Itroductio to Astrophysics Tutorial : Polytropic Models Iair Arcavi 1 Summary of the Equatios of Stellar Structure We have arrived at a set of dieretial equatios which ca be used to describe the structure of a star i equilibrium. The equatios ca be writte either as fuctio of the radius coordiate r or the mass coordiate m: Hydrostatic Equilibrium: dr Gm = ρ r dm = Gm 4πr 4 Cotiuity: dm dr = 4πr ρ dr dm = 1 4πr ρ Radiative Trasfer: (if eergy is trasferred oly by radiative diusio) dt dr = 3 4ac κρ T 3 F 4πr dt dm = 3 4ac κ T 3 F (4πr ) Thermal Equilibrium: df dr = 4πr df ρq dm = q These structure equatios are supplemeted by: P = P I + P e + P rad = R µ I ρt + P e + 1 3 at 4 κ = κ 0 ρ a T b q = q 0 ρ m T Solvig these equatios is very dicult, sice they are o-liear, coupled ad have two-poit boudary coditios. However, we ca gai isight ito the structure of stars by aalyzig the equatios without solvig them, ad by usig simple models. Oe such model is called the Polytropic Model. 1
The Polytropic Model Note that the rst pair of equatios is coected to the secod pair by the equatio of state. However, if the pressure is idepedet of temperature, the the rst pair of equatios is separated from the secod pair. Multiplyig the equatio of hydrostatic equilibrium by r /ρ ad dieretiatig with respect to r, we d: ( ) d r = G dm dr ρ dr dr Usig the cotiuity equatio o the right had side gives: ( ) 1 d r r = 4πGρ dr ρ dr Now eters the model. We cosider equatios of state of the form: P = Kρ γ where K ad γ are costats. This is kow as a polytropic equatio of state, ad we have see that it pops up with degeerate gases ad adiabatic processes. We dee the polytropic idex as: γ = 1 + 1 so a degeerate gas has idex = 1.5 for the o-relativistic case ad = 3 for the ultra relativistic case. Usig this model i the structure equatio we costructed gives: ( ) ( + 1) K 1 d r dρ 4πG r = ρ dr dr With the boudary coditios: ρ 1 ρ (R) = 0 dρ dr = 0 r=0 (where the secod coditio comes from the equatio of hydrostatic equilibrium). The solutio ρ (r) is called a polytrope, ad is uiquely deed by K, ad R. It is coveiet to dee dimesioless variables θ ad ξ by: ρ = ρ c θ r = αξ where ρ c is the cetral desity ad α = ( + 1) K 4πGρ 1 c
Our equatio the becomes the Lae-Emde equatio: with boudary coditios: ( 1 d ξ ξ ) = θ θ ξ=0 = 1 = 0 ξ=0 For < 5, θ (ξ) decreases mootoically, ad the radius of the star ca be foud by lookig for the rst zero of θ (usually umerically), called ξ 1 (the R = αξ 1 ), sice at the surface ρ = 0 so θ = 0. There are oly three cases where a aalytical solutio to the Lae-Emde equatio exists: 1. = 0, θ 0 = 1 ξ 6, ξ 1 = 6, γ =. = 1, θ 1 = si ξ ξ, ξ 1 = π, γ = 1 3. = 5, θ 5 =, ξ 1 =, γ = 6 1+ξ /3 5 I case 1, we have P = P c θ, ad this is applicable for a icompressible uid (sice = 0 meas costat desity). Case 3 is ot physical because the desity ever reaches zero. The total mass of a star ca be foud by: M = ˆR 0 4πr ρdr = 4πα 3 ρ c Substitutig from the Lae-Emde equatios we have: which gives: The mea desity ρ = M = 4πα 3 ρ c ˆξ 1 0 ˆξ 1 0 d M = 4πα 3 ρ c ξ 1 ξ θ ( ξ ) M 4 3 πr3 ca be show to be related to the cetral desity by: ρ c = D ρ ξ1 3
with: [ 3 D = ξ 1 ξ1 ] 1 This eables us to d a relatio betwee the mass ad the radius: ( ) 1 ( ) 3 GM R = M R [( + 1) K] 4πG with: M = ξ 1 R = ξ 1 ξ1 It is iterestig to see what happes i the special cases = 1 ad = 3. For = 3, the mass is idepedet of the radius: M = 4πM 3 ( K πg while for = 1 we ca d a radius idepedet of mass: ) 3/ ( ) 1/ K R = R 1 πg Geerally, for betwee these values, we ote that: R 3 1 M 1 The more massive the star, the smaller (ad hece deser) it gets! Fially, we would like to d a expressio for the cetral pressure, which ca be doe by substitutig the cetral desity ito the equatio of state P = Kρ 1+ 1 : which ca be writte as: P c = (4πG) 1 + 1 ( GM M ) 1 ( R R P c = (4π) 1/3 B GM /3 ρ 4/3 c ) 3 ρ +1 c where B icludes all the -depedet coeciets. It turs out that B varies quite slowly with, implyig that the above expressio is almost uiversal, ad therefore will be useful later. 3 The Chadrasekhar Mass Stars which are domiated by degeeracy pressure, ca be described by a polytropic equatio of state with idex = 1.5 ad K = K 1 (from the previous tutorial). White dwarfs are a example of such 4
stars (havig roughly the mass of the su but the radius of the earth, givig them a mea desity of ρ 10 5 g/cm 3 ). For such stars the mass-radius relatio is: R M 1/3 ad so the average desity icreases with the mass as: ρ MR 3 M For icreasig mass, the, the star becomes deser, ad so ultimately its electros become relativistic. I this case, = 3, for which the mass-radius relatio gives a mass that is idepedet of the radius: M = 4πM 3 ( K πg ) 3/ Pluggig i K = K from the relativistic degeerate equatio of state, gives the Chadrasekhar Mass: M Ch = M ( 3 1.5 4π Substitutig all the costats, this ca be writte as: hc Gm 4/3 H M Ch = 5.83 µ M e ) 3/ µ e This is the maximal mass a stable coguratio ca have. For a white dwarf composed mostly of He ad heavier elemets, for which we ca take µ e =, we d M Ch = 1.46M. 5
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