Section 11. Timescales Radiation transport in stars

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Shon Mills
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1 Section 11 Timescales 11.1 Radiation tanspot in stas Deep inside stas the adiation eld is vey close to black body. Fo a black-body distibution the photon numbe density at tempeatue T is given by n = T [m ] (11.1) so the aveage photon enegy is E = U/n T [J photon 1 ] (11.2) (fom eqtn. 7.26). The coe tempeatue of the Sun is T c K, whence E =.5 kev (i.e., photons in the X-ay egime). Light escaping the suface of the Sun (T eff 5770K) has a mean photon enegy 10 smalle, in the optical. The souce of this degadation in the mean enegy is the coupling between adiation and matte. Photons obviously don't ow diectly out fom the coe, but athe they diuse though the sta, tavelling a distance of ode the local mean fee path, l, befoe being absobed and e-emitted in some othe diection (a `andom walk'). The mean fee path depends on the opacity of the gas: l = 1/(κ m ρ) (11.) whee κ m is the mass opacity 1, with units m 2 kg 1. 1 Opacity can also be expessed in length units 67

2 Afte n sc scatteings the distance tavelled is, on aveage (it's a statistical pocess), n sc l. Thus to tavel a distance R we equie n sc = ( R l ) 2. (11.4) Sola-stuctue models give an aveage mean fee path l 1 mm (justifying LTE!); with R m, n sc The total distance tavelled by a (ctitious!) photon tavelling fom the cente to the suface is n sc l m ( R!), and the time to diuse to the suface is (n sc l)/c y. [Moe detailed calculations give y; why? Natually, thee ae egions within the Sun that have geate o lesse opacity than the aveage value, with the lagest opacities in the cental 0.4R and in the egion immediately below the photosphee. Because of the `squae oot' natue of the diusion, a egion with twice the opacity takes fou times longe to pass though, while a egion with half the opacity takes only times shote; so any non-unifomity in the opacity inevitably leads to a longe total diusion time.] 11.2 Stella timescales A numbe of additional chaacteistic timescales can be established: Dynamical timescale `Hydostatic equilibium' appoach If we look at the Sun in detail, we see that thee is vigoous convection in the envelope. With gas moving aound, is the assumption of hydostatic equilibium justied? To addess this question, we need to know how quickly displacements ae estoed; if this happens quickly (compaed to the displacement timescales), then hydostatic equilibium emains a easonable appoximation even unde dynamical conditions. We can wite an equation of motion, ρ a = ρ g + dp d (11.5) 68

3 whee g is the acceleation due to gavity and a = d2 dt 2 is the nett acceleation. As the limiting case we can `take away' gas-pessue suppot (i.e., set dp/d = 0), so ou equation of motion becomes Integating, d 2 dt 2 = Gm() 2. = 1 2 gt2 (fo v 0 = 0) (11.6) but g = Gm()/ 2, so, identifying the time t in eqtn with a dynamical timescale, we have 2 t dyn = Gm(). (11.7) Depatues fom hydostatic equilibium ae estoed on this timescale (by gavity in the case of expansion, o gas pessue in the case of contaction). In the case of the Sun, 2R t dyn = 7 min. GM (If you emoved gas-pessue suppot fom the Sun, this is how long it would take a paticle at the suface to fee-fall to the cente.) `Viial' appoach The `hydostatic equilibium' appoach establishes a collapse timescale; as an altenative, we can conside a pessue-suppot timescale. Noting that a pessue wave popagates at the sound speed, this dynamical timescale can be equated to a sound-cossing time. The sound speed is given by ( ) kt c 2 S = γ µm(h) (11.8) (whee γ = C p /C v, the atio of specic heats at constant pessue and constant volume). Fom the viial theoem, 2U + Ω = 0 (10.21) 69

4 with and U = = V V ktn() dv 2 2 kt ρ µm(h) dv (10.16) (11.9) M Gm() Ω = dm() (10.20) 0 Gm() = ρ() dv (11.10) V Fom eqtns , 11.9 and kt µm(h) = Gm() ; that is, fom eqtn. 11.8, γ c2 S = Gm() Fo a monatomic gas γ = 5 /, giving c 2 S = 5 9 Gm() and the sound cossing time is t = 9/5 = c S Gm() (11.11) (which is within 10% of eqtn. 11.7) Kelvin-Helmholtz/Themal Timescales Befoe nuclea fusion was undestood, gavitational contaction was consideed as a possible souce of the Sun's luminosity. 2 The time ove which the Sun's luminosity can be poweed by this mechanism is the Kelvin-Helmholtz timescale. 2 Recall fom Section 10.7 that half the gavitational potential enegy lost in contaction is adiated away, with the emainde going into heating the sta. 70

5 The available gavitational potential enegy is but Ω = R 0 Gm() dm() (10.20) m() = 4 π ρ so dm() = 4π 2 ρ d and Ω = R 0 G 16π2 4 ρ 2 () d π2 Gρ 2 R 5 (assuming ρ() = ρ(r)). The Kelvin-Helmholtz timescale is theefoe t KH = Ω( ) L. (11.12) (11.1) which fo ρ = kg m, Ω = J is t KH 10 7 y. The Kelvin-Helmholtz timescale is often identied with the themal timescale, but the latte is moe popely dened as t Th = U( ) L, (11.14) which (fom the viial theoem) is 1 /2t KH Nuclea timescale We now know that the souce of the Sun's enegy is nuclea fusion, and we can calculate a coesponding nuclea timescale, t N = fmc2 L (11.15) At the time that this estimate was made, the fossil ecod aleady indicated a much olde Eath ( 10 9 y). Kelvin noted this discepancy, but instead of ejecting contaction as the souce of the Sun's enegy, he instead chose to eject the notion of evolution. 71

6 whee f is just the faction of the est mass available to the elevant nuclea pocess. In the case of hydogen buning, expessed as a faction this `mass defect' is 0.007, so we might expect t N = 0.007M c 2 L y. Howeve, in pactice, only the coe of the Sun about 10% of its mass takes pat in hydogen buning, so its nuclea timescale fo hydogen buning is y. Othe evolutionay stages have thei espective (shote) timescales. 72

Stellar Structure and Evolution

Stellar Structure and Evolution Stella Stuctue and Evolution Theoetical Stella odels Conside each spheically symmetic shell of adius and thickness d. Basic equations of stella stuctue ae: 1 Hydostatic equilibium π dp dp d G π = G =.