Radius of the Neutron Star Magnetosphere during …Radius of the neutron star magnetosphere during...

Astronomy Letters, 2017, Vol. 43, No. 11, p. 706–729. Translated from Pis’ma v Astronomicheskij Zhurnal.

Radius of the Neutron Star Magnetosphere during Disk Accretion

E.V. Filippova*1, I.A. Mereminskiy1, A.A. Lutovinov1, S.V. Molkov1

S.S.Tsygankov2,1

1Space Research Institute, Moscow, Russia2Tuorla observatory, University of Turku, Finland

Received 29.11.2016

Abstract — The dependence of the spin frequency derivative ν of accreting neutron stars with a strongmagnetic field (X-ray pulsars) on the mass accretion rate (bolometric luminosity, Lbol) has been investigatedfor eight transient pulsars in binary systems with Be stars. Using data from the Fermi/GBM and Swift/BATtelescopes, we have shown that for seven of the eight systems the dependence ν can be fitted by the modelof angular momentum transfer through an accretion disk, which predicts the relation ν ∼ L

6/7bol . Hysteresis

in the dependence ν(Lbol) has been confirmed in the system V 0332+53 and has been detected for thefirst time in the systems KS 1947+300, GRO J1008-57, and 1A 0535+26. The radius of the neutron starmagnetosphere in all of the investigated systems have been estimated. We show that this quantity variesfrom pulsar to pulsar and depends strongly on the analytical model and the estimates for the neutron starand binary system parameters.

Key words: X-ray pulsars, neutron stars, accretion.

INTRODUCTION

X-ray pulsars are a natural laboratory for investigat-ing the disk accretion of matter, because the structureof the inner disk region and its interaction with theneutron star magnetosphere affect the pulsar spin pe-riod. It follows from observations that the pulsationperiods are subjected to variabilities in a wide rangeof characteristic time scales, from hours to years, dueto changes in the regime and rate of accretion onto theneutron star (Nagase, 1989; Bildsten et al., 1997; Lu-tovinov et al., 1994; Lutovinov & Tsygankov, 2009).One of the tools for investigating the accretion regimeis to analyze the dependence of the neutron star spinfrequency derivative on the mass accretion rate, ν(M).The accretion regime is determined by the radius ofthe neutron star magnetosphere that is specified as

rm = ξrA, where rA =(

µ4

2GMM2

)1/7

is the Alfven ra-

dius determined by the equality of the magnetic pres-sure and the pressure of accreting matter falling spher-ically symmetrical from infinity to a dipole, ξ is a di-mensionless parameter. If the magnetospheric radiusof the star is larger than its corotation radius, then thesystem is in the ”propeller” regime (Illarionov & Sun-yaev, 1975), where the neutron star flings away the ac-creting matter, thereby losing its angular momentumand spinning down. The propeller effect is quite diffi-

*e-mail: [email protected]

cult to detect, and by now this has been done only fora few X-ray pulsars (Campana et al., 2008; Tsygankovet al., 2016a,c; Lutovinov et al., 2017). If the magne-tospheric radius is smaller than the corotation radius,then the neutron star, as a rule, spins up through theangular momentum accretion of the infalling matter.There exist several models that describe the change inthe spin frequency of the neutron stars due to the inter-action of their magnetosphere with the accretion disk.This problem has been intensively studied since theearly 1970s, but the first complete solution was pub-lished in Ghosh & Lamb (1979). These authors con-sidered a model where the magnetic field lines of theneutron star thread a standard accretion disk. Tworegions are formed in this case: the interaction mustcause the neutron star to spin up at radii smaller thanthe corotation radius, because the Keplerian angularvelocity of the disk is larger than the stellar rotationrate, and to spin down at larger ones. The dependenceof the neutron star spin frequency derivative on the ac-cretion rate in this model looks like ν ∼ Mβ , whereβ = 6/7.

If a magnetic diffusion in the disk is not large enoughfor the field lines to slide in the azimuthal direction,then the magnetic field lines connecting the star andthe disk will twist as a result of different rotation ratesof their bases. This will give rise to a toroidal fieldwhose pressure will push away the field lines in sucha way that they will become almost radially directed.

706

arX

iv:1

710.

0164

6v1

[as

tro-

ph.H

E]

4 O

ct 2

017

Radius of the neutron star magnetosphere during disk accretion 707

In this case, the field lines of opposite directions canbe separated by a current sheet where their reconnec-tion can occur (Aly, 1985; Lynden-Bell & Boily, 1994;Uzdensky, 2002). Lovelace et al. (1995) proposed amodel where the field lines became open as a resultof their interaction with the disk and formed a stableconfiguration without undergoing reconnection, whichallowed the wind to take away some angular momen-tum from both the star and the disk. In the mostrecent paper Parfrey et al. (2016) considered a modelthat results from the relaxation of the model describedin Lovelace et al. (1995), where the field lines passingthrough the accretion disk are pushed beyond the lightcylinder as a result of the magnetic diffusion.

The influence of conditions in the inner disk on thedependence ν(M) was considered by Ghosh & Lamb(1992), who showed that (1) β = 0.925 for a radiation-dominated disk, (2) β = 0.15 for a two-temperature,optically thin in the vertical direction gas-dominateddisk in the case of electron cooling through Comp-tonization of soft photons from the external source, or(3)β = 0.76 in the case of electron cooling throughbremsstrahlung.

It follows from analytical calculations that duringaccretion through a standard disk ξ can be as ∼ 0.5(Ghosh et al., 1977; Kluzniak & Rappaport, 2007) as &1 (Wang, 1996). Numerical calculations show that therelative size of the neutron star magnetosphere is closerto ∼ 0.5 (Long et al., 2005; Bessolaz et al., 2008; Zanni& Ferreira, 2013). Recently, Chashkina et al. (2017)have shown that for a radiation-dominated disk thepredicted value of ξ is larger than that for a standarddisk by a factor of 1.5-2, with the absolute value ofthe magnetospheric radius being independent of themass accretion rate and the exponent β = 1. Thus,not only β but also the magnetospheric size dependson the accretion rate in other disk states. Changesin both parameters affect the evolution of the neutronstar spin frequency derivative.

Be systems are suitable objects for investigating thevariability of the accreting neutron stars rotation rate,because they exhibit a fairly regular outburst activ-ity, when the accretion rate changes by several ordersof magnitude. These systems demonstrate two typesof outbursts. The type I outbursts are periodic, co-incident with the time of periastron passage, and areassociated with an enhancement of the mass accre-tion rate as the companions approach each other. Theoutbursts of this type are weak (a maximum lumi-nosity Lmax ' 1037 erg s−1) and short (lasting sev-eral days). The type II outbursts are irregular, longer(several weeks or months), and more powerful withLmax > 1038 erg s−1. The nature of the latter isnot completely clear, but it is presumably related tothe evolution of the decretion disk around the Be star

(Negueruela & Okazaki, 2001; Okazaki & Negueruela,2001; Moritani et al., 2013; Martin et al., 2014; Mona-geng et al., 2017). As a rule, the neutron star spins upduring outbursts and spins down between them (see,e.g., Postnov et al., 2015).

