Plummer model: Difference between revisions

Revision as of 01:28, 14 May 2020

Revision as of 16:42, 14 November 2021

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For the Plummer profile: <math>R_{1/2}=a</math>.

The escape velocity at any point is

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>v_{\rm esc}(r)=\sqrt{-2\Phi(r)}=\sqrt{12}\,\sigma(r) ,</math>

TheFor bound orbits, the radial turning points of anthe orbit is 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_0}{E} R^2 - \left(\frac{L^2}{2E} + a^2\right) R - \frac{GM_0a^2}{E} = 0,</math>