Part II. Cosmologia Eduard SalvadorA partir del 29 de març (despatx 7.01)

ASTROFÍSICA I COSMOLOGIACoordinador de l’assignatura:

Eduard Salvador e.salvador@ub.edu

Part I. Astrofísica Blai SanahujaInstitut de Ciències del Cosmos i

Departament de Física Quàntica i Astrofísica Facultat de Física (despatx 7.02 part nova)

Heliospheric Physics and Space Weather groupblai.sanahuja@ub.edu www.am.ub.edu/~blai/

Astrophysics → Stellar Physics

● Stellar physics: the research field concerning the formation, evolution, interior and the atmospheres of stars. The understanding of the birth, evolution and death of stars requires the application of almost all branches of physics: mechanics, thermodynamics, radiation theory, hydrodynamics, relativity, atomic physics, nuclear physics and so on.

“Most probably this is the first timethat students have the opportunityto discover that complex structuresand processes do occur in theNature, for the understanding ofwhich all the different branches ofthe physics must be invoked andcombined” (D. Prialnik, 2010).

● Una estrella és un cos que satisfà dues condicions:‒ Està lligat per la seva pròpia gravetat.

Té forma esfèrica, donat que la gravetat és un camp amb simetriaradial (esdevé esferoïdal si apareixen forces axisimètriques).

‒ Radia energia subministrada per una font interna.Aquesta font d’energia usualment és energia nuclear alliberada per reaccions de fusió y, en curts períodes, és energia potencial gravitatòria alliberada per contracció o col·lapse.

→ Una estrella ha d’evolucionar ja que a mida que allibera energia, es produeixen canvis interns que afecten la seva estructura o la seva composició: és la evolució estel·lar.

● La evolució de les estrelles és ‘molt lenta’, en conseqüència‒ l’estudi observacional és bàsicament poblacional ‒ l’estudi teòric és evolutiu.

● Una estrella queda caracteritzada per la massa i la composició química (+ la rotació, el camp magnètic, la binarietat, etc.).

● Goals‒ To gain understanding of the physical processes in stars, how

they evolve and what parameters their evolution depends on.

‒ To learn how to interpret observational characteristics of stars in terms of the underlying physical parameters.

‒ To learn about the main features of the structure of the Sun.

‒ To gain understanding of how stars of different mass evolve, and what end products are produced.

‒ To learn what causes planetary nebulae and supernovae.

‒ To understand what types of stars produce stellar remnants such as white dwarfs, neutron stars, black holes.

Note: After these lectures you will not be able to sit down and write a code For calculating stellar models. But they will provide an understanding about what “makes a star work”, and how it relates to the underlying physics.

This is a three-month course, there is no room to address all physical issues relevant to stellar evolution (even scarcely the basic ones).

Text books

● Lecture notes, pick them up from UB Virtual

● B.W. Carroll and D.A. Ostlie. An introduction to modernastrophysics. Pearson International 2nd Edition (2007) (CO)

● O.R. Pols. Stellar structure and evolution. AstronomicalInstitute of Utrecht (2011). Notes based to a large extent on the (KW) book. Available at (OP)http://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/

− R. Kippenhahn, A. Weigert and A. Weiss. Stellar structure andevolution. 2nd Edition .Springer Verlag (2012) (KW)

− V. Bosch. Notes on Astrophysics and Cosmology course: Astrophysics part. (2013). UB/DFQA

− D. Prialnik. Theory of Stellar Structure and Evolution. Cambridge University Press (2010)

1 Introducció

1.1 Propietats observables de les estrelles

1.2 El Sol: un cas ben (?) conegut d’estrella

1.3 L’interior solar (de no fàcil accés)

Complement. Física estel·lar: uns detalls històrics

1.1 Observable properties of stars (C07 and C03)

● Basic parameters to compare theory and observations

‒ Mass (M)

‒ Luminosity (L): Total energy radiated per second

‒ Radius (R)

‒ Effective temperature (Te, Tef or Teff):

Temperature of a black body of the same radiusas the star that would radiate the same amount of energy

L = S(R) B(Te) = 4R2 Te4 (B: radiative transfer flux)

‒ Chemical composition: mass fraction of hydrogen (X), helium (Y) and ‘metals’ (Z) X + Y + Z = 1

● A star (and its evolution) is basically characterized by its mass and its chemical composition

● Els interiors estel·lars no són observables.