In this paper we investigate the dependence ν(Lbol)for eight X-ray pulsars in order to determine the ex-ponent β in the dependence ν(M) and to obtain con-straints on the magnetospheric size from the best ob-servational data available to date. In Section ”DataAnalysis” we describe the sample of pulsars that wasused in this paper and the criteria that were appliedto produce it. In Section ”Size of the Neutron StarMagnetosphere” we present the best fits to the depen-dence ν(Lbol) for several models of angular momentumtransfer during the accretion of matter onto a neutronstar that were used to determine the radius of the neu-tron star magnetosphere. In Section ”Estimating theError in the Magnetospheric Radius” we estimate therange of admissible magnetospheric radii related to theuncertainty in measuring the neutron star mass andradius and the distance to the system. The obtainedresults are discussed in Section ”Conclusions”.

DATA ANALYSIS

In this paper we selected the pulsars in Be systemssatisfying the following criteria:

• the demonstration of type II outbursts, becausesuch outbursts allow to measure the dependenceν(Lbol) in a wide range of luminosities and, con-sequently, mass accretion rates;

• there are Fermi/GBM measurements (Meeganet al., 2009) of the neutron star spin frequencyduring the outbursts;

• the orbital parameters of the system needed tocorrect the neutron star spin frequency for theorbital motion are known;

• a cyclotron line is detected in the energy spec-trum of the source, which allows to estimate themagnetic field of the neutron star;

• there are measurements of the distance to the sys-tem.

As a result, we obtained a list of eight objects, whichis given in Table 1. This table also provides the param-eters of the binary systems and neutron stars used inour analysis and gives references to the papers wherethese parameters were estimated.

The neutron star magnetic field was determinedfrom the cyclotron line energy using the formula

B12 = Ecycle/11.6× (1 + z),

ASTRONOMY LETTERS Vol. 43 No. 11 2017

708 Filippova et al.

where B12 is expressed in 1012 Gauss, Ecycle is the cy-clotron energy in keV, and z is the gravitational red-shift. We used a simple approximation that the X-rayemission is generated near the surface of a neutron starwith standard parameters MNS = 1.4M and RNS =106 cm; therefore, z = (1−2GMNS/c

2/RNS)−1/2−1 =0.3. The stellar magnetic moment was calculated fromthe formula µ = BR3/2.

Determining the Bolometric Luminosity

To determine the bolometric luminosity of a source,we used the Swift/BAT data in the 15-50 keV energyband (Barthelmy et al., 2005). By applying the bolo-metric correction, we recalculated the flux in this en-ergy band to the energy range 0.1-100 keV, which maybe deemed bolometric for an accreting neutron starwith a good accuracy. The BAT measurements wererecalculated to the luminosity via the known countrate for the Crab Nebula (0.22 counts cm−2c−1) andthe flux of 1.4×10−8 erg cm−2c−1,

L = 4πd2(

1.4× 10−8 kbolRBAT0.22

), (1)

where d is the distance to the source and RBAT is theBAT count rate.

The bolometric correction kbol and its luminositydependence were calculated for each source separatelyby the method proposed by Tsygankov et al. (2017).For this purpose, apart from the BAT data, we usedthe MAXI data in the 2-10 keV energy band (Mat-suoka et al., 2009) and the Swift/XRT data in the0.3-10 keV energy band processed by means of the On-line Service (Evans et al., 2009) for KS 1947+300. Todetermine the dependence of the bolometric correc-tion on the 15-50 keV flux, we constructed the follow-ing dependences of the flux ratio in different energybands: (F2−10 + F15−50)/F15−50 for the MAXI dataand (F0.3−10+F15−50)/F15−50 for the Swift/XRT data(Fig. 1). When constructing these dependences, weused the Swift/BAT data in which the source detectionsignificance exceeded 2σ. To determine the normal-ization of the dependence of kbol = F0.1−100/F15−50

on the 15-50 keV flux, we used either the data onthe broadband spectrum during the source outburstactivity from the literature or the broadband spec-tra constructed from the Swift/XRT and Swift/BATdata directly for the outbursts being investigated (for1A 0535+26 and V 0332+53). The derived depen-dences of the bolometric correction on the 15-50 keVflux are indicated in Fig. 1 by the dashed line. Thesedependences were used in the subsequent analysis. Forthe system 2S 1553-542 we failed to perform such ananalysis, because the Swift/BAT and MAXI sourcedetection significance during the outburst was insuf-ficient, (2 − 5)σ. The bolometric correction for this

source was calculated from the data on the broadbandspectrum from Tsygankov et al. (2016b). The derivedbolometric corrections (Table 2) lie within the range2-3.5, in good agreement with the shape of the spectrafor X-ray pulsars in a wide energy range (Filippovaet al., 2005).

Determining the Timing Characteristics

For two systems, GRO J1008-57 andXTE J1946+274, the publicly available GBM datawere not corrected for the orbital motion of the binarysystem. Therefore, we did this by ourselves using astandard technique (Argyle, 2004) and the followingorbital parameters: the eccentricity e = 0.68, theprojected orbital semimajor axis a sin i = 530 lt-s, thelongitude of periastron ω = −26, the orbital periodPorb=249.48 days, and the time of periastron passageτ=54424.71 MJD for GRO J1008-57 (Coe et al., 2007;Kuhnel et al., 2013) ; e = 0.246, a sin i = 471.2 lt-s,ω = −87.4, Porb=172.7 days and τ=55514.8 MJDfor XTE J1946+274 (Marcu-Cheatham et al., 2015).

