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 K_jk is the kinetic energy tensor, defined as
1 / / (3) K_jk = - | dx | dv f(x,v) v_j v_k . 2 / /Note that K_jk is symmetric, K_jk = K_kj, and that its trace is the total kinetic energy of the system.
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 nu(x) is also the source of the gravitational field, it follows that W_jk is symmetric, that its trace is the potential energy U, that for a spherical system W_jk = delta_jk U / 3, and that for a system flattened along the z direction W_xx/W_zz > 1 (BT87, Chapter 2.5).
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 k and j, is one-half of the second time derivative of the moment of inertia tensor,
/ (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 epsilon = 1 - c/a is the ellipticity of the model. The actual form of q(epsilon) is straightforward but tedious to write out; see BT87.
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 zaxis, 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 M is the total mass and v_0 is the mass-weighted rotation velocity (BT87, Chapter 4.3(b)). Allowing for the possibility that the random dispersion in the z direction is different from the dispersions in the x and y directions, the KE tensor associated with the random motion is
(r) 2 [ 1 0 0 ] (10) 2 K = M sigma_0 [ 0 1 0 ] , [ 0 0 (1-delta) ]where sigma_0^2 is the mass-weighted random velocity in the x direction, and delta parametrizes the velocity anisotropy.
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 exact for the somewhat idealized galaxy model adopted here.
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 m_i, x_i, and v_i are masses, positions, and velocities of particles and delta(...) is Dirac's delta function?
12. What is the KE tensor for a tumbling prolate galaxy with an anisotropic velocity distribution aligned with the long axis?
Last modified: February 2, 1995