● Cal construir models d’interiors estel·lars: les prediccions dels models poden ser verificades comparant-les amb les propietats observades de les estrelles individuals o en col·lectius (per exemple, en els cúmuls)

Propietat Determinada per Interval de valors

Massa, M Estrelles binàries, 10–1 M – 102 M

o de g (espectre) i R

Lluminositat, L Magnitud aparent i distància 10–5 L – 106 Lo espectre (classe lluminositat)

Temperatura Espectre continu ~103 K – 105 Kefectiva, Te o tipus espectral

Radi, R L i Te 10–2 R – 103 Ro interferometria i distància

Edat Cúmuls estel·lars i teoria 0 – 1010 anys

Composició Línies espectralsquímica

~10-3 Z – 5Z

Most of the stellar electromagnetic radiation is emitted by the photosphere, usually considered as the stellar surface. Example: the solar atmosphere, Te = 5800 K

(Adapted from E. Avrett, Smithsonian Astrophysical Observatory)

Te Tmin

Density (g cm-3) ~3 10-7 ~3 10-9 ~2 10-14 [~1010 part. cm-3]

Solar spectrum shifted by a factor 1012 at 1 mm to accommodate for the large dynamical spectral range.

● Irradiància solar, distribució espectral:

● Constant solar:

Potència per unitat d’àrea rebuda a 1 AU:

k = 1360.8 (± 0.5) W m–2

(NASA/SORCE, DAVOS/TSI)

‒ Variació amb la distància

Màxim (solstici d’hivern): 1412 W m–2

Mínim (solstici d’estiu): 1321 W m–2

‒ Variació (periòdica) amb el cicle solar: ± 3.5 %

● Solar irradiance: spectral distribution

Top. The spectrum of total solar irradiance, compared with a 5770 K black body radiator.

Blue dotted line: spectrum reaching the surface of the Earth.

Bottom. Spectral variability of the irradiance: fractional difference between the values at solar maximum and at minimum, over all wavelengths (Lean 1991)

Liège Solar Spectral Atlas http://fermi.jhuapl.edu/liege/s00_0000.html

http://ljr.bagn.obs-mip.fr/ljr/observing/spectrum.html

U II 385.95 nm line, as observed in the old star CS 22892-52 (VLT Kueyen + UVES, 4 h; ESO, Cerro Paranal, 2000)

● Stellar spectraRelevant parameters: Z, Te and

surface gravity (gs = GM/R2)

● Classificació espectral (CO8.1)

‒ Developed at Harvard Observatory in early 20th century (HD and HDE catalogues), based on lines which are mainly sensitive to stellarsurface temperatures rather than actual compositional differences, gravity or luminosity.

Important absorption lines are: The Hα line (656.3 nm), the H and K CaII doublet (396.8 and 393.3 nm) and the CaI line (422.7 nm), HeI and HeII lines, Fe I and Fe II lines, the TiO and several metal lines (~431 nm).

Homework: (a) look for ‘line strength’ definition. (b) Why the Hα line is stronger for A2 stars than for O8 stars, is spite that T(O8) > T(A2)?

● Formally, the equivalent width is

Where F0 represents the intensity of continuum on either side of the absorption (or emission) feature, while Fλ represents the intensity across the entire wavelength range of interest.

● A level diagram of the H atom. The energy difference between two adjacent levels decrease with level. The energy of level n is given by En = ‒13.6 / n2 eV, thus a H atom is close to ionization at about level 5 or 6. The Balmer series involves transitions up from or down to level 2; e.g., Hβ

means a level 2–4 (absorption) or 4–2 (emission) transition.

