Stellar Physics lecture 6: stellar evolution 辜品高 [email protected] 2014/4/16, 18 1 pre-main-sequence stars vertical: Hayashi track H- opacity acts as a Calvet 2004 thermostat horizontal: Henyey track, radiative core develops Stellar birth line: corresponds to the upper envelope of pre main sequence stars on the HR diagram, which also corresonds to the D burning phase (Stahler 1988) http://adsabs.harvard.edu/abs/2004adjh.conf..521C 2 Age of a young cluster 3 depletion of protostellar disks around sun-like proto-stars Near-Infrared excess comes from micron- sized, hot (about 900K) dust grains. This may imply that gas giant planets form in ~ a few million years. cf. solar-type stars spend Gyrs on main sequence. Causes: accretion, photoevaporation, disk winds? 4 radiative stars Mass - Luminosity (M - L) relation for Radiative Stars 2 4 4 16r dT 1 T L* L(r) 2 3 dr R* R* Hydrostatic equilibrium : GM kT M * M * ~ T , ~ 3 R mu R* R* M 3 so, L * 3 High - mass star (electron scattering dominated): L M * M 3 Low - mass star (b - f, f - f transitions : T 3.5 ) : L * M 11/ 2 R1/ 2 * * 4 If stars are so massive (M 140M sun ) that radiation pressure (P aT /3, es ) dominates : 1 dP 4aT 3 dT GM L 4acT 3 dT cGM F dr 3 dt r 2 4r 2 3 dr r 2 4cGM L M (Eddington Luminosity, independent of R es because both radiation pressure force and gravitational force 1/ r 2 , so they cancel) These M-L scaling laws, though oversimplified, are useful for main-sequence and horizontal tracks in the H-R disgram. 5 main sequence GM 2 M 140M ,aT 4 (i.e. due to radiation pressure support) sun c R4 M 9 T 16 CNO c c R19 M 10 total generation rate M CNO R19 Eddington luminosity L M In thermal equilibrium, the above two equal R M 9/19 M 0.5 M 2M M 20M (T due to pressure support) sun sun c R 16 17 16 M M M CNO cTc R3 R R19 M 18 total generation rate M CNO R19 luminosity L M 3 In thermal equilibrium, the above two equal R M 15/19 0.5M sun M 2M sun : appreciable outer convection zone M 0.5M sun : completely convective The above scaling approaches for radiative stars fail. computer modeling R M 0.9 6 Hertzprung-Russell diagram 7 stellar types ionized metals Hydrogen ionized Neutral Helium Helium molecules 8 stellar spectra http://frigg.physastro.mnsu.edu/~eskridge/astr101/dwarfs.gif 9 interior of man-sequence stars CNO cycles is extremely sensitive to temperature large opacity κ energy generation concentrated due to partially ionized toward the center H (b-f transition), also density > ad increases with decreasing R M mass M 1 R3 M 2 dT dr early-type star late-type star (O, B, A) (G, K, early M) http://en.wikipedia.org/wiki/Stellar_structure 10 evolutionary track of a solar-mass star asymptotic giant branch (AGB) horizontal branch (HB) red giant branch (RGB) The Sun gets brighter on the main sequence RT X 2 T 4 , P pp H Although X H decreases with time in the H - burning core, increases with time the H - burning core is then compressed, increasing and T, which in turn compensates the offset of the decrease in X H and in fact enhances pp! Because of the large luminosities on the red giant and asymptotic giant branches, the exhaustion of the majority of the fuel takes only 10% of the time that the main-sequence required to exhaust a 11 minority amount. from main sequence to subgiant 12 Schönberg-Chandrasekhar limit end of the main-sequence: core H exhaustion a He core surrounded by a H-rich envelope In what condition does the inner He core start to gravitationally contract? Ans: Schönberg & Chandrasekhar limit Assume that the He core is an ideal gas, radiative and almost in thermal and hydrostatic equilibrium. L 0 & L(m 0) 0 L 0 in the core m T 3L 0 isothermal core m 64 2r 4acT 3 Applying the virial theorem to the core but with a surface pressure from its envelope: RM dP Gm 3 cc2 Gmdm 4Rc P(Rc ) 3 P4r dr dr r 2 00r 2 3 RTc GM c 4Rc P(Rc ) 3 M c ( 2Ei Eg ) given Mc c Rc 3M RT GM 2 Pc q>q P(R ) c c c sc c 4R3 4R4 P c c c q=qsc env 94 R4G T 4 4GM P(R ) c at R c c c max 12 (4G )4 M 2 c 9RT c c c q<qsc Moreover, we may assume at the bottom of the envelope unstalbe stable M 2 M T 4 P (R ) ~ * ,T ~ T (R ) ~ * P(R ) c env c 4 c env c c 2 solution solution R* R* M * 2 2 M 0.65 Hence, P (R ) P(R ) q c q 0.37 env 0.37 0.1 env c c max M sc 4 / 3 * c Rc 13 Core contraction & H burning shell When