2.1: Basic Assumptions

Any rational structure must have a beginning, a set of axioms, upon which to build. In addition to the known laws of physics, we shall have to assume a few things about stars to describe them. It is worth keeping these assumptions in mind for the day you encounter a situation in which the basic axioms of stellar structure no longer hold. We have already alluded to the fact that a self-gravitating plasma will assume a spherical shape. This fact can be rigorously demonstrated from the nature of an attractive central force so it does not fall under the category of an axiom.

However, it is a result which we shall use throughout most of this book. A less obvious axiom, but one which is essential for the construction of the stellar interior, is that the density is a monotonically decreasing function of the radius. Mathematically, this can be expressed as \[\rho(\mathrm{r}) \leq<\rho>(\mathrm{r}) \quad \text { for } r>0,\label{2.1.1}\] where \[\mathrm{<\rho>(r) / M(r) /\left[4 \pi r^3 / 3\right],}\label{2.1.2}\]

and \(\mathrm{M(r)}\) is the mass interior to a sphere of radius r and is just \(\int 4 \pi \mathrm{r}^2 \rho \mathrm{dr}\). In addition, we assume as a working hypothesis that the appropriate equation of state is the ideal-gas law. Although this is expressed here as an assumption, we shall shortly see that it is possible to estimate the conditions which exist inside a star and that they are fully compatible with the assumption.

It is a fairly simple matter to see that the free-fall time for a particle on the surface of the sun is about 20 min. This is roughly equivalent to the dynamical time scale which is the time scale on which the sun will respond to departures from hydrostatic equilibrium. Most stars have dynamical time scales ranging from fractions of a second to several months, but in all cases this time is a small fraction of the typical evolutionary time scale. Thus, the assumption of hydrostatic equilibrium is an excellent one for virtually all aspects of stellar structure. In Chapter 1 we developed an expression for hydrostatic equilibrium [(equation \ref{1.2.28}], where the pressure gradient is proportional to the potential gradient and the local constant of proportionality is the density. For spherical stars, we may take advantage of the simple form of the gradient operator and the source equation for the gravitational potential to obtain a single expression relating the pressure gradient to \(\mathrm{M(r)}\) and \(\rho\).

The source equation for the gravitational potential field is also known as Poisson's equation and in general it is \[\nabla^2 \Omega=4 \pi \mathrm{G} \rho,\label{2.1.3}\]

which in spherical coordinates becomes \[\frac{d}{d r}\left(r^2 \frac{d \Omega}{d r}\right)=4 \pi G \rho r^2\label{2.1.4}\]

Integrating this over r, we get \[\frac{d \Omega}{d r}=\frac{G}{r^2} \int_0^r 4 \pi r^2 \rho d r=\frac{G M(r)}{r^2}\label{2.1.5}\]

Replacing the potential gradient from equation \ref{1.2.28}, we have \[\frac{d P}{d r}=-\frac{G M(r) \rho(r)}{r^2}\label{2.1.6}\]

This is the equation of hydrostatic equilibrium for spherical stars. Because of its generality and the fact that virtually no assumptions are required to obtain it, we can use its integral to place fairly narrow limits on the conditions that must prevail inside a star.

In equation \ref{2.1.2}, we introduced a new variable M(r). Note that its invocation is equivalent to invoking a conservation law. The conservation of mass basically requires that the total mass interior to r be accounted for by summing over the density interior to r. Thus, \[M(r)=\int_0^r 4 \pi r^2 \rho d r\label{2.1.7}\]

or its differential form \[\frac{d M(r)}{d r}=4 \pi r^2 \rho\label{2.1.8}\]