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Lecture 1 Overview Time Scales, Temperature-Density Scalings

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Lecture 1 Overview Time Scales, Temperature-Density Scalings I. Preliminaries The life of any star is a continual struggle between the force of gravity, seeking to reduce the star to a point, and pressure, which holds it up. Stars are gravitationally Lecture 1 confined thermonuclear reactors. So long as they remain non-degenerate and have not Overview encountered the pair instability, overheating leads to Time Scales, Temperature-density expansion and cooling. Cooling, on the other hand, leads to contraction and heating. Hence stars are generally stable. Scalings, Critical Masses The Virial Theorem works. But, since ideal gas pressure depends on temperature, stars must remain hot. By being hot, they are compelled to radiate. In order to replenish the energy lost to radiation, stars must either contract or obtain energy from nuclear reactions. Since nuclear reactions change their composition, stars must evolve. Central Conditions The Virial Theorm implies that if a star is neither too at death the degenerate nor too relativistic (radiation or pair dominated) iron cores of massive stars 2 GM are somewhat MN kT (for an ideal gas) R A degenerate GM T N AkR 1/3 ⎛ 3M ⎞ R for constant density ⎝⎜ 4πρ ⎠⎟ GM 2/3 So ρ1/3 T 3 N Ak ρ ∝ T That is as stars of ideal gas contract, they get hotter and since a given fuel (H, He, C etc) burns at about the same temperature, more massive stars will burn their fuels at lower density, i.e., higher entropy. Four important time scales for a (non-rotating) star to adjust its The free fall time scale is ~R/vesc so 1/2 structure can be noted. The shortest by far is the time required R ⎛ R3 ⎞ τ = to approach and maintain hydrostatic equilibrium. Stars not in esc v ⎝⎜ 2GM ⎠⎟ a state of dynamical implosion or explosion maintain a balance esc 1/2 between pressure and gravity on a few sound crossing times. ⎛ 3 ⎞ ⎝⎜ 8πGρ⎠⎟ The sound crossing time is typically comparable to the free fall time scale. dP GM ρ The number out front depends upon how the = − dr r2 time scale is evaluated. The e-folding time for the GM ρ density is 1/3 of this P cent R 1/2 and 1/2 1/2 ⎛ 1 ⎞ 446 ⎛ P⎞ ⎛ GM ⎞ τ HD ⎜ ⎟ sec cs = γ ⎜ ⎟ ⎝ 24πGρ ⎠ ρ ⎝⎜ ρ ⎠⎟ ⎝ R ⎠ 1/2 ⎛ GM ⎞ v esc ⎝⎜ R ⎠⎟ which is often used by the nuclear astrophysicist So the escape speed and the sound speed in the deep interior are comparable The thermal diffusion coefficient (also called the thermal The second relevant adjustment time, moving up in diffusivity) is defined as scale, is the thermal time scale given either by radiative ⎛ conductivity ⎞ K diffusion, appropriately modified for convection. D = ⎜ ⎟ = ⎝ heat capacity⎠ CP ρ This is the time it takes for a star to come into and where K appears in Fourier's equation maintain thermal steady state, e.g., for the energy generated heat flow = -K ∇T in the interior to balance that emitted in the form of radiation For radiative diffusion the "conductivity", K, is given by at the surface. 4acT 3 K = (see Clayton 3-12) 3κρ where κ is the opacity (cm2 g−1), thus nb. These two time scales and 3 R2 , HD therm ⎛ 4a cT ⎞ τ = sometimes place limits on the accuracy D = 2 therm ⎜ 3κC ρ ⎟ Dtherm with which jobs run in one code can ⎝ P ⎠ be remapped into another (e.g., 1D -> 3D). −1 −1 where CP is the heat capacity (erg g K ) can also be important KH D here thus has units cm2 s−1 Note that ⎛ c ⎞ ⎛ aT 4 ⎞ ⎛ c ⎞ ⎛ radiation energy content ⎞ D ≈ = ⎝⎜κρ ⎠⎟ ⎜ ρC T ⎟ ⎝⎜κρ ⎠⎟ ⎝⎜ total heat content ⎠⎟ ⎝ P ⎠ Possible point of confusion. The radiative diffusion time If radiation energy density is a substantial fraction of the internal vs the thermal time. The former is the time for a photon energy (not true in the usual case), D ~ c/κρ with κ the opacity, to diffuse out 2 2 and taking advantage of the fact that in massive stars electron scattering R R κρ τ ~ ~ 2 −1 rad dominates so that κ = 0.2 to 0.4 cm g , the thermal time scale then mfpc c scales like 2 2 The latter is the time for an appreciable fraction of the R κρ ⎛ 0.2R ⎞ ⎛ 3M ⎞ M τ rad ≈ ~⎜ ⎟ ⎜ 3 ⎟ ∝ heat content to diffuse out. It is longer if most of the c ⎝ c ⎠ ⎝ 4π R ⎠ R heat is in the ions, not the radiation however, massive stars have convective cores so the ⎛ total thermal energy in star ⎞ thermal time is generally governed by the diffusion τ ~τ thermal rad ⎜ radiation content of star ⎟ time in their outer layers. Since the dimensions are still ⎝ ⎠ (several) solar radii while the densities are