Optimal Plane Change

[Wcmille]

If a spacecraft is travelling in a circular orbit around a planet with a gravitational parameter of G at radius R1, and wishes to make a 90 degree planes change to a circular orbit of R2, what is the optimal burn strategy?

For example, a simple inclination burn followed by a circularization burn might be correct. If the gravity is very high, it might be correct to push the orbit far away from the planet do the plane change, and then come back. Since there are two burns, there are opportunities to spread the inclination change across both burns. How can an optimal split be made?

 

[Norcalplanner]

25 minutes ago, Wcmille said:

If a spacecraft is travelling in a circular orbit around a planet with a gravitational parameter of G at radius R1, and wishes to make a 90 degree planes change to a circular orbit of R2, what is the optimal burn strategy?

For example, a simple inclination burn followed by a circularization burn might be correct. If the gravity is very high, it might be correct to push the orbit far away from the planet do the plane change, and then come back. Since there are two burns, there are opportunities to spread the inclination change across both burns. How can an optimal split be made?

 

There's actually three burns using the preferred technique, which would be: burn 1 - mostly raising Ap (far beyond R2) with a little bit of plane change; burn 2 - at Ap, conduct nearly all of the remaining plane change while raising Pe to R2; burn 3, at R2 Pe, lower Ap and conduct last little bit of plane change.  Unfortunately I'm not well versed in the math, so I'm afraid I can't give you precise numbers.  My gut says the split would be no more than 7 or 8 degrees of inclination change each on burns 1 and 3, with the majority of the change (74 to 76 degrees) taking place in burn 2. 

[Plusck]

IIRC, any plane change exceeding 43° (or was it 47°... sorry can't remember!) is most efficiently done by raising Ap to the very edge of the SOI and doing the plane change there.

As @Norcalplanner says, burning slightly off prograde to add a percentage of the plane change is best practice and most efficient. It's also what satellite-emplacing orbital stages do (though they never go so far out). However, if you're in a relatively low orbit and you are really going to go all the way to the SOI edge (huge time hog, but certainly most efficient for a 90° plane change) then I wouldn't bother since even a 5% loss on the prograde burn will exceed the cost of doing the whole 90° change out there.

Edited August 28, 2016 by Plusck

[Norcalplanner]

3 minutes ago, Plusck said:

IIRC, any plane change exceeding 43° (or was it 47°... sorry can't remember!) is most efficiently done by raising Ap to the very edge of the SOI and doing the plane change there.

As @Norcalplanner says, burning slightly off prograde to add a percentage of the plane change is standard practice. It's what satellite-emplacing orbital stages do. However, if you're in a relatively low orbit and you are really going to go all the way to the SOI edge (huge time hog, but certainly most efficient for a 90° plane change) then I wouldn't bother since even a 5% loss on the prograde burn will exceed the cost of doing the whole 90° change out there.

I was thinking no more than 10 degrees of normal/anti-normal added to the pro-grade burn, since that will only cost around 1.5% of your prograde delta V while still creating a 17% inclination change vector.  I know just enough about math to be dangerous... :-)

[OhioBob]

41 minutes ago, Norcalplanner said:

My gut says the split would be no more than 7 or 8 degrees of inclination change each on burns 1 and 3

About 2 degrees.

Edited August 28, 2016 by OhioBob

[Plusck]

16 minutes ago, Norcalplanner said:

I was thinking no more than 10 degrees of normal/anti-normal added to the pro-grade burn, since that will only cost around 1.5% of your prograde delta V while still creating a 17% inclination change vector.  I know just enough about math to be dangerous... :-)

Sure - sorry I edited my post immediately (but apparently not before you quoted it...) because it sounded a bit like I was disagreeing with you, which was not my intention.

I tried to clarify a bit that my suggestion to "not bother" was really talking about starting at in a low orbit: 950 m/s to SOI edge followed by a 40 m/s plane change. A higher starting orbit will both reduce the total burn and increase the plane-change cost, so adding 8-10 a few* degrees on the first and third burns will meaningfully reduce the total cost.

*this edit was certainly not made to try to sound more knowledgable after the fact...

