Elliptical galaxies. II

This chapter is a bit harder, with more mathematics than I would like to use. I do not expect you to be able to reproduce all these derivations, only the ones stressed in the Summary. A little note on notation: if in one of the following equations an index occurs twice in a given term, like the index $i$ in $x_i y_i$, then you have to sum over $i$. So $x_i y_i\equiv \sum_i x_iy_i=\hbox{$\bf x$}\cdot\hbox{$\bf y$}$ is a short hand for the scalar product of $\hbox{$\bf x$}$ and $\hbox{$\bf y$}$.

We've seen that the stars in spiral disks are all on nearly circular orbits. Although some of the smaller Es sometimes rotate as well, the angular momentum of stars is not sufficient to balance gravity. So we need to look for another mechanism to prevent collapse. Basically, it's the velocity dispersion of the stars that balances gravity, much like it is a pressure gradient in the gas that balances gravity in a star. For this reason, Es (but also spiral bulges, and globular clusters for example) are called hot stellar systems.

The Jean's equations relate the stellar velocity dispersion, $\sigma_{ij}^2=\langle (v_i-\langle v_i\rangle)(v_j-\langle v_j\rangle)\rangle$ with the gravitational potential of the system, and are the stellar analogue of the equation of hydrostatic equilibrium $\nabla\Phi=-\nabla p/\rho$ in a star. They do not describe the properties of the orbits of a single star, but rather assume one can use averaged properties. To investigate whether such an approach makes sense, I first need to introduce the concept of relaxation time.