Homologous Stars: Simple Scaling Relations

Transcription

1 Homologous Stas: Simple Salig elatios Covetig the equatios of stella stutue fom diffeetial to diffeee equatios, effetively doig dimesioal aalysis ad assumig self-simila solutios, we a extat simple geeal elatios betwee lumiosity, mass, adius, ad mea moleula weight fo vaious assumptios oeig the EOS ad opaity laws. These elatios otai the essee of stella stutue. The equatio of hydostati equilibium ad ρ M/ 3 yield ad theefoe fo a ideal gas (T µp ρ ) P GM 4 T µgm. If we assume the sta is adiative, the lumiosity, L, sales as whee κ ρ T α. 3 M L κρ ( ) 3 ( GMµ M ( ) T 4 µ α+4 3 α M α+3 ) α ( ) 4 µgm / Fo Kames opaity ad a ideal gas EOS, α = 3.5, = 1, esultig i ( ) M L µ Fo Thomso opaity ad a ideal gas EOS, α =, =, ad we have The last two equatios ae lassi esults. L µ 4 M 3. If the sta is adiatio-pessue domiated, P P T 4 GM, ad if Thomso opaity 4 (ostat) pedomiates, we have L 3 GM M 5 M aothe lassi esult, this time fo vey massive stas. s 1

2 But what of (M)? Fo steadily buig stas, this is obtaied by settig the lumiosity above equal to the themoulea powe i the oe. If the speifi ulea powe, ε, goes as ε ρ u 1 T s the we a use the ρ M/ 3 ad T µgm/ elatios to obtai: µ α+4 3 α M α+3 M ( ) u 1 ( ) s M GMµ 3. If we set s = (CNO buig), u =, = 1, ad α = 3.5, we fid M.73, ot fa off. Howeve, this model fo (M) is quite ude ad full umeial alulatios ae waated. s

3 POLYTOPES Polytopes ae self-gavitatig gaseous sphees that wee, ad still ae, vey useful as ude appoximatio to moe ealisti stella models. Popeties of polytopes ae thooughly desibed i a lassial, ad vey old, textbook: A Itodutio to the Study of Stella Stutue by S. Chadasekha (1939, Dove editios: 1958, 197). We assume that a spheial sta is i a hydostati equilibium. Its stutue is desibed by two odiay, fist ode diffeetial equatio: dp d = GM ρ, (poly.1a) dm d = ρ. These may be ombied to a sigle, seod ode Poisso equatio: (poly.1b) ( ) 1 d dp = Gρ. (poly.) d ρ d We assume that thee is a polytopi elatio betwee pessue ad desity: P = Kρ 1+ 1, (poly.3) whee K ad ae eal, positive ostats, ad is alled a polytopi idex. We itodue dimesioless vaiables: ρ = ρ θ, P = P θ +1, = αξ, (poly.4) whee ρ is the etal desity, θ is polytopi tempeatue, α is a legth ostat defied as α = 1 K ( + 1)ρ G, (poly.5) ad ξ is a ew adius like vaiable. Combiig equatios (poly.-5) we obtai the Poisso equatio i dimesioless vaiables: This is kow as the Lae-Emde equatio fo polytopi stas. ( 1 d ξ ξ dθ ) = θ. (poly.) The two bouday oditios fo the eq. (poly.) ae at the ete: θ = 1, dθ =, at ξ =. (poly.7) As both oditios ae at the same poit, they ae i fat iitial oditios. Theefoe, fo evey value of a polytopi idex thee is oly oe solutio of eq. (poly.). The solutio has a physial meaig as log as θ >=. The sufae of a polytopi sta is at ξ = ξ 1, whee θ =, ad aodig to eqs. (poly.4) desity ad pessue go to zeo. Thee ae thee aalyti solutios kow: s 3

4 =, θ = 1 ξ, ξ 1 =.45, = 1, θ = si ξ ξ, ξ 1 = π 3.14, ) 1/ = 5, θ = (1 + ξ, ξ 1 =. 3 (poly.9) The ase = oespods to iompessible fluid, i.e. ρ = ρ = ost, P = P θ, ad it equies ewitig eq. (poly.) i a diffeet fom. I this ase pessue vaishes at the sufae, but desity is the same thoughout the sta. This solutio is a ude appoximatio to the iteio stutue of a plaet like Eath. The ase = 5 is also speial, as the adius of this sta is ifiite. It is possible to show that all polytopes with > 5 have ifiite adii. This meas that oly solutios with < 5 have a sufae. The two ases most iteestig fo eal stas have = 1.5 ad = 3, ad ufotuately these do ot have aalyti solutios. A solutio of eq. (poly.) depeds o oe paamete oly, the polytopi idex. If stella stutue is appoximated with a polytope with a give idex, the two salig paametes ae eeded to expess the stutue i physial uits, like.g.s. Fo example, the two paamete may be K (whih is elated to etopy) ad etal desity; o stella mass ad stella adius; o K ad the stella mass; ad so o. If we kow the umeial o aalytial solutio of the Lae-Emde equatio (poly.) the we may expess the total stella adius, ad the total stella mass M as follows: = αξ 1 = [ K G ] 1/ + 1 ρ 1 ξ 1, (poly.1a) ξ 1 M = ρd = α 3 ρ ξ θ = (poly.1b) = α 3 ρ [ K = G ξ [ d ( ξ dθ )] = ] 3/ ρ 3 The last two equatios may be ombied to obtai [ ξ dθ ]. ξ=ξ 1 whee 3 1 M = K, GN (poly.11a) N ()1/ + 1 ( [ ξ dθ ] ) 1 ξ=ξ 1 ξ 3 1, (poly.11b) is a dimesioless umbe that depeds o the polytopi idex oly. We may also alulate the aveage desity of a sta to be ρ av 3M [ 3 = ρ 3 ξ1 3 ξ dθ ], (poly.1) ξ=ξ 1 ad theefoe ρ ρ av = [ 3 ξ 3 1 ξ dθ Fially, we may alulate the etal pessue to be s 4 ]ξ=ξ1. (poly.13)

