, the gravitational redshift. This shift can be observed in sdB binariesand can be used to estimate the surface gravity of the sdB component as


Table 3.9: The atmospheric parameters obtained from the flux calibrated spectraof BD+293070, BD+341543 and Feige 87.

Parameter BD+293070 BD+341543 Feige 87E(B − V ) (mag) 0.013 0.002 0.000Flux ratioa 1.353 1.337 2.252

MS componentTeff (K) 6026 ± 300 5715 ± 300 5675 ± 250log g (dex) 4.26 ± 0.30 4.11 ± 0.30 4.23 ± 0.35[Fe/H] 0.08 ± 0.25 -0.37± 0.25 -0.39 ± 0.25

sdB componentTeff (K) 25380 ± 990 36640 ± 810 27270 ± 500log g (dex) 5.54 ± 0.18 6.13 ± 0.16 5.47 ± 0.15Heb -2.63+0.48

−1.26 -1.49 ± 0.13 -2.56+0.22−0.50

Cb -3.27 > -4.64 > -3.77 >Nb -2.79 > -3.28 > -3.69 >Ob -1.93 > -3.72 > -2.89 >

(a) Flux ratio FsdB / FMS in wavelength range 6720 - 6800 Å, (b) Abundance given aslog(nX/nH)

described in Sect. 2.4.

The measured differences in system velocity for the three systems are:

γsdB − γMS =

1.59± 0.97 km s−1 (BD + 293070)1.01± 0.52 km s−1 (BD + 341543)2.05± 0.51 km s−1 (Feige 87)

(3.3)

Using the averaged values for the surface gravity of the MS components as givenin Table 3.10, the estimated surface gravities of the sdB components of thesesystems are:

log(g)sdB =

5.92+0.18

−0.42 dex (BD + 293070),5.45+0.14

−0.34 dex (BD + 341543),5.93+0.11

−0.36 dex (Feige 87).(3.4)

The asymmetrical error on the surface gravity is calculated using a Monte Carlosimulation taking into account the error on the difference in system velocityand the errors on the parameters of the MS component. The resulting surfacegravities correspond with the results from the other methods presented in thischapter.

ABSOLUTE PARAMETERS 85

0.2

0.4

0.6

0.8

1.0

1.2

sdB

MS

BD+293070

0.2

0.4

0.6

0.8

1.0

Flu

x

sdB

MS

BD+341543

3500 4000 4500 5000 5500 6000 6500 700

Wavelength (A)

0.0

0.2

0.4

0.6

0.8

1.0

sdB

MS

Feige 87

Figure 3.6: The flux calibrated B&C spectra observed with the Bok telescope(black full line) of BD+293070, BD+341543 and Feige 87 together with thebest fitting binary atmosphere models from the XTgrid code (red dotted line).The decomposition of the model in the MS and sdB component is plotted inrespectively green and blue.

3.6 Absolute parameters

Combining the results of the SED fit, the spectral analysis, gravitational redshiftand the orbital parameters derived from the radial velocity curves, the absolutedimensions of BD+293070, BD+341543 and Feige 87 can be determined.With the assumed canonical sdB mass of MsdB = 0.47 M, the inclination ofthe systems can be derived from the reduced mass determined in Sect. 3.2.2.This inclination can be used to calculate the semi-major axis of the systems.The atmospheric parameters (Teff , log g) determined with the four differentmethods (SED fitting, derived from the iron lines, spectral fitting and from the


Table3.10

:Fu

ndam

entalp

rope

rtiesforbo

ththemain-sequ

ence

(MS)

andsubd

warf

(sdB

)compo

nent

ofBD

+29 307

0,BD

+34 1543an

dFe

ige8

7.The

sdB

masswa

sfix

edto

thecano

nicalv

alue

.

BD+29 3070

BD+34 1543

Feige8

7Pa

rameter

MS

sdB

MS

sdB

MS

sdB

P(d)

1254±

5972±

2938±

2T

02453966±

102451519±

112454232±

21e

0.15±

0.01

0.16±

0.01

0.11±

0.01

ω1.88±

0.09

1.58±

0.07

0.08±

0.15

γ(km

s−1 )

−57.79±

0.36

32.10±

0.06

32.90±

0.08

q0.37±

0.01

0.57±

0.01

0.55±

0.01

a(R)

585±

5447±

4441±

3i

(o)

73±

343±

179±

3E

(B−V

)0.011±

0.030

0.005±

0.030

0.006±

0.030

d(pc)

226±

37207±

30376±

44[Fe/H

]0.05±

0.16

−0.26±

0.08

-0.50±

0.18

K(km

s−1 )

6.22±

0.25

16.7±

0.5

5.91±

0.07

10.31±

0.15

8.40±

0.11

15.15±

0.21

M(M

)1.26±

0.06

0.47±

0.05

0.82±

0.07

0.47±

0.05

0.84±

0.07

0.47±

0.05

logg

(cgs)

4.33±

0.50

5.69±

0.45

4.18±

0.40

5.84±

0.35

4.36±

0.42

5.58±

0.34

R(R

)1.28±

0.97

0.16±

0.10

1.21±

0.48

0.14±

0.06

1.02±

0.50

0.18±

0.07

Teff

(K)

6190±

420

26200±

3000

6100±

300

36600±

3000

5980±

325

27300±

2700

L(L)

2.03±

0.25

11.19±

3.0

1.83±

0.25

30.13±

4.0

1.19±

0.20

16.84±

3.0

V0(m

ag)

10.81±

0.10

11.66±

0.15

10.72±

0.10

11.11±

0.15

12.50±

0.10

12.40±

0.15

MV

(mag)

4.04±

0.10

4.89±

0.15

4.14±

0.10

4.54±

0.15

4.62±

0.10

4.52±

0.15

vsin

i(km

s−1 )

52±

5/

17±

4/

8±

3/

ABSOLUTE PARAMETERS 87

gravitational-redshift) correspond well within their errors. The final values forthese parameters are the averages of the three methods weighted by their errors.The radii of the components are calculated from the mass and surface gravity.The system velocity is set to the system velocity of the MS component as thesdB component is subjected to significant gravitational redshift as discussed inSection 3.5.

The luminosity of both components can be calculated using L = 4πσR2T 4eff . The

apparent V magnitudes are obtained directly from the SED fitting procedure,while the absolute magnitude can be obtained by integrating the best fit modelSEDs over the Johnson V band, and scaling the resulting flux to a distance of 10pc. The distance to the system is then calculated from log d = (mV −MV +5)/5.For BD+341543 the distance obtained in this way corresponds within errorswith the distance obtained from the Hipparcos parallax 236 ± 80 pc. Theabsolute dimensions of all three systems are summarized in Table 3.10.

The proper motions of BD+293070, BD+341543 and Feige 87 as measured byvan Leeuwen (2007) are:

(µα, µδ) = (−6.29, 23.92)± (1.00, 1.55) mas yr−1, (3.5)

(µα, µδ) = (34.46,−61.08)± (2.28, 1.42) mas yr−1, (3.6)

(µα, µδ) = (14.12,−65.51)± (1.49, 1.82) mas yr−1. (3.7)

Using the method of Johnson & Soderblom (1987), these numbers together withthe measured value of γ, can be used to compute the galactic space velocityvector with respect to the local standard of rest from Dehnen & Binney (1998).Resulting in:

(U, V,W )LSR = (−42.7,−27.2,−6.9)± (3.5, 1.6, 2.0) km s−1, (3.8)

(U, V,W )LSR = (−13.0,−69.9, 28.1)± (0.9, 4.8, 2.3) km s−1, (3.9)

(U, V,W )LSR = (92.5,−31.0, 92.4)± (9.5, 5.9, 6.1) km s−1. (3.10)

for respectively BD+293070, BD+341543 and Feige 87. U is defined as positivetowards the galactic center. Following the selection criteria of Reddy et al.(2006) based on the V component of the proper motion, all systems are boundto the Galaxy, and belong to the thin or thick disk population. The differencebetween the thin and the thick disk populations is that the thick disk populationis older than the thin disk, and consequently has a lower metal abundance.There is also a difference in kinimatics beteween the thin and the thick disk.


3.7 Conclusions

Using both literature photometry and observed spectra, detailed astrophysicalparameters of BD+293070, BD+341543 and Feige 87 have been established.The long time-base spectroscopic observations made it possible to determineaccurate periods, and to solve the orbit of both the MS and sdB componentin all three systems. The atmospheric parameters were determined with threedifferent techniques: based on the spectral energy distribution, using iron linesin the disentangled MS spectra, and from low-resolution flux calibrated spectra.Furthermore, the measured gravitational-redshift could be used to derive thesurface gravity of the sdB components. The results obtained with these differentmethods agree within their errors.

The sdB components of all three systems are consistent with a canonical post-core-helium-flash model with a mass around 0.47 M. However, the orbitsof these systems are clearly eccentric, as opposed to what the current theorypredicts. The periods found here correspond with the updated version of theBPS studies of Chen et al. (2013), although a significant degree of atmosphericRLOF must be included in order to reach the ∼1300 d period of BD+293070.Furthermore, Chen et al. (2013) find a metallicity – orbital period relation,where the orbital period will decrease with decreasing metallicity. The observedperiods of BD+293070, BD+341543 and Feige 87 (1254 d, 972 d, 938 d) formetallicities (solar, about half solar, about 1/3 solar) follow the metallicityrelation, but significantly exceed the predictions of Chen et al. (2013). The sameconclusion can be reached for PG1104+243 (P = 755 d, ∼1/3 solar, see Sect. 2).We assume that the metallicity measured for the companion is comparable tothe initial Z for the system, which ignores any contamination during RLOF, andcan therefore be considered as an upper limit to the initial Z. All four systemscall for the inclusion of atmospheric RLOF, also at lower metallicities. Whilethe sample is too limited yet to make conclusions with respect to the perioddistribution, the long periods found in this paper indicate that an additionalmechanism such as atmospheric RLOF is required for all the systems, not justthose with periods >1100 d, as suggested by Chen et al. (2013).

BD+293070, BD+341543 and Feige 87 are part of an ongoing long-termobserving program of sdB + MS binaries with HERMES at Mercator. Theremaining systems will be presented in the next chapter, together with asummary of the three other long period sdB binaries published in the literature.

4The observed period-eccentricity distribution of

wide sdB binaries

4.1 Introduction

In this chapter we will assemble all orbital data of sdB binaries in wide systemsto study the distribution of orbits and eccentricities. Nine systems are alreadypublished at the time of the writing. Four presented in the previous chapters, twopresented in the conference proceedings of the sixth meeting on hot subdwarfs2013 (Sect 4.2), one system analysed by Deca et al. (2012) (Sect. 4.3), andtwo more systems analysed by Barlow et al. (2013)(Sect. 4.5). Two more sdBstars are being monitored with the Mercator telescope and we obtained enoughspectra to present their orbital periods and eccentricities (Sect. 4.4). Based onthe parameters of those 11 systems, the period-eccentricity distribution of widesdB binaries will be discussed (Sect. 4.6).

4.2 Balloon 82800003 and BD−7o5977

The observations and analysis of Bal 82800003 and BD−7o5977 were presentedin the proceedings of the 6th Meeting on Hot-Subdwarf Stars and RelatedObjects in Tucson, Arizona (Vos et al. 2014). In this section the analysis isrepeated including the most recent observations.

Both Bal 82800003 and BD−7o5977 have been observed with the HERMES

89

90 THE OBSERVED PERIOD-ECCENTRICITY DISTRIBUTION OF WIDE SDB BINARIES

spectrograph (Raskin et al. 2011) attached to the 1.2m Mercator telescopelocated on La Palma. 41 high-resolution spectra were obtained fromBal 82800003 between July 2009 and June 2014. BD−7o5977 was observedbetween July 2009 and December 2014, resulting in 43 spectra. The derivation ofthe radial velocities from the spectra is performed in identical way as explainedin Sect. 2.2.1. For the MS component, a cross correlation with a G2-type maskis used, while for the sdB component, the radial velocities are based on the HeI5875 Å line. Spectra that are taken within an interval of 5 days are merged toincrease the signal-to-noise ratio. The orbital parameters are determined byfitting a Keplerian orbit to the radial velocities, and the errors are based on aMonte-Carlo simulation with 5000 iterations.

The spectral-energy distribution (SED) of the systems based on literaturephotometry is used to determine the effective temperature and surface gravity ofboth components. The fitting procedure for the SEDs is explained in Sect. 2.3.2.Again the mass ratios derived from the spectroscopic observations are used tolimit the amount of input parameters. For both Bal 82800003 and BD−7o5977,there are nine photometric observations available.

Bal 82800003

In the spectroscopic observations of Bal 82800003, the MS component is clearlyvisible, and accurate radial velocities can be obtained. An example of thecross-correlation function (CCF) for the MS component is given in Fig. 4.1.Of the sdB component only the He I 5875 Å line is visible in the spectrum.This line is of high enough quality to derive reasonable radial velocities. Theaverage radial velocity error for the MS component is 0.07 km s−1, while forthe sdB component it is 0.9 km s−1. After fitting a Keplerian orbit to theradial velocities, the orbital parameters are determined. The average O-C valuefor the MS and sdB components are respectively 0.1 km s−1 and 1.2 km s−1.There is one clear outlier in the radial velocities of the sdB component. Thatparticular spectrum has a low S/N, and the RVs derived from it are not usedin the determination of the orbital parameters. The complete orbit is wellcovered, and accurate orbital parameters could be derived. Bal 82800003 hasan orbital period of 1244 ± 3 days, and an eccentricity of 0.17 ± 0.01. Thespectroscopic parameters are summarized in Table 4.1, and the radial-velocitycurves are plotted in the left panel of Fig. 4.2.

Based on the SED fit, the effective temperature and surface gravity of bothcomponents can be derived. The sdB component is rather cold with Teff,sdB= 22500 ± 3500 K and log gsdB = 5.12 ± 0.40 dex. The MS component hasTeff,MS = 5970 ± 350 K and log gMS = 4.10 ± 0.35 dex, which are expected for

BALLOON82800003 AND BD−7O5977 91

−200 −150 −100 −50 0 50 100 150

0.00

0.02

0.04

0.06

0.08-15.5 +- 0.3 km/s

fwhm ≃ 21 km/s

−200 −150 −100 −50 0 50 100 150

0.0

0.1

0.2

0.3

0.4-6.3 +- 0.2 km/s

fwhm ≃ 10 km/s

RV (km/s)

CC

F

Figure 4.1: An example CCF of hermesvr in blue for Bal 82800003 (left) andBD−7o5977 (right) together with the best fitting Gaussian profile in red. Thecentral velocity of the fitted profile is shown in green, while the full width athalf maximum (fwhm) is given in magenta.

such a star. The probabilities for the MS parameters follow a strongly peakedGaussian distribution, indicating that the parameters are well constrained. Forthe sdB component there is a clear correlation between Teff and log g, as is thecase in all previously discussed systems. The system has a reddening of E(B-V)= 0.01+0.025

−0.01 mag, which is consistent with the maximum reddening E(B-V)max= 0.035 derived from the dust maps of Schlegel et al. (1998). The SED andbest fitting model of Bal 82800003 is plotted in the left panel of Fig. 4.3.

BD−7o5977

BD−7o5977 is different from the other systems described before, as it has asubgiant companion instead of a MS star. As is expected, the HERMES spectraare dominated by the narrow lines of the subgiant. Leading to very accurateradial velocities for this component. A CCF for the MS component is shownin Fig. 4.1 The average radial velocity error for the subgiant is 0.05 km s−1.The He I line at 5875 Å is very shallow, but is still the only useful sdB linein the spectrum. The derived RVs for the sdB component have therefore alow accuracy, with an average RV error of 2.5 km s−1. Again the completeorbit is well covered by the observations and the period and eccentricity can bedetermined with high accuracy, P = 1269 ± 2 days and e = 0.16 ± 0.01. Theaverage O-C value between the best fitting Kepler curve and the observations forthe MS and sdB components are respectively 0.1 km s−1 and 1.5 km s−1. The


spectroscopic parameters are summarized in Table 4.1, and the radial-velocitycurves are plotted in the middle panel of Fig. 4.2.

The subgiant component is clearly visible in the SED, where its luminositydominates the SED up to ∼ 4000 Å. Fitting the SED results in Teff,sdB = 29000± 4000 K and log gsdB = 5.02 ± 0.40 dex for the sdB component, and Teff,comp= 4720 ± 250 K and log gcomp = 2.86 ± 0.35 dex for the subgiant companion.The subgiant parameters are well constrained, as is indicated by the Gaussiandistribution of the probabilities. The sdB parameters are less constrained, butthe correlation between the effective temperature and surface gravity is lessstrong than was the case for Bal 82800003. The reddening of the system israther high at E(B-V) = 0.034 ± 0.034 mag, but is badly constrained and is inagreement with the dust map results of Schlegel et al. (1998). The SED andbest fitting model of BD−7o5977 is plotted in the middle panel of Fig. 4.3.

4.3 PG 1018−047

The binary PG1018−047 was the first long period sdB binary that was studiedin detail. Deca et al. (2012) used spectroscopic observations taken by fivedifferent telescopes over a time period of ten years, amounting to a total of 125spectra. The analysis of the spectra resulted in an orbital period of 759 ± 6days assuming a circular orbit, or 755 ± 5 days assuming an eccentric orbitwith e = 0.24 ± 0.05. The obtained spectra did not allow to definitely decideon the eccentricity of the binary orbit.

To increase the accuracy of the orbital parameters, 20 more spectra coveringmost of the orbit were obtained with the UVES spectrograph. These spectraare reduced using the standard settings of the UVES REFLEX pipeline version2.6. To obtain the radial velocities of both components, a cross correlationwith template spectra is used. This is possible as the sdB component is mainlyvisible in the blue part of the spectrum, and the MS component is only visiblein the red part of the spectrum. For the sdB component, a high-resolutionLTE template spectrum with Teff = 30 000 K and log g = 5.50 dex from thegrids of Heber et al. (2000) is used on wavelength range 3910 - 4721 Å. For thecompanion a high-resolution spectrum with Teff = 4700 K and log g = 4.80 dextaken from the BlueRed library1 (Bertone et al. 2008) is used on wavelengthrange 6095 - 6460 Å.

Contrary to the previously analysed systems, the sdB component of PG1018−047has many visible lines in the blue part of the spectrum that are not influenced

1The BlueRed library of high-resolution synthetic stellar spectra in the optical can befound at http://www.inaoep.mx/~modelos/bluered/documentation.html

http://www.inaoep.mx/~modelos/bluered/documentation.html

PG1018−047 93

Table4.1:

Spectroscopicorbitalsolutionforb

oththesubd

warf

B(sdB

)com

pone

ntan

dthecompa

nion

ofBa

l828

0000

3,BD−7o5977

andPG

1018−047.

