point, requiring all 4 curves to be continuous Discontinuity in P

Transcription

1 . Solutio Methods a Classical approach Basic equatios of stellar structure + boudary coditios i Classical (historical method of solutio: Outward itegratio from ceter, where l = m = ward itegratio from surface, where P= Pphot, T = Tphot Fittig two resultig sets of m, l, P, T curves at itermediate poit, requirig all curves to be cotiuous Discotiuity i P i T i m i l ifiite force ifiite radiative flux ifiite desity ifiite eergy geeratio Discotiuities possible, ad allowed i other quatities Example: covective core ad radiative evelope Nuclear reactios: compositio of core chages with time Covectio: uiform compositio throughout core o chage i evelope P = P At r core U : V T = TW = discotiuity i discotiuity i κ Similarly: F F K = rad rad κ κ K discotiuity i Presece of discotiuities crucial for stellar evolutio This example applies to MS stars with M>.M H H rad (see d

2 b Schwarzschild variables Defie the followig dimesioless variables: π p GM P t Nk GM T q m l =, =, =, f =, x = M L Write: κ = κ s T, ε = ε λ T The the equatios of stellar structure are ( ( ( dq = x p dx t dp qp = dx tx df λ+ λ = Dx p t dx mass cotiuity hydrostatic equilibrium thermal equilibrium ( a ( b dt C fp + = + s+ dx xt dt Γ dp = t dx Γ p dx radiative eergy trasport r adiabatic covective eergy trasport Where: Nk C = C s = F ac H κ s G K (, + + ( π F G M D= D = H G ( λ, ε λ + ( π Nk L KJ L M These cotai depedece o: - physical costats - two scale factors which are fuctios of L,, M ad s+ + λ+ λ s s + Boudary coditios: x = : q = f = x = : p= p, t = t phot phot Most applicatios: use (covective zero surface coditios The differetial equatios (-( ca be itegrated by stadard umerical techiques (e.g., uge-kutta schemes ad require the fittig of core ad evelope solutios

3 c variats ad the (U,V-plae Fittig of core- ad evelope itegratios is coveietly doe usig the dimesioless (homology ivariat variables: U d m l π r local desity = = = dl r m mea iterior desity dl P Gm gravitatioal eergy V = = = dl r r P iteral eergy W d l l πr ε local eergy geeratio rate = = = dl r l mea eergy productio iside r dl P e + = = = effective polytropic idex d l T Limitig values at ceter: U =, V =, W = at surface: U =, V, W = ward ad outward itegratios must be matched at some itermediate poit, but required smoothess of fit depeds o whether is cotiuous or discotiuous (cf a cotiuous Solutio must be: - cotiuous i (U,V,W space - with cotiuous slope discotiuous At fittig poit: F HG F HG F HG U V U KJ F = H G i K J V KJ F = H G i K J out out W W εkj F = H G ε K J i out

4 Fittig ofte doe outside regio of eergy geeratio: W= suffices to cosider the (U,V-plae f core or evelope is polytropic: fit to a sigle curve elatio to Schwarzschild variables U px q V W D xp λ+ t λ + s+ qt Γ =, =, =, ( e + rad =, ( e + ad = + qt tx f C fp Γ (U,V,W, e + are defied as logarithmic derivatives, ivariat uder certai scale trasformatios (, e These are ofte applied to Schwarzschild variables, separately i core ad evelope fittig coveiet i U,V (+W space d Cowlig model We will see i 5 that mai sequece stars with M>.M geerate their eergy via the CNO-cycle: ε steep fuctio of T eergy geerated i cetral core Expect: covective core & radiative evelope (differet possible ε(r= for r>r c = radius of core This is the Cowlig model

5 Mechaical structure For r rc solve eqs (, (, (b: polytrope of idex =/ rc r solve eqs (, (, (a: umerical solutio This is coveietly doe usig Schwarzschild variables: - tegrate iwards for chose C - Whe e drops to.5: core is reached: this defies r c - Fit, i (U,V plae to curve of =/ polytrope: possible for oly C-value This shows that o kowledge of ε is eeded! s But eed to kow κ = κ T for evelope itegratios differet model obtaied for each choice of ad s Thermal equilibrium z λ+ λ Eergy geerated = eergy radiated eq(: = D x p t dx For each choice of λ ad this gives the value of D required by thermal equilibrium Expressios for C ad D the give M=M(, or L=L(M, KW e Scalig of stellar models (homology Mechaical structure Two stars i HE, with masses M ad M, radii ad r r Poits r ad r homologous whe: = = x m m Two stars homologous whe at all homologous poits M = M = q m( x M r( q =, =, ( x ( x M F M = Px ( Px ( mx ( M rq ( M H G = F F KJ H M K H G KJ For P=P( homology requires P~ γ i.e., the star is a polytrope Temperature ( x M deal gas Tx ( = Tx ( ( x M T-profiles have same form oly if / is idepedet of x Homologous stars with P = Pgas + Prad have differet temperature profiles (e.g. stadard model of g 5

