# Design of return trajectory from lunar polar orbit to Earth's atmosphere

Given a flight path angle, altitude and velocity at re-entry into the Earth's atmosphere, how do I go about designing the return trajectory from a polar orbit at 17km altitude about the lunar surface to intercept with the Earth at the aforementioned requirements ?

'Fundamentals of astrodynamics' discourages using patch conics... Is there any other way ?

• related: how to best patch my conics? – Russell Borogove May 30 at 17:02
• According to this paper patched conics were used for the Voyager and Galileo missions as a starting point for numerical simulation. That is: solve your trajectory using patched conics, take that solution into a full N-body simulation, and make small corrections to it until it's acceptable for your purposes. – Russell Borogove May 30 at 17:07

'Fundamentals of astrodynamics' discourages using patch conics...

While I would also discourage you from relying on patched conics I would certainly encourage you to try using patched conics as a very useful, educational exercise!

Patched conics is a methodology based on Keplerian orbits. Remember that in this case the Earth and Moon move in elliptical Keplerian orbits around a common center of mass which is almost 5000 kilometers away from the center of the Earth!

If you feel you can neglect the rotation of the Earth-Moon system around the Sun for a few days, then you can certainly try this method as a first approximation, but choose your conics carefully and make sure that when you patch between an Moon orbit and an Earth orbit that you get the relative motion correct.

Is there any other way?

Yes there certainly are!

A strategy I would suggest would be to use some patched conic solver to generate an initial orbit and the impulsive maneuver, then try to port that to a numerical simulation in the Earth-Moon system.

It's not easy, it take some 3D thinking and math to translate it to the Earth-Moon system, but you can then see how far off the patched conics solution is in the real word. Then by fine-tuning the initial orbit and the impulsive burn you can converge on a real numerical solution.

Once that works, you can add the motion of the Earth-Moon system around the sun and see once again how far off you are, and tweak further to recover your atmospheric reentry.