Stellar Structure and Evolution

Transcription

1 Stella Stutue and Evolution Spin 7 ain topis Basi equations of stella stutue Physis of stella matte Stella evolution fom main sequene to the final staes inludin synthesis of the elements eommended book Kippenhahn & Weiet (1989; KW) Hansen, Kawale & Timble (5; HKT) Bakound/supplemental: see sepaate list Exam ateial disussed in letues & poblems Optional: `siptie on topi to be aeed upon f simple pefet laws uniquely ule the Univese, should not pue thouht be apable of unovein this pefet set of laws without havin to lean on the uthes of tediously assembled obsevations? Tue, the laws to be disoveed may be pefet, but the human bain is not. Left on its own, it is pone to stay, as many past examples sadly pove. n fat, we have missed few hanes to e until new data feshly leaned fom natue set us iht aain fo the next steps. Thus pillas athe than uthes ae the obsevations on whih we base ou theoies; and fo the theoy of stella evolution these pillas must be thee befoe we an et fa on the iht tak. atin Shwazshild (1958) 1

2 1. Some fats about stas a) ass, luminosity, adius, & effetive tempeatue Sun = am = m L = e/se ρ = 1.86 m/m O p L= π σ Teff Teff, O= 578 K Stas Appaent manitudes olos Stella atmosphee models L, T eff, Cetain binaies LL O ρ ρ O 1 8. O 1 Teff O K b) The Hetzspun-ussell Diaam Disovey of the Hetzspun-ussell diaam a entuy ao showed that fo most stas the absolute manitude V and effetive tempeatue T eff (as deived fom its spetal type SP) ae oelated V ost stas on the main sequene, some on the iant banh, and one lone outlie (white dwaf) Spetal Type

3 H- diaam fo the neaby stas Luminosities based on appoximate distanes: satte in V Usin oan-keenan spetal types: diseteness in T eff H- diaam fo the Sola Neihbohood V deived fom paallax π obtained by HPPACOS fo neaby stas with V<1 V- olo is funtion of T eff, Unlike spetal type SP, it is a ontinuous quantity

4 ) Hetzspun-ussell diaams fo sta lustes All stas at almost same distane: an use V instead of V Use olo (B-V o V-) instead of SP: ontinuous quantity ain sequene, tun-off, subiants, iants, hoizontal banh, asymptoti iant banh, and white dwaf sequene Deep H- diaam of a lobula luste AGB stas HB Giants Exteme HB Blue stales WDs Subiants S

5 NGC 697 HST/ACS 5

6 The Hyades ndividual paallaxes oeted fo depth of the luste by usin the pope motions seula paallaxes These ae a fato moe auate than π T eff fom atmosphei models fo speta the uently most auate absolute H- diaam fo any sta luste de Buijne et al. 1 A&A 67, 111 Composite H- diaam fo stalustes Aim is to undestand: Position of stas in this diaam Evolution of stas in this diaam Diffeenes between luste HD s Natue of Cepheids Sandae 1957 ApJ 15, 5 6

7 d) Thee Kinds of H- Diaams Absolute manitude V vesus spetal type SP Absolute manitude V vesus a olo, e.., B-V Bolometi luminosity vesus effetive tempeatue: lo L vesus lo T eff ( physial H diaam) m = 5lo d 5+ A V V V d = p= AU π π Distane modulus; A V extintion Tionometi distane d follows fom paallax π Absolute bolometi manitude bol and luminosity L: = + BC = lol L bol V O with BC the bolometi oetion e) ass-luminosity elation Fo etain binaies (e.., doublelined elipsin vaiables) possible to detemine individual masses (e.. Poppe 198 AA&A atin & inad 1998 AA, 585) eent measuements: L = L O b b b O O O O O O 1 O O O 7

8 f) ntenal stutue Helioseismoloy Study of osillations of Sola sufae Povides pobe of intenal stutue Extemely auate Sola model Fational eo in sound speed P/ρ Asteoseismoloy dem fo othe stas, but sufae not spatially esolved ) Nuleosynthesis Cosmi abundanes of most of the elements podued by nuleosynthesis in stas (exept H, He, and Li, Be, B) ass numbe of element 8

9 h) Some questions What is the intenal stutue of stas? What auses the mass-luminosity elation? What sets the ane of stella masses? What eneates diffeent lasses of stas in H Diaam? What ae the final staes of stella evolution? How do stas podue the heavy elements? Why do some stas pulsate? What additional poesses ou in binay stas? To be answeed by applyin the laws of physis i) Outline of ouse Deivation of fou equations of stella stutue - ass ontinuity ( ) - Hydostati equilibium ( ) - Themal equilibium ( ) - Eney tanspot by adiation ( 6) o onvetion ( 8) equied physis - Themodynamis ( ) - Equation of state inludin deeneay, and intenal eney ( 5) - Opaity of stella mateial ( 7) - Nulea eney eneation ( 9, 1) Solution methods and simple models ( 11-1) Oveview of stella evolution ( 15-) 9