Bal82800003

BD−7o5977

PG1018−047

Parameter

MS

sdB

MS

sdB

MS

sdB

P(d)

1244±

31269±

2752±

2T

02454110±

142454955±

32456696±

16e

0.17±

0.01

0.16±

0.01

0.05±

0.01

ω6.00±

0.05

5.43±

0.01

1.61±

0.14

q0.62±

0.03

0.49±

0.05

0.66±

0.04

i(o)

17±

121±

245±

2γ(km

s−1 )

-14.9±

0.1

-13.3±

3-8.6±

0.1

-2.4±

0.5

37.8±

0.2

38.4±

0.1

K(km

s−1 )

2.41±

0.05

3.9±

0.5

2.63±

0.05

5.4±

0.5

6.9±

0.4

10.4±

0.1

asin

i(R)

58.3±

1.0

94.5±

4.0

65.1±

0.5

132±

12103.0±

5.9

155.1±

1.3

a(R)

198±

4.0

321±

9182±

5.0

370±

20146.5±

3.4

220.5±

7.8

Msin

3i(M)

0.02±

0.01

0.01±

0.01

0.04±

0.01

0.02±

0.01

0.25±

0.01

0.16±

0.02

M(M)

0.76±

0.10

0.47

0.96±

0.09

0.47

0.71±

0.04

0.47

adeno

testhesemi-m

ajor-axisof

theorbit.

The

inclinationis

determ

ined

assumingacano

nicals

dBmassof

0.47

M.The

quoted

errors

are

thestan

dard

deviationfrom

theresultsof

5000

iterations

inaMon

teCarlo

simulation.


−20

−15

−10

−5

−101 0

.00.2

0.4

0.6

0.8

1.0

−202

−15

−10

−50510

−101 0

.00.2

0.4

0.6

0.8

1.0

−202

30

35

40

45

50

−2

−1012 0

.00.2

0.4

0.6

0.8

1.0

−0.6

−0.3

0.0

0.3

0.6


Pha

se

Bal

8280

0003

BD

−7

5977

PG

1018

−04

7

Figu

re4.2:

The

spectroscopicorbitals

olutions

forBa

l828

0000

3(le

ft),B

D−7o59

77(m

iddle)

andPG

1018−04

7(right).

Top:

thebe

stfittin

gradial-velocity

curves

(solid

line:

MS,

dashed

line:

sdB)

,and

theob

served

radial

velocitie

s(blue

filledcircles:

compa

nion

,green

open

triang

les:

sdB)

.The

measuredsystem

velocity

ofbo

thcompo

nentsis

show

nby

ado

tted

line.

Middle:

resid

uals

oftheMScompo

nent.Bo

ttom

:resid

uals

ofthesdB

compo

nent.

PG1018−047 95

103

104

2MA

SSB

AL

LO

ON

CO

USI

NS

JOH

NSO

NST

RO

MG

RE

N

104

−0.2

−0.1

0.0

0.1

5.4

5.6

5.8

6.0

6.2

4.0

4.1

4.2

MS

20

22

24

4.9

5.0

5.1

5.2

5.3

sdB

104

104

−0.2

0.0

0.2

0.4

4.7

4.8

4.9

2.7

2.8

2.9

3.0

MS

26

27

28

29

30

31

32

4.9

5.0

5.1

sdB

102

103

104

104

−0.0

1

0.0

0

0.0

1

3.6

4.0

4.4

4.8

5.2

4.0

4.1

4.2

4.3

MS

26

28

30

32

34

36

5.0

5.1

5.2

5.3

sdB

λ·Flux(erg/s/cm2) O−C

Wav

elen

gth

(A)

logg(dex)

Teff

(kK

)

Bal

8280

0003

BD

−7

5977

PG

1018

−04

7

Figu

re4.3:

Thespectral

energy

distrib

utionan

d(O

-C)f

orBa

l828

0000

3(le

ft),B

D−7o59

77(m

iddle)

andPG

1018−04

7(right).

Thebe

stfittin

gmod

elis

plottedin

red.

Theph

otom

etric

observations

areshow

nin

circle

s,wi

ththeho

rizon

tal

errorba

rindicatin

gthewidth

oftheph

otom

etric

band

.In

thebo

ttom

theconfi

denc

eintervalson

theeff

ectiv

etempe

rature

andsurfa

cegravity

forbo

thcompo

nentsis

show

n.


by the companion. The radial velocities of the sdB component will have a higheraccuracy than those of the companion that is only visible in the more noisy redpart of the spectra. The average error on the sdB RVs is on the order of 0.25 kms−1, while for the companion it is 1.0 km s−1. Based on the best fitting Keplercurve, the period is 752 ± 2 days, and the eccentricity is 0.05 ± 0.01. Using theeccentricity test of Lucy & Sweeney (1971), the orbit is significantly eccentric (p< 0.0007, higher as three sigma). The average deviation between the observedvelocities and the fit for one observation is 0.03 km s−1 for both the sdB andcompanion. The spectroscopic parameters are summarized in Table 4.1, andthe radial-velocity curves are plotted in the right panel of Fig. 4.2.

There are seven photometric observations available for PG1018−047 in theliterature. Even though this is a low number, they cover both the red part(2MASS) and the blue part (Strømgren) of the SED. Using the mass ratio derivedfrom the radial-velocity curves, the effective temperature and surface gravitiesof both components are: Teff,sdB = 29100 ± 3000 K, log gsdB = 5.20 ± 0.30 dex,Teff,comp = 4360+200

−700 K and log gcomp = 4.14+0.10−0.20 dex. The sdB component is

well constrained in the sense that there is very little correlation between theTeff and log g in comparison with e.g. Bal 82800003 and BD−7o5977. For thecompanion the probability distribution is elongated towards lower Teff and log g,resulting in the non-symmetrical errors.

4.4 EC 11031−1348 and TYC2084−448−1

Both EC11031−1348 and TYC2084−448−1 have been observed since 2011 withthe HERMES spectrograph attached to the Mercator telescope. EC 11031−1348and TYC2084−448−1 are of similar magnitude as the other systems, but theircool companions are rapidly rotating, and subsequently have a lower contrastbetween the lines and the continuum. The obtained radial velocities are lessaccurate than the already published systems.

EC11031−1348

There are a total of 41 spectra available of EC11031−1348, obtained betweenFebruary 2011 and March 2015. The average S/N of these spectra is around 25.The cool companion of EC 11031−1348 is a fast rotator, which is clearly visiblein the CCF function plotted in Fig. 4.5. To obtain the radial velocities from theCCFs, a rotationally broadened profile is used instead of the standard Gaussianfitting. The average error on the MS component is ∼0.7 km s−1. For the sdBcomponent there is as usual only one useful line (He I 5875 Å), but it is very

EC11031−1348 AND TYC2084−448−1 97

Table 4.2: Spectroscopic orbital solution for both the subdwarf B (sdB)component and the companion of EC11031−1348 and TYC2084−448−1.

EC11031−1348 TYC2084−448−1Parameter MS sdB MS sdBP (d) 1072 ± 15 1060 ± 10T0 2455369 ± 20 2454625 ± 20e 0.13 ± 0.05 0.14 ± 0.05ω 6.16 ± 0.06 0.50 ± 0.10q 0.38 ± 0.07 0.52± 0.05i (o) 47 ± 8 74± 5γ (km s−1) -13.8 ± 1.2 -9.7 ± 1.5 -16.5 ± 0.5 -15.1 ± 0.5K (km s−1) 5.1 ± 0.7 13.3 ± 0.9 7.7 ± 0.25 14.7 ± 0.4a sin i (R) 106 ± 20 277 ± 30 160 ± 5 304 ± 8a (R) 144 ± 20 376 ± 30 166 ± 5 317 ± 8M sin3 i (M) 0.49 ± 0.09 0.19 ± 0.05 0.79 ± 0.05 0.42 ± 0.05M (M) 1.22 ± 0.2 0.47 0.90 ± 0.05 0.47v sin i (km s−1) 69 ± 1 / 45 ± 1 /

a denotes the semi-major-axis of the orbit. The inclination is determined assuming a canonicalsdB mass of 0.47 M. The quoted errors are the standard deviation from the results of 5000iterations in a Monte Carlo simulation.

weak. The average error on the radial velocities of the sdB component is ∼1.7km s−1. Fitting a Keplerian orbit to the RVs of the cool companion resultsin a reasonable fit. The O-C values average around 0.9 km s−1, and there isno visible trend in the residuals. The RVs of the sdB component have a largerspread, with an average O-C value of 1.5 km s−1.

The parameters obtained for EC11031−1348 are less reliable than for thesystems presented before, but the period and eccentricity of the systems couldbe established with a reasonable accuracy. At P = 1072 ± 15 days and e =0.13 ± 0.05, EC 11031−1348 is roughly in the middle of the period-eccentricitydistribution of our observed sample. The spectroscopic parameters are given inTable 4.2, while the phase folded radial-velocity curves are plotted in Fig. 4.4.

TYC2084−448−1

TYC2084−448−1 was observed 39 times between February 2011 and March2015. The average S/N for a spectrum is slightly higher as for EC11031−1348,around 25. The lines of the cool companion in TYC2084−448−1 are lessbroadened than those of EC11031−1348, and also the He I 5875 Å sdB line


−30

−20

−10

0

−3

0

3

0.0 0.2 0.4 0.6 0.8 1.0

−3

0

3

−30

−25

−20

−15

−10

−5

0

−2

0

2

0.0 0.2 0.4 0.6 0.8 1.0

−3

0

3

Rad

ialv

eloc

ity

(km

/s)

Phase

EC 11031−1348 TYC 2084−448−1

Figure 4.4: The spectroscopic orbital solutions for EC11031−1348 (left) andTYC2084−448−1 (right). Top: the best fitting radial-velocity curves (solid line:MS, dashed line: sdB), and the observed radial velocities (blue filled circles:companion, green open triangles: sdB). The measured system velocity of bothcomponents is shown by a dotted line. Middle: residuals of the MS component.Bottom: residuals of the sdB component.

is much more visible. This results in more accurate radial velocities for bothcomponents. The average RV error for the cool companion and the sdB arerespectively 0.6 km s−1and 1.1 km s−1, less than half of those of EC 11031−1348.After fitting a Keplerian orbit to the RVs, the average residuals are 1.6 km s−1

and 2.2 km s−1 for the cool companion and the sdB component respectively.

The phase coverage of the observations is very good, and there is no visible trendin the residuals of any of the two components. For both components reliablespectroscopic parameters could be presented, which are given in Table 4.2. Theorbital period of the system is 1060 ± 10 days placing this system in the middleof the period range of the wide sdB binaries. The same goes for the eccentricityof 0.14 ± 0.05. Only roughly 1 orbital period was covered by the observations,and further monitoring of this system will increase the accuracy of the period

PG1449+653 AND PG1701+359 99

−200 −150 −100 −50 0 50 100 150

0.00

0.01

0.02

0.03-18.2 +- 0.9 km/s

vrot sini = 69 +- 1 km/s

−200 −150 −100 −50 0 50 100 150

0.00

0.01

0.02

0.03-11.2 +- 0.6 km/s

vrot sini = 45 +- 1 km/s

RV (km/s)

CC

F

Figure 4.5: An example CCF of hermesvr in blue for EC11031−1348 (left)and TYC2084−448−1 (right) together with the best fitting rotational profilein red. The central velocity of the fitted profile is shown in green, while therotational velocity (vrot sin i) is given in magenta.

and eccentricity. The RV curves and the best fitting solution are plotted in theright panel of Fig. 4.4.

4.5 PG 1449+653 and PG1701+359

PG1449+653 and PG1701+359 are part of a sample of long period sdO-Bbinaries that is being monitored by Barlow et al. (2013). Both targets areobserved with the Hobby-Eberly Telescope (HET) in the periods 2005-2008 and2012-2013. In total there are 20 spectra available for PG1449+653 and 29 forPG1701+359.

To determine the spectral type of the cool companions, different spectra of MSstandards varying from F0 to K7 were used in the fxcor routine of iraf, andthe best fit is determined by the Tonry-Davis ratio (Tonry & Davis 1979). Thisresults in a best fitting spectral type of G0V for PG1449+653 and K0V forPG1701+359. The radial velocities of the cool companion are determined usinga cross correlation with an observed standard spectrum of the given spectraltypes. The radial velocities of the sdB component are determined by self-crosscorrelation with the highest S/N spectra of the He I 5875 Å line. This techniquewas only successful for PG1449+653. Barlow et al. (2013) indicate the lowerS/N of the spectra of PG1701+359, and the intrinsically weaker He lines as thereason for this method not working for this system.


Figure 4.6: The spectroscopic orbital solutions for PG1449+653 (top) andPG1701+359 (bottom). RV measurements for the cool companion obtainedwith the HET/MRS, HET/HRS and MMT are shown in filled circles, squaresand triangles respectively. Open circles indicate the RVs for the sdB component.The best fitting orbital solutions are shown in blue full line for the cool companionand red dashed-dotted line for the sdB component. Residuals from the fit areshown in the lower portions of the panels. Figure taken from Barlow et al.(2013).

The orbital parameters were determined using the rvlin software (Wright &Howard 2009) and the uncertainties were determined using the rvlin-drivebootstrapping technique (Wang et al. 2012). The revised Lucy-Sweeney test(Lucy 2013) was used to determine whether the orbits are significantly eccentricor not.

PG1449+653

The main sequence RVs are accurately determined with average errors of 0.24km s−1 for the HET spectra and 1.4 km s−1 for the MMT spectra. The errors

PG1449+653 AND PG1701+359 101

Table 4.3: Spectroscopic orbital solution for both the subdwarf B (sdB)component and the companion of PG1449+653 and PG1701+359 obtained byBarlow et al. (2013).

Parameter PG1449+653 PG1701+359± circular fit eccentric fit

P (d) 909 ± 2 734 ± 3 738 ± 3T0 2453675 ± 35 2453269 ± 60 2453869 ± 60e 0.11 ± 0.02 0 0.07 ± 0.04ω 3.03 ± 0.25 / 5.20 ± 0.45q 0.64 ± 0.06 / /γ (km s−1) -135.5 ± 0.2 -120 ± 0.2 -121 ± 0.2KMS (km s−1) 8.2 ± 0.3 3.9 ± 0.2 3.6 ± 0.2KsdB (km s−1) 12.8 ± 1.1 / /

on the RVs of the sdB companion are of the order of 3-7 km s−1. Only the MSRVs were used to derive the orbital parameters of the system. It has an orbitalperiod of 909 ± 2 days, and is significantly eccentric with e = 0.11 ± 0.03. TheRVs of the sdB and MS components were used to derive the mass ratio of thesystem. The spectroscopic parameters of PG1449+653 are given in Table 4.3,and the RVs with best fitting orbit are plotted in Fig. 4.6.

PG1701+359

The RV errors for the cool component of PG1701+359 range around 0.25 kms−1 for the HET spectra and 1-2 km s−1 for the MMT spectra. As mentionedbefore Barlow et al. (2013) could not derive RVs for the sdB component, thusno mass ratio could be determined for this spectrum. Both an eccentric orbitand a circular orbit have been fitted to the data, the best fit results in bothcases are given in Table 4.3. Even though the eccentric fit is slightly better,the eccentricity test of Lucy (2013) indicate that the orbit of PG1701+359 iseffectively circular. The final period is then determined at P = 734 ± 3 days,placing PG1701+359 very close to PG1104+243 in period-eccentricity space.Barlow et al. (2013) does mention that the phase coverage of PG1701+359 israther bad, with most observations taken on the ascending slope, and extraobservations are necessary to determine definitely if the system is circular oreccentric. The radial velocities with best fitting circular orbit are plotted inFig. 4.6.


Table 4.4: The observed periods and eccentricities of all known long period sdBbinaries sorted on ascending orbital period. See also Fig. 4.7

Object name Period (days) EccentricityPG1701+359 734 ± 5 0.00 ± 0.07PG1104+243 753 ± 3 0.04 ± 0.02PG1018−047 759 ± 3 0.05 ± 0.01PG1449+653 909 ± 5 0.11 ± 0.04Feige 87 938 ± 2 0.11 ± 0.01BD+341543 972 ± 2 0.16 ± 0.01TYC2084−448 1060 ± 10 0.14 ± 0.05EC11031−1348 1072 ± 15 0.13 ± 0.05Bal 8280003 1244 ± 3 0.17 ± 0.01BD+293070 1254 ± 5 0.15 ± 0.01BD−75977 1269 ± 2 0.16 ± 0.01

4.6 Period-eccentricity distribution

We determined nine orbits and these, supplemented with two more orbits fromthe literature we have eleven orbits, which can be compared to what theorypredicts. In Fig. 4.7, the eccentricity of all known analysed wide sdB systems isplotted in function of their orbital period.

Considering all known wide sdB binaries, it is clear that almost all of thesesystems have significantly eccentric orbits. The only system that has a circularorbit is PG1701+359. However, as mentioned in Sect. 4.5 this system can befitted with an eccentric orbit with e = 0.07 as a best fit. Barlow et al. (2013)report a circular orbit as the Lucy (2013) eccentricity test decides against aneccentric orbit. However, they indicate that there are not enough observationson the slopes of the RV curves to rule out an eccentric orbit completely. Theother two sdB binaries with periods around 750 days were first reported to becircular, but after re-analysing them using spectra taken over a longer timeframe, and in case of PG1018−047 with a higher S/N, both are found to havelow but nevertheless significant non-zero eccentricities.

A first important observation from Fig. 4.7 is that there is a cut-off at orbitalperiods around ∼ 700 days. No systems wit shorter periods are observed, eventhough there is no observational bias for this. On the contrary, if shorter periodsystems would exists, they would be easier to find than those with longer orbitalperiods. This cut-off is caused by the stability of RLOF on the RGB. Systemswith shorter period will enter a CE phase, and end up as short period binaries.

PERIOD-ECCENTRICITY DISTRIBUTION 103

700

800

900

1000

1100

1200

1300

1400

Peri

od(d

ays)

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

Eccentricity

PG

1701

+35

9

PG

1104

+24

3

PG

1018

−04

7

PG

1449

+65

3

Feig

e87

BD

+34

154

3

TY

C20

84−

448−

1B

D+

29 3

070

EC

1103

1−13

48

BD

−7

5977

Bal

8280

003

Figu

re4.7:

The

orbitalp

eriodan

deccentric

ityof

allk

nownan

alysed

widesdB

bina

ries.

System

sob

served

bydiffe

rent

research

grou

psareplottedin

diffe

rent

colours:

Voset

al.(20

12,2

013,

2014

)in

dark

blue

,Decaet

al.(20

12)in

red,

Barlo

wet

al.(2012)in

green,

andtheun

publish

edsystem

sin

light

blue

.


This cut-off can be used to test stability criteria for RLOF.

In Fig. 4.7 there is a clear trend visible of increasing eccentricity with increasingorbital period. The eccentricity of the non circular systems varies from 0.04at P ≈ 750 days, to 0.17 at P ≈ 1300 days. A Pearson test (Pearson 1896)for the correlation between period and eccentricity yields corr(P, e) = 0.88with a p value of 0.00029, indicating that the correlation between period andeccentricity is significant at the three sigma level. An alternative interpretationis that there is not a linear increase in eccentricity with increasing orbitalperiod, but that there are two separate populations. One population at shortorbital periods with low eccentricities, and a second population at higher orbitalperiods (P > 950 d), with higher eccentricities. However, there are not enoughobservations to confirm that.

