Editing Plummer model (section)

== Description of the model ==
[[Image:Plummer rho.png|thumb|right|220px|The density law of a Plummer model]]

The Plummer 3-dimensional density profile is given by
<math display="block">\rho_P(r) = \frac{3M_0}{4\pi a^3} \left(1 + \frac{r^2}{a^2}\right)^{-{5}/{2}},</math>
where <math>M_0</math> is the total mass of the cluster, and ''a'' is the '''Plummer radius''', a scale parameter that sets the size of the cluster core. The corresponding potential is
<math display="block">\Phi_P(r) = -\frac{G M_0}{\sqrt{r^2 + a^2}},</math>
where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]]. The velocity dispersion is
<math display="block">\sigma_P^2(r) = \frac{G M_0}{6\sqrt{r^2 + a^2}}.</math>

The isotropic distribution function reads
<math display="block">f(\vec{x}, \vec{v}) = \frac{24\sqrt{2}}{7\pi^3} \frac{a^2}{G^5 M_0^4} (-E(\vec{x}, \vec{v}))^{7/2},</math>
if <math>E < 0</math>, and <math>f(\vec{x}, \vec{v}) = 0</math> otherwise, where <math display="inline">E(\vec{x}, \vec{v}) = \frac{1}{2} v^2 + \Phi_P(r)</math> is the [[specific energy]].