Intuition for the velocity ellipsoid

Question

The velocity ellipsoid is a quantity that comes up in the kinematic modelling of N-body systems, e.g. stars in a galaxy. The velocity dispersion tensor describes the local distribution of velocities at each point:

\begin{align}\sigma_{ij}^2({\bf x})&=\frac{1}{\nu({\bf x})}\int {\rm d}^3{\bf v}(v_i-\bar{v}_i)(v_j-\bar{v}_j)f({\bf x},{\bf v})\\ &=\overline{v_iv_j}-\bar{v}_i\bar{v}_j\end{align}

If one chooses a coordinate system where $\boldsymbol{\sigma}^2$ is diagonal, then the diagonal elements can be thought of as the semi-axis lengths of an ellipsoid aligned along the diagonalizing coordinates.

The above is summarized from 4.1.2 in B&T 2008.

1. Is the velocity ellipsoid necessarily global to the system, or can the orientation & aspect of the ellipsoid change with $\bf x$?
2. Related, must the coordinate system in which $\boldsymbol{\sigma}^2$ is diagonal be cartesian, or could this be e.g. a spherical coordinate system? Then I suppose the ellipsoid would rotate around depending on angular position (supposing a spherically symmetric system, rotation would be the only change)?
3. Is there a heuristic interpretation of the ellipsoid? E.g. if the longest axis is along the $\hat{\bf e}_i$ direction, is this the axis of maximal dispersion in the velocities? And/or are there other useful heuristics?

Answer 1

1. Through the function $f$, the velocity dispersion tensor depends on the location $\mathbf{x}$

2. The velocity dispersion tensor is a symmetric tensor, hence its eigenvectors form an orthonormal basis. But, locally, i.e. depending on $\mathbf{x}$, you may have different orientations (resp. eigenvectors). This has nothing to do with coordinate systems. You need to distinguish carefully between base vectors, the eigenbasis and coordinate systems.

3. I believe that the eigenvalues of this tensor give the deviation of the velocity of the stars near $\mathbf{x}$ compared to some average. Large eigenvalues mean a large variance.