● Boltzmann excitation equilibrium equation

The Boltzmann equation gives the fraction of atoms (ions) of a given element that has an electron excited to a given energy level, x,

(CO8.1, p. 209)

Ni,x : number atoms, or ions, ionized to the degree i and excited to energy level x (i = I, neutral state; i = II, once ionized, etc.)

gi,x : statistical weight of the energy level x

Ni : total number of atoms, or ions, ionized to the degree i,

ui : partition function,

Χi,x : excitation potential of the level x ( → Χi,∞ : ionization potential)

● Saha ionisation equilibrium equation (Saha, 1920)

The Saha equation gives the fraction of atoms ionized to a given degree, as function of temperature, electron density and ionization energy of the atoms,

Ni : total number atoms ionized to the degree i

Ni+1: total number atoms ionized to the degree i + 1

Ne : number density of free electrons

ui and ui+1: partition functions for Ni and Ni+1, respectively

Χi,∞: ionization potential (e.g., 13.60 eV for H+; 54.42 eV for He++; 0.75 eV for H−)

Meghnad Saha(1893‒1956)

[From E. Novotny, ‘Introduction to stellar atmospheres and interiors’, NASA. (Oxford Univ. Press, 1973)]

When ionization takes place, the number of particles of the gas changes with other physical properties of the gas and the adiabatic exponent changes too. This is of particular importance to the stability of stars,

− Fraction of particles in ionization stage

● Colour-magnitude diagramsConstructing an H-R diagram requires measuring accurate Te for many stars, thus, spectra and model atmospheres are needed.‒ Magnitudes are measured at different wavelength bands

(e.g., mv at V band of the standard UBVRI-photometric system).‒ Calculate M, the absolute magnitude of a star: its apparent

magnitude, m, if the star were placed at a 10 pc:

m – M = 5 log(d/10) [d in pc]

‒ To compare with theory, it is more useful to determine the bolometric magnitude, Mbol, the absolutemagnitude that the star would have if measured with a bolometersensitive to all wavelengths.

‒ Then, Mbol – Mbol, = −2.5 log(L/L)

with Mbol, = +4.74 [Bolometric correction, BC; Mbol = Mv + BC(T)]

(CO3.6)

● Diagrama HR (Color‒Magnitud, Temperatura‒Lluminositat)

Ejnard Hertzsprung (1915): Relació entre el tipus espectral o color

→ temperatura] i la magnitud absoluta [ → lluminositat].

Norris Russell (1914): Primer diagrama color‒magnitud de cúmuls.

(CO8.2)

● Luminosity classesIa: luminous supergiantsIb: less luminousII: bright giantsIII: normal giantsIV: subgiantsV: main sequence D: white dwarfs

H-R diagram with 22000 stars from the Hipparcos catalog and 1000 from the Gliese catalog of nearby stars (Hipparcos 1989−1993)

The Gaia mission (launch: 29/12/2013), halo orbit at the Sun-Earth L2 point.Science start in July 2014. First catalogue release: September 2016. 2D map with ~2109 stars (1% of the Milky Way).

Play a little bit with star evolution in a theoretical H-R diagram at http://spiff.rit.edu/classes/phys230/lectures/star_age/evol_hr.swf

● Chemical composition of stars

Deduced from:− The photospheric composition by studying the

solar and stellar spectra. The information provided often is incomplete.

[Stars in globular clusters can be 10 to 1000 times metal deficient compared to Sun, while having the same H and He abundances]

Notation:

Iron is often used asa proxy for metallicity:

− The solar system abundances. They give reasonable agreement between analysis of solar spectrum and laboratory studies of meteorites (carbonaceous chondrites).

The vast majority of normal stars have similar composition to thatof the Sun (‘quasi-solar’ composition). Typically, − by number: H: 90%; He:10%; other elements: 1%

− by mass (ρelement /ρ): X = 0.70; Y = 0.28; Z = 0.02

− HD140283 (d = 58 pc) Age: 14.46 (± 0.31) 109 years

Z = 8 10−4 Z[Fe/H] = −2.40

− HE1523−0901 (d = 2300 pc) Age: 13.8 109 years (~0.8 M)

[Fe/H] = −2.95(2nd generation of population II?)

− HE0107−5240A halo giant with [Fe/H] = −5.3

→ Z = ~10−7 Z(1st population III discovered?)

[Universe: 13.77± 0.06 Gyr]● Stellar populations

‒ Population I 0.5 Z < Z < Z. In spiral arms, young stellar clusters, mainly in the galactic disc.Age ≤ 109 years.