When all systems are considered, there are no observed systems with aneccentricity over e = 0.17. This seems to indicate that there is a maximumeccentricity that can be reached by the interaction mechanisms at play in thesebinaries.

Continued monitoring of these systems is necessary to decrease the one sigmauncertainty on the parameters. Especially to further constrain the eccentricitiesof the orbit, which requires a good phase coverage of the orbit. Extendingthe monitoring program to include new systems is essential to test the lowercut-off in period, and to distinguish between the interpretation of a continuousdistribution with higher eccentricities at higher orbital period, or the existenceof two separate populations.

Current theory predicts only circular orbits for these long period sdB binariesformed through the stable RLOF channel. During the red giant phase of the sdBprogenitor the tidal forces between the sdB progenitor and its companion shouldcircularize the orbit very efficiently (Zahn 1977), and the further evolutionprovides very few possibilities to re-introduce eccentricity to the orbit. Decaet al. (2012) proposed the hierarchical triple merger scenario of Clausen &Wade (2011) for the possibly eccentric sdB + K system PG1018-047, wherethe K-type companion would not have been involved in the evolution of thesdB component. However, such a scenario seems too unlikely to be observedfrequently.

This eccentricity problem is not new, it has been observed in different types ofsystems and there are various possible solutions presented in the literature. Inthe second part of this thesis three such eccentricity pumping processes will bediscussed in the context of wide sdB binaries.

IITesting eccentricity pumping

processes with MESA

105

5Modules for Experiments in Stellar Astrophysics

In this chapter the stellar evolution code Modules for Experiments in StellarAstrophysics (MESA)1 will be described. A very brief introduction to the stellarpart is given in Sect. 5.1, while the physics included in the binary module ofMESA are summarized in Sect. 5.2. In this latter section also all necessaryadditions to model the eccentricity pumping mechanisms discussed in the nextchapter are listed. The goal of this chapter is to briefly describe the MESAcode. In the next chapter a parameter study will be performed on the processesrelevant for eccentricity pumping. Sect. 5.2 is part of the appendix of the articlein chapter 6.

5.1 MESA

MESA is a 1D stellar evolution code based on the original EZ stellar evolutioncode (Eggleton 1971; Paxton 2004). It was developed by a team of theoretical andcomputational astrophysicists over a period of six years, and is still continuouslyimproved based on input from the scientific community. We have used version7211 dated 31 October 2014. MESA is developed with six key cornerstones inmind as outlined in the MESA Manifesto2:

Openness MESA is written in Fortran and the open-source code is availablefor any researcher both to use and modify.

1MESA is available on: http://mesa.sourceforge.net/2available on: http://mesa.sourceforge.net/assets/mesa_manifesto.pdf

107

http://mesa.sourceforge.net/

http://mesa.sourceforge.net/assets/mesa_manifesto.pdf

108 MODULES FOR EXPERIMENTS IN STELLAR ASTROPHYSICS

Modularity MESA consists of independent modules for both the physicalprocesses and the numerical computations that can be used independentlyfrom each other.

Applicability MESA is capable of calculating stellar and binary evolutionmodels in a wide range of environments.

Modern techniques MESA makes use of state of the art numerical ap-proaches.

Microphysics up-to-date wide ranging, flexible and independent microphysicsmodules.

Performance MESA is designed to run fast, even on personal computers,using multi-core architectures. Evolving a solar like star from the pre-MSto the WD stage takes roughly 4 hours on an average laptop. The binarymodels used in the next chapter take roughly 15 hours to evolve theprimary from MS through the core He flash to He-shell burning.

The capabilities of the code are wide, including mass loss, mass gain, doublediffusion, gravitational settling, radiative settling and levitation, rotation,magnetic fields, and many more. For asteroseismology MESA is coupled with twooscillation codes, ADIPLS (Christensen-Dalsgaard 2008) and GYRE (Townsend& Teitler 2013). The computational methods implemented in MESA allowthe code to consistently evolve stellar models through complicated phases intheir evolution, including the He core flash in low-mass stars and advancednuclear burning in massive stars. The stellar evolution modules of MESA areextensively described in the two instrument papers: Paxton et al. (2011, 2013).In the next two sections we introduce shortly some micro- and macropyhiscalinput included into the MESA code.

5.1.1 Microphysics

The values for physical constants are taken from the CODATA database (Mohret al. 2008), while values for solar parameters are taken from Bahcall et al.(2005). The solar abundances on several scales come from Anders & Grevesse(1989); Grevesse & Noels (1993); Grevesse & Sauval (1998); Lodders (2003) andAsplund et al. (2005).

The density-temperature (ρ−T) tables used by the equation-of-state (EOS)module of MESA are based on the 2005 update of the OPAL EOS tables (Rogers& Nayfonov 2002), while the SCVH tables (Saumon et al. 1995) are used in

BINARY PHYSICS IN MESA 109

the low ρ−T range. Outside the predefined ρ−T range, the HELM (Timmes &Swesty 2000) and PC (Potekhin & Chabrier 2010) EOSs are used.

The MESA opacity tables are constructed by combining electron conductionopacities with radiative opacities. The electron conduction opacities are takenfrom Cassisi et al. (2007). The radiative opacities are from Ferguson et al.(2005) and from OPAL, both the original list (referred to as type i, Iglesias &Rogers 1993) and the updated list (type ii, Iglesias & Rogers 1996).

The nuclear reaction networks are derived from publicly available codes3. Thisincludes a basic network of eight isotopes 1H, 3He, 4He, 12C, 14N, 24Mg and ex-tended networks covering hot CNO reactions, α-capture chains, (α,p)+(p,γ) reac-tions and heavy-ion reactions (Timmes 1999). The pp_cno_extras_o18_ne22.netreaction net used for the sdB progenitor in Chapter 6 includes nuclear reactionsfor the basic isotopes, extended with reactions for 2H, 7Li, 7Be, 8O, 14O, 18O19F, 18Ne, 19Ne, 22Ne and 22Mg.

5.1.2 Macrophysics

Both the standard mixing length theory (MLT) of convection as describedin (Cox & Giuli 1968, chap. 14) and the modified MLT of Henyey et al.(1965) are implemented. Convective overshoot mixing is implemented followingHerwig (2000). MESA also offers the possibility to include semi convection andthermohaline mixing.

The atmospheric boundary conditions can be obtained using three differentmodels: direct integration, interpolation in model atmosphere tables, and a“simple recipe” based on a user specified optical depth in the constant opacitysolution of radiative diffusion. The standard setting used in Chapter 6 integratedthe hydrostatic balance equation of Eddington (1926) to an optical depth ofτs=2/3.

5.2 Binary physics in MESA

The binary module of MESA is under continuous development. In the originalbinary module all orbits were circular by default. I added the necessaryframework to evolve eccentric orbits to the binary module of mesa, and extendedit with several physical processes necessary for this research. All changes to theoriginal binary module are listed below:

3http://cococubed.asu.edu/code_pages/codes.shtml

http://cococubed.asu.edu/code_pages/codes.shtml


• Inclusion of the framework necessary to track eccentric orbits

• Integration over phase to determine the average mass loss per year duringRLOF.

• Accretion of the mass lost in stellar winds. To calculate the accretionrates, the Bondi-Hoyle formalism as described in Hurley et al. (2002) isused.

• Tidally enhanced mass loss. Wind mass loss can be enhanced by the tidalinfluence of the companion star. The Companion Reinforced AttritionProcesses (CRAP) mechanism of Tout & Eggleton (1988) is used tocalculate this.

• Phase-dependent mass loss during RLOF or phase-dependent mass lossthrough stellar winds can possibly increase the eccentricity of the orbit.The formalism of Soker (2000) and Eggleton (2006) is used to determinethe change in eccentricity due to mass lost to infinity and mass accretedby the companion.

• Circumbinary (CB) disks. Due to Lindblad resonances, CB disk-binaryinteractions can change the orbital period and eccentricity. The formalismof Artymowicz & Lubow (1994) and Lubow & Artymowicz (1996) is usedto calculate the CB disk-binary interaction.

A list of the processes implemented in the binary module is given in the followingsections. We limited ourselves to listing the physical processes. For the exactsoftware implementation, we refer the reader to the open-source code, andfuture instrument papers. Most of the binary additions summarized above, willbe added to MESA in the future. Until then, a copy of the binary module usedhere will be made available4.

In this section we will refer to the primary as the mass-losing star or donor star,using the subscript d. The companion is also called the accretor, even when itwon’t accrete mass, using subscript a.

5.2.1 Evolution of the orbital parameters

The orbital angular momentum of a binary system is:

Jorb = Md ·Ma

√Ga(1− e2)Md +Ma

. (5.1)

4The binary module is available on: http://ster.kuleuven.be/~jorisv/MESA_binary

http://ster.kuleuven.be/~jorisv/MESA_binary


Where a is the binary separation, and e is the eccentricity of the orbit. Md

and Ma are respectively the mass of the donor and the accretor, and G is thegravitational constant. The evolution of the binary separation is obtained fromthe time derivative of Eq. (5.1):

a

a= 2 Jorb

Jorb− 2Md

Md− 2Ma

Ma+ Md + Ma

Md +Ma+ 2 ee

1− e2 . (5.2)

In MESA the, the mass loss and accretion rates, the change in orbital angularmomentum, and the change in eccentricity are computed, and then used toupdate the orbital separation.

The change in mass of both components depends on the mass lost in stellarwinds, the mass lost through Roche-lobe overflow (RLOF), and the fraction ofmass that is accreted by the companion. For the donor star the net mass loss is

Md = Mrlof + Mwind,d − εwind,a · Mwind,a, (5.3)

while for the accretor

Ma = Mwind,a − εwind,d · Mwind,d − εrlof · Mrlof . (5.4)

Where εwind is the fraction of the wind mass loss of the other star that is accretedand εrlof is the fraction of the RLOF mass loss accreted by the companion. Theamount of mass lost to infinity is then:

M∞ = (1− εrlof) · Mrlof + (1− εwind,d) · Mwind,d

+ (1− εwind,a) · Mwind,a. (5.5)

The change in orbital angular momentum has several components: mass lost toinfinity from both Roche-lobe overflow and stellar winds, takes away angularmomentum. Further angular momentum is lost due to gravitational waveradiation and magnetic braking. A final component of the angular momentumloss is angular momentum change due to resonances between the binary and apotential circumbinary (CB) disk. Note that the angular momentum change dueto a stellar wind is split up in the angular momentum change due to the actualmass loss (Jwind) , and the angular momentum change due to the couplingbetween the star and its magnetic wind (Jmb). The total change in angularmomentum is then:

Jorb = Jrlof + Jwind + Jgr + Jmb + Jdisk. (5.6)

In the wide binary models considered here, the strongest contributions to Jorbwill come from mass loss during Roche-lobe overflow and the interaction betweenthe binary and a circumbinary disk.


The evolution of the eccentricity is governed by the tidal interactions betweenboth stars, the effect of phase dependent mass loss/transfer which can increaseor decrease the eccentricity based on the mass ratio and the interaction betweenthe binary and a circumbinary disk. Furthermore there is the circularizing effectof gravitational waves, which in our case is negligible:

e = etides + eml + edisk + egr. (5.7)

The relation between the orbital period and the separation is:

P = 2π

√a3

G(Md +Ma) . (5.8)

All physical mechanisms relevant to the terms given in equations (5.3 - 5.7) areexplained in the following subsections.

5.2.2 Roche-lobe overflow

The instantaneous Roche-lobe radius of the donor star (RL,d) at a specificmoment in the orbit is approximated by adapting the fitting formula for theeffective Roche-lobe radius of Eggleton (1983) by allowing the binary separationto depend on the orbital phase (θ):

RL,d(θ) = 1− e2

1 + e cos(θ) ·0.49 q2/3

d a

0.60 q2/3d + ln(1 + q

1/3d )

. (5.9)

Where a is the semi-major axis, e the orbital eccentricity. The mass ratios qdand qa are defined as:

qd = Md

Ma, qa = Ma

Md. (5.10)

The Roche-lobe radius of the accretor can be obtained by inverting the massratio in Eq. 5.9, thus replacing qd with qa.

Mass loss is implemented in MESA following the prescription of Ritter (1988)and Kolb & Ritter (1990), assuming a stationary isothermal subsonic flow ofgas through the inner Lagrange point, reaching sound velocity near L1. Thisformalism differs between mass loss from the optically thin region (Rd ≤ RL),and that from the optically thick region (Rd > RL). Mass-loss from the formerregion is given by:

Mthin = M0 · exp(Rd −RL,d

HP,L1

), (5.11)


where Rd is the donor radius and HP,L1 is the pressure scale height at the innerLagrange point (L1), which can be linked to the pressure scale height at thephotosphere of the donor HP,ph as:

HP,L1 = HP,ph

γ(q) = 1γ(q) ·

PphR2d

ρphGMd. (5.12)

Here Pph is the pressure at the donor photosphere and ρph is the photosphericdensity of the donor. γ(q) is a factor that depends on the mass ratio as:

γ(q) =

0.954 + 0.025 log10(q)− 0.038 log2

10(q) 0.04 < q < 1.

0.954 + 0.039 log10(q) + 0.114 log210(q) 1 ≤ q ≤ 20.

(5.13)

M0 is the mass loss rate when the donor star just fills its Roche-lobe:

M0 = 2π√e

(RTeff,d

µph,d

)3/2 R3L,d

GMdρph,d F (qd). (5.14)

µph,d and ρph,d are the mean molecular weight and density at the donorphotosphere, Teff,d is the effective temperature of the donor and R is the idealgas constant. e is the base of the natural logarithm and not the eccentricity.F (q) is a function of the mass ratio depending on the Roche geometry, given by(see also Meyer & Meyer-Hofmeister 1983):

F (q) = q√g2(q)− (1 + q) g(q)

(a

RL,d

)3, (5.15)

where g(q) is:g(q) = q

x3L

+ 1(1− xL)3 . (5.16)

Where xL is the distance between the centre of mass of the donor and the L1point in units of the binary separation. An approximation of this distance isgiven by Frank et al. (2002):

xL = 0.5− 0.227 log10(q). (5.17)

In the case that the stellar radius significantly overfills its Roche-lobe, theRoche-lobe will lie in the optically thick region of the donor. The mass loss isthen calculated via:

Mthick = M0 + 2πF (q)R3

L,d

GMd

∫ Pph

PL1

F3(Γ1)

√RTµdP. (5.18)


Where M0 is given in Eq. (5.14), PL1 and Pph are respectively the pressureat L1 and at the stellar photosphere. Both the temperature T and the meanmolecular weight µ depend on the pressure. F3 is a function that depends onthe adiabatic exponent Γ1, and is given by:

F3(Γ1) =√

Γ1

(2

Γ1 + 1

) Γ1 + 12Γ1 − 2

. (5.19)

The mass-loss equations depend on the eccentric anomaly through the Roche-lobe radius. The time steps that MESA uses in the evolution are much longerthan one orbital period, thus to obtain the average mass loss rate to be used ina time step, the above equations are integrated over the orbit.

RLOF is not necessarily conservative. To allow for mass to be lost from thesystem the formalism of Tauris & van den Heuvel (2006) and Soberman et al.(1997) is used. This system describes three fractions (α, β, δ) to lose mass fromthe system:

• α: mass lost from the vicinity of the donor as a fast wind (Jeans mode).This is modelled as a spherically symmetric outflow from the donor starin the form of a fast wind. The mass lost in this way carries the specificangular momentum of the donor star.

• β: mass lost from the vicinity of the accretor as a fast wind (Isotropicre-emission). A flow in which matter is transported from the donor to thevicinity of the accretor, where it is ejected as a fast isotropic wind. Masslost in this way carries the specific angular momentum of the accretor.

• δ: mass lost from a circumbinary coplanar toroid. The radius of thecoplanar toroid is determined by γ as Rtoroid = γ2a.

The accretion efficiency of RLOF is then given by:

εrlof = 1− α− β − δ. (5.20)

These mass-loss fractions are also shown in Fig. 5.1

When the mode of mass loss is known, the change in angular momentum can becalculated. Mass accreted by the companion will not change the total angularmomentum of the system, thus only fractions α, β and δ will have an influenceon J . The total effect of mass loss through Roche-lobe overflow on the changein angular momentum is given by Tauris & van den Heuvel (2006):

Jrlof = α+ βq2 + δγ(1 + q)2

1 + q

Mrlof

Md· Jorb. (5.21)


−1 0 1

−1

0

1

L1L2 L3RGMS

ǫ

α

β

δ

Figure 5.1: The mass-loss fractions as described by Tauris & van den Heuvel(2006) and Soberman et al. (1997). α: mass lost from the vicinity of the donor.β: mass lost from the vicinity of the accretor. δ: mass lost from a CB toroidwhich in this figure has a radius equivalent to the location of L2. ε: fraction ofaccreted mass. See also Sect. 5.2.2


5.2.3 Wind mass loss

The amount of mass lost through stellar winds is determined by any of thewind loss prescriptions defined in the stellar part of MESA. The binary moduleoffers the possibility to boost this mass loss using the Companion ReinforcedAttrition Process (CRAP) mechanism (Tout & Eggleton 1988), also known astidally enhanced mass loss. In this model it is assumed that tidal interactionsor magnetic activity are responsible for the enhancement of the wind. Theenhancement is expected to have a similar dependence on radius over Roche-lobe radius as the torque in a tidal friction model, and there is an expectedsaturation when co-rotation is reached (in their model when R = RL/2). Thephase dependent form of this equation is (Siess et al. 2014):

Mwind(θ) = MReimers ·

1 +Bwind ·min

[(R

RL(θ)

)6,

126

]. (5.22)

The factor Bwind is estimated by Tout & Eggleton (1988) to be of the orderBwind ≈ 104, but can vary significantly depending on which system needs to beexplained.

Part of the mass lost due to stellar winds is accreted by the companion. Inthe case of fast winds (vwind vorb) the accretion fraction is given by theBondi-Hoyle mechanism (Hurley et al. 2002). Here we give the equations for thedonor, those of the accretor can be obtained by switching the d and a subscripts:

εBH,d = 1√1− e2

(GMd

v2wind,a

)2αBH

2a21

(1 + v2)3/2 . (5.23)

The velocities are:

v2 = v2orb

v2wind,a

, (5.24)

v2orb = G(Md +Ma)

a, (5.25)

v2wind,a = 2βW

GMa

Ra. (5.26)

The wind velocity used here is set proportional to the escape velocity fromthe stellar surface. Based on observed wind velocities in cool super giants(vwind = 5− 35 km s−1) (Kučinskas 1998), βW is taken to be 1/8(Hurley et al.2002). The free parameter αBH = 3/2 based on Boffin & Jorissen (1988).


