1. Hydostatic Equilibium and Stella Stuctue 1.1. The Isothemal Sphee and Stella Coes We peviously discussed tuncated isothemal sphees and thei stability as applied to clouds in the intestella medium ( Bonno-Ebet sphees ). A simila question of stability aises in the case of stella evolution. When hydogen-buning has exhausted the supply of hydogen in the coe of a sta, an inet helium coe emains. This coe has no enegy souce, and hence will become isothemal the tempeatue will be constant at the value set by the inne edge of the hydogen-buning shell suounding this coe. This coe will have the stuctue of a tuncated isothemal sphee. Students sometimes suppose that isothemal egions in stas will have constant density, but this is not the case. The density must incease towad the cente to satisfy the equation of hydostatic equilibium. While the sta is buning hydogen in its coe, the tempeatue is highest at the cente. Fo an ideal gas, this tempeatue gadient helps suppot the sta, while fo an isothemal coe, the density gadient alone must suppot the stella mass, and the density gadient steepens. As hydogen buning in the suounding shell adds mass to the inet helium coe, the isothemal sphee extends futhe fom the cente and the cente/edge density atio inceases. Just as in the case of the Bonno-Ebet sphees, if we conside a fixed coe mass M c, and decease the coe adius R c, we find that the pessue at R c eaches a maximum P max and then deceases beyond that point the coe is unstable. Thus a solution is possible only if the pessue of the envelope is below that maximum pessue. We futhe find that P max deceases with inceasing coe mass as Mc 2. This means that as an isothemal coe gows, it will each a point of instability the coe will apidly contact and heat, and this may lead to the onset of helium buning. This limit to the atio of the coe mass to the total stella mass is known as the Schönbeg-Chandasekha limit (sometimes Chandasekha-Schönbeg limit). It is appoximately ( ) 2 M c M.37 µenv whee µ env is the mean molecula weight of the envelope (.66), and µ coe that of the coe ( 1.33), so that M c /M.9. Note: This in not the same as the Chandasekha mass, the mass limit fo a white dwaf. That is fo mateial which is degeneate, while the stella coe discussed hee is of lowe density and behaves as an ideal monatomic gas. µ coe 1
1.2. A Lowe Limit to the Cental Pessue in Stas Let us ecall the equation of hydostatic equilibium and the equation fo M : dp = GM ρ and d 2 Let s combine these equations dp = dp ( d = dm d dm GM ρ 2 dm d = 4π 2 ρ ) ( 1 ) 4π 2 ρ to obtain the equation of hydostatic equilibium in tems of the vaiable M : dp dm = GM 4π 4 Now integate this expession fom = to = R: P(R) P() dp = G 4π The pessue at the suface, P(R), is zeo so P(R) M M dm 4 P() dp = P(R) P() = P c Now, since 1/ 4 1/R 4, we have the inequality P c > G 4πR 4 M M dm and thus P c > G ( ) 2 ( ) 4 M2 M = 4.49 1 14 R dyne cm 2 8π R 4 M R This is not a vey close limit (it s odes of magnitude too small), but it does show that the cental pessues of stas must be quite high. We can actually get a bette estimate by assuming that the density though the sta is constant. That assumption leads to P c = 3GM2 8π R 4 (You might ty poving this.) That this is a lowe limit equies demonstating the (easonable) poposition that a sta whee the density is not constant, but inceases inwads, must have a highe cental pessue. The mean density of a sta is < ρ >= M/V = 3M/4πR 3, which fo the Sun woks out to < ρ > = 1.41 gm cm 3. If we wite the cental pessue of ou constant density sta as P c = 3GM2 = G ( ) M 3M = G M 8π R 4 2 R 4πR 3 2 R ρ 2