‒ Population II 0.01 Z < Z < 0.1 Z. High velocity stars in the galactic halo and in globular clusters.Age ~1010 years.

‒ Population III (hypothetical) Z ~0. First generation of stars? Possibly with very different properties (massive stars leading to relatively massive black holes?).

− HD 140283: A Star in the solar neighborhood that formed shortlyafter the Big Bang (E. Howard et al. Astroph. J. 2013).

Magic atomic number of nucleons have a higher average binding energy per nucleon than expected. They are more stable than the others against nuclear decay: 2, 8, 20, 28, 50, 82 and 126 (4He, 16O, 40Ca, 48Ca, 48Ni, 56Ni and 208Pb).− Evidence for ongoing nucleosynthesis:

◦ Detection (1953) of technetium in red giants (AGB stars)Tc has no stable isotopes and τ (98Tc) = 4.2 106 years.

◦ Detection of 26Al decay τ (26Al) = 7 105 years.◦ Supernova light curves; i.e., SNe Ia powered by 56Ni decay.

● Standard abundance distribution (SAD) of elements

● A good place where to look for: Star clusters

Stars all at same distance • Dynamically bound• Same age• Same chemical composition• Can contain 103 –106 stars

NGC3603 (Hubble Space Telescope) NGC3293 Open cluster 47 Tuc Globular cluster

→ Differences are mainlydue to initial mass

− Globular vs. Open clusters

Globular Open

− MS turn-off points in similar position. Giant branch joining MS

− Horizontal branch from giant branch to above the MS turn-off point

− Horizontal branch often populated only with variable RR Lyrae stars

− MS turn off point varies massively, faintest is consistent with globulars

− Maximum luminosity of stars can get to Mv −10

− Very massive stars found in these clusters

The differences are interpreted due to age – open clusters lie in the disk of the Milky Way and have large range of ages.

Globular clusters are (all) ancient, with the oldest tracing the earliest stages of the formation of Milky Way (~12 109 yrs)

− Example: HR diagram of the globular cluster M55

High-mass stars evolved onto the giant branch

Turn-off point reveals the age of the cluster

Low-mass stars still on the main sequence

d = 5.4 kpcM = 2.7 105 M

Age = 12.3 106 yr

Horizontal branch

Treball obligatori:lectura de la secció13.3 de (CO; p. 474, 2na edició). Tracta de cúmuls estel·lars,poblacions i isòcrones

‒ Les proto-estrelles més massives es contrauen més ràpidament que les poc massives arribant abans a l’inici de la seqüencia principal, la ZAMS (Zero AgeMain Sequence).

ZAMS

● Stellar radii

‒ Radii of ~600 stars measured (accuracy <1.5 %) by optical interferometry, eclipsing binaries or lunar occultation.

‒ Angular diameter of sun at distance of 10 pc:

= 2R/10 pc = 5 10−9 radians = 10−3 arcsecHST resolution: ~0.05 arcsec → difficult to measure R directly

Sirius (α Canis Majoris)

Sirius A (α CMa A) A1 V d = 2.63 pc L = 26.1 LR = 1.711 R M = 2.14 M

Te = 9900 K Edat: 2−3 108 anys

Sirius B (α CMa B) DA2

P = 50.09 anysL = 0.026 LR = 0.0084RM = 1.05−0.98M

Te = 27000 KEdat: 2−3 108 anys

● Masses estel·lars (CO7.2, p. 49 o 189)

‒ F. Bessel (1844) va demostrar la existència d’una companya novisible, a partir dels canvis observats en el seu moviment propi.

‒ A.G. Clark (1892) va observar Siri B per primera vegada.

[(M1 + M2)/M = a3/P2 on a = a1+a2 i M1 a1 = M2 a2]

M ‒ L relation of MS stars

We must understand the M−L and the L−Tef

relation theoretically.

Models must reproduce observations.

This means that the MS on the HR diagram is a function of mass: from bottom to top of MS, stars increase in mass

L/L (M/M)ν ν = 3.5

But the slope changes at extremes.

● Mass–Luminosity relation for the Main Sequence (MS) stars

● Stellar mass distribution

Most of the mass in stars is locked up in low-mass stars (brown dwarfs maybe), while most of the luminosity comes from high mass stars.