an isothermal He core is built up to the Schönberg & Chandrasekhar limit (i.e. q > qsc ≈ 0.1M*), the core cannot maintain quasi-hydrostatic equilibrium but starts to contract. the star leaves the main-sequence The He core contracts on the KH timescale. The weight of the envelope increases the pressure at the base of the envelope to maintain the quasi- hydrostatic equilibrium. The increase of the pressure and temperature to ignite H burning in the shell at the base of the envelope. H burning shell drops He “ash” into core, which adds to its gravity and tendency to contract even more. Weight and therfore pressure of H burning shell becomes greater which makes H fusion occurs faster. The gain in heat by the envelope is larger than what is lost from the photosphere. The envelope expands (i.e. due to pdV work) even as the core contracts. The star evolves to a subgiant. If this expansion occurs while the envelope is mostly radiative, then the star must follow a more-or-less horizontal track in the H-R diagram. It finally evolves to the Hayashi line. As stars ascend their Hayashi tracks to the red giant branch, their envelopes becomes increasing convective. 14 evolutionary tracks on H-R diagam Roughly speaking, evolutionary tracks of stars in the H-R diagram either go horizontal (if the envelopes are radiative) or they do vertical (if the envelopes are convective). Evolutionary tracks in the H-R diagram therefore never go, for example, at a diagonal (in either branch of an X). 15 Ascent to red giant subgiant phase (horizontal tracks): luminosity roughly const. increase radius decrease effect temperature, He core mass exceeds qsc, contracts on the KH timescale finally encounter the Hayashi line (vertical tracks): effect temperature almost const., luminosity increases and the red giant expands 16 Age of a stellar cluster turn-off point from the main sequence cluster age 17 age of globular clusters Vandenberg et al. 2002 need distance candles distance modulus: (m M )V 5log d(pc) 5 AV B V M B MV E(B V ) (B V )observed (B V )intrinsic A V 3.2 E(B V ) best fit: age ~ 13.5 Gyrs almost as old as the universe isochrones 18 RGB & He flash at the tip contracting radiative core becomes increasingly dense, eventually making electrons in it more degenerate. (He “white dwarf” inside a star) the H burning shell deposits He “ash” into the core. 2 5/3 surface gravity of the core (GMc/Rc Mc ) increases with Mc, and therefore raises T and ρ of the H burning shell enhance nuclear-energy generation in the shell, raise T of the shell and core larger luminosity enters the base of the convective envelope star expands tremendously and approaches the “tip” of the RGB, some mass is lost from the stellar convective envelope core becomes hot enough (~108 K) to ignite He fusion into C in a flash (triple alpha reactions) He flash: the extra energy release raises the local temperature, but a slight increase in temperature adds little to the pressures that are already present in the form of the electron degenerate pressure. However, the rate of He burning near T ~108 K scales as T40. Thus, even a modest increase in T creates a vastly enhanced rate of nuclear burning. More nuclear burning means higher temperatures, which means more nuclear burning. The He flash occurs on time scales that are still appreciably longer than the sound-crossing time scale in the inner regions of the star; thus, the helium flash produces only relatively slow expansional motions. The expansion “lifts” the degeneracy of the core. In the H-R diagram, the star now descends to the 19 horizontal branch. Electron degeneracy of the core a gas of free electrons begins to become noticeably degenerate when 1/ 2 h2 3 nede Broglie 2, where de Broglie . also obtained from F kT 2mekT 3 2 2 2 F 2 8 F 2 8 h 3 with F mec xF / 2 for an N.R. gas and n f ( )4p dp p dp xF . h3 0 h3 0 3 mc The dividing line between non - degenerate and degenerate conditions : 3/ 2 h2 n T 3/ 2 2 4.831015 cm-3 K-3/2. e crit 2mek 4 For a pure He plama, ne . So the contracting He core becomes degenerate when 2mu 3/ 2 2 -3 Te c 9.410 g cm , crit 1.5107 K 7 where Te 1.510 K is the estimated temperature that H burning in the shell might be expected to hold the core at when the post - main sequence core begins its gravitational contraction.