= Solution
The particle count is the enclosed mass divided by the particle mass $m$:
$$
N(\xi)=\frac{4\pi\rho_ca^3}{m}\int_0^\xi s^2e^{-\Psi(s)}\,ds.
$$
Integrating the <isothermal Lane-Emden equation> gives
$$
\int_0^\xi s^2e^{-\Psi(s)}\,ds
=\left[s^2\Psi'(s)\right]_0^\xi=\xi^2\Psi'(\xi),
$$
where regularity makes the lower endpoint vanish. Thus
$$
\boxed{N(\xi)=\frac{4\pi\rho_c}{m}
\left(\frac{K}{4\pi G\rho_c}\right)^{3/2}\xi^2\Psi'(\xi).}
$$
The prefactor is a mass divided by a mass, and the remaining factor is dimensionless. In particular, at fixed $K$ the dependence on central <mass density> is $N\propto\rho_c^{-1/2}f(\xi)$, with $f(\xi)=\xi^2\Psi'(\xi)$. Both changes must be retained when comparing equilibria at fixed $N$.
Back to article page
