Lecture 5-2: Polytropes. Literature: MWW chapter 19

Transcription

1 Lecture 5-2: Polytropes Literature: MWW chapter 9!"

2 Preamble The 4 equatios of stellar structure divide ito two groups: Mass ad mometum describig the mechaical structure ad thermal equilibrium ad eergy trasport describig the temperature structure. These two sets of equatios are liked through the equatio of state. Uder certai circumstaces, pressure is idepedet of temperature ad the solvig the equatios of stellar structure simplify eormously. Defie: P = Kρ γ = Kρ +/ with the polytropic idex 2

3 a) Decouplig examples: i) Stars supported by electro degeeracy: P e = K r ρ 5/3 o-relativistic Fermi electro gas ( = 3 / 2) P e = K r ρ 4/3 relativistic Fermi electro gas ( = 3) The costats, K r & K r, deped oly o µ e & atomic physics ii) Covective eergy trasport & radiatio pressure uimportat:! # " d logt d log P P = & % ad = ad P T / ad ρ µm u kt = K c ρ /( ad ) Now, the costat, K c, depeds o the star's boudary coditios. For a completely ioized gas, ad = 2 5, ad = 3 / 2 3

4 iii) Ratio of (ideal) gas to radiatio pressure is costat: P = P gas β = " P = 3 % ' # a & /3 ρ µm u β kt & P = P rad ( β) = at 4 3( β) 4/3 " k % " β % ' ' # µm u & # β 4 & /3 ρ 4/3 = K β ρ 4/3 = 3 The value of the costat, K b, depeds ow o b. Cosider: rad = P T dt dp = 6πcG 3P 4aT 4 dp rad dp = 6πcG 3P κl(r) at 4 M(r) P κl(r) P rad M(r) dp rad dp = 6πcG κl(r) M(r) with Kramer's law opacity, κ T 3.5 ad pp-chai ε pp L M T 3.5 4

5 b) Defiitio If pressure is a fuctio of desity oly, P = P ρ, the eqs () & (2) ca be treated separately. Use r as idepedet variable: dm(r) = 4π r 2 ρ(r) dr dp(r) = Gm(r)ρ(r) dr r 2 " # % r 2 d dr r 2 ρ(r) dp dr ( ) = 4π Gρ(r) () (Poisso eq) Boudary coditios: (i) (ii) either or r = : dp dr = r = R : P = r = : P = P c (zero coditios) (R: value of r where P=) 5

6 c) Lae-Emde equatio Write: Defie dimesioless fuctio with K = polytropic costat = polytropic idex γ = polytropic expoet by P = P c θ + + ( +) with P c = Kρ c ad () Defie dimesioless radius with r = Emde legth ξ via r = r ξ r 2 = The () becomes the Lae-Emde equatio Boudary coditios: Cofiguratio has radius: R = r ξ with ξ first zero of θ Solutios P = Kρ + = Kρ γ θ =θ(r) ξ = : θ = ad 4π G ρ = ρ c θ P c ρ c 2 θ (ξ): Lae, Ritter, Emde, 6 Chadrasekhar, dθ dξ = ξ 2 r 2 d dθ dr r2 dr = θ ( +)K 4π G ρ c d dξ ξ 2 dθ dξ = θ

7 d) Solutios = : θ (ξ) = 6 ξ 2 ξ = 6 Homogeeous sphere =: θ (ξ) = siξ ξ = 5 : θ 5 (ξ) = " + 3 ξ 2 # ξ = π ξ /2 = % ' & For other : umerical solutio of Lae-Emde equatio θ (ξ) Properties: decreases mootoically with icreasig < 5: ξ < 5: cofiguratio exteds to ifiity Series expasio: θ (ξ) = 6 ξ ξ 4 Plummer sphere ξ > (8 5) 52 ξ 6 + ( ) ξ

8 e) Physical properties Possible to obtai much isight i the properties of polytropes without umerically solvig the Lae-Emde equatio Radius:!( R = r ξ = + )P c # 2 & " 4π Gρ c % /2 ξ = N k µ! 2 ( +)T c # & " 4π GP c % /2 ξ Mass: r ξ ξ ξ m(ξ) = 4π r 2 3 ρ(r)dr = 4πρ c r ξ 2 θ 3 d # dξ = 4πρ c r dξ ξ 2 dθ & % (dξ = 4π r 3 ρ c ξ 2 dθ dξ ' dξ!( M = 4π + )K # & " 4π G % 3/2 3 2 ρ c ( ξ 2 dθ + * - ) dξ, ξ=ξ This is fiite for 5 Mass-radius relatio: elimiate from expressios for R ad M: K = + ( 4π ) ρ c { ξ + " # θ! } % ξ=ξ GM 3 R 8

9 Lae-Emde Solutios 9

10 Ratio of mea ad cetral desity: ρ(ξ) = 3m(ξ) 4π r 3 ξ = 3 3 ξ ρ dθ c dξ ρ c = # ξ & % 3 θ! (ξ)' (ξ=ξ ρ Cetral pressure: P c = Kρ c + = Cetral temperature: use T = T c θ 4π ( +) "!θ! (ξ)# P = K 'T + = T c = µk N k ρ c ξ=ξ ad assume ideal gas Mass-radius-chemical compositio relatio: ideal gas: K ' = P c T = + ( N k ) c ( +! + θ! (ξ)* + ) # & ξ " G % 4π 2 GM 2 R 4 GM ( +) " # ξ θ! (ξ) % R ξ=ξ + ξ=ξ µ N k P = K 'T + µ + M R 3

11 Recap: For a give polytrope idex,, we have a set of models that are specified by two parameters, P c ad r c, (or equivaletly M ad R) which yield the costat K ad the ru of desity ad temperature as well as the mass ad radius (or equivaletly, the cetral desity ad pressure).

12 Costats ad ratios that appear i above expressios Compare with slide 6 Lecture 2 Results for =, ad 5 follow from aalytical solutios give earlier Other values derived from umerical solutios 2

13 f) Potetial eergy Write Φ = gravitatioal potetial. The H.E. dp ρ dr = dφ dr P = Kρ + dp ρ dr = ( +) d dr P ' & ) % ρ ( *,, +,, - Φ = ( +) P ρ + Φ I = ( +) P ρ GM R where P/ρ θ Also: E g = Ω = 2 R Φdm = P ( +) 2 R ρ dm 2 GM R R dm R = 2 GM ( +) P dv 2 2R = 2 GM ( +)Ω 6 2R so that we obtai: Ω = 3 GM 2 5 R 3