then we can equate this to the pessue fom the ideal gas law, cancel the ρ, and solve fo T c to obtain G M 2 R ρ = P c = N Ak k T c = G µ M 2N A k R = 11.5 16 µ ρ T c, ( M M ) ( ) R R K With µ =.66, this gives us T c = 7.6 1 6 K fo the sun; the actual value is almost twice as high (T c = 14.4 1 6 K). But emembe that the constant density model only gives us a lowe limit. 1.3. The Viial Theoem fo Stas The volume inside adius is just V = 4 3 π3. Let us multiply both sides of the dp/dm vesion of the hydostatic equilibium equation by V to obtain and integate this ove the whole sta V dp = G 3 M dm P(R) P() V dp = 1 3 M GM dm Now, it tuns out that the integal on the ight hand side is 1/3 the gavitational potential enegy of the sta. We can see this if we conside the wok against gavity we need to expend if we dispese the sta by lifting off successive layes of mass dm. The wok equied to lift one laye dm fom to is dw = dm foce distance = dm [ GM 2 d = dm GM ] = dm GM and this must be integated ove all the mass shells to obtain the enegy needed to dispese the sta. Thus the negative of this quantity is the gavitational potential enegy Ω: Ω = M GM dm and the equation above becomes P(R) P() V dp = 1 3 Ω. On the left hand side of this equation we employ integation by pats. 3
V dp = [P V ] R P dv Since the pessue is zeo at = R and the volume V is zeo at =, the fist tem on the RHS vanishes and we ae left with R P dv = 1 3 Ω. Now fom ou ealie discussion of the themal popeties of gases, we ecall that the pessue is elated to the intenal enegy by P = (γ 1) u. Theefoe R P dv = (γ 1) R u dv = (γ 1) U, whee U is the total intenal (themal) enegy of the sta. Thus we aive at The Viial Theoem: Ω = 3 (γ 1) U Conside the case elevant to most stas, whee the mateial behaves as an ideal, monatomic gas. Then γ = 5/3, (γ 1) = 2/3 and the viial theoem becomes Ω = 2 U o U = 1 2 Ω. It is also infomative to look at the total (themal + gavitational) enegy E total : E total = Ω + U = 1 2 Ω = U. It is no supise that E total <, since stas ae gavitationally bound. By contast, conside the case whee γ = 4/3. (Recall that this is the case fo adiation. Thee ae othe examples as well, such as elativistic degeneate mateial.) Then γ = 4 3 = (γ 1) = 1 3 = Ω = U = E total =! It s not just that we can t make a gavitationally bound object out of adiation alone; this also implies that vey massive stas, whee most of the pessue is due to adiation not gas, ae only weakly bound, as E total will appoach zeo. So how can we estimate the gavitational potential enegy Ω? Let us stat again with ou constant density appoximation. It is easy to see that in this case (M /M) is popotional to the atio of volumes: ( ) 3 M = M fo ρ = const. R 3 4
Then ( 2 dm = 3M Ω = 3GM2 R 6 So we can wite ou esult as ) d = R 3 R Ω = q GM2 R Ω = 4 d = 3GM2 R 6 R ( G 3 M R 3 [ 1 5 5 ] R ) 3M 2 R 3 d = 3 GM 2 5 R whee, fo ρ = constant, q = 3 5 It tuns out that the fom given above is quite geneal, with the value of q depending on the distibution of matte within the sta. The minimum value fo q is the constant density value of 3/5; a moe typical value might be q 1.5. Fo the density distibution of a polytope (discussed in the next section) we have the emakable esult that q = 3/(5 n), whee n is the index of the polytope. To look at numeical values, we can intoduce the sola mass and adius: ( ) 2 ( ) 1 M R Ω = 3.8 1 48 q eg When a sta of mass M and adius R is fomed, Ω is the amount of gavitational enegy eleased; half of this enegy goes into themal enegy that the sta needs to suppot itself, while the othe half must be adiated away. (Well, not quite half: some enegy is used dissociating H 2 molecules and then ionizing the gas.) Going back to the viial theoem, U = 1 Ω, we see we have an estimate fo the 2 aveage themal enegy of the Sun: U 2 1 48 eg. Dividing by the Sun s volume, V = 4πR 3 3 = 1.41 1 33 cm 3, the enegy pe unit volume is < u >= U/V = 1.4 1 15 eg cm 3. Now the themal enegy is elated to the tempeatue by < u >= 3nk < T 2 >. The numbe of paticles is n = N A ρ/µ, so can wite < T >= 2µ < u 3N A ρk >. Plugging in the sola values, we find < T > 5 1 6 K. So without much physics beyond the viial theoem we ae able to establish that the inteio tempeatue of the Sun is millions of degees. The viial theoem has many uses which we cannot go into hee, but it is woth noting that it can be applied to pat of a sta: in ou deivation we just integate out to some adius = R s athe than the suface = R. Then we find that M R Ω + 3 (γ 1) U = 3P s V s, whee Ω and U efe to the pat of the sta within adius R s, P s is the pessue at that adius and V s the enclosed volume. 5