● Initial mass function (IMF)

The IMF, ξ(m), gives the distribution ofstellar masses immediately after those stars have formed. ξ(m) Δm: number of stars that a given time are been born with

initial masses between m and m + Δm.

− For stars in the solar neighborhood, E.E. Salpeter (1955) givesξ(m) = ξ0 m−γ where γ = 2.35

(ξ0 = 0.05, sets the local stellar density) − Caveat: could Salpeter’s law be considered ‘universal’?

(CO12.1)

− Salpeter’s IMF implies that : ◦ most of the stars (by number) are low mass stars◦ most of the mass in stars resides in low mass stars◦ after a burst of star formation, most of the luminosity comes from high mass stars

Observations point that Salpeter’s IMF is only valid for stars with m > 0.5 M, since it diverges when extrapolated to arbitrarily low mass stars.

So the IMF should ‘flatten’ at lower masses, but the exact form of the low mass IMF is ill defined.There are other IMFs →

Chabrier’s IMF

● Use of stellar IMFs in extragalactic physics or cosmology‒ Comparison of different IMFs by plotting the normalized mass

fraction versus mass; i.e., the integral under each curve is unity (assumed valid for 0.1 M < m < 120 M).

The Kennicutt, Kroupa01, BG03 and Chabrier IMFs are better for reasonable mass-to-light ratios and galaxy colours.The BG03 analysis favoured a slope shallower than Salpeter’sat the high mass end, based on constraints from local luminosity densities and cosmic star-formation history.

Relevance …?

1.2 El Sol: el cas més ben (?) conegut• G2 V Població I• Esfera de gas en rotació, autogravitant i termoregulada. • La influencia gravitatòria dels planetes és menyspreable

i la força centrípeta a l’equador << força gravitatòria (<<10-5)

• L = 3.83 1026 W M = 1.99 1030 kgR = 6.96 108 m Te, = 5780 K

• Rotació diferencialPeríode mitjà: 27.3 dies

25.4 dies a l’equador, 34.6 dies als pols

Inclinació respecte a la eclíptica: ~7°

• Edat: (4.57 ± 0.05) 109 anys

• Composició: 92.7% H, 7.3% He i traces de metalls (O, C, Fe, etc; factor < 10−3)

Fracció màssica: X = 0.747, Y = 0.236, Z = 0.017

• Densitat mitjana: 1400 kg m–3

• Densitat superficial: 2.07 10–5 kg m–3

• Pèrdua de massa − radiació EM (L/c2): ~4 109 kg s–1

− vent solar: ~1 109 kg s–1

• Gravetat superficial, g: 274 m s–1 (~27g)• Camp magnètic fotosfèric:

~30000 nT a l’equador (~3 G)~10000 nT als pols

• Irradiància solar a 1 AU: 1361.0 ± 2.5 W m–2

• Calculats (SSM)Temperatura central: 1.56 107 KDensitat central: 1.50 105 kg m–3

Pressió central: 2.29 1016 N m–2 (Pa)Composició central: X = 0.34, Y= 0.64, Z = 0.02

• Edat: 4.55 109 anys

[1 T = 104 G ; imant ~100 G; electroimant ~1.5 104 G; 1 G = 105 ηT; <B> ~0.5 G; estrella de neutrons 1012 G]

EIT/SOHO (17.1 nm Fe IX)

[

The PMOD composite TSI plotted in different colors for the different originating experiments

Solar constant: Construction of a composite Total Solar Irradiance (TSI) time series from 1978 to present

[Physikalisch-Meteorologisches Observatorium Davos-World Radiation Center (www.pmodwrc.ch/pmod.php?topic=tsi/composite/SolarConstant)]

‒ M: (1.9889 ± 0.0003) 1030 kg [error experimental en G (± 0.0013 %)]

‒ Pèrdua de massa: per radiació solar (L/c2) i pel vent solar (pot haver estat més intens anteriorment)

‒ Radi solar, R: (6.960 ± 0.001) 108 m◦ mesura del diàmetre angular del disc solar; dependència del

mètode (punt d’inflexió de la intensitat emergent, τ ≈ 0.004) ◦ mesures en intervals espectrals estrets: radi variable