5 whee P = Kρ +1 GM = W 4, (poly.14a) W ( 3 ρ ρ av ) +1 N. (poly.14b) The equatios (poly.11a) ad (poly.14a) a be obtaied with a muh simple appoximate aalysis. We eplae the diffeetial equatios (poly.1a) ad (poly.1b) with the oespodig algebai equatios: ρ ρ av M 3, P GM ρ GM 5, P GM 4. Eqs. (poly.15) may be ombied with (poly.3) to obtai the mass-adius elatio: (poly.15) 3 1 M K G. (poly.1) Of ouse, with this appoximate aalysis we aot obtai the umeial values of N ad W. We must have the solutio of the Lae-Emde equatio i ode to have those oeffiiets. The esults of umeial itegatios ae give by Chadasekha i his textbook. We shall fequetly eed all those oeffiiets fo the two impotat ases: fo = 1.5, whih oespods to a adiabati sta suppoted by pessue of o-elativisti gas, ad fo = 3, whih oespods to a adiabati sta suppoted by pessue of ulta-elativisti gas. We have ρ /ρ av ξ 1 N W kt µh GM /8 π - 1. π /3 π Most of the symbols i the table ae self-explaatoy. The last olum gives the dimesioless umbe fo polytopes suppoted by pessue of pefet gas, with P = k µh ρt, ad allows a alulatio of the etal tempeatue T. Notie, that kt is oughly themal eegy pe patile, while µhgm/ is gavitatioal eegy pe patile, ad the dimesioless umbes i the last olum,.539 ad.854, ae equal to the atios of these two eegies fo the two polytopi models. The mass adius elatio fo a sta with = 1.5 is give as while fo = 3 we have M 1/3 = K, = 1.5, (poly.17a).44 G ( ) 1.5 K M =, = 3. (poly.17b).339 G These mea that the adius of a sta with = 1.5 is smalle if the sta is moe massive, while a sta with = 3 has its mass uiquely detemied by the value of K ostat, while its adius is ot estited by eithe K o M. Clealy, = 3 is a vey speial ase, ad it has may astophysial appliatios. Thee is a iteestig ad useful expessio fo a gavitatioal potetial eegy of a polytopi sta: Ω M GM dm = 3 GM 5. (poly.18) s 5

6 We shall deive it usig the oditio of hydostati equilibium (eq. poly.1a) ad the polytopi elatio give with eq. (poly.3), ad also i the fom: dp ρ = K + 1 ( ) ρ 1 P 1 dρ = ( + 1)d. (poly.19) ρ Gavitatioal potetial eegy is defied as eegy equied to emove all stella mass, shell afte shell, all the way to ifiity. We shall dive the eq. (poly.18) itegatig by pats ad usig eqs. (poly.1a), (poly.1b), (poly.3), ad (poly.19). As we shall be hagig itegatio vaiables may times we shall use symbols ad s to idiate ete ad the sufae, i.e. the limits of the itegals. Ω GM dm = Ω = Ω. [ GM = Gd ( ) M = ] s 1 GM d = M ρ dp = M d ( ) P = ρ [ ] s P M + 1 ρ + 1 P ρ ρd = + 1 [ P 3 d ( 3) = 3 P3 ] s GM ρd = GM dm = P ρ dm = The last lie of eq. (poly.) gives the desied aswe, i.e. Ω = 3 GM 5. 3 dp = (poly.) s

7 A simple, appoximate estimate of the total o aveage ulea buig ate i a sta a be obtaied i the otext of polytopes if oe assumes that the buig ate is ε = ε ( ρ ρ ) u 1 ( T T ) s, whee the subsipt efes to the etal values, that whee is the polytopi idex, ad that ( ) ( ) ρ T θ, ρ T Pluggig i, we deive ad θ e ξ. ε ε dm M 3.3 (3u + s) 3/ = 3. ε ε dm M.4 (1.5u + s) 3/ = 1.5. Note that fo lage values of s this aveage ulea buig ate is oespodigly smalle tha the etal value. This idiates that the degee of etal oetatio of the buig is lage whe the depedee of ε o tempeatue is steep. Oe a show that ude suh iumstaes, oe buig is moe likely to be ovetive. Suh is the ase fo the mai-sequee buig of stas a bit moe massive tha the Su, fo whih hydoge bus by the CNO yle ad ot the pp hai. s 7