EQUATIONS OF STELLAR STRUCTURE General Equations

1 EQUATIONS OF STELLAR STRUCTUREG eneral EquationsWe shall consider a spherically symmetric, self-gravitating star. All the physical quantities willdepend on two independent variables:radiusandtime, (r, t). First, we shall derive all the equationsof STELLAR STRUCTURE in a General , non spherical case, but very quickly we shall restrict ourselves tothe spherically symmetric microscopic properties of matter at any given point may be described with density , tem-peratureT, and chemical composition, the abundances of various elementsXi, withi= 1,2, as many values as there are elements. All thermodynamic properties and transport coeffi-cients are functions of ( , T, Xi).

2 In particular we have: pressureP( , T, Xi), internal energy perunit volumeU( , T, Xi), entropy per unit massS( , T, Xi), coefficient of thermal conductivity perunit volume ( , T, Xi), and heat source or heat sink per unit mass ( , T, Xi). All the partialderivatives ofP,U, , and are also functions of ,T, andXi. Using these quantities the first lawof thermodynamics may be written asT dS=d(U ) P 2d ,( )If there are sources of heat, , and a non-vanishing heat flux~F, then the heat balance equation maybe written as TdSdt= div~F .( )The heat flux is directly proportional to the temperature gradient:~F= T.

3 ( )The equation of motion (the Navier-Stockes equation of hydrodynamics) may be written asd2~rdt2+1 P+ V= 0,( )where the gravitational potential satisfies the Poisson equation 2V= 4 G ,( )withV 0 whenr .In spherical symmetry these EQUATIONS may be written as1 P r+ V r+d2rdt2= 0,( )1r2 r(r2 V r)= 4 G ,( )F= T r,( )1 11 r2 (r2F) r= TdSdt,( )It is convenient to introduce a new variable,Mr:Mr r 04 r 2 dr ,( )which is the total mass within the radiusr, and another variable,Lr:Lr 4 r2F,( )which is the luminosity, the total heat flux flowing through a spherical shell with the radiusr,and also =4acT33 1 ,( )where is the coefficient of radiative opacity (per unit mass) ,cis the speed of light, andais theradiation constant.

4 The last equation is valid if the heat transport is due to the definitions and relations ( ) we may write the set of EQUATIONS ( ) in a morestandard form:1 P r+GMrr2+d2rdt2= 0,( ) Mr r= 4 r2 ,( ) T r= 3 Lr16 acT3r2,( ) Lr r= 4 r2 ( TdSdt),( )This system of EQUATIONS is written in a somewhat inconvenient way, as all the space derivatives( / r) are taken at a fixed value of time, while all the time derivatives (d/dt) are at the fixed masszones. For this reason, and also because of the way the boundary conditions are specified (we shallsee them soon) , it is convenient to use the massMrrather than radiusras a space-like independentvariable.

5 Therefore, we replace all derivatives / rwith 4 r2 / Mr, and we obtain P Mr= GMr4 r4 14 r2 2r t2,( ) r Mr=14 r2 ,( ) T Mr= 3 Lr64 2acT3r4,( ) Lr Mr= T S t.( )The set of EQUATIONS ( ) describes the time evolution of aspherically symmetric star with agiven distribution of chemical composition with mass,Xi(Mr), provided the initial conditions andthe boundary conditions are specified. If the time derivative in the equation ( ) vanishes thenthe star is inhydrostatic equilibrium. If the time derivative in equation ( ) vanishes thenthe star is inthermalequilibrium. Notice, that we always assume that throughoutthe star thematter and radiation are inlocal thermodynamic equilibrium, LTE, no matter if the star as awhole is in hydrostatic or in thermal equilibrium.

6 From now on we shall consider stars that are inthe hydrostatic equilibrium, we shall assume that the time derivative in the equation ( ) isnegligible 2 Boundary ConditionsWe shall consider now the boundary conditions. At the STELLAR center the massMr, the radiusr,and the luminosityLr, all vanish. Therefore, we have theinner boundary conditionsr= 0,Lr= 0,atMr= 0.( )In most cases we shall be interested in STRUCTURE and evolution of a star with a fixed total massM. At the surface, whereMr=M, the density falls to zero, and the temperature falls to a valuethat is related to the STELLAR radius and luminosity.

7 The proper outer boundary conditions requirerather complicated calculations of a model STELLAR atmosphere. We shall shall adopt a very simplemodel atmosphere within the Eddington approximation, which means we shall use the diffusionapproximation to calculate the temperature gradient not only at large optical depth, but also atsmall optical depth. The Eddington approximation also means that the surface temperature is21/4 times lower than the effective temperature. Theouter boundary conditionsare = 0,T=To=(L8 R2 )1/4,atMr=M,( )where is the Stefan-Boltzman constant. Notice, that the so calledeffective temperature of a staris defined asTeff (L4 R2 )1/4= 21/4To.

8 ( )At the STELLAR center we have two adjustable parameters: thecentral density c, and the centraltemperatureTc. At the STELLAR surface there are other two adjustable parameters: the STELLAR radiusR, and the STELLAR luminosityL. These four parameters may be calculated when the differentialequations of STELLAR STRUCTURE are solved. Notice, that only two of those parameters,RandLaredirectly observable. Also notice, that the EQUATIONS for spherically symmetric stars (10 or 11) may bederived without considering the General case, but startingwith simple geometry of thin, sphericallysymmetric shells, and balancing mass, momentum and energy across those EquationsLet us consider now an even simpler case: a star which may be described with the equationsin which all time derivatives may be neglected, a star that is in the hydrostatic and thermalequilibria.

9 Now, that we have no time dependence, the STELLAR STRUCTURE depends on one space-likevariable only, which we may choose to be radiusr, or the massMr. Now, we have four ordinarydifferential EQUATIONS :dPdMr= GMr4 r4,( )drdMr=14 r2 ,( )1 3dTdMr= 3 Lr64 2acT3r4,( )dLrdMr= .( )These have to be supplemented with the boundary conditions ( ) and ( ) , as well as the totalstellar massM, and the distribution of all elements with the mass,Xi(Mr). We have four ordinarydifferential EQUATIONS , four boundary conditions, and fourparameters to be found:Tc, c,R, it looks like the problem described with the eqs.

10 ( ) , ( ) , ( ) has a uniquesolution. In fact, the so called Vogt-Russell theorem claims just that. However, this is not were found numerically first, and only later the astronomers noticed that thereis no mathematical basis for the Vogt-Russell theorem . The initial value problems usually have aunique solution. However, if the boundary conditions are specified at the two different locations, inour case at the STELLAR center and at the STELLAR surface, then there may be no solutions, or in generalthere may be many solutions. We shall find some examples lateron. Nevertheless, the Vogt-Russell theorem served a useful purpose, it explained the nature of the Main Sequence, a linear sequenceof STELLAR models with the total STELLAR mass being the parameter that varied along the numerical STELLAR models demonstrate that in this case there is indeed a unique solutionfor a large range of STELLAR masses, with all stars being chemically homogeneous, their luminositygenerated by nuclear burning of hydrogen into fact that stars are luminous, they are radiating away some energy, implies that theymust change in time.