Bi-eliptic transfer windows

[Captain Sierra]

Yes, even vets need help from time to time.

It probably goes without saying that the phase angles for a standard Hohmann transfer are vastly different than those for a bi-eliptic. The question is: "What should the phase angle be and how to I quickly guesstimate it?"

If it matters to anyone, this is for a mission to Neidon of OPM. When the Semi-major axis differential between starting body and target body becomes that large, this kind of transfer isn't as bad as it usually is when compared to a traditional Hohmann transfer orbit.

[Superfluous J]

I've never given this any thought (and I've had more than my fair share of scotch tonight) but it seems to me that with a bi-elliptic transfer you don't need to worry about transfer windows so much, because you are in total control of your transfer time by means of your apoapsis. A very small change in your apoapsis will greatly affect your next periapsis, so you can pretty much eject any time you want.

[Yasmy]

If the starting or final orbits are not very circular, the math is a bit hairy. Here's how you do the simple case of circular orbits.

With a Hohmann transfer, you encounter your target 180 degrees away from starting position. It takes time T = pi sqrt(a³/mu) to go 180 degrees, where the semi-major axis is a = (r₁ + r₂)/2, where r₁ is the initial radius, and r₂ is the final radius. The target's orbital period is T₂ = 2 pi sqrt(r₂³/mu), so in time T, the target planet travels T/T₂ of an orbit, 360 * T/T₂ degrees. Thus the phase angle is 180 - 360 * T/T₂.

The math is the same for a bi-elliptic transfer. It simply takes another step. Let's say the intermediate step has radius r. Then the total transfer time is T_b = pi sqrt(((r₁+r)/2)³/mu) + pi sqrt(((r+r₂)/2)³/mu). Now instead of meeting your target 180 degrees away from your starting position, you meet it 360 degrees away. Thus the phase angle is 360 (1 - T_b/T₂) degrees.

Like 5H said, you can often pick r to make the phase angle more or less whatever you want, including whereever the target is at the moment relative to your starting location.

[FancyMouse]

I think bi-elliptic transfer has its transfer window at AN/DN - that's solid twice a year regardless of target inclination/period, and if inclination is close enough, then pretty much any time would work.

[Geschosskopf]

10 hours ago, Captain Sierra said:

Yes, even vets need help from time to time.

It probably goes without saying that the phase angles for a standard Hohmann transfer are vastly different than those for a bi-eliptic. The question is: "What should the phase angle be and how to I quickly guesstimate it?"

If it matters to anyone, this is for a mission to Neidon of OPM. When the Semi-major axis differential between starting body and target body becomes that large, this kind of transfer isn't as bad as it usually is when compared to a traditional Hohmann transfer orbit.

I'll answer your actual question in a minute but for the nonce, why use this to get to Neidon?  At the best configurations, it already takes like 17 years to get there the normal way so why add considerably more time to the flight?  A bi-elliptic to Neidon requires an orbit analogous to that of a long-period comet   Capturing at Neidon isn't that big a burn (a bit less than at Jool) even when arriving at a very steep crossing angle to its path because you're going so slow way out there.

But back to your question.....  With a bi-elliptic transfer, you first raise your Ap way out there above the target.  Then you wait until you get there and raise your Pe up to the target's altitude, planning to meet the target at your Pe.  Therefore, you will be intercepting the target after a complete orbit of the sun, with the point of interception more or less on the same line as between Kerbin and the sun when you left home.  Obviously, some radial +/- burns at Ap will slide your Pe around in an arc relative to the sun so you've got some wiggle room, but interception will be in say a 45^-wide pie slice centered on that line.

Therefore, the phase angle between Kerbin and Neidon would be that which will put Neidon more or less on this target line by the time you get back to it yourself.  This is a function of your orbital period in your intermediate high-Ap orbit and Neidon's orbital period.  Thus, to develop a way to eyeball this quickly, you need some info first.  I'd proceed as follows:

1. Learn Neidon's orbital period, which I think you can get in-game on the map view.  Once you know this, you can mentally picture how far Neidon will move in some fraction of that time.
2. Do a test mission where you send a ship from Kerbin to an Ap well out beyond Neidon, however far you want (or can afford with available dV).  Just make it an altitude you can easily remember so you can use it again for the real flight later.  Once out of Kerbin's SOI, note how long it will take you to reach your Ap and write this down..
3. Warp ahead until you're at your Ap, then raise your Pe to Neidon's level.  Note the time shown to reach your new Pe and add this to the time it took to reach your Ap.  This is the travel time it will take for your trip.
4. Determine the ratio of your ship's travel time to Neidon's orbital period.  This will tell you what fraction of its orbital path Neidon will move during your trip.
5. Once you know how far Neidon will move, you know what the phase angle will be and can eyeball it on the map view.

Example:  Let's assume that your travel time is 1/3 of Neidon's orbital period.  Therefore, you want to leave Kerbin when Kerbin is about 120^ ahead of Neidon. when compared to the sun.

That should be close enough for government work.  When doing the actual trip, I'd be sure to match planes with Neidon on the way out to Ap.  This will make getting the actual intercept much easier when I get to Ap.  At Ap, I would burn to raise my Pe to Neidon's orbit and use whatever radial +/- I needed to slide the Pe along Neidon's orbit to get the intercept.

This is, essentially, how I get from an Eve intercept to Gilly, except there I don't do any math or worry about phase angles.  If it takes me several orbits of Eve to synch up with Gilly, no problem, it's only a few weeks.  But with Neidon, waiting to synch up will take decades if not centuries so it's best to figure the phase angle.

1 hour ago, FancyMouse said:

I think bi-elliptic transfer has its transfer window at AN/DN - that's solid twice a year regardless of target inclination/period, and if inclination is close enough, then pretty much any time would work.

This works for going to Moho but I wouldn't do this for Neidon.  Moho's orbital period is so low that you don't care about WHEN you get there and can wait a few orbits to synch up.  But Neidon is too slow for this to be practical.

Edited January 31, 2016 by Geschosskopf

[Snark]

8 hours ago, Geschosskopf said:

I'll answer your actual question in a minute but for the nonce, why use this to get to Neidon?  At the best configurations, it already takes like 17 years to get there the normal way so why add considerably more time to the flight?

Yah, it's a looooong trip.  I "solve" this problem by being terminally impatient, and launch with far more dV than needed, so that I cut the trip down to ~3 years.  Yes, that's insanely un-economical.  Yes, it requires some creative approach to reentry design.  But it nicely solves the launch window problem, just take off when Kerbin is traveling in Neidon's direction and fire away. 

[Geschosskopf]

1 hour ago, Snark said:

Yah, it's a looooong trip.  I "solve" this problem by being terminally impatient, and launch with far more dV than needed, so that I cut the trip down to ~3 years.  Yes, that's insanely un-economical.  Yes, it requires some creative approach to reentry design.  But it nicely solves the launch window problem, just take off when Kerbin is traveling in Neidon's direction and fire away. 

It's a long way to Planet Neidon
It's a long way to fly
It's a long way to Planet Neidon
To the very edge of the sky
So long to sunny Kerbol
You'll barely see him there
It's a long, long way to Planet Neidon
But you'll love it, I swear