I am currently studying the Jeans equation for a system in spherical equilibrium. Since the distribution function $f(x,v)$ can be written in depency of energy and angular momentum $f(E, L)$ it seems to follow that most of the velocity moments disappear:

$$ \tag{1} \left<\upsilon_r\right> = \left<\upsilon_\vartheta\right> = \left<\upsilon_\varphi\right> = 0 $$


$$ \tag{2} \left<\upsilon_r \upsilon_\vartheta\right> = \left<\upsilon_r \upsilon_\varphi\right> = \left<\upsilon_\vartheta \upsilon_\varphi\right> = 0. $$

Where $f(x,v)$ denotes the number density of particles in phase space and $\left<Q\right> = \frac{m}{\rho(x)} \int Q\,f(x,v) \text{d}^3v$,$\;\;$ $\rho(x) = m \int f(x,v)\text{d}^3 v$.

I don't find it intuitive why these velocity moments are zero. What ist the mathematical way to get these results?


Jeans equation is just an analog to the Euler equations.

In the limit you describe, the results of your Equation 1 imply you are working in the center of momentum frame, i.e., the bulk flow rest frame. The results of equation 2 state the pressure tensor can be diagonalized, i.e., there is no viscosity. Viscosity arises from off-diagonal terms in the pressure tensor.

I have not heard the term "spherical equilibrium" before but I assume you are implying spherial symmetry? Regardless, in the bulk flow rest frame there is no first velocity moment, i.e., it is zero. The lack of off-diagonal terms implies there are no stress/strains in the flow, e.g., azimuthal flow will not be transported across a radial plane.


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