The angular momentum loss due to wind mass loss, assuming a sphericallysymmetric wind, is:

Jwind,d = Mwind,d

(Ma

Md +Maa

)2 2πP

√1− e2. (5.27)

For the total change in angular momentum of the binary due to stellar winds,the contribution of the accretor need to be added and Jwind = Jwind,d + Jwind,a.

5.2.4 Phase-dependent mass loss

When mass loss, whether due to stellar winds, RLOF or other reasons, is notconstant during the binary orbit, it will have an effect on the eccentricity ofthe system. There are multiple processes which can induce a periodicity in themass loss and mass accretion. For example, the mass loss might be caused bystellar pulsations, which can transfer more mass at the periastra that coincidewith a maximum stellar radius. The methods implemented in MESA are phase-dependent wind mass loss through tidal interactions, and phase-dependentRLOF on eccentric orbits. This will boost mass-loss near the periastron passagewhile during apastron the star is completely inside its Roche-lobe. The effectof phase dependent mass-loss on the orbital eccentricity was studied by Soker(2000) based on the theoretical work of Eggleton (2006, Ch. 6.5).

In calculating the effect on the eccentricity we have to distinguish between masslost to infinity, and mass that is accreted by the companion. Assuming isotropicmass-loss, the change in eccentricity for mass lost from the system is given by:

elost(θ) = |M∞(θ)|Md +Ma

(e+ cos θ), (5.28)

where θ is the true anomaly, and M∞(θ) is the mass lost from the system at aspecific phase θ during the orbit. From this equation one can see that constantmass-loss will not change the eccentricity, if however the fraction of mass lostnear periastron is significantly larger than during the remainder of the orbit,Eq. 5.28 predicts a positive e.

The effect on e of mass accreted by the companion, again under the assumptionof isotropic mass-loss, is given by:

eacc(θ) = 2|Macc(θ)|(

1Md− 1Ma

)(e+ cos θ), (5.29)

where Macc(θ) is the mass accreted by the companion at a specific phase θduring the orbit. As mass transfer is expected near periastron, the eccentricitywill increase if Md < Ma, which is required in the case of stable mass transfer.


To obtain the total change in eccentricity in one orbit, Eq. 5.28 and 5.29 areintegrated over the orbit:

eml =∫θ

[ elost(θ) + eacc(θ) ] dθ. (5.30)

5.2.5 Tidal forces

The gravitational forces acting on the components in binary systems inducea deformation of their structure, and creates tidal bulges. These bulges arelagging behind, thus the gravitational attraction generates a torque on thosebulges, which forces the synchronisation and circularisation of the stellar andorbital rotation. The orbital parameters change because the stars rotationalenergy is dissipated into heat. There are two cases one has to consider: starswith convective envelopes and stars with radiative envelopes. In the former, thekinetic energy of tidally induced large scale currents is dissipated into heat byviscous friction of the convective environment. In radiative stars it is mainlyradiative damping of gravity modes that functions as dissipation process. SeeZahn (2005) for a review on tidal dissipation.

We use the formalism developed by Hut (1981) to calculate the effect of tideson circularisation and synchronisation. This formalism, developed based on theweak friction model, results in (where we again give only the equations relevantfor the donor star, and those of the accretor are obtained by switching the dand a subscripts):

etides,d = −27(k

T

)dqd(1 + qd)

(Rd

a

)8e

(1− e2)13/2(f3(e2)− 11

18(1− e2)3/2f4(e2)Ωω

)yr−1, (5.31)

for the circularisation, while the synchronisation is given by:

Ωtides,d = 3(k

T

)d

q2dr2g

(Rd

a

)8 √G(MdMa)/a3

(1− e2)6(f2(e2)− (1− e2)3/2f5(e2) Ω√

G(MdMa)/a3

)yr−1. (5.32)


Here f2−5 are polynomials in e:

f2(e2) = 1 + 152 e

2 + 458 e

4 + 516e

6, (5.33)

f3(e2) = 1 + 154 e

2 + 158 e

4 + 564e

6, (5.34)

f4(e2) = 1 + 32e

2 + 18e

4, (5.35)

f5(e2) = 1 + 3 e2 + 38e

4. (5.36)

Furthermore k/T depends on the star being convective or radiative. In the caseof a convective envelope, k/T is given by Hurley et al. (2002, eq. 30-33), basedon Rasio et al. (1996): (

k

T

)conv

= 221fconv

τconv

Menv

Myr−1. (5.37)

Where τconv is the eddy turnover time-scale (the timescale on which the largestconvective cells turn over):

τconv = 0.4311(MenvRenv(R−Renv/2)

3L

)1/3, (5.38)

and fconv is a numerical factor depending on the tidal pumping time-scale Ptid:

fconv = min[

1,(

Ptid

2 τconv

)2], (5.39)

1Ptid

=∣∣∣∣ 1Porb

− 1Pspin

∣∣∣∣ . (5.40)

Menv and Renv are respectively the mass in the outer stellar convective zone,and the stellar radius at the base of that zone. L is the luminosity.

In the case of a radiative envelope, the damping is caused by a range ofoscillations that are driven by the tidal field. k/T is then (Hurley et al. 2002,eq. 42-43 based on Zahn 1975, 1977):(

k

T

)rad,d

= 1.9782 · 104MdR2d

a5 (1 + qa)5/6E2 yr−1, (5.41)

where E2 is a second order tidal coefficient which can be fitted with (Seeappendix B of Siess et al. 2013):

E2 = 1.592 · 10−9M2.84d . (5.42)


The total effect on the change in orbital eccentricity is then obtained bycombining the tidal forces on both components:

etides = etides,d + etides,a. (5.43)

5.2.6 Circumbinary disks

Circumbinary disks (CB disks) can form around binaries during the Roche-lobe overflow phase, if part of the mass can leave the system through theouter Lagrange points and form a Keplerian disk around the binary. The CBdisk – binary resonant and non-resonant interactions have been described byGoldreich & Tremaine (1979) and Artymowicz & Lubow (1994), by using alinear perturbation theory. There are two main assumptions in these models;the first is that the disk is thin (0.01 < H/R < 0.1, where H and R arerespectively the thickness and the half angular momentum radius of the disk).The second assumption is that the nonaxisymmetric potential perturbationsare small around the average binary potential.

The effect of the disk – binary resonances on the orbital parameters has beenthe subject of many studies. In MESA we follow the approach of Artymowicz& Lubow (1994) and Lubow & Artymowicz (1996). The model of Lubow &Artymowicz (1996) for small and moderate eccentricities (e < 0.2) is based onthe result of smooth particle hydrodynamic (SPH) simulations, which showthat for a disk in which the viscosity is independent of the density, the torqueis independent of resonance strength and width. The variations in orbitalseparation are caused by inner and outer Lindblad resonances5, which areorbital resonances in which the epicyclic frequency (frequency at which aradially displaced fluid parcel oscillates) is a multiple of a forcing frequency.They are given by (Lubow & Artymowicz 1996):

a

a= −2l

m· JD

JB· 1τv. (5.44)

Where l and m are integer numbers indicating which resonance has the strongestcontribution. JD and JB are respectively the angular momentum in the disk andin the binary, and τv is the viscous evolution timescale, which is the timescaleon which matter diffuses through the disk under the effect of the viscous torque.It is given by:

1τv

= αD

(H

R

)2Ωb, (5.45)

5Named after the Swedish astronomer Bertil Lindblad


where αD is the viscosity parameter of the disk and Ωb is the orbital angularfrequency. With a disk viscosity of αD = 0.1, the viscous timescale is typicallyon the order of 105 yr.

To determine the angular momentum of a Keplerian disk, one needs to knowthe surface mass distribution σ in the disk. JD is then given by:

JD =∫A

r · σ · v dA =∫ Rout

Rin

r · σ ·√G(Md +Ma)

r2 π r dr. (5.46)

Here the Keplerian velocity of an element in the disk at distance r from thecentre of mass, when neglecting the mass of the disk is: v =

√G(Md +Ma)/r.

Rin and Rout are the inner and outer boundaries of the disk. We assume thesurface distribution in the disk to depend on the radius to the power of a freeparameter δ:

σ(r) = Dc

rδ. (5.47)

The distribution constant Dc depends on the total disk mass and δ. With thisdistribution the disk angular momentum can be calculated:

JD = 2π Dc√G(Md +Ma)

∫ Rout,i

Rin

r3/2−δ dr, (5.48)

which, depending on parameter δ, has the following solutions:

JD =

2πDc

√G(Md +Ma)

5/2− δ

(R

5/2−δout,i −R

5/2−δin

), δ 6= 5/2.

2πDc√G(Md +Ma) (lnRout,i − lnRin) , δ = 5/2.

(5.49)

As shown by the SPH simulations of Lubow & Artymowicz (1996), only theinner part of the disk plays a role in the disk – binary interactions. Thus in theprevious equations the angular momentum is only calculated from the regionbetween Rin and six times the binary separation (Rout,i = 6a), see also Fig. 5.2.

To calculate the distribution constant in Eq. (5.47), one has to know thetotal mass in the disk (Mdisk). The total disk mass can then be related to thedistribution constant as:

Dc = Mdisk

2π∫ RoutRin

r1−δ dr, (5.50)


CM Rin Rout,i = 6 · a Rout = 250AU

Distance

−1.0

−0.5

0.0

0.5

1.0

Hei

ght

Interaction No Interaction

a

Figure 5.2: Schematic representation of the CB disk model. CM is the centreof mass of the binary, a is the separation of the binary. The total disk mass islocated in the filled area, while only the red region interacts with the binary.When the disk angular momentum (JD) is calculated in eq. 5.49, to obtain theinteraction effects, this is only the angular momentum in the red region. Seealso Sect. 5.2.6

which depending on δ has two solutions:

Dc =

1

2π(2− δ)Mdisk

R(2−δ)out −R(2−δ)

in, δ 6= 2.

12π

Mdisk

lnRout − lnRin, δ = 2.

(5.51)

The main difference between the implementation here and that of Dermineet al. (2013) is that the disk surface distribution parameter δ in our model isconsistent throughout the calculation of the disk angular momentum and thedistribution constant Dc, while Dermine et al. (2013) used δ = 2.5 in the former,while δ = 1 was used in the latter.

We use two processes to determine the inner radius of the CB disk. Theinteraction between the binary and the disk will clear the inner part of the disk.SPH simulations performed by Artymowicz & Lubow (1994) show that theinner radius depends on the Reynolds number of the disk gas (< = (H/R)−2α1)and the eccentricity. A fitting formula of their results was derived by Dermine


et al. (2013):Rin,SPH = 1.7 · 3

8 log (<√e) AU, (5.52)

where e is the orbital eccentricity. The second limiting factor on the inner radiusis the dust condensation temperature. Based on Dullemond & Monnier (2010,Eq. 1-12), and adding an offset for the binary separation we obtain:

Rin,dust =√

Ld + La

4π σbol T 4cond

+ a(MaLd +MdLa)(Md +Ma)(Ld + La) . (5.53)

Where Ld and La are respectively the luminosity of the donor and accretor,and σbol is the Stefan-Boltzmann constant. Tcond is the dust condensationtemperature which we take at 1500 K. The last factor in Eq. (5.53) is theaverage binary separation weighted by the luminosity of both components. Theactual inner radius of the disk is the maximum of that determined from SPHsimulations and the dust condensation radius:

Rin = max [Rin,SPH , Rin,dust ] . (5.54)

There are no direct observations of the outer radius of a CB disk around a post-AGB or post RGB binary. Post-AGB and post-RGB disks are likely similar toprotoplanetary disks, thus it can be assumed that the surface mass distributionof a CB disk does not follow the same behaviour along the whole radius. Theinner part of the disk follows σ(r) ∼ r−δ, δ = 1− 2, while the outer part has anexponential decline in σ. For the CB disk-binary interaction only the inner partof the disk is important. The outer disk radius here represents the radius wherethe exponential drop in mass distribution starts, and the maximum disk massrepresents the mass in this inner part of the disk. We assume an outer diskradius of Rout = 250 AU, see for example Bujarrabal et al. (2007, 2013, 2015).Keep in mind that this outer radius is not the same as the Rout,i = 6a radiusthat is used in calculating the effective disk angular momentum in eq. 5.49 (See also Fig. 5.2).

The change in eccentricity due to Lindblad resonances can be given as a functionof a/a, and depends on the eccentricity. For small eccentricities this can becalculated analytically. In the range e ≤ 0.1√αD them = l resonance dominates,while in the region 0.1√αD < e ≤ 0.2 the m = 2, l = 1 resonance is dominant.For small e the analytical form of e is (see also Lubow & Artymowicz 2000):

edisk = 1− e2

e+ αD100e

(l

m− 1√

1− e2

)· aa, e ≤ 0.2. (5.55)


0.0 0.2 0.4 0.6 0.8 1.0

Eccentricity

0

2

4

6

8

10

12

14

16

e/a

Exact

Extrapolation

Figure 5.3: The dependence on orbital eccentricity of the eccentricity pumpingeffect of a circumbinary disk. For low eccentricity the dependence is analytical(blue full line), for higher eccentricities, the dependence is extrapolated (reddashed line). Above an eccentricity of e = 0.7, there is no more eccentricitypumping. The maximum eccentricity pumping is around e = 0.03. See alsoSect. 5.2.6

For average eccentricities the above equation is extrapolated according to anefficiency decreasing with 1/e:

edisk =(c4e

+ c5

)· aa, 0.2 < e ≤ 0.7. (5.56)

Where the constants are c4 = −1.3652 and c5 = −1.9504 so that Eq. 5.56smoothly connects to Eq. 5.55, while going to 0 at e = 0.7. For largereccentricities resonances that damp the eccentricity start to dominate, thusedisk = 0 in the range e > 0.7 (e.g. Roedig et al. 2011).

The change in angular momentum due to disk - binary interactions is:

Jdisk = Jorb ·(a

2a −eedisk

1− e2

), (5.57)

where a/a is given by Eq. (5.44).

The mass feeding the CB disk is the fraction of the mass lost from the binarysystem through the outer Lagrange point during RLOF. The CB disk mass-lossrate is determined by the maximum mass in the disk Mdisk,max, and the life


time of the disk τdisk. Both are input parameters in our model. The rate inwhich the disk loses/gains mass is then:

Mdisk = δ |Mrlof | −Mdisk,max

τdisk. (5.58)

In this implementation the disk will continue existing for a period τdisk, afterRLOF stops. During this period τdisk, the disk mass will decrease linearly.

5.2.7 Gravitational wave radiation and magnetic braking

Two processes that are mainly of interest in close binaries are angular momentumloss through gravitational waves and magnetic breaking. We give them herefor completeness as they are implemented in MESA, and will have an effect(although negligible) on the orbital evolution.

Angular momentum loss through gravitational wave radiation is important inclose binaries like coalescing neutron stars. The gravitational wave formalismcan be derived from the weak-field approximation of general relativity (Landau& Lifshitz 1971; Hurley et al. 2002), and its validity was demonstrated inthe Hulse-Taylor pulsar (Taylor & Weisberg 1989). The change in angularmomentum due to gravitational waves for eccentric orbits under the assumptionof point masses is then:

Jgr

Jorb= −32G3

5 c5MdMa (Md +Ma)

a41 + 7/8 e2

(1− e2)5/2 s−1, (5.59)

where c is the speed of light in vacuum. The effect on the orbital eccentricity isgiven by:

egr

e= −32G3

5 c5MdMa (Md +Ma)

a419/6 + 121/96 e2

(1− e2)5/2 s−1. (5.60)

For stars with very small separations the effect of gravitational waves will be acontinuous spiral-in, and a circularisation of the orbit if that would not havehappened yet due to tidal interactions. As we only consider long orbital periods,these effects will not have any influence on the models presented in this paper.

Like gravitational radiation, magnetic braking is efficient in close binaries witha radiative core. Angular momentum can be lost from the system via magneticbraking of the tidally coupled primary by its own magnetic wind. Based onobservations and the Skumanich (1972) breaking law, Rappaport et al. (1983)derived:

Jmb = −3.8× 10−30MdR4min(Rd, RL,d)γmb ·

(2πP

)3s−1. (5.61)


Where γmb is the magnetic breaking index as defined in Rappaport et al. (1983),which is taken to be γmb = 3. This formalism is derived under the assumptionthat the magnetic breaking is independent of possible mass transfer, and thatthe mass losing star is forced to co-rotate with the orbital revolution due totidal forces.

In the next chapter we will apply these modules to test if we can model theobserved periods and eccentricities of sdB stars in wide orbits.

6Testing possible eccentricity pumping processes


Testing eccentricity pumping mechanisms to modeleccentric long-period sdB binaries with MESA

J. Vos, R.H. Østensen, P. Marchant, H. Van Winckel

Author contributionsThe parameter study of the different eccentricity pumping processes was performed by J. Vos.The development of the binary module of MESA was started by P. Marchant, while theaddition of several physical processes to the binary module of MESA necessary to analyseeccentric systems and including eccentricity pumping was done by J. Vos (see Chapter 5 for adetailed list). R.H. Østensen and H. Van Winckel contributed to the discussion of the results,and the final text.

ABSTRACT

Context. Hot subdwarf-B stars in long period binaries are found to be oneccentric orbits, even though current binary evolution theory predicts thoseobjects to be circularised before the onset of Roche-lobe overflow (RLOF).Aims. We aim to find binary evolution mechanisms that can explain these

127

128 TESTING ECCENTRICITY PUMPING PROCESSES

eccentric long period orbits, and reproduce the currently observed period-eccentricity diagram.Methods. Three different processes are considered; tidally enhanced wind massloss, phase dependent RLOF on eccentric orbits and the interaction between acircumbinary disk and the binary. The binary module of the stellar evolutioncode MESA (Modules for Experiments in Stellar Astrophysics) is extendedto include the eccentricity pumping processes. The effects of different inputparameters on the final period and eccentricity of a binary evolution model aretested with MESA.Results. The end products of models with only tidally enhanced wind massloss can indeed be eccentric, but these models need to lose too much mass,and invariably end up with a helium white dwarf that is too light to ignitehelium. Within the tested parameter space, no sdBs in eccentric systems areformed. Phase dependent RLOF can reintroduce eccentricity during RLOF, andcould help to populate the short-period part of the period-eccentricity diagram.When phase dependent RLOF is combined with eccentricity pumping via acircumbinary disk, the higher eccentricities can be reached as well. A remainingproblem is that these models favour a distribution of higher eccentricities atlower period, while the observed systems show the opposite.Conclusions. The models presented here are capable of explaining the period-eccentricity distribution of long period sdB binaries, but further theoreticalwork on the physical mechanisms is necessary.

6.1 Introduction

In Part I of this thesis the observed properties of all known wide sdB binarieshave been extensively discussed. The orbital parameters were compared tothe theoretical predictions in Sect. 4.6, where it became clear that the maindiscrepancy between the observations and theory is the predicted circularisationof the orbits. In the current theoretical models, all long period sdB binaries arecompletely circularized, while almost all of the observed systems have significanteccentricities varying from e = 0.04 to 0.17. This eccentricity problem isnot entirely new, and a few possible solutions have been proposed, althoughnone have been applied to the case of sdB binaries. There are three potentialmechanisms that can create eccentric orbits described in the literature.

