Solutions of the Equations and Simple Models

Science is based on quantitative results. In astrophysics, these are obtained from observations and models. The key to progress is the close comparison of the two.

As the equations of stellar evolution have in general no analytical solutions, one solves them numerically, especially more than opacities and nuclear reaction rates are given as data tables. The numerical models or simulations of stellar evolution made great advances since the 1960s, thanks to the development of computers.

Many properties found by the numerical models can be derived analytically in a simplified and approximate way. Often, the analytical developments were made after the numerical models. The knowledge of the analytical relations is enlightening for the physical understanding of the astrophysical processes; this is why when possible we present them.

24.1 Hydrostatic and Hydrodynamic Models

Hydrostatic models apply when evolution is slow with respect to the dynamical timescale (1.28).

24.1.1 Hydrostatic Models and Vogt-Russel Theorem

We collect here the basic equations of equilibrium in Eulerian and Lagrangian forms (Table 24.1). These are the equations of hydrostatic equilibrium (1.6), of continuity (1.12), of energy equilibrium (3.40) and of energy transport, either radiative (3.17) or convective (5.57). In the interiors, convection is present if Vrad > Vad (Sect. 5.3), otherwise the transfer is radiative. The equation of state, opacities and nuclear reactions are expressed by the functions

Q = Q (P, T,Xi), K = K(P, T,Xi), £ = £(P., T,Xi) , (24.1)

*This chapter may form the matter of a basic introductory course.

A. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and 593

Astrophysics Library, DOI 10.1007/978-3-540-76949-1-24, © Springer-Verlag Berlin Heidelberg 2009

Table 24.1 Basic equations in Eulerian and Lagrangian forms

dP

GMr

dP

GMr

dr

r2

dMr '

4nr4

dMr dr

- 4nr2 q

~dMr ~~

4kq r2

dLr dr

- 4nr2Q (£ + £grav - V

~dMr ~~

= (£ + £grav — £v)

rad :

dr

3kq Lr 4acT3 4nr2

~dMr ~~

GMrT v " 4nr4PVrad

conv :

dr

r2 - 1 T dP r2 P dr

~dMr ~~

GMrT v " 4nr4PVad

generally given by numerical tables for different abundances X;. We have a system of four first-order differential equations with four unknowns Mr, P, T, Lr as a function of r (Eulerian case), if egrav = 0. If not, egrav has to be evaluated from the time derivative (3.64) with respect to previous models.

The chemical composition of the model needs to be specified; it normally results from previous evolution (Sect. 25.1.2). The simplest case is chemically homogeneous models, typically with hydrogen, helium and heavy element mass fractions X = 0.70, Y = 0.28 and Z = 0.02 (standard composition). The homogeneous composition is that of stars at the beginning of H burning on the zero-age main sequence (ZAMS) (only a few light elements are modified in pre-MS evolution, Sect. 20.7). The "best" initial solar composition is X = 0.720, Y = 0.266 and Z = 0.014 (Sect. 7.2; Appendix A.3). The four equations need boundary conditions. A zero-order approximation, e.g., in Eulerian form at r = R, is (see also Sect. 24.1.3)

We have four conditions with 3 free parameters R, L and M. Let us suppose that we integrate the four equations starting from the surface. When the center is reached, i.e., at r = 0, one should normally also have Lr = 0 and Mr = 0. This is not automatically the case for an arbitrary choice of R, L and M at the surface. This means that for a given M the other two surface parameters have to be adjusted until the integration leads simultaneously to r = 0, Lr = 0 and Mr = 0 when the center is reached. The two additional conditions at the center reduce the number of free parameters at the surface from three to one. Usually, one chooses the stellar mass M.

This leads to the Vogt-Russel theorem: The properties of a star of a given composition and in equilibrium are entirely determined by its mass M. The theorem is generally applied to chemically homogeneous stars; however we stress that this is not a necessity. There have been a number of rather academic discussions as to whether the Vogt-Russel theorem always applies. Indeed, there are cases where a minute change of composition may produce major changes of the overall parameters (such as the evolution from red supergiants to WR stars). However, thermal equilibrium is generally not satisfied in these cases. The Vogt-Russel theorem implies the existence of relations such as L = L(M, composition) and R = R(M, composition).

24.1.2 Hydrodynamic Equations

In a radially pulsating star or in rapid evolutionary phases (as in the protostellar phase or in the pre-supernova stage), the departures from hydrostatic equilibrium, i.e., the terms with r, are significant and Eq. (1.14) must be used. The time t appears as an independent variable in addition to Mr in the Lagrangian form, which is always used here. One introduces the velocity v as a new variable and the full set of equations becomes dv A 2 dP — = -4nr2 dt

dMr dr dt:

3 Kg LrP 16acnr2T4 (dP/dr)

The derivative (d/dMr) is taken at a fixed t and (d/dt) at a fixed Mr. The first equation is Euler's equation. The equation of energy equilibrium (24.6) requires some explanations. The expression of egrav is (3.64)

Mr where U is the specific internal energy U = aT4/g + (3/2)(kT/(pmu) continuity equation (1.1) gives which is now used in dg ~dt

Mr dt

Tr v

dr dg dr

Thus, egrav becomes

0 0

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