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The radial turning points of an orbit characterized by [[specific energy]] <math>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>R^3 + \frac{GM}{E}R^2 - \left(\frac{L^2}{2E} + a^2 \right) R - \frac{GMa^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>\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 Henon units defined as <math>\underline{E}=Er_V/(GM)</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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