Equations for calculating infalling orbit intercept?

[jedidia]

So, I'm currently devising a very simple trajectory solver for my game. The difficulty is that the game should actually calculate trajectories and maneuvers based on the character's skill, which is a bit more tough than just giving the a player a couple of sliders to fiddle with their DV themselves and see where they end up.

So far the going is tough, but steady. I've got two-tangent transfers (Hohmann) figured out completely, both outwards and inwards, and I have a generic solution for calculating outwards one-tangent burns. For the most part, the equations from http://www.braeunig.us/space/orbmech.htm have been invaluable.

However, I'm hitting a snag with inwards one-tangent burns, because said website doesn't cover them. I'm also facing the problem that I don't really know what I'm doing, just dropping equations I don't understand into a context which I halfways do. I understand enough of what's going on to say that equations 4.67 to 4.71 (still on this website) are not going to work just like that when applied to what is effectively the inverse problem (an orbit coming from a higher apoapsis intersecting an orbit with a higher periapsis), but I don't understand them enough to have any chance at rearranging them.

If anybody could give me that set of 5 equations from 4.67 to 4.71 adapted for the problem of a higher orbit intersecting a lower while falling in, I would be extremely grateful. I figure that at least 4.67 and 4.68 need to be rearranged, the rest might actually work out once those two return the correct result, but I'm far from sure of that...

Note: Origin and target orbits can be considered circular for this. I'm not going to make my life unnecessarily complicated...

 

[MontBlanc2012]

Ok, not entirely clear on what you want - but it seems to be the following: let's suppose that you start in a circular (outer) orbit and at some point execute a retrograde burn (tangent to your orbit) at some arbitrary point on that orbit and assuming that the transfer orbit intersects with some known inner (circular orbit), how does one compute the magnitude and direction of the burn (relative to the tangent vector of the transfer burn) at the point of intersection needed to enter into the circular orbit. Is that basically it?

 

[jedidia]

how does one compute the magnitude and direction of the burn (relative to the tangent vector of the transfer burn) at the point of intersection needed to enter into the circular orbit. Is that basically it?

Not just the the burn, I also need the true anomaly of the spacecraft when it intersects the inner orbit (or the time it needs to get there from apoapsis... one can be used to calculate the other). I need that to calculate the time of the transfer burn so I actually hit the target at the intersection. But that's essentially it, yes.

 

[MontBlanc2012]

OK, see if the attached file is any use. I haven't rigorously checked for errors, but I've done a couple of spot checks and it seems to work.

 

[jedidia]

Oh wow, that's very detailed! Thanks a lot! I'll be over there plugging all that into my code for a while...

 

[jedidia]

Thank you very much! I have not implemented the whole DV thing yet, but I've gone through my code with your equations at hand and managed to correct it so I now get valid target intercepts when running it against orekit, so the hard part is done.
All in all, the equations I was using were correct, but your detailed instructions lead me to the error I had in there:

But we actually want the time-of-
flight from the retrograde burn which was at orbital apoapsis. So, we will need to take off half an orbital
period off the time-of-flight from orbital periapsis to compensate for this

That was essentially my core problem That, and me not knowing that the equation for calculating the eccentric anomaly from true anomaly gives a quadrant-relative angle, not an absolute angle, a fact I realised when I implemented this correction and realised that I was still getting the same eccentric anomaly even though my true anomaly now looked fine.

Again, thank you very much! You've saved me a whole lot of time!

 

[MontBlanc2012]

No problem. If I had had to guess from your original post, I would have been inclined to think that nuances of inverse trig functions and using Kepler's equation to solve for the time-of-flight would have been the culprits - they certainly have caused me a lot of angst over the years. In resolving your problem, it seemed easier if I just set out the logical flow of a solution from start to finish in the hope that this might trigger an 'aha!' moment. Which it seems to have done. So, great - job done!

Just one other comment though. The 'theta' calculated at the end of my note is not the burn angle of the entry burn into the circular orbit. That would be the calculated as the angle between the computed burn vector (DV') and the prograde direction on the transfer orbit (vr, vt). The 'theta' is merely the angle between the prograde direction and a local horizon (essentially a pitch of your prograde direction as you approach the target orbit). I wasn't sure which of these you wanted. Just a note of caution.