For the system KS 1947+300 the time dependenceof the period derivative P constructed from the GBM1

(the periods were corrected for the orbital motionusing the orbital parameters from (Galloway et al.,2004)) exhibits features with a characteristic periodequal to the orbital one (Fig. 2). Such a behaviormay point to a deviation of the true orbital parame-ters at the time of our observations from those usedto correct the neutron star spin frequency, i.e., to thefact that the orbital parameters could slightly changesince their determination in 2004. Using the data un-corrected for the orbital motion, we improved the or-bital ephemeris of the binary system at the time ofour observations. The pulsation period could changeas due to the Doppler shift caused by orbital motionso due to the accretion of angular momentum duringthe outburst. As has been noted in the Introduction,the spin frequency derivative is proportional to the ac-cretion rate at some power at first approximation 6/7,

P ∼ R6/7BAT . That is a form that we used to find a

solution for P measured by GBM and corrected forthe orbital motion for a set of different orbital sys-tem parameters. In reality, it is difficult to improveall parameters of the binary system based on the datafor only one outburst. Therefore, we varied only oneof them, the time of periastron passage τ , to whichthe correction procedure is most sensitive. As a re-sult, we found that the best solution corresponds toτ = 56582.941 MJD, which differs from the predic-tions made in accordance with the orbital parametersfrom (Galloway et al., 2004) by several days (Fig. 2).It can be seen from the figure that the dips in the timedependence of P from the original data correspond to

1https://gammaray.nsstc.nasa.gov/gbm/science/pulsars


https://gammaray.nsstc.nasa.gov/gbm/science/pulsars


Table 1. The list of Be systems investigated in this paper. The sources are arranged in order of increasing pulsation period

Source d, kpc Ecycle, keV B, 1012

GaussPspin, c Porb, days References

1 4U 0115+634 8 ± 1 16 1.8 ± 0.1 3.61 24.32 1,2

2 V 0332+53 7 (6-9) 28.9 3.2 ± 0.02 4.37 33.833 3,4

3 RX J0520.5-6932 50 ± 2 31. 3.5 ± 0.1 8.04 23.93 5,6

4 2S 1553-542 20 ± 4 23.5 2.9 ± 0.2 9.28 31.345 7,8

5 XTE J1946+274 9 ± 1 38.3 4.3 ± 0.3 15.75 172.7 9,10

6 KS 1947+300 10.4 ± 0.9 12 1.37 ± 0.08 18.81 40.415 11,12

7 GRO J1008-57 5.8 ± 0.5 78 8.7 ± 0.3 93.5 249.48 11,13

8 1A0535+26 2 (1.3-2.4) 42.9 4.8 ± 0.1 103.5 111.1 14,15

[1] Negueruela & Okazaki (2001); [2] Tsygankov et al. (2007); [3] Negueruela et al. (1999); [4] Tsygankov et al.(2006); [5] Inno et al. (2013); [6] Tendulkar et al. (2014); [7] Lutovinov et al. (2016); [8] Tsygankov et al. (2016b);[9] Verrecchia et al. (2002); [10]Doroshenko et al. (2017); [11] Riquelme et al. (2012); [12] Fuerst et al. (2014);[13] Bellm et al. (2014); [14] Steele et al. (1998); [15] Maitra & Paul (2013).

4U 0115+634

Fig. 1. The filled circles indicate the dependence of the flux ratio (F2−10 + F15−50)/F15−50 ((F0.3−10 + F15−50)/F15−50 forKS 1947+300) on the 15-50 keV flux during the outbursts of the systems being investigated in this paper. The solid line representsthe best-fit model. The diamonds indicate the flux ratio (F0.1−100/F15−50) derived from a spectral model or the analysis ofbroadband spectra. The dashed line indicates the flux dependence of the bolometric correction, which are the above best-fit modelsscaled to the measured flux ratios F0.1−100/F15−50. These dependences were used in the subsequent analysis.



V 0332+53

RX J0520.5-6932

Fig. 1 — Contd.



XTE J1946+274

KS 1947+300

Fig. 1 — Contd.



GRO J1008-57

1A 0535+26

Fig. 1 — Contd.



Fig. 2. Time dependence of the period derivative P for KS 1947+300 during the outburst being investigated here. The filled circlesindicate the dependence that was constructed from the data corrected for the orbital motion using the ephemeris from Gallowayet al. (2004). The crosses indicate the dependence constructed using the orbital parameters improved in our paper. The verticallines indicate the times of periastron passage: in accordance with the orbital parameters from Galloway et al. (2004) (dotted lines)and according to the parameters obtained in this paper (dashed lines).

the improved times of periastron passage, where maxi-mum deviations from the true neutron star spin periodderivatives are expected in the case of using incorrectorbital parameters.

As it is mentioned above both the Doppler shift andthe intrinsic pulsar frequency change related to the ac-cretion of matter should be taken into account duringcorrection the frequency for the orbital motion. Toa first approximation, the pulsation frequency changedue to accretion may be deemed proportional to theflux, ν ∼ F β . On the one hand, such an approach putsinformation into the dependences ν(Lbol) constructedin this paper, but, on the other hand, the quality ofthe data (their scatter and the error in ν) makes thedependence insensitive to this law. We checked thisassertion using XTE J1946+274 as an example, forwhich the orbital parameters are provided in the liter-ature for two cases: β = 6/7 and 1 (Marcu-Cheathamet al., 2015). The difference in ν for different β turnedout to be considerably smaller than the measurementerrors, while fitting the dependence by a power lawgave the same exponent for both cases.

The GBM data are rarer than the BAT data, i.e.,one pulsation period measurement corresponds to sev-eral flux measurements. The longest time step betweenGBM observations we used is 3.5 days. To obtain the

sets of interested parameters (t;L;ν), we took the mea-surements of the frequency ν (corrected for the orbitalmotion) from the Fermi/GBM data. For each pair ofsuccessive νi, νi+1 measurements made at times ti, ti+1

and spaced less than 3.5 days apart we selected all ofthe daily averaged Swift/BAT observations that fellinto this time interval; there can be j = 0, 1, 2, 3 suchpoints. If only one observation fell into the above in-terval, then ν = (νi+1 − νi)/(ti+1 − ti) was assignedto the observed luminosity L. If there were two points(with recording times tbj , tbj+1) and if both points layon one side of tmean = (tj+1+tj)/2, then we wrote theset ((tbj+1 + tbj)/2;Lw; ν), where Lw is the weightedmean. If, alternatively, tbj , tbj+1 lay on different sidesof tmean or if there were three points, then, in thiscase, a straight line was drawn through them by theleast squares method (given the errors in the lumi-nosity) and then the interpolated value Linterp wasdetermined at tmean.

The errors in the luminosity were calculated as asum of the error in the weighted mean and the half ofluminosity measurement range. This provided phys-ically motivated results both in the regions of lowfluxes, where the error in the weighted mean domi-nates, and at the outburst maxima, where averagingseveral points with a high significance leads to an un-



Table 2. The dependence of the bolometric correction on theflux in 15-50 keV energy range, F in erg s−1cm−2

Source kbol

4U 0115+634 3.7 − 1.5 × 108F

V 0332+53 2.7 + 5.1 × 107F

RX J0520.5-6932 2.5

2S 1553-542 3.5

XTE J1946+274 1.9

KS 1947+300 2.2

GRO J1008-57 2.2 − 2.1 × 107F

1A 0535+26 0.657F−0.0645

derestimation of the actual luminosity range in whichν was measured. Figure 3 shows the time dependencesof the luminosity, spin frequency, and spin frequencyderivative for the outbursts investigated in this paper.

