Integrating the collisionless Boltzmann equation over all velocities
*and* positions yields a simple relationship, the tensor virial
theorem, which links a galaxy's shape to its kinematics. Analysis of
the observations shows that bright elliptical galaxies are not flattened
by their rotation.

To derive the tensor virial equation, multiply the CBE by `v_j
x_k` and integrate over all velocities and positions (BT87, Chapter
4.3). We have already done the integral over all velocities in Eq. 4
of last lecture; thus

/ d | ___ (1) -- | dx x_k nu v_j = dt | / / / | d _______ | dPhi - | dx x_k ---- (nu v_j v_i) - | dx x_k nu ---- , | dx_i | dx_j / /where the time derivative has been moved outside the integral on the LHS. We will come back to this term shortly, but it is worth noting that for a system in equilibrium the integral is constant, and thus the LHS is zero. The first term on the RHS may be simplified using the divergence theorem:

/ / | d _______ | _______ (2) - | dx x_k ---- (nu v_j v_i) = | dx nu v_j v_k = 2 K_jk . | dx_i | / /Here

1 / / (3) K_jk = - | dx | dv f(x,v) v_j v_k . 2 / /Note that

The second term on the RHS is the *potential energy tensor*,

/ | dPhi (4) W_jk = - | dx x_j nu ---- . | dx_k /Under the assumption that the stellar mass density

Using the symmetry of `K_jk` and `W_jk`, Eq. 1 becomes

/ 1 d | ___ ___ (5) - -- | dx nu (x_k v_j + x_j v_k) = 2 K_jk + W_jk . 2 dt | /The LHS, explicitly symmetrized over

/ (6) I_jk = | dx nu x_j x_k . /Putting everything together finally gives the tensor virial equation,

1 d d (7) - -- -- I_jk = 2 K_jk + W_jk . 2 dt dt

Remark: the trace of Eq. 7 is the more familiar scalar virial theorem; see BT87, Chapter 4.3 for a discussion.

Consider a slightly idealized model of an oblate galaxy with
density contours which are similar, concentric spheroids. The short
axis of the model is aligned with the `z` axis of the
coordinate system. From BT87, Chapter 2.5, we know that the potential
energy tensor is diagonalized in this coordinate system and that

W_xx (8) W_xx = W_yy , ---- = q(epsilon) > 1 , W_zzwhere

The motions of stars within the model may always be split into a
net *streaming* motion and a *random* dispersion with
respect to the streaming motion at each point. The total kinetic
energy tensor is just the sum of the tensors for these separate
motions. Assuming that the only streaming motion is rotation about
the `z`axis, the associated KE tensor is

(s) 1 2 [ 1 0 0 ] (9) 2 K = - M v_0 [ 0 1 0 ] , 2 [ 0 0 0 ]where

(r) 2 [ 1 0 0 ] (10) 2 K = M sigma_0 [ 0 1 0 ] , [ 0 0 (1-delta) ]where

Applying the tensor virial theorem, we have

1 2 2 - M v_0 + M sigma_0 W_xx K_xx 2 (11) q(epsilon) = ---- = ---- = ------------------- , W_zz K_zz 2 (1-delta) M sigma_0which may be rearranged to give

v_0 1/2 (12) ------- = (2 (1-delta) q(epsilon) - 2) . sigma_0This is the promised relationship between shape and kinematics. Note that this equation is

The application of Eq. 12 is illustrated by a couple of examples:

If `v_0 = 0`, then the velocity anisotropy is

1 (13) 1 - delta = ---------- . q(epsilon)

If `delta = 0`, then the rotation velocity is

1/2 (14) v_0 = sigma_0 (2 q(epsilon) - 2) .

The observational application of these relationships is somewhat
complicated by projection effects (see BT87, Chapter 4.3(b)).
However, for `delta = 0` the effects of projection on the
*apparent* ellipticity and rotation of a galaxy is such as to
very nearly obey Eq. 14. Thus the observational result that
low-luminosity E galaxies fall along the predicted curve in the
`v_0/sigma_0` vs. `epsilon` diagram, whereas
high-luminosity ellipticals scatter *below* this line, implies
that the former are *consistent* with isotropic rotation, while
the latter *must* be flattened by velocity anisotropy.

Due date: 2/9/95

10. If `f(x,v;t)` is non-negative everywhere in phase space
at `t = 0`, prove that it must also be non-negative everywhere
at later times. Note: if you write more than two lines, you're
working too hard!

11. What is the KE tensor for a system with

-- N \ 3 3 f(x,v) = | m_i delta(x - x_i) delta(v - v_i) , / -- i=1where

12. What is the KE tensor for a tumbling prolate galaxy with an anisotropic velocity distribution aligned with the long axis?

Joshua E. Barnes (barnes@zeno.ifa.hawaii.edu)

Last modified: February 2, 1995