Stellar Modeling. 1 The Equations of Stellar Structure. Stellar Astrophysics: Stellar Modeling 1. Update date: October 21, 2010

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1 Stellar Astrophyscs: Stellar Modelng Stellar Modelng Update date: October, 00 The Equatons of Stellar Structure We consder the modelng of stars n hydrostatc and thermal equlbrum (.e., tme-dependent processes are gnored for now). From the modelng, we hope to know how such global propertes as the lumnosty and radus of a star depend on ts mass and ntal chemcal composton. To do so, we need to solve a set of four dfferental equatons as we dscussed n Chapter, whch are equvalent to a fourth-order dfferental equaton: The mass conservaton hydrostatc equaton energy equaton dr = dm r 4πr ρ, () dp dm r = GM r 4πr 4, () dl r dm r = ɛ, (3) and the heat transfer method (radatve, conducton, and/or convecton), or the form of the model dlnt dlnp. (4) To mplement the specfc heat transfer, compute rad = 3 P κ 6πac T 4 L r GM r, (5) whch assumes that the transfer s all due to the radaton (the conducton s neglected here). Then we may set = rad f rad ad (6) for pure dffusve radatve transfer or conducton, or = ad f rad > ad (7) when adabatc convecton s present locally as n a mxng length theory. Four boundary condtons are requred to close the system. For smplcty, we choose zero condtons whch are r = L r = 0 at the center (M r = 0), and ρ = T = 0 at the surface (M r = M). Here M was specfed beforehand.

2 Stellar Astrophyscs: Stellar Modelng Of course, we assume that we already know the mcroscopc consttuent physcs, as we have dscussed;.e., the quanttes P, E, κ, and ɛ as functons of ρ, T, and X, where X s shorthand for composton. These quanttes should be avalable on demand ether n analytc and numercal forms. It should be noted that the avalablty of the quanttes does not guarantee that a soluton to the equatons always exsts or unque! Before gong nto how the above stellar structure equatons are solved n practce, we frst consder a smplfed case, whch s both useful practcally and llumnng. Polytropc Equatons of State and Polytropes We defne a polytropc stellar model (or polytrope) to be one n whch the pressure s gven by P (r) = Kρ +/n (r) (8) where both the polytropc ndex n and K are constant. Ths model allows us to avod dealng wth both the heat transfer and thermal balance. We have encountered such power law before; e.g., the EoS for a zero temperature, completely degenerate electron gas (e.g., P e = ( ρ µ e ) 5/3 dyn cm f non-relatvstc). The polytrope s also a good approxmaton for certan ( types of ) adabatc convecton zones. For lnt a regon wth effcent convecton,.e., = ad = = /Γ. If Γ s assumed lnp ad constant, then P (r) T Γ /(Γ ) (r). (9) If n addton, the gas s deal, then P (r) ρ Γ (r). For a polytrope, we can derve from the hydrostatc and mass conservaton equatons (n Eucldean coordnates) the followng ( ) d r dp = G dm r = 4πGρ (0) r dr ρ dr r dr Now perform the transformatons to make the the equaton dmensonless: ρ(r) = ρ c θ n (r) and r = r n ξ, we then have the Lane-Emden equaton: ( d ξ dθ ) n = θ ξ n n () dξ dξ where P c s defned (from the EoS) as P c = Kρ +/n c and rn = (n + )P c. The solutons are 4πGρ c called Lane-Emden solutons and denoted by θ n (ξ).