SIZE OF THE NEUTRON STARMAGNETOSPHERE

During accretion the neutron star spin-up is deter-mined mainly by the angular momentum that is trans-ferred by the accreting matter; ν = Nacc/2πI, whereI is the moment of inertia of the neutron star taken tobe 1045 g cm2 and Nacc = M

√GMξrA during accre-

tion through a disk (Frank et al., 1985). SubstitutingM = LbolRNS

GMNSyields a well-known dependence for disk

accretion (in the case where the magnetosphere-diskinteraction occurs only near the inner disk boundary):

ν = 3.6× 10−13ξ1/2L6/736 µ

2/730 (MNS)−3/7R

6/7NSI

−145 , (2)

where MNS is expressed in units of M, and RNS isin 106 cm.

To determine how much the exponent in the ob-served dependence ν(Lbol) can differ from 6/7, thedata shown in Fig. 3 were fitted by a simple powerlaw ν = ALβ , where the exponent and the normal-ization were free parameters. We fitted the data bythe least squares method by taking into account theerrors in the neutron star spin frequency derivative.All errors are specified at a level of one standard devi-ation (1σ). If several outbursts were recorded fromthe system, then the data were fitted both for alloutbursts simultaneously and for each individual out-burst. The best fits to the data by the law ν = ALβ

for all outbursts of the source are presented in Table3 (second column) and Fig. 4 (solid lines). A hystere-sis behavior is observed in the systems KS 1947+300and V 0332+53 during their outbursts. We fitted

the increasing and decreasing (in luminosity) branchesseparately and found that for KS 1947+300 on bothbranches the exponent is virtually the same; the in-creasing branch lies systematically above the decreas-ing one. For V 0332+53 the increasing branch is belowthe decreasing one; βincr = 1.4± 0.2 on the increasingbranch and βdecr = 3.3 ± 0.7 on the decreasing one.It can be seen from Fig. 3 that a hysteresis behaviorof ν(Lbol) during some outbursts of 1A 0535+26 andGRO J1008-57 is also observed. In both systems theincreasing branch lies below the decreasing one; fittingthe branches separately for GRO J1008-57 gives expo-nents that coincide, within the error limits, both be-tween the branches of one outburst and from outburstto outburst. In 1A 0535+26 during the second out-burst the exponent on the increasing branch slightlydiffers (βincr = 1.5± 0.1) from that on the decreasingone (βdecr = 1.0± 0.1).

It follows from Fig. 4 and Table 3 that most ofthe systems have an exponent close to the predictionν ∼ L6/7. The system V 0332+53, for which the ex-ponent turned out to be about two, is an exception.Subsequently, to improve the statistical properties ofthe dependence ν(Lbol), the data for all outbursts fromone system were fitted simultaneously.

Fitting the data by Eq. (2) allows to estimate thesize of the neutron star magnetosphere under assump-tion that the magnetosphere-accretion disk interactionoccurs only near the inner disk boundary. The resultsare presented in Table 4, the second column (ξacc).The best fit to the data by Eq. (2) is indicated in Fig.4 by the dashed line.

In order to estimate the size of the neutron star mag-netosphere, we also used several analytical models thattook into account the interaction of the magnetospherewith different parts of the accretion disk and containedan explicit dependence on the magnetospheric radius.

The Models with a Neutron Star Magnetic FieldThreading an Accretion Disk

The first most complete analytical model where thestellar magnetosphere-accretion disk interaction wasconsidered was published in Ghosh & Lamb (1979).Subsequently, Wang (1987) showed the internal incon-sistency of this solution, while Wang (1995) and Li& Wang (1995) published their calculations for self-consistent models. These models use the so-called fast-ness parameter ω = Ωs/ΩK(rm) = (rm/rco)

3/2, whereΩs is the stellar spin frequency, ΩK is the Keplerian ro-tation frequency at radius rm, and rco = (GM/Ω2

s)1/3

is the corotation radius of the star. Thus, the totaltorque is n(ω)Nacc, where the dependence of n(ω) onthe fastness parameter is determined by the configu-ration of the interaction of the neutron star magneticfield with the accretion disk. Wang (1995) derived the



60 65 70 75 80 85 90 95 1000

20

40

60

80

100

120

L bol, 1

036

erg

/s

60 65 70 75 80 85 90 95 100276.684

276.688

276.692

276.696

Frequency

, m

Hz

60 65 70 75 80 85 90 95 100

MJD-55660

10

5

0

5

10

15

ν, 1

0−

12 H

z s−

1

20 40 60 80 100 120

Lbol, 1036 erg/s

5

0

5

10

ν, 1

0−

12 H

z s−

1

4U 0115+634

0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

140

160

L bol, 1

036

erg

/s

0 5 10 15 20 25 30 35 40 45276.680

276.684

276.688

276.692

276.696

Frequency

, m

Hz

0 5 10 15 20 25 30 35 40 45

MJD-57305

4

2

0

2

4

6

8

ν, 1

0−

12 H

z s−

1

20 40 60 80 100 120 140

Lbol, 1036 erg/s

2

0

2

4

6

8

ν, 1

0−

12 H

z s−

1

4U 0115+634

Fig. 3. Evolution of the parameters of the systems during their outbursts. For each system the figure shows on the left, from topto bottom: the light curve, the time dependence of the spin frequency and the spin frequency derivative (the vertical dashed linesmark the times of periastron passage); the observed dependence ν(Lbol) is shown on the right. The arrows indicate the directionin which the system passes from point to point with time.



0 20 40 60 80 100 1200

50

100

150

200

250

300

L bol, 1

036

erg

/s

0 20 40 60 80 100 120228.495

228.505

228.515

228.525

228.535

Frequency

, m

Hz

0 20 40 60 80 100 120

MJD-57175

6

4

2

0

2

4

6

8

ν, 1

0−

12 H

z s−

1

40 60 80 100 120 140 160 180 200 220 240 260 280

Lbol, 1036 erg/s

6

4

2

0

2

4

6

8

ν, 1

0−

12 H

z s−

1

V 0332+53

0 20 40 60 80 100 1200

50100150200250300350400

L bol, 1

036

erg

/s

0 20 40 60 80 100 120124.300

124.400

124.500

124.600

124.700

Frequency

, m

Hz

0 20 40 60 80 100 120

MJD-56630

0

10

20

30

40

50

60

ν, 1

0−

12 H

z s−

1

100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Lbol, 1036 erg/s

15

20

25

30

35

40

45

50

ν, 1

0−

12 H

z s−

1

RX J0520.5-6932

Fig. 3 — Contd.