10 . Spheial stas a) ass-ontinuity equation density at adius ρ( ) mass enlosed inside m () z dm m () = () d = () 1 d π ρ π ρ b stella adius total mass of sta mean density inside mean density of sta ρ() ρ = = m( ) m () = π = π (KW 1) b) Gavitation Gavitational aeleation inside a spheial body (KW 1) Gm() = () = = dφ d with Φ the avitational potential, and Φ sufae Chek of auay of spheial appoximation G = otation peiod of the Sun is 7 days entifual aeleation at equato: v / Gavitational aeleation at equato: G / atio is x 1-5 so otation unimpotant, and sta an be teated auately as a sphee 1

11 ) Hydostati equilibium Small ylinde of thikness d, sufae 1m mass ρ d Newton s equation of motion: 1 dp Gm() && = ρ d with P the pessue Equilibium if and only if dp Gm() ρ() = af d && = fo all This is the equation of hydostati equilibium, whih equates the pessue adient to the avitational foe Often useful to employ m=m() as vaiable, not. Then: af 1 af (KW.1-.) d 1 = dm π ρ dp Gm = dm π Solutions of (1) and () exist fo the speial ase These ae so-alled baotopi stas, and inlude the famous polytopes with P= Kρ γ but also the white dwafs, both of whih we study late ( 1; ) But eneally P = P( ρ, T) with T the tempeatue; this is alled the equation of state; it also depends on omposition P = P( ρ) Example: deal as has P ρ N k T μ = N o is Avoado s numbe = k is Boltzmann s onstant = e/k μ is the mean moleula weiht (see b) 11

12 d) Time sales Conside aain Newton s equation: 1 dp Gm() && = ρ d Two exteme ases: P= : && = = : τ ff = Fee-fall τ ff time sale = : P ρ && = : τ l = Explosion exp ρ τ expl P υ sound time sale n H.E. both tems ontibute equally (but with opposite sin), so we an wite τ expl = τ ff. Usin G/ P G / ( f), we find: 1 τ ff = τ expl = τ hydo = G Gρ This is the hydodynamial timesale Poblem: Solve && = Gm()/ with = at t=t, and show that = is eahed fo t = π / Gρ n Sola units: τ hydo F = 16 H G O F KJ H / 1 / O K seonds Use ane of mass and adii fo stas se τ 1days hydo This is simila to the peiods of pulsatin stas Stas ae in hydostati equilibium Poblem: What is t hydo fo a neuton sta? Can a pulsa be a pulsatin neuton sta? 1

13 e) Potential eney E E = the amount of wok needed to bin stella matte fom infinity to the pesent onfiuation Let m() be pesent inside adius, and add mass dm between and + d. Then: de z Gm = d dm = Gm dm E z z Gm 1 = dm = dm Φ inteate by pats; twie n hydostati equilibium: z z z E = π dp = π P 1π P d = PdV = z P dm ρ We shall see in b that E E i, the intenal eney of the sta f) Ode of manitude estimates ean pessue Cental pessue z z z G m z z 1 1 P = Pdm = mdp = G π P = P( ) P( ) = dp = π m dm dm Potential eney ean av. aeleation ean tempeatue (fo ideal as) E G m dm = = z z z 1 dm G = = = z m dm 1 T Tdm 1 μ P μ = = dm = Nk ρ N k E z 1

14 The inteals on the iht-hand side ae all of the fom σν, σ ν / Gm z ν z π ν/ σ ν/ dm G ρ m dm = =F H K whee we have used = m()/ π ρ() Assume the density ρ() does not inease outwads, Then: dρ( )/ d and ρ( ) = ρ ρ( ) = ρ π t follows that: σ G G ν σν, σ + ν σ + ν + 1 σ + 1 a f a f ν with defined by = π ρ Physial intepetation Conside a mass distibution with total mass, adius and abitay ρ(), whih does not inease outwads () Conside two elated onfiuations with ρ() onstant (&): Then: P P P E E E T T T 1

15 Speifially, this ives: G π G 8π G 5 G G P π G 5 G μ G μ G T 5Nk 5Nk G P 8π E Numbes G 16F O = dyne m H G. / G G μ Nk G 8F = H G O O O e O m / se = F H G 7F = 9. 1 H G KJ F H KJ F H KJ F H OKJ F H O K K K K O μ K Enomous pessues, densities and tempeatues! 15