I'm creating a simulation for orbiting bodies with my primary challenge being calculating the true anomaly of a trajectory with respect to time. I have figured out circular, elliptical, and parabolic orbits, but i'm struggling with hyperbolic.

Everything I have found in my searches provides equations relating true anomaly v, to hyperbolic anomaly F, but nothing to determine F on its own.

The closest i've found to a solution is from this site. With the equation But deriving a solution was a little beyond me.

Can anyone help me find or derive the equations to calculate the true anomaly of a hyperbolic trajectory with respect to time? Thanks!


The solution is the same as the elliptic case, except: $$ t = \sqrt\frac{a^3}{GM} (\theta - e\sin(\theta))$$ becomes: $$ t = \sqrt\frac{a^3}{GM} (e\sinh(\theta)-\theta)$$

and $$ \nu = 2 \arctan\left( \sqrt{ \frac{1-e}{1+e} } \tan\left( \frac \theta 2 \right) \right) $$ becomes: $$ \nu = 2 \arctan\left( \sqrt{ \frac{e+1}{e-1} } \tanh\left( \frac \theta 2 \right) \right) $$


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