Soker (2000) based on theoretical work of Eggleton (2006) proposed themechanism of tidally-enhanced wind mass-loss to allow the eccentricity ofthe orbit to increase. In this framework, the wind mass-loss is increased dueto the tidal influence of a companion in an eccentric orbit, and the effect is

INTRODUCTION 129

present before the system comes into contact. It reduces the tidal forces bykeeping the sdB progenitor within its Roche lobe, while the phase-dependentwind mass-loss can increase the eccentricity. This mechanism is also knownas Companion Reinforced Attrition Process (CRAP). Siess et al. (2014) usedthis mechanism to successfully explain the orbit of the highly eccentric He-WDbinary IP Eri.

A second possible mechanism is that of phase-dependent mass loss duringRLOF. Similar to the phase-dependent mass loss in the CRAP mechanism, avarying mass-loss rate in a binary that is larger during periastron than apastron,can increase the eccentricity of the orbit. The difference with the previousmechanism is that it is active during RLOF. This mechanism was used byBonačić Marinović et al. (2008), who used a model with enhanced mass lossfrom the AGB star due to tidal interactions and a smooth transition betweenthe wind mass loss and the RLOF mass loss regimes to explain the eccentricpopulation of post-AGB binaries.

The third method of increasing the eccentricity of binaries is by interaction of thebinary with a circumbinary (CB) disk. The motivation to include this processin the context of sdB stars, is the observational finding that stable circumbinarydisks are commonly observed around evolved post-AGB binaries (e.g. de Ruyteret al. 2006; Hillen et al. 2014, and references therein). The longevity of thesedisks are corroborated by the strong processing of the dust grains as attestedby the infrared spectral dust-emission features (e.g. Gielen et al. 2008, 2011)and the millimetre continuum fluxes that indicate the presence of large grains(de Ruyter et al. 2005), while interferometric techniques are needed to resolvethem (see e.g. Hillen et al. 2014, and references therein). The Keplerian rotationhas, so far, only been spatially resolved in two objects (Bujarrabal et al. 2005,2015). Recent surveys of the Large and Small Magellanic Clouds (van Aarleet al. 2011; Kamath et al. 2014) show that a significant fraction of post-AGBstars show the distinctive near-IR excess indicative of a stable disk. One ofthe results of Kamath et al. (2014, 2015) is that a significant population ofpost-RGB stars were identified with circumstellar dust likely in a disk as well.The disk evolution will determine the infrared life time of the systems andhence the detectability. While there is ample observational evidence that disksare common in evolved binaries, we assume here that disks were also presentduring the RGB evolution of the sdB progenitor. Dermine et al. (2013) exploredthe eccentricity-pumping mechanisms of CB disks in post-AGB binaries, basedon theoretical work and smooth-particle-hydrodynamic (SPH) simulations ofArtymowicz & Lubow (1994) and Lubow & Artymowicz (1996). An issue withthe results of Dermine et al. (2013) is that there was a mismatch betweenthe disk mass distributions used to derive different parts of the interactionmechanisms.


In this chapter we will explore these three methods to find formation channelsthat can explain the eccentricity of sdB binaries, using the stellar/binaryevolution code MESA. In the case of CB disks that are formed during RLOF,the eccentricity pumping effect of the disk is combined with the effect of phase-dependent RLOF. We do not aim to perform a binary-population-synthesisstudy, but to explore in a limited number of initial conditions, the efficiencyof the different processes. Our aim is to describe the effect of the modelparameters on the final period and eccentricity of the binary, and to discoverwhich areas in the period-eccentricity diagram can be covered by the threephysical prescriptions.

The MESA evolution code is explained in Sect. 6.2. The modelling methodologyis explained in Sect. 6.3, while the different models are presented in Sects. 6.4,6.5 and 6.6 for respectively the CRAP models, phase-dependent RLOF modelsand models containing a CB disk. The obtained period-eccentricity distributionis discussed in Sect. 6.7, and a summary and conclusion is given in Sect. 6.8.

6.2 MESA

Modules for Experiments in Stellar Astrophysics (MESA) is an open-source state-of-the-art 1D stellar evolution code, which amongst others, includes a binarymodule to compute evolutionary tracks of binary stars. The stellar evolutionmodules of MESA are summarized in Chapter 5. The binary interaction physicsincluded in MESA is extensively described in Sect. 5.2. A reason to work withthis evolution code is the availability as open source. Moreover, the codecalculates stellar models of low-mass stars through the helium flash. As anexample of binary tracks leading to post-RGB evolution, the evolutionary tracksin the HR diagram for a late hot flasher and an early hot flasher are displayedin panels B and C of Fig. 6.1.

6.2.1 Stellar input parameters

As the main focus of this contribution is the study of binary evolution processes,we will use standard parameters for the evolution of the individual stellarcomponents. These include a standard atmospheric boundary condition atan optical depth of τ = 2/3, a mixing length parameter of αMLT = 2 anddefault opacity tables (OPAL type i). For the sdB progenitor we used anextended version of the standard nuclear networks to include all reactions forhydrogen and helium burning (pp_cno_extras_o18_ne22.net). The initialcomposition is X = 0.68, Y = 0.30 and Z = 0.02. Furthermore we used a

MESA 131

3.6

3.9

4.2

4.5

4.8

5.1

0 1 2 3 4 5 6 7 8

logg(dex)

3.6

4.0

4.4

4.8

3.6

4.0

4.4

4.8

3.6

4.0

4.4

4.8

log

Teff

(K)

AB

CD

Figu

re6.1:

HR

diagram

oftheevolutionof

thesdB

progenito

rfordiffe

rent

initial

perio

ds.Heignitio

nis

indicated

with

aredsqua

re.Pa

nelA

:P=

600d,

thedo

nors

tarloses

toomuchmasst

oignite

Hean

den

dsup

asacoolingwh

itedw

arf(M

d=

0.448M).

Pane

lB:P

=650d,

alate

Hefla

sher

(Md=

0.456M).

Pane

lC:P

=750d,

anearly

He

flasher

(Md=

0.466M).

Pane

lD:P

=800d,

The

core

istoomassiv

e,an

dthedo

norignitesHeon

thetip

ofthe

RGB

(Md=

0.929M).

Seesection6.5fordiscussio

n.


Reimers wind on the RGB with ηreimers = 0.7 (Reimers 1975), and a Blöckerwind scheme on the post-EHB with ηblocker = 0.5 (Blocker 1995). These arestandard values in MESA. All models are calculated from the pre-main sequencetill the WD cooling curve, with the RGB taking the longest computation time.The sdBs resulting from these standard settings have higher surface gravitiesand effective temperatures than the observed sdBs. This has been reportedbefore, for example by Østensen et al. (2012); Østensen (2014). We are awareof the discrepancy, but solving this is beyond the scope of this work.

6.3 Modelling methodology

The focus of this chapter is the effect of the different eccentricity-pumpingmechanisms on the evolution. Three methods are investigated in the followingsections. Tidally-enhanced wind mass-loss is considered separately from theother two mechanisms. The main argument is that tidally-enhanced windmass-loss increases the wind mass loss so that the donor star never fills itsRoche lobe. The effects of firstly phase-dependent RLOF alone (Sect. 6.5), andsecondly phase-dependent RLOF in combination with a CB disk (Sect. 6.6)are described. As the CB disk is created from mass lost during RLOF, theeccentricity-pumping effects of phase-dependent RLOF will also be presentwhen the CB disk-binary interaction starts. We do not consider the CB disk-binary interactions separately, but assume the disk is formed by mass lost in theprevious phase. For the phase-dependent RLOF models and the CB-disk models,a default model is used to show the evolution of several binary parameters.Subsequently the effect of initial binary parameters and method dependentparameters are discussed. The complete range in period and eccentricity thatcan be covered using any of the eccentricity-pumping mechanisms is discussedseparately.

In the case of phase-dependent RLOF, and the CB disk-binary interactionmodels, tidal forces will have circularised the orbit before the onset of RLOF.However, if the eccentricity is zero, the eccentricity-pumping effect of phase-dependent mass loss and CB disk-binary interactions will be zero as well.Therefore, we impose a lower limit on the eccentricity of 0.001. How thisminimum eccentricity can influence the final orbital parameters is discussed inSects. 6.5.2 and 6.6.2. A physical argument for enforcing a minimum eccentricitycan be justified from the convective envelope of the red giant, where thegravitational quadrupole moment of convective cells can introduce eccentricityin the orbit. See for example Lanza & Rodonò (2001); Kissin & Thompson(2015) This minimum eccentricity is applied in all models, also those with onlytidally-enhanced wind mass-loss. Thus if we refer to circularised models, this

TIDALLY ENHANCED WIND MASS LOSS 133

effectively means models that reached the minimum eccentricity of e = 0.001.

For each eccentricity-pumping method a substantial number of models withvarying initial parameters and process-depending parameters are calculated.The process-dependent parameters and their ranges are discussed in separatesections. The initial binary parameters (period, eccentricity and mass of bothcomponents) are determined based on the type of systems that we want toproduce. We are focusing on the long period sdB binaries, thus initial orbitalperiods vary between 500 and 900 days. The sdB progenitor mass variesbetween 1.0 and 1.5 M. The initial companion mass varies between 0.8 and1.45 M, and is always lower than the sdB-progenitor mass. Apart from thetidally-enhanced wind mass-loss models, all models circularise completely beforethe eccentricity-pumping mechanisms become effective. Changing the initialeccentricity for these models then only affects the initial orbital momentum ofthe binary, thus changing ei has a similar effect as changing the initial period.

Only a small subsection of this parameter space results in binaries containingan sdB component. In the discussion of the effect of the process-dependentparameters, only the models containing an sdB component are used. Theselection criteria for sdBs are based on the stellar mass at He ignition, whichhas to be below 0.55 M, and the absence of a hydrogen burning shell. In ourmodels, there is a clear mass gap between the sdB models and the models thatignite He on the RGB. The latter having final masses of Md & 0.7M, whilethe sdB models have final masses of 0.45 . Md . 0.49M. Models that don’tignite He are obviously not sdBs either.

6.4 Tidally enhanced wind mass loss

Siess et al. (2014) models the long-period eccentric system IP Eri by usingtidally-enhanced wind mass-loss in combination with the eccentricity-pumpingeffect of phase-dependent mass loss. By increasing the wind mass loss, theradius of the donor can be kept small in comparison to its Roche-lobe, thus thetidal forces that circularise the system are weaker. By using the mechanism ofTout & Eggleton (1988), the wind mass loss depends on the orbital phase, withit being stronger at periastron than apastron. This difference between the masslost at periastron and apastron can increase the orbital eccentricity, dependingon the mass ratio of the system and the fraction of the wind mass loss that isaccreted by the companion.

The enhanced-wind-mass-loss model depends on only one parameter, Bwind,which Tout & Eggleton (1988) estimate at Bwind = 104 to explain the massinversion in the pre-RLOF system Z-Her. The wind-mass-loss rate as a function


of the orbital phase θ is:

Mwind(θ) = MReimers ·

1 +Bwind ·min

[(R

RL(θ)

)6,

126

], (6.1)

where R is the stellar radius and RL(θ) is the Roche-lobe radius at phase θ. Adetailed description of the wind mass loss and accretion fractions in MESA isgiven in Section 5.2.3. The strength of the eccentricity pumping depends on themass-loss rate and the fraction that is accreted by the companion. We followthe model proposed by Soker (2000) to calculate the change in eccentricity:

eml =∫θ

[|M∞(θ)|Md +Ma

+ 2|Macc(θ)|(

1Md− 1Ma

)](e+ cos θ) dθ, (6.2)

where Md and Ma are the donor and accretor mass, M∞ is the mass lost atphase θ to infinity, Macc is the mass lost at phase θ accreted by the companionand e is the eccentricity. Mass lost to infinity increases the eccentricity. Massthat is accreted, drives a change in eccentricity that is positive only if the donormass is less than the accretor mass. See Section 5.2.4 for a detailed descriptionof this mechanism.

In Fig. 6.2, the eccentricity as a function of the mass of the donor star is plottedfor a binary model with initial period of 600 days, a donor mass and companionmass of 1.2 and 0.8 M, and different values of Bwind: 100, 5 000 and 10 000.As can be seen, this method can result in a significant final eccentricity of theorbit after the RGB evolution of the primary, if the enhancement parameter issufficiently large. For the system shown in Fig. 6.2, the circularisation is avoidedwhen Bwind & 5000, while an sdB star is only formed when 10 . Bwind .100. The maximum wind-mass-loss rates in these systems are: Mwind,max =10−6.5, 10−6.2, 10−6.0M yr−1 for the systems with respectively Bwind = 100,5 000 and 10 000. For comparison, the wind-mass-loss rate for the same systemwithout enhancement would be Mwind,max = 10−6.8M yr−1.

When comparing all calculated models, we find that the amount of mass thatthe donor needs to lose to maintain an eccentric orbit is so high that the finalcore mass is too low for He ignition. The donor star then ends its life asa cooling He-WD on an eccentric orbit. To reach a final donor mass largeenough to ignite He, the mass loss has to be so low that the system completelycircularises on the RGB. Therefore, the sdB binaries formed in this channel areall circularised. The final mass of the donor changes with the initial parameters(orbital period, donor and accretor mass, wind accretion fraction), but noneof the parameter combinations resulted in an eccentric system containing ansdB companion. Siess et al. (2014) gives a good overview of the effect of theinitial binary parameters and the wind-enhancement parameter on the finalparameters of the binary system.

PHASE DEPENDENT RLOF 135

0.40.50.60.70.80.91.01.11.2

Donor mass (M⊙)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Ecc

entr

icit

y

Bw = 1·102

Bw = 5·103

Bw = 1·104

Figure 6.2: The eccentricity as a function of donor mass for models withdifferent values for the wind enhancement parameter Bwind: 100 (solid blue),5 000 (dashed red) and 10 000 (dot-dashed green). The initial period is 600 days,and the initial donor and companion masses are 1.2 + 0.8 M. For discussionsee section 6.4.

We conclude that in the parameter regime considered here, the progenitors ofsdBs do not evolve with strongly enhanced mass loss on the RGB. To form ansdB by tidally-enhanced wind mass-loss, the enhancement of the wind needsto be small, so essentially no eccentricity pumping occurs. Furthermore, ifthere is significantly enhanced wind mass-loss, the sdB progenitor will lose itshydrogen envelope in a stellar wind, and does not undergo RLOF during itslater evolution.

6.5 Phase dependent RLOF

When there is no strongly enhanced wind mass loss during the RGB, the sdBprogenitor eventually fills its Roche-lobe and starts RLOF. If this mass losshappens on a slightly eccentric orbit, the mass-loss rate will not be constantover the orbit, and the mass loss can have an eccentricity-pumping effect. Thestrength of the eccentricity pumping will depend on the mass-loss rate, and


the fraction that is accreted by the companion, in the same way as in thetidally-enhanced-mass-loss mechanism.

6.5.1 Model and input parameters

In this section we describe behaviour of the mass lost from the donor star in ourstudy of phase-dependent RLOF. This mass can be accreted by the companionstar or is lost from the system to infinity. To describe the mass lost from thesystem to infinity, the method of Tauris & van den Heuvel (2006) is used. Adetailed description of that mechanism is given in Section 5.2.2. The totalmass lost from the system is subdivided in three fractions. 1) from around thedonor star (α), which carries the angular momentum of the donor. 2) masstransferred to the vicinity of the companion through the inner Lagrange point,and lost from around the companion as a fast wind (β). This lost mass carriesthe angular momentum of the companion star. 3) through the outer Lagrangepoint (δ) which is modelled as a circumbinary toroid with radius 1.25 times thebinary separation (Rtoroid = γ2a, γ = 1.12, based on Pennington 1985). Theactual fraction of the mass loss accreted by the companion is defined by theaforementioned fractions as: ε = 1− α− β − δ.

The default system has an initial period of 700 days, initial eccentricity of0.3, an sdB progenitor mass of 1.2 M and a companion mass of 1.0 M.The companion star is on the main sequence during the relevant part of theevolution, thus radiative tidal dissipation is assumed. The sdB progenitor is onthe RGB when the tidal forces are strongest, and the dissipation mechanismis that of a convective star. A maximum value for the RLOF mass-loss rate isset at 10−2M yr−1, the mass-loss fractions are α = β = 0.35, δ = 0.30 andthe location of the outer Lagrange point is RL2 = 1.25 a, thus γ = 1.12. Theaccretion fraction onto the companion star is zero in this default model. Theseparameters are summarised in Table 6.1. Only the parameters mentioned inthe text are changed.

As it is not known when exactly a common envelope would start to form weassume that RLOF is stable in our models. To this end, we have capped themass-loss rate during RLOF at 10−2M yr−1 and applied an ad hoc upperlimit on the Roche-lobe overfilling of max(R/RL) = 1.25, meaning that modelsin which the donor star radius exceeds 1.25 · RL are discarded. The latterlimit is also imposed as our physical model breaks down at high Roche-lobeoverfilling. The limit on the RLOF mass-loss rate of 10−2M yr−1 is chosen dueto numerical considerations. The effect of the imposed mass-loss limit on thefinal period eccentricity distribution is discussed in section 6.7. It is still possiblethat in these circumstances a CE would form during RLOF. However, there is


Table 6.1: Standard parameters of the binary models. For the models focusingonly on phase-dependent RLOF, the disk mass is set to 0. The symbol columnrefers to the symbols used in the figures.

Parameter Symbol ValuesdB progenitor mass (M) Md 1.2companion mass (M) Ma 1.0period (d) P 700eccentricity e 0.3minimum eccentricity / 0.001

Tidal forcesdissipation type sdB-progenitor / Convectivedissipation type companion / Radiative

Mass lossmaximum M (M yr−1) / 10−2

fraction lost around companion α 0.35fraction lost around donor β 0.35fraction lost through L2 δ 0.30location of L2 γ 1.12accreted fractiona ε 0

CB diskmaximum mass (M) Mdisk 0.01life time (yr) τ 105

viscosity αD 0.01mass distribution σ(r) r−1

(a) The accreted fraction is defined by the other mass loss fractions as: ε = 1 − α− β − δ.

no good formalism to describe red giants that enter RLOF, but observationaldata indicates that CE evolution must exist both with and without orbitalspiral-in.

6.5.2 Parameter study

Eccentricity evolution

The evolution of the default model is plotted in Fig. 6.3. There are three eventsindicated on the figure, a: eml > etidal, b: eml < etidal and c: log(etidal) <


-15, where eml is the eccentricity pumping of the mass loss in phase-dependentRLOF and etidal is the circularisation due to stellar tides. These events indicatetwo phases in the period-eccentricity evolution of the binary. The time scalebetween these phases differs on the figure, and within one phase the time scaleis linear. Underneath the figure the duration of each phase is given.

