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In a hyperbolic orbit/trajectory, if you know the distance traveled, and the initial true anomaly in the orbit, how do you figure out the new true anomaly? You know the mass, eccentricity, axis values and other misc information as well. I cannot find any resources on this and cannot figure it out.
This only needs to be in 2 dimensions.
Thank you
Yes, it's messy arc length calculus.
We can start with the relation between radius and true anomaly, which is rendered here in a form applicable to all conic section orbits by using the periapsis $p$:
$r=\dfrac{p(1+e)}{1+e\cos\theta}$
This is to be combined with the arc length differential
$(\dfrac{ds}{d\theta})^2=r^2+(\dfrac{dr}{d\theta})^2$
Direct substitution leads to
$\dfrac{ds}{d\theta}=\dfrac{p(1+e)\sqrt{(1+e\cos\theta)^2+e^2\sin^2\theta}}{(1+e\cos\theta)^2}$
$=\dfrac{p(1+e)\sqrt{1+e^2+2e\cos\theta}}{(1+e\cos\theta)^2}$
Unless $e=0$ (circular orbit) or $e=1$ (parabolic orbit), integrating this requires elliptic functions, and that is a bridge too far for me. More practical for your problem seems to be leaving the relation in differential form, updating the derivative with each timestep.
