Lecture 10

The Tensor Virial Theorem
Just as we took velocity moments of the collisionless Boltzmann
equation (CBE) to obtain the Jeans equations, we can now take
spatial moments of the CBE. If we multiply the CBE by x k and
integrate over space we obtain
∫
∫
∂(νv j )
x k d 3 x = −
dt
∂(νv i v j )
x k d 3 x −
∂x i
∫
νx k
∂Φ
∂x j
d 3 x. (7)
The second term on the right hand side can be identified with the
Chandrasekhar potential energy tensor, W. The first term
on the right hand side can be rewritten using the divergence
theorem:
∫
∫
∂(νv i v j )
x k d 3 x = − δ ki νv i v j d 3 x = −2K kj , (8)
∂x i
where we have defined the kinetic energy tensor K by
K jk ≡ 1 2
∫
νv i v j d 3 x. (9)
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