1.4. Polytopic Stella Models A pope undestanding of stella stuctue daws on many diffeent aeas of physics,e.g., themonuclea eactions, the atomic physics of stella opacity, etc. Howeve, we can go a good bit futhe using the pinciples of hydostatic equilibium. Thee is a athe beautiful mathematical theoy developed ove a centuy ago, the theoy of polytopic models. Ou discussion of the isothemal sphee was a special case of such models. Recall equation (26) that we deived thee: [ 1 d 2 2 d ρ ] dp d = 4π G ρ. In geneal, P = P(ρ, T) and the equation above must be coupled with an equation fo T. But thee ae cases whee we may assume that P = P(ρ) only. This is the case with the so-called polytopes, which ae the solutions which follow when P ρ γ, whee γ is a constant. (In some but not all cases γ is the atio of specific heats we have used befoe.) It is usual to define a polytopic index n, which is elated to gamma by γ = 1+(1/n). Then the pessue can be witten P = K ρ 1+ 1 n whee K is anothe constant. We now eplace ρ by a new vaiable θ defined by the elation ρ = ρ c θ n, whee ρ c is the density at the cente of the sta. The pessue then becomes P = Kρ n+1 n c θ n+1. We also intoduce a dimensionless vaiable ξ in place of the adius by = aξ. With the ight choice of a (see, e.g. Choudhui, p 133), ou equation becomes ( 1 d ξ 2dθ ) = θ n. ξ 2 dξ dξ This is known as the Lane-Emden equation. It is a second-ode diffeential equation, and needs two bounday conditions to define a solution. These conditions ae θ = 1 and dθ/dξ = at the cente ξ =. If n =, 1 o 5, thee ae analytic solutions: fo example, fo n = 1, θ(ξ) = (sin ξ)/ξ is a solution. Othewise, the equation must be integated numeically. As with the case of the isothemal sphee (which, by the way, can be consideed the n = case), the numeical integation cannot be stated at ξ = because of the 1/ξ 2 tem. Again, we use a seies expansion nea the oigin: θ(ξ) = 1 1 6 ξ2 + n 12 ξ4 n(8n 5) 1512 ξ 6 + As the index n vaies, we obtain a seies of models with inceasing cental mass concentation (the n = case is in fact the constant density model). Hee we show some solutions nomalized to the same adius and cental density. 6
Fig. 1. Density Distibution as a Function of Polytopic Index 7
Fig. 2. Log Density Distibution as a Function of Polytopic Index 8
Befoe astonomes leaned to make eal stella models, polytopes wee quite impotant. As shown in the next figue, whee I have plotted the actual density distibution of the Sun compaed with an n = 3 polytope, thee is a eal similaity. While polytopes ae only of histoical inteest fo the study of main sequence and giant stas, they still have thei use. Fo example, white dwaf stas of low mass (non-elativistic) look like polytopes of index n = 1.5 (γ = 5/3), while with inceasing density, as the electon degeneacy becomes elativistic, the stuctue of a white dwaf appoaches that of an n = 3 (γ = 4/3) polytope. 1.5. The Equations of Stella Stuctue The pope constuction of a stella model equies the simultaneous solution of fou diffeential equations. These basic equations ae (Maoz, p 42; Choudhui, p 71): dp d = GM 2 ρ dm = 4π 2 ρ d dt d = 3L κ ρ 4π 2 4ac T 3 dl = 4π 2 ρ ǫ d In addition, we need what is called the equation of state. This is the elation between pessue, tempeatue and density. In stas like the sun, this would be the ideal gas law: P = N Ak µ ρ T, while fo massive stas we have to add a tem T 4 due to adiation pessue. The equation of state becomes even moe complicated fo the coes of stas in the late stages of stella evolution, when the quantum effects known as degeneacy set in. We ae familia with the fist two of these fou equations. The thid equation fo the tempeatue gadient dt/d has a lot of physics behind the quantity κ, which epesents the mean opacity of the stella mateial. It is a complex function of tempeatue, density and chemical composition. The fouth equation tells us how the luminosity L flowing outwad inceases due to enegy geneation by nucleosynthesis. The symbol ǫ epesents the enegy geneation pe gam, and is an extemely stong function of tempeatue. 9
Fig. 3. The Sola Density Distibution Compaed to an n=3 Polytope 1