I've been trying to model the orbit of a comet of arbitrary mass around our sun from having a known initial condition with $r_0, v_0$ with angle $\alpha$ at $t_0$ around our sun.

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I've been able to derive the radius in the form

$r=$ ${c}\over{\epsilon \cos(\phi + \delta)+1}$

With c, and $\epsilon$ corresponding to their meanings in conic sections (effective radius and eccentricity).

My problem is almost done, I just wanted to know if there was a geometric relationship between $\delta$ and $\alpha$ I could exploit? There is an energetics way of solving this but a geometric method to find the tangent line as a function of angle from foci seems like something that should be solvable.


The complement of $\alpha$ is known as the heading angle ($\psi$). The supplement of $-\delta$ is known as the true anomaly ($\nu$).

$ \psi = \frac{\pi}{2} - \alpha $

$ \nu = \pi + \delta $.

The heading angle as a function of the true anomaly and eccentricity is given by $$ \psi = \arctan\left( \frac{ e \sin(\nu) }{ 1 + e \cos(\nu)} \right) $$

More equations for the heading angle here.

  • $\begingroup$ you are awesome $\endgroup$ – Skyler Jan 30 '15 at 9:17

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