0 20 40 60 80 1000

20

40

60

80

100

L bol, 1

036

erg

/s

0 20 40 60 80 100107.700

107.740

107.780

Frequency

, m

Hz

0 20 40 60 80 100

MJD-57040

5

0

5

10

15

20

ν, 1

0−

12 H

z s−

1

20 40 60 80

Lbol, 1036 erg/s

0

5

10

15

ν, 1

0−

12 H

z s−

1

2S 1553-542

0 10 20 30 40 50 60 7005

10152025303540

L bol, 1

036

erg

/s

0 10 20 30 40 50 60 7063.420

63.440

63.460

63.480

Frequency

, m

Hz

0 10 20 30 40 50 60 70

MJD-55340

5

0

5

10

15

20

ν, 1

0−

12 H

z s−

1

20 40

Lbol, 1036 erg/s

0

5

10

15

ν, 1

0−

12 H

z s−

1

XTE J1946+274

Fig. 3 — Contd.



0 20 40 60 80 100 1200

20

40

60

80

100

120

140

L bol 1

036

erg

/s

0 20 40 60 80 100 12053.100

53.200

53.300

Frequency

, m

Hz

0 20 40 60 80 100 120

MJD-56560

0

5

10

15

20

25

30

ν, 1

0−

12 H

z s−

1

20 40 60 80 100 120 140

Lbol, 1036 erg/s

5

10

15

20

25

30

ν, 1

0−

12 H

z s−

1

KS 1947+300

0 10 20 30 40 50 600

20

40

60

80

100

120

L bol, 1

036

erg

/s

0 10 20 30 40 50 6010.665

10.675

10.685

10.695

Frequency

, m

Hz

0 10 20 30 40 50 60

MJD-56235

2

0

2

4

6

8

10

ν, 1

0−

12 H

z s−

1

20 40 60 80 100

Lbol, 1036 erg/s

2

0

2

4

6

8

10

ν, 1

0−

12 H

z s−

1

GRO J1008-57

Fig. 3 — Contd.



0 10 20 30 40 500

20

40

60

80

100L bol, 1

036

erg

/s

0 10 20 30 40 5010.700

10.704

10.708

10.712

10.716

Frequency

, m

Hz

0 10 20 30 40 50

MJD-57025

2101234567

ν, 1

0−

12 H

z s−

1

20 40 60 80

Lbol, 1036 erg/s

2

0

2

4

6

8

ν, 1

0−

12 H

z s−

1

GRO J1008-57

0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

L bol, 1

036

erg

/s

0 10 20 30 40 50 609.655

9.665

9.675

Frequency

, m

Hz

0 10 20 30 40 50 60

MJD-55150

864202468

10

ν, 1

0−

12 H

z s−

1

20 40 60 80

Lbol, 1036 erg/s

5

0

5

10

ν, 1

0−

12 H

z s−

1

1A 0535+26

Fig. 3 — Contd.



0 10 20 30 40 500

10

20

30

40

L bol, 1

036

erg

/s

0 10 20 30 40 509.672

9.676

9.680

9.684

9.688

Frequency

, m

Hz

0 10 20 30 40 50

MJD-55600

6

4

2

0

2

4

6

8

10

ν, 1

0−

12 H

z s−

1

20 40

Lbol, 1036 erg/s

4

2

0

2

4

6

8

ν, 1

0−

12 H

z s−

1

1A 0535+26

Fig. 3 — Contd.

Table 3. The exponent in ν = ALβ . Column 2 gives the fit for all outbursts from the source being investigated; column 3 givesthe fit for the first outburst; column 4 gives the fit for the second outburst, if it exists. In the case where a hysteresis behavior ofthe dependence ν(Lbol) was observed during the outburst, the exponents on the increasing (βincr) and decreasing (βdecr) branchesare given separately

Source β βIout βIIout

βincr βdecr βincr βdecr

4U 0115+634 1.0(2) 0.9(7) 0.9(2)

V 0332+53 1.8(5) 1.4(2) 3.3(7)

RX J0520.5-6932

0.7(2)

2S 1553-542 0.6(2)

XTE J1946+274 0.9(2)

KS 1947+300 1.0(1) 1.0(1) 1.1(1)

GRO J1008-57 0.78(3) 0.9(1) 0.8(1) 0.7(1)

1A 0535+26 0.94(3) 0.91(5) 1.5(1) 1.0(1)



Table 4. The size of the neutron star magnetosphere for severalmodels describing the dependence ν(Lbol)

Source ξacc ξwang ξli ξparf

4U 0115+634 0.1 0.1 0.04 0.3

V 0332+53 0.02 0.01 0.01 0.2

RX J0520.5-6932 0.8 0.6 0.3 –

2S 1553-542 0.9 0.7 0.3 –

XTE J1946+274 2.2 1.8 1.1 –

KS 1947+300 2.2 1.7 0.9 –

GRO J1008-57 0.1 0.1 0.04 0.15

1A 0535+26 0.4 0.3 0.2 –

formula:

n(ω) = (7/6− 4/3ω + 1/9ω2)(1− ω)−1.

The sizes of the neutron star magnetosphere ξwang cal-culated from this formula are given in Table 4, thethird column.

Li and Wang (1995) showed that the dependencecan also be in the form:

n(ω) = 1 +20(1− 31/16ω)

31(1− ω).

The sizes of the neutron star magnetosphere ξli de-rived in this model are given in Table 4, the fourthcolumn. This approximation gives a magnetosphericsize is about half that from the formula derived byWang (1995). The estimate of the magnetosphericradius for V 0332+53 within the models from Wang(1995) and Li andWang (1995) is purely formal in na-ture, because, strictly speaking, they do not describethe data (Fig. 4).

Using the model from Li and Wang (1995), we es-timated the magnetospheric size separately on the in-creasing and decreasing branches of the dependenceν(Lbol) for KS 1947+300, GROJ1008-57, and 1A0535+26 during outbursts exhibiting hysteresis. ForKS 1947+300 ξ ' 0.7 on the increasing branch andξ ' 1.0 on the decreasing one. For GRO J1008-57and 1A 0535+26 the magnetospheric size remains ap-proximately the same from branch to branch and fromoutburst to outburst and is equal to the value obtainedby fitting all data for the source simultaneously.

The Model with Open Neutron Star Magnetic FieldLines

In the model proposed by Parfrey et al. (2016) somefraction of the field lines of the neutron star magneto-sphere ζ are assumed to become open as a result of itsinteraction with the accretion disk, which leads to an

efficient neutron star spin-down. In our calculationswe assumed that ζ = 1 (Parfrey et al., 2017). The to-tal angular momentum being transferred to the star isdefined by the formula for the case where the magneto-spheric radius rm is smaller than the corotation radiusrco. This leads to the following luminosity dependenceof the neutron star spin frequency derivative:

ν = 3.6× 10−13L6/736 ξ

1/2M−3/7NS R

6/7NSµ

2/730 −

− 7.2× 10−14ξ−2/PspinL4/736 µ

6/730 M

−2/7NS R

4/7NS

(3)

Ratio of the contributions from the spin-up andspin-down torques during an outburst shows that thismodel gives results different from the simple accre-tion model for three pulsars from our sample, 4U0115+634, V 0332+53, and GRO J1008-57, as ra-tio lies within the ranges ∼1.4-3, 0.86-1.4, and 1.2-6,correspondingly. In the remaining cases, the spin-uptorque due to the accretion of matter exceeds consider-ably the spin-down one, and this model gives the samemagnetospheric radius as does the model that takesinto account only the accretion of matter (Eq. (2)).The fifth column in Table 4 gives the magnetosphericsizes ξparf obtained within this model. For V 0332+53we fitted the increasing and decreasing branches of thedependence ν(Lbol) separately. The inferred relative-magnetospheric radii coincide, within the error limits,with one another and with the value obtained by fit-ting both branches simultaneously.

