Correcting errors to inject to another body

[Wayfare]

So, after travelling to Vall, I found my ship in a 700,000km parking orbit around Jool waiting for the return window to Kerbin. I chose this orbit to avoid unwanted encounters with any of Jool's many moons during the long wait. Because I was launching from Vall I chose to go wider out, as I didn't want to waste delta-v by going deeper inside Jool's gravity well. Good plan, no?

Well, no.

I've come to rely on the amazing Protractor to time my injection burns. However, there was no way for me to match up the angles due to my orbit having a whopping four day period. Obviously, over a distance this big, even a small deviation at exit becomes a big one at the expected rendezvous. So I tried and tried and tried but eventually gave up hoping for a rendezvous simply by getting the angles as close to ideal as possible.

Next I started fudging with some different angles and simply seeing where that would take me. I took measurements of how many days behind/ahead of Kerbin I'd arrive on different trajectories and tried to correct by burning more or less Delta-V during exit. It didn't work out (even after I realized that a year on Kerbin is only 106 days... d'oh). But it did lead me to the solution: I made an as-close-to-perfect-as-possible exit burn, timed the error, and burned that many days later. This still took a few tries on account of my long orbital period, but after two days of work I finally cracked it and brought my Kerbals home safe and sound.

My question is this: is there a smarter way to compensate for not being able to make a perfect exit burn? This whole trial-and-error thing was fun (and sometimes frustrating, but fun) but there must be some rule of thumb / napkin math way to figure out how you should adjust your exit burn if you can't get the planets angled perfectly. Right?

[Yeomans]

In kerbal space program, most people only think of prograde, retrograde, normal+, and normal- burns, but sometimes fiddling with your burn parameters can help you achieve a better result, if at the cost of a little more fuel. During a burn, and during the planning of a burn, there are 3 factors you must consider: The delta-v you are trying to achieve, the angle of the burn, and the time of the burn.

More delta-v is a higher orbit. The further you are from the center of mass in an orbit, the slower you move. Ergo, burning a bit longer will put you in a higher orbit, and you will then slow down, which is good when you want the target to "catch up" to you.

The angle of the burn is important. Because angling a burn is actually a point on a 3-dimensional plane, I will try my best to make it less complicated. You have a prograde, which is the most efficient method of achieving a higher orbit. You can however then move towards the radius+ and radius- positions (facing or facing away from the object). These burn directions have the advantage of shifting your periapsis and apoapsis towards or away from the object without changing your delta-v, you then have the retrograde burns, which decrease your delta-v and lower your orbit.

As far as inclination changes go, normal+ and normal- burns change the inclination. You will notice many bodies in the system are inclined. You have to account for the inclination at burn, and adjusting it can bring you closer or farther away from the target.

As far as the time of burn, the time where the easiest path can be taken is always preferred. Doing a Hohmann transfer orbit is recommended, but burning before or after the specified time can change your destination relative to the target, which is useful as well.

Ultimately, there are many things you must consider, and frankly, I would experiment with transferring from the moon to minmus and back to get a handle on things before interplanetary flight.

Good luck!

[Wayfare]

Thanks for the reply Yeomans. I get the basics of adjusting an orbit, though doing some more practice runs around the Mun sounds like a good idea.

What I'm asking for, though, is some guidelines on adjusting an interplanetary transfer when you're unable to burn in your perfect window. This window is determined by the planetary phase angle of the bodies you're travelling between, and the ejection angle at which you conduct your burn (modified for thrust/weight ratio). What modifications to a burn would you have to make if one or both of these angles cannot be achieved perfectly?

In my situation, I noticed that changing the ejection angle had fairly disastrous effects in terms of delta-v expended. So I decided to try and compensate for a planetary phase angle that was either 2 degrees short or 3 degrees over the ideal angle. Now I was expecting both situations to require me to expend more delta-v, catching Kerbin either a few days before or after the perfect intercept point. But try as I might, even the slightest adjustment resulted in over- or undershoots off by as much as forty days.

My final fudge worked, but I get the feeling I should have been able to solve it with some fairly simple maths, in terms of: "If your planetary phase angle is off by X degrees, adjust your ejection angle by X degrees to compensate."

[Yeomans]

Well, that's a bit more complicated. The ejection angle calculated is the angle at which you have to achieve the escape velocity. So if you aren't right on at first, keep thrusting and you should at some point intersect (according to the patched conics). But to readjust your ejection angle based on a different phase angle, that's very very complicated. But a good rule of thumb is "for every degree off, change your burn angle towards the planet by a tenth of the offset" (I hope that makes sense).

If you aren't quite sure what to do during the ejection burn, and your relatively close to an encounter, you can always do mid-course corrections!

[Wayfare]

Cheers, I'll give that rule of thumb a go if when I get my crew into another pickle like that

The biggest problem I found in trying to get back from Jool was that the slight inclination of its orbit led to several hundreds of millions of kilometers of "vertical" error on the intercept - and since I couldn't realistically alter that inclination until I was at a proper angle to the sun, I had to sit through a lot of maximum-warp transits to find out if my latest change to the exit path had done the trick...

[Yeomans]

Yeah, generally speaking, you want to align your inclination with Jool when you are launching from kerbin, because then it's just like flying from the mun to minmus.