Phase a–b: The sdB progenitor is close to completely filling its Roche-lobe(R/RL = 0.97), and the mass loss rate reached 2 · 10−4M yr−1, at whichpoint the eccentricity-pumping forces are more effective than the tidalforces. Due to the increasing mass-loss rate, the eccentricity pumpingstays stronger than the tidal forces, even though the latter increase due tofurther Roche-lobe overfilling (maxR/RL = 1.10). The eccentricity startsincreasing, and reaches a value of 0.060 by the end of this phase, an increaseby a factor 60 compared to the assumed minimum eccentricity. Duringthe mass-loss phase the period increases as well, from 852 to 885 days.When the star starts shrinking again, the mass-loss rate decreases as wellas the Roche-lobe-filling factor, until event ‘b’, when the tidal forces againbecome stronger than the eccentricity pumping (M = 10−5.1M yr−1,R/RL = 0.85). This phase of strong mass loss only takes 325 years.

Phase b–c: When the RLOF diminishes, the tidal forces take over, and thebinary starts circularising again for another 13400 years. During thisphase the sdB progenitor is contracting, thus the tidal forces weaken.The eccentricity diminishes from 0.060 to 0.052, while the period slightlyincreases to 888 days. In this system, the eccentricity increased by roughlya factor 50. The evolution is plotted to the point where etidal < 10−15,after which there is no more significant change in eccentricity or period.

Effect of minimum eccentricity

The effect of the implied minimum eccentricity of emin = 0.001 on the finalperiod and eccentricity of the RLOF models is important. By decreasingemin, the final systems will decrease in eccentricity, and, at a certain thresholddepending on other input parameters, the eccentricity pumping forces cannotovercome the tidal forces anymore. For the standard RLOF model this happenswhen the minimum eccentricity is decreased to emin ≈ 0.0001. By increasingemin the final eccentricity increases as well.

In Fig. 6.4, the RLOF standard model is compared to a model with lower eminof 0.0005, and a model with a higher emin of 0.005. All other parametersremained unchanged. The final eccentricities of the adapted models are 0.047and 0.078 respectively for the emin = 0.0005 and emin = 0.005 models, compared


−14

−12

−10

log(

J/J

)

−14

−13

−12

−11

log(

e)

mass loss

tidal

0.00

0.02

0.04

0.06

e

a b c825

850

875

P(d

)

A

B

C

D

Time325 yr 13400 yr

Figure 6.3: Time evolution of several binary properties during the RLOF phase.Three different events are indicated on the X-axes, (a): eml > etidal, (b): eml <etidal and (c): log(etidal) < -15. The time scale on the plot differs between thephases, but is linear within each phase. The duration of each phase is shownunder the figure. Panel A: the change in angular momentum. Panel B: massloss through RLOF. Panel C: the tidal forces (blue) and eccentricity pumpingdue to mass loss (red). Panel D: the orbital eccentricity. Panel E: the orbitalperiod. See section 6.5.2 for discussion.


to ef = 0.052 for the standard model. The change in orbital period for themodel with lower minimum eccentricity is only 1 day, while the model witha higher minimum eccentricity ends up on period of 933 days, 45 days longerthan the standard model. In Sect. 6.6.2 it is shown that this dependence onemin is significantly diminished when CB disks are included in the models.

Effect of initial period and companion mass

The effect of the initial period and companion mass is plotted in Fig. 6.5-A.With increasing initial period, the final period of the sdB binary increases aswell, while at the same time the eccentricity decreases. A similar effect is visiblewhen increasing the companion mass. The closer the companion mass is to thedonor mass of 1.2 M, the longer the final orbital period, and the lower theeccentricity. We first discuss the effect of the initial binary parameters, as theyclearly illustrate the effect of the orbital period on the eccentricity pumpingforce that is also essential in the discussion of the process dependent parameters.

To explain these effects, the evolution of several parameters for models withdifferent companion masses of 0.8, 1.0 and 1.15 M are shown in Fig. 6.6.These parameters are plotted as a function of the donor mass instead of time,so that the different models can be more easily compared. The connectionbetween eccentricity and orbital period is found in the change in mass-loss rateduring RLOF. If the period of the system increases, the Roche-lobe overfillingis lower, varying from R/RL < 1.16 for the Ma = 0.8 M system to R/RL <1.09 and 1.07 for Ma = 1.0 and 1.15 M respectively. This lower overfilling ofthe Roche lobe leads to a lower mass-loss rate and a shorter total time duringwhich mass is lost at the maximum rate. In the models described here, themaximum mass-loss rate is the same, but the time during which it is sustaineddiffers from roughly 25 years for the 0.8 M companion to 20 years for the 1.0M companion and 15 years for the 1.15 M companion. The difference intime scale seems small, but is significant when the mass-loss rate is 10−2 Myr−1. This diminishes the total eccentricity pumping force, even though thetime during which it overpowers the tidal forces is longer for the 1.15 M model(∼ 380 yr) than for the 0.8 M model (∼ 200 yr). Due to the lower Roche-lobeoverfilling, the tidal forces also decrease with increasing orbital period, with afactor 10 difference between the 0.8 M model and the 1.15 M model. Evenso, there is a net effect of lower eccentricity enhancement in the model with theheaviest companion.

In Fig. 6.5 only the period and mass range of models that resulted in an sdBbinary are shown. In a system with a specified donor and accretor mass, theeffect of the initial period on the final mass is important. If the orbital period


−14

−13

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e)t

idal

emin

0.0050.0010.0005

−14

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log(

e)M

L

0.00

0.03

0.06

0.09

e

a b c

850

900

P(d

)

A

B

C

D

Time

360 yr 13100 yr325 yr 13300 yr300 yr 13400 yr

Figure 6.4: Time evolution of several binary properties during the RLOF phasefor the standard model (blue full line), a model with emin = 0.0005 (red dashedline) and a model with emin = 0.005 (green dashed-dotted line). Three differentevents are indicated on the X-axes, (a): eml > etidal, (b): eml < etidal and (c):log(etidal) < -15. The time scale differs between the phases, but is linear withineach phase. The duration of each phase for each model is shown under thefigure. Panel A: the tidal forces. Panel B: eccentricity pumping due to massloss. Panel C: the eccentricity. Panel D: the orbital period. See section 6.5.2for discussion.


700 800 900 1000 11000.02

0.04

0.06

0.08

0.10

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0.14

Pi = 650 − 700d

Pi = 650 − 750d

Pi = 650 − 750d

800 1000 1200 1400 16000.00

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0.06

α = 0.8 β = 0.0

α = 0.0β = 0.8

400 600 800 1000 1200 14000.00

0.05

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0.15

0.20

0.25

γ =

0.9− 1.3

δ = 0.1− 0.3

880 900 920 940 960 980 1000

0.04

0.05

0.06

0.07

0.08

ǫ = 0− 20%

Period (days)

Ecc

entr

icit

y

A B

C D

Figure 6.5: The effect of the initial and mass-loss parameters of the RLOFmodels on the final period and eccentricity of the orbit. Panel A: Models withdifferent initial periods (indicated by the black arrows) and different companionmasses: Ma = 0.8 (blue stars), Ma = 1.0 (red dots) and Ma = 1.15 (greensquares). Panel B: Models with different mass loss fractions α and β, for twocompanion masses: 1.0 M (red circles) and 1.15 M (blue squares). The blackarrow indicates decreasing α and increasing β, δ is constant at 0.2. The sum ofthe mass loss fractions is unity for each model. Panel C: Models with differentmass loss fractions δ: 0.1, 0.2 and 0.3, with γ = 1.12 (red squares) and differentvalues for γ: 0.9, 1.0, 1.2 and 1.3 with δ = 0.3 (blue dots). These models haveno accretion, thus α = β = (1 − δ)/2. Panel D: The effect of the accretedfraction for two companion masses and accretion fractions of 0%, 10% and 20%.Ma = 1.0 M (red circles) and Ma = 1.15 M (blue squares). See Sects. 6.5.2and 6.5.2 for discussion.


is too short, the mass loss is too strong, and the donor star will loses too muchmass to ignite helium, and ends up on the He-WD cooling track. The lowerlimit on helium ignition in MESA is around 0.45 M, slightly depending oncomposition. If the initial orbital period is too high, the mass-loss rate duringRLOF will be too low, and the donor ignites helium on the RGB and hence nosdB is created. In Fig. 6.1 the HR diagram of four models with a donor andcompanion mass of 1.2 + 1.0 M and initial periods of 600, 650, 750 and 800days are shown. The system with the shortest orbital period (panel A) ends upwith a donor star mass of 0.448 M, this is just below the He-ignition limit,and the donor star ends its evolution as a He white dwarf. The system withthe 650 day initial period (panel B) ends with a donor star of 0.456 M andignites helium on the He-WD cooling track, which is called a late hot flasher.With an even longer period of 750 days, the final donor mass is 0.465 M. Thedonor ignites helium shortly after departing of the RGB, during the evolutionat constant luminosity, an early hot flasher. If the period is increased further,to 800 days, the donor is too massive, and ignites helium on the RGB when itstill has a total mass of 0.929 M, after which mass loss is stopped until thedonor finishes core-He burning and enters the AGB phase.

Effect of mass-loss fractions and accretion

The different ways to lose mass to infinity change the final orbital parametersof the system. These parameters mainly influence the amount of angularmomentum that is removed from the system with the lost mass. The effect ofthese mass-loss fractions and the location of the outer Lagrange point is shownin figures 6.5-B and 6.5-C. In Fig. 6.5-B, models for two different companionmasses 1.0 and 1.15 M at a constant δ = 0.2 are plotted, while changing thefractions of mass lost around the donor (α) and companion (β). There is noaccretion in these models (α+ β + δ = 1). When most mass is lost from aroundthe donor the resulting period will be lower than when the mass is lost fromaround the companion star. This is easily explained by Eq. 5.21. The massthat is lost from around the donor star will thus carry more angular momentumwith it, which will result in a shorter orbital period. This change in periodwill influence the change in eccentricity by altering the mass-loss rates duringRLOF, similar as was explained in Sect.6.5.2. From a certain threshold periodthat depends on the initial and mass-loss parameters, the eccentricity pumpingis smaller than the tidal forces, and the orbits stay circularised. The effect ofthese two mass-loss parameters on the period is large, of the order of severalhundred days. By changing from most mass lost around the donor to mostmass lost around the companion, the final period can double.

By increasing the fraction of mass that is lost through the outer Lagrange point


0.9

1.0

1.1

R/R

L

−4

−3

−2

log(

M)

−13

−12

−11

−10

log(

e)t

idal

0.40.50.60.70.80.91.0

Md (M⊙)

−13

−12

−11

−10

log(

e)p

ump

A

B

C

D

Figure 6.6: Comparison of three models with different values for the companionmass: 0.8 M (solid blue), 1.0 M (dashed red) and 1.15 M (dot-dashedgreen). Panel A: the Roche-lobe overfilling factor. Panel B: the mass loss rateduring RLOF in M yr−1. Panel C: change in e due to tidal interaction inlog(s−1). Panel D: change in e due to mass loss in log(s−1). The noise on e isnumerical. Continued in next figure . . .


0.0

0.3

0.6

0.9

e·1

010

(s1)

0.4

0.6

0.8

1.0

1.2

Md/M

a

0.00

0.03

0.06

0.09

e

0.40.50.60.70.80.91.0

Md (M⊙)

750

800

850

900

P(d

)

E

F

G

H

Figure 6.6: Continued from previous figure. Panel E: net total change ofeccentricity. The noise on e is numerical. Panel F: mass ratio. Panel G: theeccentricity. Panel H: the orbital period in days. See section 6.5 for discussion.


(δ), more angular momentum is lost than if that mass was lost from aroundeither of the binary components. The decrease in final period with increasingδ then follows directly from Eq. 5.21. Increasing the size of the circumbinarytoroid representing the location of the outer Lagrange point (γ) has the sameeffect, for exactly the same reason. Fig. 6.5-C shows the effect of changing δand γ. The models with varying δ have no accretion, thus α = β = (1− δ)/2.The increase in the eccentricity with increasing δ and γ is directly related tothe decrease in the orbital period, and thus a higher mass-loss rate. The twomodels on Fig. 6.5-C with γ values of 1.2 and 1.3 result in binaries with veryhigh Roche-lobe-overfilling factors (R/RL > 1.6). We note that our descriptionbreaks down at high Roche-lobe-overfilling values, and the models in Fig. 6.5-Cwith high values for γ suffer from extrapolation. The effect of γ is only given asan indication, as we keep it at 1.12 to represent the expected location of theouter Lagrange point. By changing δ by only a small amount, the final periodcan again change drastically, of the order of hundreds of days. Similar to theα and β fractions, there is a certain threshold for δ under which the orbit willstay circularised. The exact threshold value depends on the other parameters.

The effect of the mass-loss fraction accreted onto the companion (ε) is shownin panel D of Fig. 6.5. By increasing the accreted fraction, the eccentricityincreases while the orbital period stays more or less constant. By accreting acertain fraction of the mass loss, the size of the Roche-lobes will differ betweenthe models. Higher accretion leads to slightly higher Roche-lobe overfilling, anda slightly higher eccentricity pumping. The period during which the eccentricity-pumping forces are stronger than the tidal forces also increases with increasingaccretion rates. By increasing the accretion, the final eccentricity can almostbe doubled, while the period remains constant.

6.6 Circumbinary Disks

The models with phase-dependent RLOF can indeed explain a certain part ofthe period-eccentricity diagram, but have problems in the high-period high-eccentricity range, and cannot reproduce the circular systems. Circumbinarydisks (CB disks) could potentially explain the high-period, high-eccentricitysystems as they add extra eccentricity-pumping forces on top of those fromphase-dependent RLOF.

CIRCUMBINARY DISKS 147

6.6.1 Model and input parameters

CB disks can form around binaries during the Roche-lobe overflow phase, ifpart of the mass can leave the system through the outer Lagrange points(mass-loss fraction δ) and form a Keplerian disk around the binary. The CBdisk–binary resonant and non-resonant interactions have been described byGoldreich & Tremaine (1979) and Artymowicz & Lubow (1994), by using alinear-perturbation theory. The effect of the CB disk–binary resonances onthe orbital parameters has been the subject of many studies. In MESA wefollow the approach outlined by Artymowicz & Lubow (1994) and Lubow &Artymowicz (1996).

The effect of the disk on the binary separation is given by:

a

a= −2l

m· JD

JB· 1τv, (6.3)

where JD is the angular momentum of the disk and JB the orbital angularmomentum of the binary. l and m are integer numbers indicating the resonancewith the strongest contribution and τv is the viscous evolution time scale (whereτv ∼ α−1

D , see Eq. 5.45). The change in eccentricity depends on the change inbinary separation as:

edisk = 1− e2

e+ α

100e

(l

m− 1√

1− e2

)· aa

(6.4)

for small eccentricities (e < 0.2) and decreases with 1/e for higher eccentricities.The derivation and implementation of these equations is given in Section 5.2.6.

The CB disk is described by its maximum mass (Mdisk), the mass distribution,the inner and outer radius, relative thickness near the inner rim (H/R), theviscosity of the matter (αD) and the total life time of the disk (τ). In ourmodel, the disk is formed by matter lost from the binary through the outerLagrange point, and the disk itself loses mass at a rate determined by its lifetime. The mass in the disk is not constant, and only the maximum disk mass isa defined input parameter. The model of Artymowicz & Lubow (1994) is onlyvalid for a thin disk. Thus, H/R = 0.1 is assumed. The inner radius of thedisk is determined based on smooth-particle-hydrodynamic (SPH) simulations,and the dust-condensation radius of the binary (see Eq. 5.52 – 5.54). Based onobservations and the assumed surface density behaviour, the outer radius ofthe disk is fixed at 250 AU (see also Section 5.2.6).

Maximum mass, life time, viscosity and distribution, are input parameters inthe model. Based on observations of post-AGB disks (Gielen et al. 2011; Hillen


et al. 2014, 2015), following parameter ranges are assumed. The total disk masscan vary between 10−4 and 10−2M. Disk life times after the end of RLOFrange from 104 to 105 years. The viscosity ranges from αD = 0.001− 0.1, withthe more likely range being αD ≤ 0.01. The surface mass distribution decreaseswith increasing distance from the centre: σ(r) ∼ rD, where −2 ≤ D ≤ −1. Inthe default model given in Table 6.1, a maximum disk mass of 10−2M, a lifetime of 105 years, a viscosity of 0.01 and a surface distribution of σ(r) ∼ r−1

are chosen.

The disk model used in this thesis is not self consistent, and conservation ofangular momentum for the CB disk - binary system is not enforced in thecode. However, in all the models presented here, the total angular momentumin the disk is at any time several orders of magnitude lower than the angularmomentum lost by the binary system, and no angular momentum is created.This is shown in Fig. 6.7 for the standard model discussed in Sect. 6.6.2. In thetop panel the total angular momentum (Jbinary + Jdisk) is plotted in functionof time. This Jtotal continuously decreases during the evolution as a result ofthe mass lost from the system. In the bottom panel the contributions of thebinary orbital angular momentum and CB disk angular momentum are shownseparately.

6.6.2 Parameter study

Eccentricity evolution

In Fig. 6.8 the evolution of the CB disk mass, the period and eccentricity, thechange in angular momentum and the change in eccentricity are plotted infunction of time. The model that is shown has the same parameters as themodel shown in Fig. 6.3, with the addition of a CB disk with a life time of 105 yr.These parameters are also given in Table 6.1. Four different events are indicatedon the figure. a: edisk > etidal, b: eml > etidal, c: eml < etidal and d: edisk< etidal. These events define three different phases in the period-eccentricityevolution of the binary. On the figure, the time of each phase is linear, but thetime scales between phases differ. The duration of each phase is plotted underthe figure.

Phase a–b: During this phase mass lost from RLOF is filling the CB disk,and the disk - binary interactions are strong enough to overcome the tidalforces, thus eccentricity starts to increase. The mass in the disk continuesto grow while the donor star continues to expand, eventually filling andoverfilling its Roche-lobe. The tidal forces continue to increase as well,and when the CB disk reaches its maximum mass after roughly 6100 years,


−4

−2

0∆

Jt

·10−

52

0 20 40 60 80 100 120

Time after disk formation (Myr)

46

48

50

52

log

J

Binary

Disk

Figure 6.7: Angular momentum in the binary system and the CB disk in unitsof gr cm2 s−1. Top panel: The total difference in total angular momentum(Jtot,initial − Jtot). Bottom panel: The evolution of the angular momentumin the binary (blue solid line) and the angular momentum in the disk (greendashed line).

the tidal forces again overtake the disk-binary interaction. By this timethe eccentricity of the system reached 0.016 and the period decreased from878 to 852 days. The change in eccentricity is solely due to the disk-binaryinteractions, but the loss of angular momentum from the binary is causedmainly by the mass loss. During the remaining ∼1000 years when edisk <etidal, there is a negligible amount of circularisation.

Phase b–c: The disk reached its maximum mass, and is maintained by RLOF.Due to phase-dependent mass loss, the eccentricity continues to increase.What happens in this phase is very similar to what is shown in Fig 6.3.However, because the binary has a higher eccentricity than that in themodel without a disk, the eccentricity pumping due to mass loss is stronger.This leads to an eccentricity of 0.146, a tenfold increase. The orbitalperiod first decreases and then increases again, reaching 880 days. Thiswhole phase lasts only 310 years.