6 adiated eergy l rad κ ( x T ( x dl[ T( x / T( x] M Equatio (a: l( x = l( x S U + ( x T ( x T V F κ dl T( x W MH K geeral l( x ad l( x will differ κ ( x F M deal gas ad idepedet of x: l( x = l( x ( x H G / κ M Geerated eergy l ge ( x ( x Equatio (: lge x l ge x M ε ε ε ( = ( l ge( x M ε( x T lge( x ε( x dx ε( x v deal gas + ε = ε λ T ad /, ε / ε idepedet of x l ge M ( x lge( x ε λ+ + λ+ S x z KJ F H K depeds strogly o, M ad CNO: λ =, = 7 d ( x dx U V W Model i TE: l ( x l ( x (permaet homology HKT.6 α β γ equires all costitutive equatios of form T s κ = κ ε ε λ χ χ T, = T, P= P T T ad: κ / κ, ε / ε, / idepedet of x α α α α The: M, T M, M, L M With: α = α α L = + λα + αt (, For radiative eergy trasport: ideal gas: ( χ + ( χ + λ + T s α =, α T = Drad Drad ( ( ( Drad = χ s χ T λ + + For covective eergy trasport T L ( Γ α =, αt =, Dcov = ( χ + χt ( Γ D D cov ge rad cov χ = χ T = Drad = s λ D = Coclusio: homology relatios useful for approximate scalig; geerally eed to compute each model separately cov Γ 6

7 f Heyey method Classical solutio methods geerate oe model at the time For costructio of series of eighborig models (evolutio! it is more coveiet to use relaxatio method (Heyey dea Start with approximate solutio (polytrope, or previous model Use equatios of stellar structure to calculate correctios terate util covergece Algorithm (see KW.; HKT 7.. Write: y = r, y = P, y = T, y = l dyi The eqs (-(: = dm f y y y y i i(,,, =,,, Divide model i K- mass shells, ad discretize + yi yi + / + / = Ai = f y + y i(,..., m m i shell boudaries shell ceter i ( Values i ceter of cell ca be foud i various ways, e.g., + / + / + + fi ( y,..., y = fi([ y + y ]/,...,[ y + y ]/ Four special equatios at ceter, sice r = m = l = Series expasios of b give: r P m π A GF π P + 8 H G π / [ m ] = A KJ / κ ( ε ε εg [ T ] [ T ] + [ ] [ m ] ac π / π G ad T T +F H m A 6 K + / / l l [ ] [ ] = P l ( ε ε + ε m = A / F H G K J = g / + F / / H K = A adiative Covective K K Oly two equatios at surface, as P, T fixed by boudary coditios ( c total of K- equatios 7

8 Solve ( by meas of a Newto-aphso techique Assume you have trial solutio y i, (e.g. previous model Sice this is approximate, the A i do ot vaish Hece we must calculate correctios δ y i, Liearize equatios ( by Taylor expasio to first order This gives K- liear relatios betwee A i ad δ y i, Write this i matrix form Fδ y, F A with H the Heyey matrix.. as or: H.. so = K K δ y H U = A U = H A A G KJ G KJ δ y A H is of block-diagoal form ca be iverted efficietly so that correctios follow ad ew solutio: yi+ = yi+ δ y The iterate util covergece; this gives full solutio H K H K,, i, Heyey method: Fast ad reliable (Appedix Choose K sufficietly large Take ito accout physical trasitios iside star Be careful with evaluatio of average shell values g Chages i chemical compositio Trasmutatio of elemets i core chagig chemical compositio model evolves o τ uc evolutioary track i the Hertzsprug-ussell Diagram Heyey method ideally suited for calculatio of sequece of models as fuctio of time, as chages are small 8

9 Start with compositio X i (i=,,, i mass elemet m, at time t The: Sice X i =, these are - idepedet differetial equatios Eqs (-(5 are the equatios of stellar evolutio Nuclear trasmutatio See 9: Also: F HG K J = X i ( 5 Xi( m, t+ Δt = Xi( m, t + Δt ( i,,..., t i= L N M Xi = mi r t i L NM X = m ε i i i t Q i k k ε ik Q ik r O QP ik O Q P with m i = mass of ucleus i r i = reactio rate for i r i = idem for i with ε i = eergy geerated per uit mass by trasmutatio i Q i = eergy geerated whe uit mass i The Q i are kow from uclear physics, ad the ε i ca be calculated at each poit i star (as fuctio of ad T; see 9 mt, [ ε = ε ] k, k Covectio Material i covective zoe is mixed: τ mix <<< τ KH << τ uc Evolutioary calculatios: Δt. τ or Δt. τ (o TE istataeous mixig of ew material over etire zoe uc KH esult: z m F HG K J + m X X X X uclear ST c i ih c i ih ( ( t m UVW Xi = Xi m dm t m m t m m t Secod term caused by possible movig of zoe boudaries See KW 8.. for additioal mixig processes 9

10 Appedix: Explicit ad implicit schemes Cosider the differetial equatio dx dt Forward differecig: (explicit Back differecig: (implicit = a bx X = a + b e bt X X + + Explicit scheme: Ustable uless bδt< X Δt X Δt = a bx X = aδt+ X ( bδt + aδt+ X = a bx+ X+ = + bδt mplicit scheme: Provides correct limit a/b whe Δt large ad/or X small Geerally more accurate