Jeans Equation Derivations

In this section of the documentation we derive a useful and commonly-used simplification of the collisionless Boltzmann equations known as the Jeans equations. We will take the following expression as our starting point, and refer the reader to Section 5.4.2 of Galaxy Formation and Evolution by Mo, van den Bosch and White (2010) for a first-principles derivation of this equation:

\[\frac{1}{\rho_{\rm sat}(r)}\frac{\rm d}{{\rm d}r}\left[\rho_{\rm sat}(r)\sigma^{2}_{r}(r)\right] + 2\beta(r)\frac{\sigma^{2}_{r}(r)}{r} = -\frac{{\rm d}\Phi(r)}{{\rm d}r}\]

In the above equation, \(\rho_{\rm sat}\) is the number density profile of an ensemble of massless tracer particles (in this case the satellite galaxies); \(\sigma^{2}_{r}\) is the second moment of the distribution of radial velocities of the satellites; \(\Phi\) is the gravitational potential; and \(\beta\equiv 1 - \frac{\sigma_{\theta}^{2}}{\sigma_{r}^{2}}\) is the so-called anisotropy profile, with \(\beta=0\) corresponding to isotropic orbits and \(\beta=1\) purely radial orbits.

With two independent unknowns, \(\beta(r)\) and \(\sigma_{r}^{2}(r)\), this differential cannot be solved without making additional assumptions.

Derivation of the isotropic Jeans equations

If we further assume that satellite orbits are isotropic, then the second term on the LHS of the above form of the Jeans equation drops out and we have:

\[ \begin{align}\begin{aligned}\begin{split}\frac{\rm d}{{\rm d}r}\left[\rho_{\rm sat}(r)\sigma^{2}_{r}(r)\right] = -\rho_{\rm sat}(r)\frac{{\rm d}\Phi(r)}{{\rm d}r} \\\end{split}\\\Rightarrow \left[\rho_{\rm sat}(r)\sigma^{2}_{r}(r)\right]_{r}^{\infty} = -\int_{r}^{\infty}\rho_{\rm sat}(r)\frac{{\rm d}\Phi(r)}{{\rm d}r}\\\Rightarrow \sigma^{2}_{r}(r) = \frac{1}{\rho_{\rm sat}(r)}\int_{r}^{\infty}\rho_{\rm sat}(r)\frac{{\rm d}\Phi(r)}{{\rm d}r},\end{aligned}\end{align} \]

where in the last equation we have applied the boundary condition that \(\rho_{\rm sat}(r\rightarrow\infty)\rightarrow 0\). This is the form of the equation used by the IsotropicJeansVelocity component model and the NFWPhaseSpace composite model.

Further Reading

There are many interesting papers on the Jeans equations. We refer the interested reader to the following papers for details: Lokas & Mamon (2000), arXiv:0002395; van den Bosch et al. (2004), arXiv:0404033; More et al. (2008), arXiv:0807.4532; Wojtak et al. (2008), arXiv:0802.0429, and references therein.