3 Stellar Astrophyscs: Stellar Modelng 3 Note that f the EoS for the model materal s an deal gas wth constant µ, θ n measures temperature T (t) = T c θ n (r), where T c = Kρ /n c (N A k/µ). Snce the Lane-Emden equaton s a second-order dfferental equaton, we need two real boundary condtons: Frst for ρ c to really be the central densty, we requre that θ n (ξ = 0) = ; Second, the sphercal symmetry at the center (dp/dr vanshng at r = 0) requres that θ n(ξ = 0) = 0, wth the resultng regular solutons called E-solutons (the abandoned dvergent solutons at center may be used n part of a star, however). The surface of a model star s where the frst zero of θ n occurs, θ n (ξ ) = 0, where ξ s the locaton of the surface. Ths nterpretaton of the soluton s not a boundary condton. Analytcal E-solutons for θ n are obtanable for n = 0,, and 5, as gven n the textbook. For example, for n = 0, ρ(r) = ρ c. Ths constant-densty sphere has the soluton (when the boundary condtons at ξ = 0 are used): Clearly, ξ = 6 /. θ 0 (ξ) = ξ 6. () Numercal methods must be used for general n. So gven n and K, we can n prncple fnd the dependence of P and ρ on ξ. However, to get the absolute physcal numbers, we need R = r n ξ, whch depends on ρ c, as shown above. These two parameters are lnked by the stellar mass, whch we wsh to specfy va M = R 0 4πr ρ(r)dr ξ = 4πrnρ 3 c ξ θndξ n 0 (3) = 4πr 3 nρ c ( ξ θ n) ξ For a gven n, ( ξ θ n) ξ s known, the above equaton then gves ρ c, and hence P c, n terms of M. A useful quantty that depends only on n s the rato ρ c < ρ > = ( ) ξ, (4) 3 θ n ξ where < ρ > s the volume-averaged mean densty of a star. For the pressure of a completely degenerate, but non-relatvstc electron gas ( ρ 5/3 ), n =.5. For the completely degenerate and fully relatvstc case, n = 3. For an deal gas convecton zone ( ρ 5/3 ), n =.5. Unfortunately, nether of these values have analytc E-functons.

4 Stellar Astrophyscs: Stellar Modelng 4 3 The Eddngton Standard Model Ths gves a smple example of the use of polytropes n makng a stellar pseudo-model, whch approxmately ncorporates the energy and radatve transfer equatons. Recall that n case of no convecton the radatve transfer equaton can be expressed as where = 3 P κ 6πac T 4 L r GM r, (5) dlnt dlnp = P dp rad 4 P rad dp. (6) Here we have ntroduced the radatve pressure P rad = at 4 /3 so that T can be replaced to get dp rad dp = κl r. (7) 4πcGM r We defne < ε(r) > L r r = εdm 0 r r M r dm (8) 0 r where the energy equaton dl r /dm r = ε s used. We further defne The transfer equaton then becomes < η(r) > < ε(r) > < ε(r) > = L r/m r L/M. (9) dp rad dp = L κ(r)η(r). (0) 4πcGM Assumng that the surface pressure s equal to zero, the ntegraton of the above equaton gves L P rad (r) = < κ(r)η(r) > P (r) () 4πcGM where the average expresson s < κ(r)η(r) >= P (r) P (r) 0 κηdp. () If we can assume that < κ(r)η(r) > vares weakly wth poston n a star, or close to a constant, as Eddngton dd, then the rato of β P rad /P s a constant and so s β. Ths constancy may be translated nto a T vs. ρ relaton as follows. If the pressure s contrbuted by deal gas plus radaton only, then P rad = P P gas = (/β )P gas = β β N A k µ ρt = at 4 /3 (3)

