How to calculate time needed for spacecraft to travel a given number of meters in elliptical orbit or hyperbolic trajectory?

We have elliptical orbit and spacecraft travelling it. I need a formula to calculate a time needed for spacecraft to travel given amount of meters starting from it's current position.

Keplerian elements of orbit are known.

If I know difference between current and future true anomaly, I can use formula of orbital period to calculate that time but how to find the future anomaly by given orbital ellipse chord length (closest point) and initial true anomaly?

And also I need the same calculations for hyperbolic trajectory

• I don't think there's any closed-form formula for arc-length along an arbitrary ellipse., and you'll likely have to result to numerical approximations. What is it that you are planning to do with this value? There may be much simpler options that give the results you want. Commented Nov 18, 2021 at 13:02
• @notovny We can replace arc length by chord length. Is it calculable? Commented Nov 18, 2021 at 13:07
• As a starting point, what you want is an incomplete elliptical integral of the second kind. Some tools (e.g., Mathematica, MATLAB) provide implementations. (Most do not.) This starting point will provide the arc length from one time to another (alternatively, with a different formulation, arc length from one true anomaly to another). That's why I wrote "starting point". You want the inverse function of an incomplete elliptical integral of the second kind. I don't know of any tool that provides that. Commented Nov 18, 2021 at 14:29