(per exemple, a 700 nm sembla que és ~0.1” més gran que a 400 nm)◦ el límit d’un interior solar se situa a τ = 2/3; en el model SSM això

vol dir ~300 km per sota del valor observacional)

‒ Variacions evolutives (des de la ZAMS): estimada en 2.4 cm/any pel radi; la lluminositat ha augmentat un 72 % fins a l’actual (factor 1.58)

‒ Inici combustió al nucli ≈ edat sistema solar = 4.55 ± 0.05 109 anys.Determinat per datació radioactiva de la desintegració 87Rb → 87Sr (τd = 4.8 1010 anys). Comparant l’abundància del 87Rb i del 87Sr respecte la del 86Sr (és estable) en mostres de diferents meteorits.

● El Sol (comentaris)

‒ The Standard Solar Model (SSM) gives (deviations < 0.1%)Tc 15.6 106 K c 1.5 105 kg m−3

c 2.3 1016 N m−2 21011 atm.

● By observation of neutrino fluxes produced during the hydrogenthermonuclear burning

Core: < 0.25 RRadiative region:

0.25 R – 0.75 RConvective region:

0.75 R – 1R‒ Core temperature prediction from

SNOΦ(8B) neutrino observations[α Tc

25]

1.3 The solar interior is not of easy access

Tc 15.7 106 K (±0.01)

What is a Standard Solar Model? David Guenther, 2010.(www.google.es/#q=what+is+the+standard+solar+model)

UV coronal emission

He II 304 nm

Fe XII 195 nm

Fe XV 208 nm

The Extreme-Ultraviolet Solar Irradiance Spectrum observed with the Coronal Diagnostic Spectrometer (CDS) on SOHO ("Sun as a star" spectrum) on 1997 May 15. The identification of most of the brightest lines are shown. Superimposed as a dotted line in each graph is the effect that a constant background would have on the two spectra, after passing through the calibration process

Solar Dynamic Observatory: Atmospheric Imaging Assembly(A. Title. Stanford/Lockheed Institute for Space Research)

SDO http://sdo.gsfc.nasa.gov/https://www.nasa.gov/mission_pages/sdo/main/index.html

SDO/AIAhttp://aia.lmsal.com/

EVE

AIA

HMI

NASA. February 2010 Geosincronous orbit

at 36.000 km

Solar Dynamic Observatory (SDO)

AIA 17.1 nm HMI Magnetogram

(http://sdo.gsfc.nasa.gov/data/)

HMI: Heliosismic and Magnetic Imager

EVE: Extreme UltravioletVariability Experiment

AIA: Atmospheric ImagingAssembly

AIA: High resolution images (1”) at 10 wavelengths(from 170 nm to 9.4 nm), each 10 seconds.

HMI 450 nm

16 november 2012

03-Mag_comb20111207

(CO14.5)

● By helioseismologic observations

R. Leighton, R.Noyes and G.W. Simon discovered (1960) that different regions of the surface of the Sun regularly ascend and descend with a periodicity of about 5 minutes (1/300 cycles/s = 3.33 mHz).

Since 1975 more than de 107 coherent oscillation modes have been identified.

They are the manifestation of acoustic waves (mainly) trapped and propagating in the inner convective zone of the Sun

− Waves main features:Coherence length: between 3000

and 20000 kmSingle mode velocity 0.1 m s−1

Luminosity variation: δL/L~10−6

‒ Doppler shift observations of spectral absorption lines, such as FeI at 557.6099 nm, using both a narrow filter (~0.001 nm) and a narrow slit that follows the solar rotation.

MDI-Heliosismo01-01

Relative power of solar p-modes (Grec et al. Nature 1980) [Human hearing frequency range: 20 Hz to 20000 Hz].

5 min = 3.33 mHz

1991 January 1

l = 20, m = 16

Dos tipus bàsics de modus d’oscil·lació:

Modes p: són fluctuacions de pressió (és la força restauradora) produïdes per ones sonores generades pels moviments turbulents a la regió convectiva del Sol. Aquestes ones acústiques assoleixen la màxima amplitud a la superfície;amb una freqüència, f, entre 1 mHz i 4 mHz, i màxima intensitat per f ~3.33 mHz.