ESTIMATING THE UNCERTAINTY IN THEMAGNETOSPHERIC RADIUS

It can be seen from Eqs. (2)-(4) that ν(Lbol) de-pends on the neutron star mass and radius. Theo-retical estimates predict that the radius of a neutronstar with a mass of 1.4M can be 14 km (Hebeleret al., 2010; Suleimanov et al., 2011), while most ofthe measured radii lie within the range 9.9-11.2 km(Ozel & Freire, 2016). It is also known from obser-vations that the neutron star masses can change from1 to 2.1M for different systems (Nice et al., 2005;Falanga et al., 2015). To get an idea about the ac-curacy of the relative magnetospheric radius estima-tions, we analyzed the dependence of the magneto-spheric radius on the uncertainty in the neutron starmass and radius and the distance to the system in themodel from Li & Wang (1995) for all systems (exceptfor V 0332+53, because it does not describe the ob-servational data for this system) and the model fromParfrey et al. (2016) for the systems for which it isapplicable. The neutron star mass was varied in therange (1 − 2)M; the neutron star radius was variedin the range (1− 1.5)× 106 cm. We analyzed how themagnetospheric size changed with neutron star mass



0 20 40 60 80 100 120 140Luminosity, 1036 erg/s

10

5

0

5

10

15

ν,10−

12Hzs−

1

4U 0115+634

0 50 100 150 200 250 300Luminosity, 1036 erg/s

6

4

2

0

2

4

6

8

ν,10−

12Hzs−

1

V 0332+53

Fig. 4. The neutron star spin frequency derivative versus bolometric luminosity during type II outbursts. The solid line indicatesthe fit to the data by a model ν = ALβ , where the exponent was a free parameter. The dashed line corresponds to the case wherethe exponent was fixed at β = 6/7.



100 150 200 250 300 350 400Luminosity, 1036 erg/s

10

15

20

25

30

35

40

45

50

55ν,

10−

12Hzs−

1RX J0520.5+6932

0 20 40 60 80 100Luminosity, 1036 erg/s

5

0

5

10

15

20

ν,10−

12Hzs−

1

2S 1553-542

Fig. 4 — Contd.



5 10 15 20 25 30 35 40Luminosity, 1036 erg/s

5

0

5

10

15

20

ν,10−

12Hzs−

1

XTE J1946+274

0 20 40 60 80 100 120 140Luminosity, 1036 erg/s

0

5

10

15

20

25

30

ν,10−

12Hzs−

1

KS 1947+300

Fig. 4 — Contd.



20 0 20 40 60 80 100 120Luminosity, 1036 erg/s

2

0

2

4

6

8

10ν,

10−

12Hzs−

1GRO J1008-57

0 10 20 30 40 50 60 70Luminosity, 1036 erg/s

8

6

4

2

0

2

4

6

8

10

ν,10−

12Hzs−

1

1A 0535+26

Fig. 4 — Contd.



Table 5. The range of possible sizes of the neutron star mag-netosphere when the uncertainty in the neutron star mass andradius and the uncertainty in the distance to the system aretaken into account

Source ξli, ξli, fix. ξparf

fix. dist. MNS , RNS

4U 0115+634 0.01-0.04 0.02-0.05 0.2-0.3

V 0332+53 – – 0.2-0.3

RX J0520.5-6932 0.1-0.4 0.3 –

2S 1553-542 0.1-0.5 0.2–0.8 –

XTE J1946+274 0.2-1.3 0.6-1.1 –

KS 1947+300 0.1-1 0.5-1. –

GRO J1008-57 0.01-0.03 0.03-0.05 0.1-0.2

1A 0535+26 0.03-0.2 0.1-0.7 –

and radius at a fixed distance to the system and howit depended on the uncertainty in the distance to thesystem (see Table 1) at typical neutron star massesand radii, MNS = 1.4M and RNS = 106 cm. For themodel from Parfrey et al. (2016) the range of magne-tospheric radii is given for the uncertainty both in thedistance to the system and in the neutron star massand radius, because this dependence turned out to beinsignificant. The inferred ranges of the neutron starmagnetosphere radius are given in Table 5.

It can be seen from Table 5 that even if the un-certainty in the distance to the system and the neu-tron star mass and radius are taken into account,the radius of the neutron star magnetosphere remainwithin a narrow range, 0.1 - 0.3, for three systems,4U 0115+634, V 0332+53, and GRO J1008-57. Forthe others the possible range of magnetospheric radiiis wide, from 0.1 to ∼ 1.

CONCLUSIONS

We carried out a systematic study of the spin fre-quency derivative of X-ray pulsars as a function of thebolometric luminosity, ν(Lbol), or the mass accretionrate onto them. This dependence is an important testfor the theories of angular momentum transfer dur-ing accretion and the structure of the inner accretiondisk. It is important to note that a proper construc-tion of the experimental dependence ν(Lbol) requiresadditional information to estimate the bolometric lu-minosity of the source and to correct the pulsationfrequency for the orbital motion, which was done inthis paper. At present, there is a large set of obser-vational data from the Fermi/GBM telescope, whichhas continuously monitored X-ray pulsars and tracedthe history of their pulsation frequency changes since

2008, and data from the Swift/BAT telescope, whichhas constructed continuous light curves of sources inthe hard X-ray band (15-50 keV) since 2005. Basedon these observations, we were able to construct thedependence ν(Lbol) for eight X-ray pulsars in Be sys-tems during type II outbursts, which allowed this de-pendence to be investigated in a wide range of massaccretion rates. We corrected the Fermi/GBM datafor the orbital motion by ourselves for in the systemsXTE J1946+274 and GRO J1008-57 using their orbitalparameters from the literature and for the source KS1947+300 using improved orbital parameters.