5 Stellar Astrophyscs: Stellar Modelng 5 ( β T (r) = β 3 a ) /3 N A k ρ /3 (r). (4) µ where P = P gas β K = = N Ak µ [ β β 4 3 a ρt β = Kρ4/3 (r), (5) ( ) ] 4 /3 NA k. (6) µ So we have a polytrope wth n = 3, whch can be readly solved numercally. 4 Numercal calculaton of the Lane-Emden equaton A convenent way to numercally solve such an equaton s to cast the second-order problem n the form of two frst-order equatons by ntroducng the new varables x = ξ, y = θ n, and z = (dθ n /dξ) = (dy/dx): y = dy dx = z z = dz dx = yn x z (7) Here we use a smple shootng method, whereby one shoots from a startng pont and hopes that the shot wll end up at the rght place; e.g., usng a Runge-Kutta ntegrator. The soluton s leap-frogged from x to x + h, where h s called the step sze. Suppose we know the values of y and z at some pont x and call these values y and z. If h s some carefully chosen, then we can use the above equatons to fnd y + and z + at x + = x + h. Care needs to be taken at the orgn, where z s ndetermnate because both x and z are equal to zero. The resoluton to ths problem s to expand θ n (ξ) n the Lane-Emden equaton n a seres about the orgn. Insertng θ n (ξ) = a 0 + a ξ + a ξ... nto the equaton, compare the coeffcents of ndvdual ξ terms, and apply the boundary condton to establsh the constants n the expanson, we get θ n (ξ) = 6 ξ + n 0 ξ4... (8) For ξ 0, fnd that z /3, whch may be used to start the ntegraton. Ths way, the calculaton may march from the orgn to the surface, when y = θ n cross the zero.

6 Stellar Astrophyscs: Stellar Modelng 6 5 Newton-Raphson and Henyey Methods A more powerful technque to solve the stellar structure equatons s the ntegraton over the model, nstead of shootng from one pont to another. Consder a second-order system as an example dy = f(x, y, z) dx (9) dz dx = g(x, y, z) wth boundary condtons on y and z specfed at the endponts of the nterval x x x N and may be generally expressed as b (x, y, z ) = 0 b N (x N, y N, z N ) = 0 (30) where y and z are y(x ) and z(x ). Assumng that f, g, b, and b are well behaved, the dfferental equatons can be cast n a fnte dfference form over a predetermned mesh n x;.e., x, x,..., x N at whch y and z are to be evaluated. For smplcty, consder that the mesh nterval s constant;.e., x + x = x for all. The equatons can hen be expressed as y + y = x (f + + f ) z + z = x (g + + g ) where f, for example, s shorthand for the functon f(x, y, z ). The above expressons then represent N equatons, whch together wth the two boundary condtons can n prncple be used to solve N varables y and z. Thus, ths s not an ntal value problem. However, the dffculty s that the varables are all mxed up among the equatons, generally n a nonlnear fashon. (3) 5. Newton-Raphson Method One way to get out of ths dffculty s to use the Newton-Raphson method: fnd the soluton by lnearzng the equatons and the boundary condtons. Suppose that we have a guessed soluton (e.g., from the shootng method) that gves y and z for all, whch generally do not satsfy the equatons. we may make correctons y y + y z z + z (3)

7 Stellar Astrophyscs: Stellar Modelng 7 so that the new y and z mght satsfy both the equatons and the boundary condtons. We now estmate the values of y and z for all by lettng the new y and z satsfy the lnearzed equatons and boundary condtons. For example, the frst equaton of Eq. 3 becomes y + + y + y y = x Some manpulaton leads to [ f + + ( ) f y x ( ) f [ f + ] z + + ) ( f y + ( ) ] f z (33) y + y x (f + + f ) = [ ( ) x f ] [ ( x f + y + [ ( ) x f + ) ] ] y + + [ ( ) ] x f z + z + (34) + Note that the left-hand sde of these equatons are zero when the dfference equatons are satsfed; that s when y and z go to zero. Smlarly, the boundary condtons can be lnearzed nto b ( or N) + ( ) b y ( or N) + ( or N) We can arrange all these equatons n a matrx form n whch, where the superscrpt T ndcates transpose; ( ) b z ( or N) = 0. (35) ( or N) M U = R. (36) U ( y, z, y, z,, y N, z N ) T (37) R = ( b, Y 3/, Z 3/,, Y N /, Z N /, b N ) T (38) where Y +/ y + y x (f + + f ) Z +/ z + z x (g + + g ); (39)

