Spitzer Mass Segregation Instability

Let us consider two populations of stars, one where the mass an individual star is $m_1$ and another with $m_2$. The total mass of stars in the first population is $M_1$, and of the second population $M_2$. We assume that individual stars from the second population are much more massive than from the first population $m_2 \gg m_1$ but that the total mass of stars from the first population dominates $M_1 \gg M_2$. The first population is assume to uniformly fill a sphere of radius $r_1$, and the second population uniformly fills a smaller sphere of radius $r_2$. The average kinetic energy of stars from the first population is

$m_1 v_1^2 \approx \frac{G M_1}{r_1}$

where $G$ is the gravitation constant. The average kinetic energy of stars from the second population is

$m_2 v_2^2 \approx \frac{G M_2}{r_2} + \frac{G M_1}{r_2} \left(\frac{r_1}{r_2}\right)^{3}$

In a thermal equilibrium, equipartition holds and both kinetic energies are equal, implying

$\frac{r_2}{r_1} \approx \frac{m_2}{m_1} \left(\frac{M_2}{M_1} + \frac{r_2^3}{r_1^3}\right) \approx \frac{m_2}{m_1} \frac{M_2}{M_1} \left(1+\frac{M_1/r_1^3}{M_2/r_2^3}\right)$

We replace the distances with the mass density of each species $\rho_i \approx M_i / r_i^3$

$\left(\frac{M_2/\rho_2}{M_1/\rho_1}\right)^{1/3} \approx \frac{m_2}{m_1} \frac{M_2}{M_1} \left(1+\frac{\rho_1}{\rho_2} \right) \Rightarrow \frac{M_2}{M_1} \left(\frac{m_2}{m_1}\right)^{3/2} \approx \frac{\left(\rho_1/\rho_2\right)^{1/2}}{\left(1+\rho_1/\rho_2\right)^{3/2}}$

The right hand side attains a maximum when both densities are similar $\rho_1 \approx \rho_2$, at which point the left hand side is also of order unity. When the left hand side is larger than unity than thermal equilibrium cannot be reached. As a result of the interaction, the cluster of heavier stars contracts, and that of the lighter stars expands.