To a first approximation, the neutron star spins upduring outbursts due to the transfer of angular mo-mentum by the accreting matter. In this case, thedependence ν(Lbol) is a power law with an exponentof 6/7. Ghosh & Lamb (1979); Wang (1995); Li &Wang (1995) showed that the interaction of the neu-tron star magnetosphere with the accretion disk alsocauses the dependence ν(Lbol) to be a power law withan exponent of 6/7. Our fitting the experimental data

by a simple power law ν ∼ Lβbol gives an exponentclose to 6/7 for most systems, with the exception of V0332+53 for which β = 1.8± 0.5. For several systemswe obtained a value of β different from 6/7 within the1σ error limits (KS 1947+300, 1A 0535+26, 2S 1553-542, and GRO J1008-57). However, the errors in νshown in Figs. 3 and 4 are statistical in nature andmay be underestimated, because the systematic mea-surement error (related, for example, to the inaccuracyof determining the orbital parameters) is disregarded,which, in turn, leads to an underestimation of the errorin the slope in the power-law dependence. Thus, theavailable data do not allow one to distinguish the de-pendences from Ghosh & Lamb (1992) for a radiation-dominated disk and a two-temperature, optically thinin the vertical direction gas-dominated disk, wherethe electrons cool through their bremsstrahlung (seethe Introduction), but allow to exclude the model ofa two-temperature, optically thin in the vertical di-rection gas-dominated disk, where the electrons coolthrough Comptonization of soft photons from the ex-ternal source.

A hysteresis in neutron star spin-up behavior duringan outburst was detected in the systems KS 1947+300,GRO J1008-57, and 1A 0535+26 and confirmed inthe system V 0332+53. In V 0332+53 the spin-upon the increasing branch is systematically higher thanthat on the decreasing one. At the same time, in KS1947+300, GRO J1008-57, and 1A 0535+26 the spin-up on the increasing branch is systematically lowerthan that on the decreasing one. For these systemswe fitted the increasing and decreasing branches ofthe dependence ν(Lbol) separately and found that inKS 1947+300 and GRO J1008-57 the exponent barely



changes and is equal to the exponent obtained by fit-ting both branches simultaneously. In 1A 0535+26βincr = 1.5±0.1 on the increasing branch and βdecr =1.0 ± 0.1 on the decreasing one. However, this differ-ence may be related to the systematic deviations dueto insufficient coverage of the outburst onset by theobservations. In V 0332+53 βincr = 1.4 ± 0.2 on theincreasing branch and βdecr = 3.3±0.7 on the decreas-ing one.

We estimated the size of the neutron star magne-tosphere in the approximation of the simple accretionmodel (Eq. (2)) and the models from Wang (1995)and Li and Wang (1995). Owing to the independentmeasurements of the magnetic field strength from acyclotron absorption line in their energy spectra, thesize of the neutron star magnetosphere was the onlyfree parameter in the models under consideration (theabsence of any deviation of the magnetic field struc-ture from a dipole configuration was demonstrated forseveral pulsars by Tsygankov et al. (2016a) and Lu-tovinov et al. (2017)). The magnetospheric size forthe model from Wang (1995) is close to the valueswithin the simple accretion model, while the magne-tospheric size is approximately half as large for themodel from Li and Wang (1995). It is important tonote that the magnetospheric radius rm < rA for mostsources, but for two objects, XTE J1946+274 and KS1947+300, it turned out to be about 2rA. (Chashk-ina et al., 2017) showed that the magnetospheric sizecould reach 2rA for a radiation-dominated disk with alarge geometrical thickness. However, at the luminosi-ties demonstrated by both systems 1037−1038 erg s−1

and magnetic moments of ∼ (0.7 − 2) × 1030 G cm3

these calculations predict an accretion disk thicknessH/R ≤ 0.03, i.e., the disk must be a standard one,and the size of the neutron star magnetosphere ∼0.3-0.5. The behavior of the dependence ν(Lbol) duringan outburst in V 0332+53 suggests that the modelsfrom Wang (1995) and Li and Wang (1995) do notwork for it; for its investigation we used the modelfrom Parfrey et al. (2016), which can also be appliedfor 4U 0115+634 and GRO J1008-57. For the sys-tems V 0332+53, KS 1947+300, GRO J1008-57, and1A 0535+26 we estimated the magnetospheric size onthe increasing and decreasing branches of the depen-dence ν(Lbol) separately and found that in V 0332+53,GRO J1008-57, and 1A 0535+26 the magnetosphericsize does not change from branch to branch, while inKS 1947+300 there is some difference: the magneto-spheric size on the increasing branch is smaller thanthat on the decreasing one by a factor of 1.4.

In conclusion, it should be noted that the used mod-els were constructed in the approximation of an alignedrotator. This is not a quite correct assumption, be-cause pulsations are observed from the neutron stars.

So far there is no complete analytical model that takesthis effect into account. In the literature there arestudies where the individual components of accretiononto an oblique rotator are considered. In particu-lar, Perna et al. (2006) considered the simplest modelof an oblique rotator where the pulsar is in the pro-peller regime and part of the matter does not leavethe system but returns back into the disk when inter-acting with the neutron star magnetosphere. Kulkarni& Romanova (2013) performed numerical simulationsof matter accretion onto a strongly magnetized mis-aligned rotator (the angle between the rotation andmagnetic axes was θ < 30) and the dependence ofthe radius of the neutron star magnetosphere on theaccretion rate ξ ∼ 0.77(µ2/M)−0.086 was derived. Forall of the pulsars from our sample ξ derived from thisformula lies within the range 0.1− 0.2.

ACKNOWLEDGMENTS

This work was supported by the Russian ScienceFoundation (grant no. 14-12-01287). We used the datafrom the Fermi/GBM observatory, the Swift ScienceData Center at the University of Leicester, and thedata from the MAXI telescope provided by RIKEN,JAXA, and the MAXI team. We wish to give specialthanks to P. Jenke for his help in working with theFermi/GBM data and A. Beloborodovy for the dis-cussion of results and useful comments.