Modes g (de gravetat): fluctuacions causades per l’expansió adiabàtica del material solar (la flotabilitat és la força restauradora). A la regió radiativa el material que es mou capa la superfície esdevé més fred i dens que el que l’envolta i retorna a la posició original (és estable front a la convecció).

‒

Són ones internes gravitatòries de baixa freqüència (< 0.4 mHz), amb una amplitud que s’esmorteix a la zona convectiva; són pràcticament inobservables a la superfície.

P-mode anatomyA p-mode is a standing acoustic wave. Each mode is described by a spherical harmonic, identified by two nodal numbers: l (degree) and m(azimuthal order; any one of the 2l+1 integers between –l and +l)*.

The harmonic degree, l, indicates the number of node lines on the surface, which is the total number of planes slicing through the Sun (0 < l ~2000; as large it is, less penetrates the mode).

The azimuthal number m, describes the number of planes slicing through the Sun longitudinally (the rest of the slicing planes are parallel to the equator).

(* l = m = 0 correspond to a pure radial pulsation)

● Description by spherical surface harmonics in polar coordinates (r, θ, φ)

Legendre polynomials

Modes p solars: Observacions (cercles) i teoria (línies) (Libbrecht 1988)

http://commons.wikimedia.org/wiki/Spherical_harmonic

● Orbit of SOHO, ACE, Wind, etc., around the Sun

Solar Heliospheric ObservatoryESA - NASA 1995

Advanced CompositionExplorer. NASA 1997

(NOAA 11/02/2015; www.nesdis.noaa.gov/DSCOVR/)

By examining more than five years of low-order acoustic modes, obtained using the GOLF and MDI instruments aboard the ESA/NASA SOHO spacecraft, the rotation rate has been inferred for the deep solar layers (error bars), mainly along the solar equator. There is uniform rotation in the radiative zone. The sound waves do not reach the central part of the energy-generating core. (A.G. Kosovichev and S. Couvidat 2010).

● Internal structure of the Sun

Observed-modeled differences in cs as a function of depth in the Sun. Fast (slow) sound waves imply a T higher (lower) than expected from theory. The "bump" just below 0.7 Rindicates the location of possible excess turbulence.

− Local sound speed, cs = (γP/ρ)−1/2

Near core: 700 km s−1

Near surface: 7 km s−1

→ The actual SSM gives values for cs that very nicely fits with those derived from heioseismologic observations: differences are smaller than 0.5 %. Since, cs is (more or less) αT1/2, the temperature inside the SSM is most probably correspondingly close to reality.

● By applying inversion techniques to thousands of measurements of p-modes frequency, helioseismology observations confirm the SSM model predictions at 99.9 confidence level:

− The sound speed in the Sun to great accuracy (deviations less than ± 0.1%). Recent measures shows the sound speedy agrees to better than 1% even in the center of the solar core (< 0.1R) and to better than 0.2% at most depths.

− The location of the boundary of the radiation-convection regions. at 0.714 R, and the density at that point, ρ = 0.186 g cm-3.

● 8B-neutrino and 7Be-neutrino flux observations (produced at depths of 0.04 R and 0.06 R, respectively) fits very well with theoretical predictions from SSM.

‒ Neutrino flux uncertainties, mostly come from nuclear physics uncertainties.

Cartoon of a possible magnetic field flux-tube scenario

But there are still differences that must be quantitatively explained.− Solar rotation and polar flows deduced from SOHO/MDI

measurements: The cutaway reveals differences in rotation speed inside the Sun. Red-yellow is faster than average while blue is slower than average (and sunspots tend to form at the edge of these bands).

GONG (NSO/NSF) and SOHO/MDI (ESA and NASA)

The GONG network of ground observatories and the SOHO spacecraft have revealed the deep pulses by monitoring for many years the sound waves passing through the Sun.

‒ “Plasma rivers" in the Sun: existence of flows of plasma about 10% faster/slow than the surrounding material (shown in red/blue). Around the level of a the tachocline(white line) flow speeds of gas circulation change markedly over 6 months. Near the surface, left of the cutaways,bands of faster/slower (red/green) gas move towards the equator.