Phase c–d: RLOF has ended, and the mass in the CB disk starts to decreaselinearly. The CB disk-binary interactions continue to increase the


0.00

0.25

0.50

0.75

1.00

Mdis

k/M

max

−14

−12

−10

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J/J

)

−13

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e)

0.0

0.1

0.2

e

a b c d

825

850

875

P(d

)

A

B

C

D

E

Time7300 yr 310 yr 95000 yr

Figure 6.8: Time evolution of several binary properties during the life timeof the CB disk. Four different events are indicated on the x-axes. a: edisk> etidal, b: eml > etidal, c: eml < etidal and d: edisk < etidal. The time scalediffers between the phases, but is linear within each phase. The duration ofeach phase is shown under the figure. Panel A: the mass in the CB disk as apercentage of the maximum CB disk mass (0.01 M). Panel B: the changein angular momentum due to mass loss (red dashed line) and the CB disk -binary interaction (green dashed dotted line). Panel C: the tidal forces (blue fullline), eccentricity pumping due to mass loss (red dashed line) and eccentricitypumping through CB disk - binary interactions (green dashed dotted line).Panel D: the eccentricity. Panel E: the orbital period. See section 6.6 fordiscussion.


eccentricity, while at the same time angular momentum is transportedfrom the binary to the CB disk, thus decreasing the orbital period again.The effect of the CB disk-binary interactions diminishes due to a decreasein disk mass while simultaneously, the resonances that are responsiblefor the eccentricity pumping become less effective at higher eccentricities.When the disk is completely dissipated, the binary has an eccentricity of0.245 and a period of 855 days.

We note that the transport of angular momentum from the binary to the diskcontinues as long as the disk lives, contrary to the change in eccentricity thathas an upper limit of e ∼ 0.7. However, no models in the parameter range weused reach this eccentricity limit.

Effect of minimum eccentricity

Contrary to the models that only have eccentricity pumping through phasedependent RLOF, models containing a CB disk are much less sensitive to changesin the minimum eccentricity. In Fig. 6.9 the standard model (emin = 10−3)is compared with models with a minimum eccentricity of emin = 10−4 andemin = 10−5 . The final period and eccentricity of the emin = 10−4 modelonly differs with respectively 2 days and 0.005 between both models, which, inlight of the other assumptions in the models, is negligible. For the model withemin = 10−5 the deviation in final period and eccentricity is respectively 3 daysand 0.01, a little larger than for the emin = 10−4 model, but still insignificant.Similar to the RLOF models, there is a lower limit on the minimum eccentricityto still obtain models with a non-zero final eccentricity. The exact lower limitdepends on other input parameters, and in the case of the standard model it isaround emin = 10−6.

To further test the effect of the minimum eccentricity on the period-eccentricitydistribution, a small set of 10 models randomly distributed in the period-eccentricity space were recalculated with a lower minimum eccentricity. Theyall show deviation in final period and eccentricity similar to those for themodels discussed above. Based on this test we conclude that the results formodels containing a CB disk do not depend strongly on the assumed minimumeccentricity.

The reason for this low dependence on the minimum eccentricity originatespartly in the eccentricity dependence of the eccentricity pumping effect of theCB disk. The pumping efficiency has a steep rise at low eccentricities, and peaksat e = 0.03 (see also Fig. 5.3). For large eccentricities it decreases in efficiency.Models with a higher minimal eccentricity will reach this tip quicker, after


which the disk contribution to the eccentricity pumping diminishes. Changingthe minimum eccentricity also changes the mass-loss rates during RLOF, theeccentricity pumping due to phase-dependent mass loss as well as the tidalforces. However, in combination all these effects seem to cancel out, and themodels reach similar final periods and eccentricities.

Effect of disk properties

There are four parameters in the CB disk model that can influence the CBdisk-binary interactions: the maximum mass in the disk, the life time of thedisk, the viscosity parameter and the assumed distribution of the mass withradius. Next to the disk parameters, the rate at which mass is fed to the diskis determined by mass-loss fraction δ. The effect of these five parameters isshown in Fig. 6.10 panels A, B, C and D. For each of maximum disk mass,viscosity, mass distribution and δ, models with five different disk life timesare shown. These life times can also be interpreted as the evolution in theperiod-eccentricity diagram after the end of RLOF. Based on the equationsthat govern the CB disk-binary interaction given in section 5.2.6 the effect ofeach parameter can be easily explained. When the disk survives longer, theinteraction with the binary will last longer, and the increase in eccentricity /decrease in period will be stronger.

By increasing the maximum disk mass as shown in panel A, the final eccentricitywill be higher, while the final period will be slightly lower. The mass-loss ratesduring RLOF are high enough that there is very little difference in time to filla disk of 0.001 M or 0.01 M. An increased disk mass will lead to a higherangular momentum of the disk, and, according to Eq. 5.44, to a higher changein binary separation and eccentricity.

The viscosity of the disk (αD) shown in panel B, also has a linear effect on thechange in binary separation and eccentricity (see Eq. 5.45). A higher viscositywill lead to a higher final eccentricity and lower period.

The effect of the mass distribution in the disk is plotted in panel C. The massdistribution will determine the mass fraction that resides close to the inner rimof the disk, where it has the strongest effect on the binary. Normally a radialmass distribution of r−1 is chosen, which takes into account that the disk isnot just flat, but also has a specific thickness that varies with the radius. Adistribution of r−2 assumes that the thickness of the disk is more constant withradius. Under that assumption, there will be more mass closer to the inner rim,and the CB disk-binary interactions will be stronger, again leading to a highereccentricity and lower period.


−14

−13

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−11

log(

e)t

idal

emin

10−3

10−4

10−5

−14

−13

−12

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e)M

L

−14

−13

−12

−11

log(

e)d

isk

0.0

0.1

0.2

e

a b c d

825

850

875

P(d

)

A

B

C

D

E

Time

7300 yr 310 yr 95000 yr7600 yr 315 yr 94000 yr8200 yr 321 yr 93000 yr

Figure 6.9: Time evolution of several binary properties during the life time of theCB disk for the standard model (blue full line) and a model with emin = 0.0001(red dashed line). Four different events are indicated on the x-axes. a: edisk> etidal, b: eml > etidal, c: eml < etidal and d: edisk < etidal. The time scalediffers between the phases, but is linear within each phase. The duration ofeach phase for each model is shown under the figure. Panel A: the tidal forces.Panel B: eccentricity pumping due to mass loss. Panel C: eccentricity pumpingthrough CB disk - binary interactions. Panel D: the eccentricity. Panel E: theorbital period. See section 6.6.2 for discussion.


930

935

940

945

950

955

960

965

970

0.1

0

0.1

5

0.2

0

0.2

5τ

=10

4-1

05yr

Mdisk

=10

−2

M⊙

Mdisk

=10

−3

M⊙

900

910

920

930

940

950

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970

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930

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4

σ(r

)∼r−

1

σ(r

)∼r−

2

Peri

od(d

ays)

EccentricityA

BC

Figu

re6.10:The

effectof

theCB

disk

andbina

rypa

rameterson

thefin

alpe

riodan

deccentric

ityof

theorbit.

All

mod

elsareplottedfor5diffe

rent

disk

lifetim

esτ=

1·1

04 ,2.

5·1

04 ,5·1

04 ,7.

5·1

04an

d1·1

05years.

Pane

lA:

Mod

elswi

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The rate at which mass is fed to the CB-disk is defined by mass-loss parameterδ. The effect of changing this parameter is shown in panel D. As the disk willreach its maximum mass earlier, the eccentricity-pumping effect of the disk willslightly increase with increasing δ. However, the main effect of this parameter isa change in orbital period. As discussed in Sect. 6.5.2, increasing δ will decreasethe orbital period because more angular momentum is lost. This results inhigher eccentricities reached during the RLOF phase.

Effect of binary properties

In panels E and F of Fig. 6.10, the effect of several binary parameters is shown.These are also discussed in the Sect. 6.5.2 and are shown here, to indicatehow strong their influence on the distribution in the period-eccentricity planecan be. It is especially the range in period space that is altered by changinginitial parameters such as companion mass and initial period. Where the disk-binary interactions have an influence of roughly five per cent on the final period,changing the initial binary parameters can result in a change of several 100days. The main influence of the CB disk is then in reaching higher eccentricities.Where the phase-dependent RLOF models could reach eccentricities of maximum0.15, the models containing a CB disk can reach eccentricities of maximum 0.34.

6.7 Period-eccentricity distribution

By varying the initial binary parameters described in Sect. 6.3 and the modeldependent parameters described in Sects. 6.5 and 6.6, a significant area of theperiod-eccentricity diagram can be covered. In Fig. 6.11, both the region coveredby models with only phase-dependent RLOF (red shade with solid border) andthat covered by models including a CB disk (green shade with dashed border)is plotted on top of the observed systems described in Sect. 4.6 (blue circles).The parameter ranges used to obtain this distribution are given in Table 6.2. Asthe models with only tidally enhanced wind mass loss cannot produce eccentricsdB binaries, they are not shown here. We see that by including eccentricitypumping processes during the evolution, we are able to form models with sdBprimaries at wide eccentric orbits. The importance of these results is not theexact boundaries of the region in the period-eccentricity diagram that can becovered by the models discussed here. We did not perform a full populationsynthesis covering a wide range of initial binary properties to investigate theboundaries. What is important is that our models can indeed form eccentricsdB binaries in a period-eccentricity region that corresponds to the observedsystems.

PERIOD-ECCENTRICITY DISTRIBUTION 157

Table 6.2: Parameter ranges used to determine the period-eccentricitydistribution of the phase dependent RLOF models and the CB-disk models.Not all parameter combinations result in an sdB binary.

Parameter RangeBinary

sdB progenitor mass (M) 1.0 - 1.50companion massa (M) 0.8 - 1.45period (d) 500 - 900

Mass lossα 0.0 - 1.0β 0.0 - 1.0δ 0.0 - 0.30γ 1.12ε 0.0 - 0.25

CB diskmaximum mass (M) 10−4 - 10−2

lifetime (yr) 104 - 105

viscosity 0.001 - 0.01mass distribution r−1

(a) The companion mass is always lower as the donor mass.

From Fig. 6.11, we can conclude that the region covered by the models withphase-dependent RLOF does not correspond with the observations. The effect ofmost parameters is to reduce the orbital period, while increasing the eccentricity,leading to an eccentricity pumping effect that is most efficient at short periods.At high orbital periods, the eccentricity pumping is too weak to overcome thecircularisation, leading to circular orbits at periods over ∼1250 days. It is thuspossible to recreate the short period systems with moderate eccentricity, butthe highly eccentric long-period systems cannot be reached.

The models including a CB disk have a larger coverage than the models withonly phase-dependent RLOF. However, even though the disk models allowmost observed systems, they suffer the same problem as the models with onlyphase-dependent RLOF. Again, the effect of almost all parameters is to increasethe eccentricity at the cost of a lower orbital period. The disk models canexplain the high period - high eccentricity region of the period-eccentricitydiagram that the models with only phase dependent RLOF cannot cover. Whena lower limit on the lifetime of a disk is set at 104 yr, the systems with periodsaround 750 days and eccentricity around e ∼ 0.05 cannot be explained.


600 800 1000 1200 1400 1600

Final Period (d)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Fin

alec

cent

rici

ty

Figure 6.11: The approximate region of the period-eccentricity diagram thatcan be explained by the RLOF models (red shaded region with solid border)and models containing a CB disk (green shaded region with dashed border).The regions overlap at the dark green-red colored region. The observed systemsare plotted in blue dots.

Another discrepancy for both models is that the circular system PG1701+359at an orbital period of 734 days cannot be reproduced, when the eccentricitypumping effects are active. The only way to model these circular systems at loworbital periods is by turning off all eccentricity pumping mechanisms. However,as explained in Sect. 4.5, this system could be eccentric with e = 0.07, butthere is insufficient orbital coverage to confirm this. If this system would indeedhave e = 0.07, then it falls into the period-eccentricity region covered by thephase-dependent RLOF models.

The period-eccentricity distribution of the two eccentricity pumping modelsseems to indicate that there is a trend of higher eccentricities at lower periods,opposite to the observed trend of higher eccentricities at higher periods. Thisis under the assumption that the method dependent parameters might varyindependent of each other and the initial binary parameters. However, it is notunreasonable that for example the mass loss fractions (α, β and δ) depend onother binary properties like the mass ratio. Such a dependence between differentparameters might indeed reproduce the observed period-eccentricity trend, but

SUMMARY AND CONCLUSIONS 159

600 800 1000 1200 1400 1600

Final Period (d)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35F

inal

ecce

ntri

city

log Mmax = −2

−2 > log Mmax ≥ −3

log Mmax < −3

Figure 6.12: The approximate region of the period-eccentricity diagramthat is covered by the RLOF and CB disk models. The different coloursindicate different ranges of the maximum mass-loss rate reached duringRLOF: log Mmax = −2.0 in red, −2.1 > log Mmax ≥ −3.0 in yellow andlog Mmax < −3.0 in green.

the models presented here can not prove nor disprove any such dependence.

Due to numerical considerations the mass-loss rate during RLOF is cappedat 10−2M yr−1. To indicate the effect this limit has on the final period-eccentricity distribution of the models, we indicate in Fig 6.12 which modelsare affected. As can be seen, only the models with the shortest final periods doreach this limit. At wider orbits, the limit is never reached. So a differentlyimposed mass-loss limit, will affect mainly the shortest periods.

6.8 Summary and conclusions

The goal of this chapter was to test if three eccentricity pumping mechanismsproposed in the literature, are consistent with the observed period-eccentricitydistribution of long-period binaries containing a hot subdwarf B star. The threeproposed mechanisms are 1. tidally enhanced wind mass loss, 2. eccentricity


pumping through phase-dependent RLOF and 3. the interaction between thebinary and a circumbinary disk formed during RLOF combined with phasedependent RLOF. The tidally enhanced wind mass loss method is not combinedwith any of the two other methods, as the point of the enhanced wind is toprevent the donor star from filling its Roche lobe, and in that way reducing thetidal forces. This would be counterproductive in the context of phase-dependentRoche-lobe overflow.

In this work we attempted to describe the effects of model parameters on therelationship between final period and eccentricity. Further work must be done inorder to combine these models with realistic distributions of starting parametersin a binary-population-synthesis context.

Creating eccentric orbits with a tidally enhanced mass-loss mechanism is basedon two processes. By increasing the wind mass loss, the radius of the donor staris kept well within its Roche lobe, thus reducing the tidal forces and maintainingthe initial eccentricity of the orbit. The tidally increased wind mass loss dependson the orbital phase. This phase-dependent mass loss will further increase theeccentricity. The models calculated with MESA indeed lead to binaries on aneccentric orbit. However, the amount of mass that the donor star needs to loseis so large that the final mass of the donor star is lower than the minimumcore mass necessary to ignite helium. All eccentric binaries formed through thischannel end up as He WDs, while all sdB stars ended on a circularised orbit asthe wind enhancement parameter needs to be very low.

The phase-dependent RLOF mechanism is based on the same principle as thephase-dependent wind mass loss. By assuming a minimal eccentricity, themass-loss rate will be higher during periastron than during apastron, and theeccentricity may increase. This method can create eccentric sdB binaries over areasonable period-eccentricity range. The maximum eccentricity tends to bearound 0.15, while the period range for eccentric sdBs is 600 to 1200 days. sdBbinaries with longer orbital period, reaching ∼1600 days can be formed butare circularized within our parameter range. This mechanism can explain theobserved systems with moderate eccentricities (0.04 < e < 0.15) on shorterorbital periods (P < 1100 d). Neither systems with higher eccentricities, norlow-period circularised systems can be formed this way.

Adding CB disks to the models with phase-dependent RLOF does furtherincrease the eccentricity of the produced binaries due to resonances betweenthe binary and the CB disk. This binary-CB disk interaction increases theeccentricity while slightly decreasing the orbital period. These models canreach eccentricities up to 0.35 when assuming reasonable values for the diskparameters, and could potentially reach far higher eccentricities if more extremevalues for maximum disk mass or surface mass distribution would be adopted.

SUMMARY AND CONCLUSIONS 161

The disk model can reproduce most of the observed sdB binaries, except thecircular short-period system.

When the models are compared to the observed sample, it is clear that phasedependent RLOF in combination with CB disks can cover a significant part of theperiod-eccentricity diagram. Part of the discrepancy between the observed andmodelled period-eccentricity diagram can be explained by insufficiently accuratemodels, while it is also possible that certain areas of the period-eccentricitydiagram have not yet been populated due to a lack of observations.

The observed trend of higher eccentricities at longer orbital periods doesnot follow directly from the method-depending parameters in the presentedmodels. In our future research we will investigate whether certain parametersare dependent on the initial or current configuration of the binary. Such adependence could potentially result in the observed period-eccentricity trend.However, the models presented here can neither prove nor disprove any suchtrend.

All three tested eccentricity pumping mechanisms are derived under certainassumptions, such as isotropic mass loss for eccentricity pumping through phase-dependent mass loss, or the approximation of a thin disk for the CB disk-binaryinteractions. Furthermore, values for several input parameters, for example themass loss fractions (α, β and γ), are badly constrained or currently unknown,and are also likely to be functions of the system parameters such as mass ratioand companion mass rather than constants.

In this chapter we have shown that we are able to form binary models with ansdB primary at wide eccentric orbits by eccentricity pumping of phase dependentRLOF and CB disks. Tidally enhanced wind mass loss is unlikely to contributeto the formation of eccentric sdBs. Small discrepancies between the observedsystems and the theoretical models remain, and will need to be addressed infuture work.

7Conclusions and future prospects

7.1 Summary and conclusions

At the start of this PhD only one wide sdB binary was studied in reasonabledetail, PG1018−047. An sdB binary with a K-type companion on a most likelycircular orbit of 760 ± 6 days. Preliminary periods of other wide sdB systemshad been published but but all assumed circular orbits as the data was notgood enough to discriminate circular versus eccentric systems. Part of thegoal of this PhD was to expand the sample of wide sdB binaries. To whichend a long-term spectroscopic observing program with the Mercator telescopewas already started in 2009. In the first part of this thesis the results of thisobserving program have been presented.

In total eight more wide sdB binaries have been added to the sample. The firstsystem to be analyzed was PG1104+243, an sdB + G0 system with an orbitalperiod of ∼750 days. Using all available spectra, it was found that this systemhas a slight but significant eccentricity of e = 0.04 ± 0.02. The spectral energydistribution from literature photometry was used to determine the effectivetemperature and surface gravity of both components by fitting model SEDs tothem. The number of free parameters was reduced by using the mass ratio as aconstraining factor. Furthermore, PG1104+243 is the first sdB binary in whichgravitational redshift is observed, and used to estimate the surface gravity ofthe sdB component.