8 Stellar Astrophyscs: Stellar Modelng 8 and fnally where ( ) ( ) b b A + B A B 0 0 C D + C D A + B A 3 B 3 ( ) ( ) b b A x B x ( ) f ( ) f, C x, D x ( ) g ( ) g. N N (40) (4) Once the soluton set U s found, then new values of y and z are obtaned by addng y and z to the correspondng old guesses. If all goes well, then the correctons decrease as the square of ther absolute values. We terate the above procedure untl y and z become suffcently small. 5. Egenvalue Problems and the Henyey Method However the above scheme needs to be slghtly modfed f the locaton of the boundary s not known before-hand. In the polytropc stellar structure we want to get, the radus ξ needs to be found as part of the soluton. To do so, we do a smple converson, x x = ξ/λ (4) where λ = ξ. After ths converson, x s wthn the closed nterval [0, ] and can be dvded nto a grd, upon whch the dfference equatons and boundary condtons can be appled. The converted f and g are generally also depend on λ; e.g., the Lane-Emden equaton becomes y = dy dx = λz z = dz dx = λyn x z. (43) Now we have one more parameter λ, or an egenvalue, to determne. Ths can be done snce we have one more boundary condton, y = 0 at the new x =, n addton to the two at the center (y =, z = 0).

9 Stellar Astrophyscs: Stellar Modelng 9 To get the soluton, we just need to add the correcton λ λ + λ n the lnearzaton of f, g, and the three boundary condtons; e.g., ( ) ( ) ( ) f f f f f + y + z + λ. (44) (z,λ) (y,λ) λ (y,z ) Each dfference equaton thus contrbutes terms n λ to the matrx algebra problem;.e., there s now an extra row n the man matrx correspondng to the addtonal boundary condton and an extra column for the extra unknown, λ. 6 The Structure of the Envelope To construct a more realstc stellar model, we need to deal wth the surface layer more carefully. Startng from the surface nward, we may model the atmosphere as descrbed n the heat transfer chapter, where we have assumed that convecton plays no role n hear transport between the true and photosphere surface, consstent wth our noton of a radatng, statc, and vsble surface (although ths s not true for the sun). In such a radatve heat transfer regon, = rad wth n whch = dlnt dlnp = 3κL 6πacGM P T = 4 (K ) n+ ( + n eff ) P n+ T (n+s+4), (45) ( ) /(n+) K 6πacGM =, (46) + n eff 3κ g L where n eff = (s + 3)/(n + ) s the effectve polytropc ndex and the opacty has been wrtten n the nterpolaton form for deal gas, κ = κ g P n T n s. At the photosphere, we have p = (K ) n+ ( + n eff ) P p n+ Tp (n+s+4). (47) It s easy to show p = /8, usng P p = g s /3κ p, g s = GM/R, and L = 4πR σteff 4 at the photosphere, as well as σ = ac/4. Next we consder the stellar envelope, whch conssts of the porton of a star that starts at the photosphere, and contnues nward, but contans neglgble mass, has no thermonuclear or gravtatonal energy sources, and s n hydrostatc equlbrum. We want to see how deep the radatve envelope may be untl the convecton takes over for heat transfer. In the radatve heat transfer regon, Eq. 45 can be re-wrtten as If n + s + 4 s non-zero, ths can be ntegrated to gve P n dp = ( + n eff )(K ) n+ T n+s+3 dt. (48) P n+ P n+ p = (K ) n+ (T n+s+4 T n+s+4 p ). (49)