less and the In the sun these differ by a factor of about 100 and the opacity about the same, the radiative time scales are somewhat thermal time is of order the Kelvin Helmholtz time 5 less than the sun ( 10 yr; Mitalas and Sills, ApJ, 401, 759 (1992)). ⎛ E ⎞ ⎛ E ⎞ ⎛ αGM 2 ⎞ ~ matter ~ matter τ thermal τ rad ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = τ KH ⎝ E ⎠ ⎝ L ⎠ ⎝ 2RL ⎠ rad rad rad http://link.springer.com/article/10.1023/A%3A1022952621810 A closely related time scale is the Kelvin Helmholtz time scale GM 2 M 5/3ρ1/3 In general a diffusion coefficient is given by τ ≈ ∝ 1/3 KH RL L i.e. R ~ (M / ρ) 1 D ~ v l 3 Except for very massive stars, L on the main sequance where v is a typical speed (light or a convective is proportional to M to roughly the power 2 to 4, and ρ decreases element) and l is a characteristic length scale (mean with M so the Kelvin Helmholtz time scale is shorter for more free path or pressure scale height) massive stars. Note that there are numerous Kelvin Helmholtz time scales for massive stars since they typically go through six stages of nuclear burning. During the stages after helium burning, L in the heavy element core is given by pair neutrino emission and the Kelvin Helmhotz time scale becomes quite short - e.g. a protoneutron star evolves in a few seconds. ρ ∝ T 3 e.g., the sun is in thermal steady state. A presupernova star is not. Final composition of a 25 M0 star: up to Si up to O up to He unburned burned burned burned up toH-burned up toNe-burned mass fraction interior mass (M0) 25 M Presupernova Star (typical for 9 - 130 M ) 1400 R 0.5 R Examples of Time Scales H, He He • In stellar explosions, the relevant time scale is the hydrodynamic one. 240,000 L • Explosive nucleosynthesis happens when HD ~ nuc • During the first stages of stellar evolution KH < nuc. The evolution occurs on a Kelvin Helmholtz time In a massive presupernova star in its outer layers, < . The C,O Si, S, Ar, Ca • nuc thermal outer layers are not in thermal equilibrium with the interior. It also Fe happens in these same stars that nuc (core) < KH (envelope). The O,Mg,Ne core evolves like a separate star. He, C • Rotation and accretion can add additional time scales. E.g. Eddington Sweet vs nuclear. Accretion vs nuclear. Convection also has its own 0.1 R 0.003 R time scale. Central Conditions for Polytropes Eq. 2-313 and hence 3 m(r) = const*r GR2 2 ∝ ρ since 1/3 1 ⎛ 4πρ ⎞ = c R ⎜ 3M ⎟ ⎝ ⎠ for constant density = 24 most stars have 1.5 < n < 3 Actually the dimensions of Y are Mole/gm and NA has dimensions particles per Mole. N k P = Σ n kT = A ρT i µ Xi= mass fraction of species “i” N Y n Z n ρ A e = e = ∑ i i N ZY = ρ A ∑ i i 0.5 < µ < 2 (The limit µ=2 is achieved as A goes to infinity and Z = A/2, i.e. electrons dominate) For advanced stages of The decline of T3/ mostly reflects the fact evolution where A > 1, most that beyond H burning the star becomes of the pressure is due to the a red giant and no longer is a single polytrope. M is essentially reduced. This more than electrons(and radiation) core compensates for the increase in µ. A decreasing also decreases T3/. n the final stages degeneracy becomes important.! T This would suggest that the P ∝ρ c c c µ ratio would increase as the star evolved and µ became greater. ρ ∝ T 3 T ∝ µρ1/3 c c CRITICAL MASSES Ideal gas plus radiation ⎛ Pideal ⎞ =⎜ ⎟ ⎝ Ptotal ⎠ P T P ∝ ideal ∝ ρ c tot β c µβ Decrease in as star evolves acts to suppress T3/. note cancellation of ρ c = 1.45 M if Ye = 0.50 neglecting Coulomb corrections and relativistic corrections. 1.39 M if they are included. For lower densities and hence degenerate cores Specifying a core mass gives a maximum temperature significantly less than the Chandrasekhar mass achieved before degeneracy supports the star. If that maximum non-relativistic degeneracy pressure gives another temperature exceeds a critical value, burning ignites solution ( = 5/3) M1 > M 2 M <M i Ch This is the well known central density mass relation for (non-relativistic) white dwarfs He core 100% He Critical Temperatures CO Core 50%C- 50% O Ne-O Core 75% O- 25% Ne Ne-O cores CO-cores He-cores These calculations give critical masses: Main Sequence Critical Masses 0.08 M Lower limit for hydrogen ignition C ~1.0 0.45 M helium ignition Ne ~1.25 7.25 M carbon ignition O 1.39 Si 1.39 All stars with main sequence mass 9.00 M neon, oxygen, silicon ignition (off center) above the Chandrasekhar mass could in principle go on to 10.5 M ignite all stages at the stellar center burn Si.
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