REFERENCES

1. Aly, J. 1985, A&A, 143, 19

2. Argyle, B. 2004, Observing and Measuring VisualDouble Stars

3. Barthelmy, S. D., Barbier, L. M., Cummings, J. R.,et al. 2005, Space Science Reviews, 120, 143

4. Bellm, E. C., Furst, F., Pottschmidt, K., et al. 2014,ApJ, 792, 108

5. Bessolaz, N., Zanni, C., Ferreira, J., Keppens, R., &Bouvier, J. 2008, Astron. Astrophys., 478, 155

6. Bildsten, L., Chakrabarty, D., Chiu, J., et al. 1997,ApJSS, 113, 367

7. Campana, S., Stella, L., & Kennea, J. A. 2008,Astrophys. J. (Letters), 684, L99

8. Chashkina, A., Abolmasov, P., & Poutanen, J. 2017,ArXiv e-prints

9. Coe, M. J., Bird, A. J., Hill, A. B., et al. 2007,MNRAS, 378, 1427

10. Doroshenko, R., Santangelo, A., Doroshenko, V., &Piraino, S. 2017, Astron. Astrophys., 600, A52

11. Evans, P. A., Beardmore, A. P., Page, K. L., et al.2009, Mon. Not. R. Astron. Soc., 397, 1177

12. Falanga, M., Bozzo, E., Lutovinov, A., et al. 2015,Astron. Astrophys., 577, A130

13. Filippova, E. V., Tsygankov, S. S., Lutovinov, A. A.,& Sunyaev, R. A. 2005, Astronomy Letters, 31, 729



14. Frank, J., King, A. R., & Raine, D. J. 1985, Accretionpower in astrophysics

15. Fuerst, F., Pottschmidt, K., Wilms, J., Kennea, J.,& et al. 2014, ApJ, 784, 40

16. Galloway, D. K., Morgan, E. H., & Levine, A. M.2004, Astrophys. J., 613, 1164

17. Ghosh, P. & Lamb, F. 1979, ApJ, 234, 296

18. Ghosh, P. & Lamb, F. K. 1992, NASA STI/ReconTechnical Report A, 93, 487

19. Ghosh, P., Pethick, C. J., & Lamb, F. K. 1977,Astrophys. J., 217, 578

20. Hebeler, K., Lattimer, J. M., Pethick, C. J., &Schwenk, A. 2010, Physical Review Letters, 105,161102

21. Illarionov, A. F. & Sunyaev, R. A. 1975, Astron.Astrophys., 39, 185

22. Inno, L., Matsunaga, N., Bono, G., et al. 2013,Astrophys. J., 764, 84

23. Kluzniak, W. & Rappaport, S. 2007, Astrophys. J.,671, 1990

24. Kuhnel, M., Muller, S., Kreykenbohm, I., et al. 2013,A&A, 555, A95

25. Kulkarni, A. K. & Romanova, M. M. 2013, Mon. Not.R. Astron. Soc., 433, 3048

26. Li, X. & Wang, Z. 1995, Ap&SS, 225, 47

27. Long, M., Romanova, M. M., & Lovelace, R. V. E.2005, Astrophys. J., 634, 1214

28. Lovelace, R., Romanova, M., & G., B.-K. 1995,MNRAS, 275, 244

29. Lutovinov, A. A., Buckley, D. A. H., Townsend, L. J.,Tsygankov, S. S., & Kennea, J. 2016, Mon. Not. R.Astron. Soc., 462, 3823

30. Lutovinov, A. A., Grebenev, S. A., Syunyaev, R. A.,& Pavlinskii, M. N. 1994, Astronomy Letters, 20, 538

31. Lutovinov, A. A. & Tsygankov, S. S. 2009,Astronomy Letters, 35, 433

32. Lutovinov, A. A., Tsygankov, S. S., Krivonos, R. A.,Molkov, S. V., & Poutanen, J. 2017, Astrophys. J.,834, 209

33. Lynden-Bell, D. & Boily, C. 1994, MNRAS, 267, 146

34. Maitra, C. & Paul, B. 2013, ApJ, 771, 96

35. Marcu-Cheatham, D. M., Pottschmidt, K., Kuhnel,M., et al. 2015, ApJ, 815, 44

36. Martin, R. G., Nixon, C., Armitage, P. J., Lubow,S. H., & Price, D. J. 2014, ApJ Lett., 790, L34

37. Matsuoka, M., Kawasaki, K., Ueno, S., et al. 2009,Publ.Astron.Soc.Jpn., 61, 999

38. Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, ApJ,702, 791

39. Monageng, I. M., McBride, V. A., Coe, M. J., Steele,I. A., & Reig, P. 2017, Mon. Not. R. Astron. Soc.,464, 572

40. Moritani, Y., Nogami, D., Okazaki, A. T., et al. 2013,PASJ, 65

41. Nagase, F. 1989, Publ.Astron.Soc.Jpn., 41, 1

42. Negueruela, I. & Okazaki, A. T. 2001, A&A, 369, 108

43. Negueruela, I., Roche, P., Fabregat, J., & Coe, M. J.1999, Mon. Not. R. Astron. Soc., 307, 695

44. Nice, D. J., Splaver, E. M., Stairs, I. H., et al. 2005,Astrophys. J., 634, 1242

45. Okazaki, A. T. & Negueruela, I. 2001, A&A, 377, 161

46. Ozel, F. & Freire, P. 2016, Ann. Rev. of Astron. andAstrophys., 54, 401

47. Parfrey, K., Spitkovsky, A., & Beloborodov, A. M.2016, ApJ, 822, 33

48. Parfrey, K., Spitkovsky, A., & Beloborodov, A. M.2017, Mon. Not. R. Astron. Soc., 469, 3656

49. Perna, R., Bozzo, E., & Stella, L. 2006, Astrophys.J., 639, 363

50. Postnov, K. A., Mironov, A. I., Lutovinov, A. A.,et al. 2015, Mon. Not. R. Astron. Soc., 446, 1013

51. Riquelme, M. S., Torrejon, J. M., & Negueruela, I.2012, A&A, 539, A114

52. Steele, I. A., Negueruela, I., Coe, M. J., & Roche, P.1998, Mon. Not. R. Astron. Soc., 297, L5

53. Suleimanov, V., Poutanen, J., Revnivtsev, M., &Werner, K. 2011, Astrophys. J., 742, 122

54. Tendulkar, S. P., Furst, F., Pottschmidt, K., et al.2014, ApJ, 795, 154

55. Tsygankov, S. S., Doroshenko, V., Lutovinov, A. A.,Mushtukov, A. A., & Poutanen, J. 2017, ArXiv e-prints

56. Tsygankov, S. S., Lutovinov, A. A., Churazov, E. M.,& Sunyaev, R. A. 2006, MNRAS, 371, 19

57. Tsygankov, S. S., Lutovinov, A. A., Churazov, E. M.,& Sunyaev, R. A. 2007, Astronomy Letters, 33, 368

58. Tsygankov, S. S., Lutovinov, A. A., Doroshenko, V.,et al. 2016a, Astron. Astrophys., 593, A16

59. Tsygankov, S. S., Lutovinov, A. A., Krivonos, R. A.,et al. 2016b, MNRAS, 457, 258

60. Tsygankov, S. S., Mushtukov, A. A., Suleimanov,V. F., & Poutanen, J. 2016c, Mon. Not. R. Astron.Soc., 457, 1101

61. Uzdensky, D. 2002, ApJ, 572, 432

62. Verrecchia, F., Israel, G. L., Negueruela, I., et al.2002, Astron. Astrophys., 393, 983

63. Wang, Y.-M. 1987, A&A, 183, 257

64. Wang, Y.-M. 1995, ApJ, 449, 153

65. Wang, Y.-M. 1996, Astrophys. J. (Letters), 465, L111

66. Zanni, C. & Ferreira, J. 2013, Astron. Astrophys.,550, A99


Radius of the Neutron Star Magnetosphere during …Radius of the neutron star magnetosphere during...

Documents

Transcript of Radius of the Neutron Star Magnetosphere during …Radius of the neutron star magnetosphere during...