4.7: Equations of Stellar Structure

Having settled the mode of energy transport, we are in a position to describe the structure of a star in a steady-state condition. This is a good time to review briefly what we have done. The equations of stellar structure arise from conservation laws and relationships developed from the local microphysics. In Chapter 1, we posed the basic problem of stellar interiors to be the description of the variation of state variables \(\mathrm{P}\), \(\mathrm{T}\), and \(\gamma\) with position in the star. For spherical stars, this amounts to indicating their dependence on the radial coordinate r. In developing that description, we introduced additional variables and their relation to the state variables so that by now our list of parameters has grown to nine members, \(\mathrm{P(r)}\), \(\mathrm{T(r)}\), \(\mathrm{p(r)}\), \(\mathrm{M(r)}\), \(\mathrm{L(r)}\), \(\varepsilon(\mathrm{r})\), \(\overline{\kappa}(\mathrm{r})\), \(\gamma(\mathrm{r})\), and \(\mu(\mathrm{r})\). To specify these parameters, we have at our disposal three conservation laws and a transport equation in addition to three functional relationships derived from the microphysics. The function \(\gamma(\mathrm{r})\) can also be specified by microphysics and is usually given by its adiabatic value. Only the variation of \(\mu(\mathrm{r})\) needs to be specified ab initio. When we move to the stage of evolving the stellar models, the chemical composition will need to be specified for the initial model since the processes of nuclear energy generation will tell us how the composition changes with time. However, we must, at least initially, specify both the composition of the star and how it varies throughout the entire star. The use of a convective theory of transport which attempts to improve on the adiabatic gradient will also introduce another parameter, known as the mixing length, which must also be specified ab initio.

The constraints posed by the conservation laws take the form of differential equations whose solution is subject to a set of boundary conditions. Below is a summary of these differential equations and their origin:

\[\begin{array}{l}
\text { (a) }& \frac{d M(r)}{d r}&=4 \pi r^2 \rho(r) & \text {conservation of mass, eq. (2.1.8) }\\
\text { (b) }& \frac{d L(r)}{d r}&=4 \pi r^2 \rho(r) \epsilon(r) & \text{conservation of energy, eq. (4.2.16)}\\
\text { (c) }& \frac{d P(r)}{d r}&=-\frac{G M(r) \rho(r)}{r^2} & \text{conservation of momentum, eq. (2.1.6) (hydrostatic equilibrium)}\\
&\frac{d T(r)}{d r}&=-\frac{3 \bar{\kappa}(r) \rho(r) L(r)}{16 \pi a c T^3(r) r^2} & \text{ radiative transport, eq. (4.2.4) }\\
\text { (d) }&\left(\frac{d T}{d r}\right)_{a d}&=-\frac{\mu m_h G M(r)}{(n+1) k r^2} &\text{convective transport, eq. (4.3.19) }\\
&(\Delta \nabla T)&=\left[\frac{L^2(r) T(r)}{C_p^2 \rho^2(r) G M(r) \pi^2 l^4 r^2}\right]^{1 / 3} &\text{eq. (4.3.18) }
\end{array}\label{4.6.1}\]

In addition to these differential equations we have the following relations from the microphysics: \[\begin{array}{ll}
\epsilon&=\epsilon[T(r), \rho(r), \mu(r)] & \text { nuclear energy production (Chap. 3) } \\
\bar{\kappa}&=\bar{\kappa}[T(r), \rho(r), \mu(r)] & \text { radiative opacity (Sec. 4.1) } \\
\gamma&=\gamma[T(r), \rho(r), \mu(r)]=\frac{5}{3} & \text { (Chap. 2) } \\
P&=P[T(r), \rho(r), \mu(r)] & \text { equation of state (Chap. 1) }
\end{array}\label{4.6.2}\]

These eight relationships and the chemical composition completely specify the structure of the star. We now turn to describing methods by which their solution can be obtained.