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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>
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For bound orbits, the radial turning points of the orbit is characterized by [[specific energy]] <math display="inline">E = \frac{1}{2} v^2 + \Phi(r)</math> and [[specific relative angular momentum|specific angular momentum]] <math>L = |\vec{r} \times \vec{v}|</math> are given by the positive roots of the [[cubic function|cubic equation]]
<math display="block">R^3 + \frac{GM_0}{E} R^2 - \left(\frac{L^2}{2E} + a^2\right) R - \frac{GM_0a^2}{E} = 0,</math>
where <math>R = \sqrt{r^2 + a^2}</math>, so that <math>r = \sqrt{R^2 - a^2}</math>. This equation has three real roots for <math>R</math>: two positive and one negative, given that <math>L < L_c(E)</math>, where <math>L_c(E)</math> is the specific angular momentum for a [[circular orbit]] for the same energy. Here <math>L_c</math> can be calculated from single real root of the [[Cubic function #The discriminant|discriminant of the cubic equation]], which is itself another [[cubic function|cubic equation]]
<math display="block">\underline{E}\, \underline{L}_c^3 + \left(6 \underline{E}^2 \underline{a}^2 + \frac{1}{2}\right)\underline{L}_c^2 + \left(12 \underline{E}^3 \underline{a}^4 + 20 \underline{E} \underline{a}^2 \right) \underline{L}_c + \left(8 \underline{E}^4 \underline{a}^6 - 16 \underline{E}^2 \underline{a}^4 + 8 \underline{a}^2\right) = 0,</math>
where underlined parameters are dimensionless in [[N-body units|Henon units]] defined as <math>\underline{E} = E r_V / (G M_0)</math>, <math>\underline{L}_c = L_c / \sqrt{G M r_V}</math>, and <math>\underline{a} = a / r_V = 3 \pi/16</math>.
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