BD+293070, BD+341543 and Feige 87 are the first sdB binaries with

163

164 CONCLUSIONS AND FUTURE PROSPECTS

higher eccentricities, ranging from 0.11 for Feige 87 to 0.16 in BD+341543.BD+293070 is one of the longest period binaries with an orbital period of∼1250 days. All three systems have been analyzed in detail, using both theHERMES spectra and literature photometry. Furthermore, flux-calibratedspectra were used to independently fit the effective temperatures, surfacegravities and compositions of both components. All parameters derived by thedifferent methods agree within their error bars.

The developed binary-SED-fitting method is useful in other binary systemsas well, as it can provide an independent determination of the atmosphericparameters of both components based on readily available literature photometry.Its accuracy is shown by comparing the results with atmospheric parametersobtained from the equivalent width of iron lines, and those obtained in thefitting of flux-calibrated spectra.

Next to these four fully analyzed systems four more systems have been observedwith HERMES. Two of those have sufficient orbital coverage to determineaccurate orbital parameters. The remaining two systems have fast rotatingcompanions, which increases the difficulty in obtaining reliable radial velocities.In combination with published orbital parameters from Barlow et al. (2013),this observational work resulted in the creation of a small but interesting sampleof wide sdB binaries that can be used to test possible formation mechanisms.

The most important conclusion from the observed period-eccentricity distribu-tion is that, contrary to binary-population-synthesis models (see for exampleHan et al. 2002, 2003), the majority of all wide sdB binaries have significantlyeccentric orbits. The only possible circularized system, PG1701+359, has anorbital period of ∼730 days. Furthermore, there is an observed trend thathigher eccentricities are detected at longer orbital periods, varying from e ≈0.04 at P ≈ 750 days to e ≈ 0.17 at P ≈ 1250 days.

In an effort to explain the eccentric orbits presented in the first part of thisthesis, the second part focused on testing eccentricity-pumping mechanisms insdB binaries using the evolution code MESA. Three such eccentricity-pumpingprocesses have been discussed in the literature and are potential candidates.1. tidally-enhanced wind-mass loss, 2. eccentricity pumping through phase-dependent Roche-lobe overflow (RLOF) and 3. the interaction between thebinary and a circumbinary disk formed during RLOF combined with phase-dependent RLOF. We explore the tree processes and computed a series ofspecific binary evolution channels.

Models using tidally-enhanced wind-mass loss do indeed form binaries on aneccentric orbit. However, the amount of mass that the donor star needs to loseis so large that its final mass is lower than the minimum core mass necessary

FUTURE PROSPECTS 165

to ignite helium. All eccentric binaries formed through this channel end up asHe white dwarfs, while all sdB stars ended on a circularized orbit as the windenhancement parameter needs to be very low in the first place to make the sdB..

Using only phase-dependent RLOF, eccentric sdB binaries in the shorter periodrange of the observed period-eccentricity distribution could be created. Themaximum eccentricity tends to be around 0.15, while the period range foreccentric sdBs is 600 to 1200 days. sdB binaries with longer orbital periods,P < 1600 days can be formed but are circularized. This process can explainthe systems with moderate eccentricities (0.04 < e < 0.15) on shorter orbitalperiods (P < 1100 d).

Adding CB disks to the models with phase-dependent RLOF does furtherincrease the eccentricity of the produced binaries due to resonances betweenthe binary and the CB disk. With this process sdB binaries with eccentricitiesup to 0.35 and periods ranging up to 1600 days can be created. The disk modelcan reproduce all but the circular short-period system.

The observed trend of higher eccentricities at higher periods does not followdirectly from the binary-evolution models we have explored. The usedeccentricity-pumping processes tend to be more effective at shorter periods.However, it is likely that several model parameters depend on the initial andcurrent configuration of the binary. Such a dependence could recreate theobserved trend.

In summary, the first part of this thesis presents the first wide sdB binarysample based on a long-term spectral monitoring of eight systems, extendedwith three other systems published in the literature. In the second partthree eccentricity-pumping processes are tested to explain the observed period-eccentricity distribution. A combination of phase-dependent RLOF and CBdisks is capable of explaining the majority of the period-eccentricity diagram.

7.2 Future prospects

The MESA models presented in this thesis do allow for almost all of theobserved systems, but cannot really predict the observed distribution, andseveral questions remain. Future work intended to increase our understandingof the binary interactions at play in sdB binaries can be subdivided in atheoretical and an observational part.

From an observational point of view the determination of upper limits onthe orbital period and eccentricity of the observed systems is interesting. Thecurrent sample seems to indicate that the eccentricity is limited to a maximum of

166 CONCLUSIONS AND FUTURE PROSPECTS

∼0.2, but if longer-period systems would be found this could change. To extendthe current sample of wide sdB binaries we submitted observing proposals forsdB binaries with unknown periods in the southern hemisphere to UVES andFEROS.

Furthermore there is the need for a more exact classification of the companionsto the long-period sdBs. An in-depth study of the spectral properties of thecompanion stars might uncover the cause of the observed trend in the period-eccentricity diagram. An example of this would be the rotational velocity of thecompanion, and the eccentricity of the orbit, as a higher rotational velocity canbe an indicator of a higher accretion fraction during RLOF (see for examplede Mink et al. 2013). As shown in Sect. 3.3.2, accurate determination of thespectral properties of both the sdB and the cool companion will be facilitatedif distances determined by GAIA become available.

Although CB disks are common in galactic post-AGB systems, they havenot yet been observed in galactic post-RGB systems. This is different in theMagellanic clouds, where using systematic infrared surveys in combinationwith low-resolution spectral surveys of both the SMC and the LMC, (Kamathet al. 2014, 2015) discovered a significant sample of dusty post-RGB stars. Thediscovery of such galactic post-RGB systems containing a CB disk would supportthe assumption that such a disk might have been present in wide-sdB-progenitorsystems, and can indeed be responsible for the observed eccentricities.

CB disks have been observed in galactic post-AGB systems, and detailed studiesshowing widespread Keplerian rotation (Bujarrabal et al. 2013, 2015), strongprocessing of dust grains (Gielen et al. 2008, 2011) and the presence of largegrains (de Ruyter et al. 2005) in the CB disks indicate that they are likely stableand long lived. However, the parameters that are important in the CB disksmodels presented in this thesis: total disk mass, mass distribution, lifetime,inner and outer radius, are still badly constraint. Interferometric observationsof these disks would help in constraining these parameters as shown in, forexample, the interferometric study of 89 Her (Hillen et al. 2014), or AC Her(Hillen et al. 2015).

Important limits of the current models were already mentioned in the conclusionsof Chapter 6. They include the basic treatment of the disk-binary interactions,and the parameterized models for mass transfer to the companion and mass lossfrom the system. Some improvements to the basic physics are already publishedin the literature but need to be implemented in binary-modeling codes. Anexample of this would be the utilized volume-equivalent Roche-lobe radius fromEggleton (1983), which has been shown to differ significantly from the realRoche lobes in eccentric systems (Sepinsky et al. 2007). Other parameterizedmodels can be improved by smoothed-particle-hydrodynamics (SPH) studies.

FUTURE PROSPECTS 167

SPH studies could for example increase our understanding of the mass flows ina binary system during RLOF, or the conditions necessary to form a CB diskand its evolution in time (see for example Dunhill et al. 2015).

The binary MESA models are especially suited to study the effect of inputparameters on the final system and stellar parameters. In future workthese should be extended with binary-population-synthesis studies utilizingan accurate treatment of the binary-interaction processes. The outcome of thiskind of BPS studies can also help in focusing the search for new sdB binaries tothose systems that can really help distinguish between different physical models.

AMesa inlists

In this appendix a few example inlists for MESA are given. In SectionA.1 anexample inlist used for single star evolution is given. This inlist is used tocalculate the models shown in the introduction and in Chapter 2. This basicinlist is used both for regular stellar evolution, and to create sdB stars withoutthe need for binary evolution. For the latter case the stars are first evolveduntil they reach the tip of the RGB after which the envelope mass is strippedoff up to a slightly higher mass than the final sdB mass. In this case the star isstripped until 0.52 M remains near the tip of the RGB, which results in ansdB star with a flash mass of 0.47 M.

In the SectionA.2 an example inlist set for binary evolution with MESA isgiven. The binary inlist set consist of an inlist for both the donor and theaccretor, and an inlist for the binary evolution part. These inlists are used inChapter 6. The inlists given here are used to evolve the standard model withboth phase-dependent RLOF, and a CB disk, as described in Section 6.6.

169

170 MESA INLISTS

A.1 Single starsSingle star (inlist)&star_job

! begin with a pre−main sequence modelcreate_pre_main_sequence_model = . t rue .pre_ms_relax_num_steps = 100

change_in i t ia l_net = . t rue .new_net_name = ’ basic_plus_fe56_ni58 . net ’

e o s_ f i l e_p r e f i x = ’mesa ’kappa_f i l e_pre f ix = ’OP_gs98 ’

! Relax the mass at t i p RGB to sdB massrelax_mass = . t rue .r e l ax_in i t i a l_mass = . f a l s e .new_mass = 0.52lg_max_abs_mdot = −100

/ ! end o f star_job name l i s t

&con t r o l s

i n i t i a l_mas s = 1 .2i n i t i a l _ z = 0.02 d0

use_Type2_opacities = . t rue .Zbase = 0.02 d0

! Stopping c r i t e r i alog_L_lower_limit = −3.5max_model_number = 25000

! WindRGB_wind_scheme = ’ Reimers ’AGB_wind_scheme = ’ Blocker ’RGB_to_AGB_wind_switch = 1d−4Reimers_wind_eta = 0 .5 d0Blocker_wind_eta = 0 .7 d0

/ ! end o f c on t r o l s name l i s t

BINARY SYSTEMS 171

A.2 Binary systemsDonor (inlist1):&star_job

change_in i t ia l_net = . t rue .new_net_name = ’ pp_cno_extras_o18_ne22 . net ’

create_pre_main_sequence_model = . t rue .

save_model_when_terminate = . t rue .save_model_filename = ’ donor .mod ’


&con t r o l s

max_number_backups = 2500max_number_retries = 10000

! s topping cond i t i on slog_L_lower_limit = −3.0d0max_model_number = 25000

varcont ro l_ta rge t = 1d−3

RGB_wind_scheme = ’ Reimers ’Reimers_wind_eta = 0 .7 d0RGB_to_AGB_wind_switch = 1d−4AGB_wind_scheme = ’ Blocker ’Blocker_wind_eta = 0 .5 d0


Accretor (inlist2):&star_job

create_pre_main_sequence_model = . t rue .

save_model_when_terminate = . t rue .save_model_filename = ’ a c c r e t o r .mod ’


&con t r o l s

max_number_backups = 2500max_number_retries = 10000

varcont ro l_ta rge t = 1d−3


172 MESA INLISTS

Binary (inlist_project):&binary_job

in l i s t_names (1 ) = ’ i n l i s t 1 ’in l i s t_names (2 ) = ’ i n l i s t 2 ’

evolve_both_stars = . t rue .

/ ! end o f binary_job name l i s t

&binary_contro l s

m1 = 1 .2 d0m2 = 1 .0 d0in i t ia l_per iod_in_days = 700i n i t i a l _ e c c e n t r i c i t y = 0 .3 d0min_eccent r i c i ty = 0.001

do_t ida l_c i rc = . t rue .circ_type_1 = "Hut_conv "circ_type_2 = "Hut_rad "do_tidal_enhance = . t rue .

mdot_scheme = "Kolb "max_abs_mdot = 1 .0d−2limit_retention_by_mdot_edd = . t rue .mass_transfer_alpha = 0.35 d0mass_transfer_beta = 0.35 d0mass_transfer_delta = 0.30 d0mass_transfer_gamma = 1.12 d0

do_enhance_wind_1 = . f a l s e .tout_B_wind_1 = 0 .0 d0do_wind_mass_transfer_1 = . t rue .wind_BH_alpha_1 = 1 .5 d0max_wind_transfer_fraction_1 = 0 .3 d0

CB_disk_max_mass = 1d−2CB_disk_lifetime = 1 .0 d5CB_disk_viscosity = 0.01CB_disk_dist r ib i t ion = 1d0

f r = 0 .01fa = 0.01fm = 0.01f j = 0 .01f e = 0 .1fdm = 0.0001

/ ! end o f b inary_contro l s name l i s t

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Curriculum Vitae

Joris [email protected]

Joris Vos was born on the 11th of September 1988 in Genk (Belgium). Heobtained his Bachelor in Physics with a minor in Computer Science (August2009) and a Master in Astronomy and Astrophysics (June 2011) at the Universityof Leuven (Belgium). He spend his second master year at the University ofCopenhagen (Denmark), where he wrote his master thesis under supervision ofProf. Jens Viggo Clausen and Dr. Roy Østensen. After finishing his master hestarted a PhD at the University of Leuven under supervision of Prof. Hans vanWinckel and Dr. Roy Østensen.

From October 2015 Joris will start a Postdoc at the University of Valparaiso(Chile) where he will work together with Prof. Maja Vuckovic on the evolutionof wide sdB binaries.

183

mailto:[email protected]

184 CURRICULUM VITAE

List of publications

Articles in refereed international journals

Journal abbreviations and impact factors (ISI JCR 2014):A&A: Astronomy and Astrophysics (4.5)ApJ: Astrophysical Journal (6.3)ApJS: The Astrophysical Journal Supplement Series (11.2)MNRAS: Monthly Notices of the Royal Astronomical Society (5.2)

Buysschaert B., Aerts C., Bloemen S., Debosscher J., Neiner C., Briquet M.,Vos J., Papics P. I., Manick R., Schmid V., Van Winckel H., Tkachenko A.,Kepler’s first view of O-star variability: K2 data of five O stars in Campaign 0as a proof-of-concept for O-star asteroseismology, accepted for publication inMNRAS, 2015.

Vos J., Østensen R. H., Marchant P., Van Winckel H., Testing eccentricitypumping mechanisms to model eccentric long period sdB binaries with MESA,A&A, 579:A49, 2015.

Van Reeth T., Tkachenko A., Aerts C., Papics P. I., Triana S. A., Zwintz K.,Degroote P., Debosscher J., Bloemen S., Schmid V. S., De Smedt K., FrematY., Fuentes A. S., Homan W., Hrudkova M., Karjalainen R., Lombaert R.,Nemeth P., Oestensen R., Van De Steene G., Vos J., Raskin G., Van WinckelH., Gravity-mode period spacings as seismic diagnostic for a sample of gammaDoradus stars from Kepler space photometry and high-resolution ground-basedspectroscopy, ApJS, 218:27V, 2015.

Hareter M., Paparó M., Weiss W., García Hernández A., Borkovits T., LampensP., Rainer M., De Cat P., Marcos-Arenal P., Vos J., Poretti E., Baglin A.,Michel E., Baudin F., Catala C., HD 51844: An Am δ Scuti in a binary showingperiastron brightening, A&A, 567:A124, 2014.

Beck P. G., Hambleton K., Vos J., Kallinger T., Bloemen S., Tkachenko A.,García R. A., Østensen R. H., Aerts C., Kurtz D. W., De Ridder J., HekkerS., Pavlovski K., Mathur S., De Smedt K., Derekas A., Corsaro E., Mosser B.,Van Winckel H., Huber D., Degroote P., Davies G. R., Prša A., Debosscher J.,Elsworth Y., Nemeth P., Siess L., Schmid V. S., Pápics P. I., de Vries B. L., vanMarle A. J., Marcos-Arenal P., Lobel A., Pulsating red giant stars in eccentricbinary systems discovered from Kepler space-based photometry. A sample studyand the analysis of KIC 5006817, A&A, 564:A36, 2014.

CURRICULUM VITAE 185

Vos J., Østensen R. H., Németh P., Green E. M., Heber U., Van Winckel H.,The orbits of subdwarf-B + main-sequence binaries. II. Three eccentric systems;BD +29o3070, BD +34o1543 and Feige 87, A&A, 559:A54, 2013.

Bloemen S., Steeghs D., De Smedt K., Vos J., Gänsicke B. T., Marsh T. R.,Rodriguez-Gil P., Remarkable spectral variability on the spin period of theaccreting white dwarf in V455 And, MNRAS, 429:3433, 2013.

Vos J., Østensen R. H., Degroote P., De Smedt K., Green E. M., Heber U.,Van Winckel H., Acke B., Bloemen S., De Cat P., Exter K., Lampens P.,Lombaert R., Masseron T., Menu J., Neyskens P., Raskin G., Ringat E., RauchT., Smolders K., Tkachenko A., The orbits of subdwarf B + main-sequencebinaries. I. The sdB+G0 system PG 1104+243, A&A, 548:A6, 2012.

Østensen R. H., Degroote P., Telting J. H., Vos J., Aerts C., Jeffery C. S., GreenE. M., Reed M. D., Heber U., KIC 1718290: A Helium-rich V1093-Her-likePulsator on the Blue Horizontal Branch, ApJ, 753:L17, 2012.

Smolders K., Verhoelst T., Neyskens P., Blommaert J. A. D. L., Decin L., VanWinckel H., Van Eck S., Sloan G. C., Cami J., Hony S., De Cat P., Menu J.,Vos J., Discovery of a TiO emission band in the infrared spectrum of the Sstar NP Aurigae, A&A, 543:L2, 2012.

Vos J., Clausen J. V., Jørgensen U. G., Østensen R. H., Claret A., Hillen M.,Exter K., Absolute dimensions of solar-type eclipsing binaries. EF Aquarii: aG0 test for stellar evolution models, A&A, 540:A64, 2012.

Articles in conference proceedings

Vos J., Eccentricity pumping through circumbinary disks in hot subdwarfbinaries, Proceedings of the POE 2015 meeting.

Beck, P. G., Hambleton, K., Vos, J., Kallinger, T., Garcia, R. A., Mathur,S., Houmani, K., Oscillating red-giant stars in eccentric binary systems,Proceedings of the CoRoT Symposium 3, Kepler KASC-7 joint meeting 2014.

Vos J., Østensen R., Van Winckel H., Long Period sdB + MS Binaries withMercator, ASPC, 481:265, 2014.

Nemeth P., Østensen R., Vos J., Kawka A., Vennes S., SD1000 Collaboration:Hunting down the Subdwarf Populations, ASPC, 481:75, 2014.

Hambleton K., Beck P., Bloemen S., Vos J., Prsa A., Kurtz D., Aerts C.,Panoramix: The Red Giant Heartbeat Star, giec.conf, 40301, 2013.

186 CURRICULUM VITAE

Gorlova N., Van Winckel H., Vos J., Østensen R. H., Jorissen A., Van EckS., Ikonnikova N., Monitoring evolved stars for binarity with the HERMESspectrograph, EAS, 64:163, 2013.

FACULTY OF SCIENCEDEPARTMENT OF PHYSICS AND ASTRONOMY

INSTITUUT VOOR STERRENKUNDECelestijnenlaan 200D box 2402

B-3001 [email protected]

http://fys.kuleuven.be/ster

ARENBERG DOCTORAL SCHOOL Faculty of Science...sequence to the white-dwarf stage will be illustrated....

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