Stellar Equations: Continuity

Thursday, Feb 23, 2017
Categories: Physics,
Series: The four stellar equations,
Tags: astrophysics, astronomy, space,

Stars essentially are balls of extremely hot, compressed, ionized gas (plasma). Their properties therefore depend on their mass, temperature, and density. Analyzing these properties gives us four equations that govern the structure of a star. The first of these is the Equation of Continuity. The equation of continuity relates the mass of a shell inside a star to the density at that distance from the center of the star.

Assuming a star is a perfect sphere, its volume is given by:

$$\begin{equation} V = \frac{4}{3}\pi r^{2} \end{equation}$$

The volume of a shell in the star a distance \(r\) from the center and of thickness \(dr\) is given by:

$$\begin{equation} dV = 4\pi r^{2} dr \end{equation}$$

And assuming that the density is radially uniform, and is given by \(\rho(r)\), the mass \(dM\) of a shell in the star a distance \(r\) from the center becomes:

$$\begin{equation}dM(r) = \rho(r) \times dV = 4\pi r^{2} \rho(r) dr \end{equation}$$

$$\frac{dM(r)}{dr} = 4\pi r^{2} \rho(r)$$

Where \(M(r)\) is the total mass enclosed at a radius \(r\) from the center of the star. This equation describes the conservation of mass in a star. That is, the sum of masses of all shells inside the star will be the mass of the star itself.