10 Stellar Astrophyscs: Stellar Modelng 0 Replacng P and P p wth and p from Eqs. 45 and 47, respectvely, the above equaton can be expressed as = + n eff + ( Tp T ) n+s+4 ( p + n eff In case that n+ and n+s+4 are both postve and as we go to deep depths n the envelope, then we have P K T +n eff (5) and ) (50) + n eff. (5) Thus as far as the nteror structure s concerned, we could just as well have used zero boundary condtons for the densty and temperature, as was done above. Examples for ths case nclude Kramer s opacty (n = and s = 3.5) and electron scatterng (n = s = 0). ( ) For deal gas wth constant composton, P = Kρ +/n eff, where K = N A k +/n(k ) /n. µ We can then have a polytropc-lke soluton n the envelope, though t does not need to be of the complete E-soluton varety. Wth approprate condtons of contnuty, one can then connect the envelope soluton to the nteror one. Specfcally, f Kramers opacty holds n the envelope, then n eff = 3.5 and = < ad = /Γ = 0.4, whch mples no convecton as we have assumed. The same s true for the electron scatterng opacty. 6. Radatve envelope structure Assumng that the envelope s radatve wth a constant n eff, as n Eq. 5, we may determne the temperature dstrbuton as a functon of the radus. Rewrte the equaton of hydrostatc equlbrum n the form dp dr = P dt T dr = GM ρ. (53) r Usng P = ρn A kt/µ to replace the pressure, we have An ntegraton of ths gves ( GMµ T (r) = + n eff N A k r ) R =.3 ( ) ( 07 K M R µ + n eff ( + n eff ) dt dr = GMµ N A kr. (54) M R ) ( ) (55) x

11 Stellar Astrophyscs: Stellar Modelng where x = r/r. It s clear that for a reasonable value of n eff (e.g., = 3.5 for Kramers opacty) and µ ( 0.6 for a Pop I star), the last term of the r.h.s of the above equaton has to be very small;.e., the temperature drops to a photosphere temperature over a radus δr % of R. We can also get a densty dstrbuton from the above temperature dstrbuton, usng P = K T +n eff for deal gas. It can be shown that for a solar mass and lumnosty, for example, traversng 5% of the total radus nward from the surface uses up only a lttle less than % of the mass, confrmng our assumpton that M r M through the envelope. 6. Convectve envelopes and stars An mportant counterexample to the above case s where the envelope opacty s due to H (n = /, and s = 9) and photosphere boundary condtons have a strong nfluence on the underlyng layers. It s also true that n cool stars, where H opacty s mportant, the underlyng layers are convectve and the above analyss does not apply. We frst check where the convecton may take place n a such envelope. For H opacty (hence n eff = 4), Eq. 50 can be expressed as (r) = 3 + ( ) 9/ T (56) 4 T eff where p = /8 s nserted. Note that snce temperature ncreases wth depth, so does. Eventually, when > ad, the stellar materal becomes convectve. For smplcty, we assume that the convecton s adabatc. Thus at depths deeper than the crtcal depth, = ad = 0.4 for deal gas (Γ = 5/3). The transton to convecton occurs at T f = (8/5) /9 T eff =.T eff. The correspondng pressure P f = /3 P p can be found from ( ) [ n+ ( ) n+s+4 P T = + ], (57) P p + n eff p T p whch can be easly obtaned from Eq. 49. For the assumed convecton, the mplyng polytrope of ndex s 3/ and P = K n=3/t 5/. (58) For a completely convectve star, K n=3/ can be related to the the mass and radus of the star as defned by the E-soluton polytrope K n=3/ µ 5/ M / R 3/ (59) Usng P f = K n=3/ T 5/ f and L = 4πσR Teff 4 to replace R, we get

12 Stellar Astrophyscs: Stellar Modelng T eff µ 3/5 (M/M ) 7/5 (L/L ) /0 (60) Therefore, such a completely convectve star has a nearly constant effectve temperature, ndependent of the lumnosty. In realty, onzaton processes and convecton, albet almost neglgble, occur n the outer layers of nearly all stars and a complete and accurate ntegraton ncludng all effects s necessary n modelng real stars. 7 Revew Key concepts: Polytropes, Lane-Emden equaton, the Eddngton Standard Model, Newton- Raphson and Henyey Methods What are the basc equatons and the boundary condtons that are needed to construct a normal stellar nteror model? What mcroscopc physcs should be mplemented n such a modelng? Why s the polytropc stellar model only a second-order dfferental equaton? Please gve two examples of the stuatons n whch a polytropc equaton of state may be used? What s the key assumpton made n the Eddngton Standard Model? Why may ths assumpton be reasonable? What s the basc approach of the Newton-Raphson or Henyey Method n solvng the stellar equatons? How s t dfferent from a smply shootng method?