Edited August 28, 2016 by Plusck

[Norcalplanner]

Just now, Plusck said:

Sure - sorry I edited my post immediately (but apparently not before you quoted it...) because it sounded a bit like I was disagreeing with you, which was not my intention.

I tried to clarify a bit that my suggestion to "not bother" was really talking about starting at in a low orbit: 950 m/s to SOI edge followed by a 40 m/s plane change. A higher starting orbit will both reduce the total burn and increase the plane-change cost, so adding 8-10 degrees on the first and third burns will meaningfully reduce the total cost.

Then again, OhioBob has spoken.  Two degrees is the magic number.

[OhioBob]

9 minutes ago, Norcalplanner said:

Then again, OhioBob has spoken.  Two degrees is the magic number.

It really depends on how high the Ap is raised.  If the Ap is raised to only a couple thousand kilometers, then you might be able to include as much as 3 degrees in burns 1 and 3.  If the Ap is raised all the way out to Mun or beyond, then probably only about 1 degree.  However, we're talking about small differences.  If you did a couple degrees when the optimum is only 1 degree, you really barely added anything to your total delta-v.

 

Edited August 28, 2016 by OhioBob

[Plusck]

4 minutes ago, Norcalplanner said:

Then again, OhioBob has spoken.  Two degrees is the magic number.

Amen.

[GoSlash27]

I've wondered about the "optimal plane change problem" for a while, but never took a serious look at it.

I took a stab at it this morning and came up with a surprising result (at least to me): It isn't worth leaving low Kerbin orbit for inclination changes less than 53.8°, but changes 53.8° or more are most efficient by running all the way up to the SoI. There doesn't seem to be any point where it is most efficient to raise apoapsis to any less than the SoI.

 What I did was look for the "break-even point", where raising the Ap, doing the plane change, and retroburning back down costs the same as doing the inclination change at low level.

2DVprograde+DVrad/degree hi = DVrad/degree low

I simplified the math a bit. DVrad/degree would normally be "2Vsin(theta/2)", but I cheated and called it "thetaV/60".

Reorganizing the equation, I was able to find close approximations of the cutoff point by %orbital velocity prograde".

Rather than finding a progressive function, I found a plateau where the global minima were low orbit and SoI.

 

I haven't had much coffee this morning, so I'm interested in seeing how I've screwed it up

 

Assuming I *haven't* screwed this up, my answer would be to raise my apoapsis all the way out to the SoI, perform my inclination change and periapsis increase, then circularize at the new periapsis. I wouldn't do *any" of the inclination change at r1 or r2.

 

Best,
-Slashy

 

Edited August 29, 2016 by GoSlash27

[Red Iron Crown]

Relevant thread for raising Ap to change inclination:

>60 degree change, raise Ap as much as possible. 60>change>39 degrees, raise Ap somewhat less (formula given in thread). <39 degrees, don't raise Ap.

[OhioBob]

57 minutes ago, Red Iron Crown said:

>60 degree change, raise Ap as much as possible.

But there's a diminishing margin of return as the Ap gets pushed higher and higher.  This means there is a practical limit if not a mathematical one.  I would say that the excessive time needed to execute the maneuver makes pushing the Ap out much past the orbit of Mun not worth the delta-v savings.  For instance, the difference between executing a 90-degree plane change at the moon's orbit is only ~65 m/s more the performing it at the SOI.  And for a 60-degree plane change the difference is negligible, only a couple m/s.

Edited August 29, 2016 by OhioBob

[Red Iron Crown]

1 minute ago, OhioBob said:

But there's a diminishing margin of return as the Ap gets pushed higher and higher.  This means there is a practical limit if not a mathematical one.  I would say that the excessive time needed to execute the maneuver makes pushing the Ap out much past the orbit of Mun not worth the delta-v savings.  For instance, the difference between executing a 90-degree plane change at the moon's orbit is only ~65 m/s more the performing it at the SOI.  And for a 60-degree plane change the difference is negligible, only a couple m/s.

That's very true. I assumed the OP was asking about optimizing for delta-V during the plane change. If time is a factor then raising Ap for plane changes can be a losing strategy (same with bielliptic transfers).

[GoSlash27]

I've plugged the numbers into my model, and they check out. Perhaps some of you folks could verify or refute them?

For a 53° inclination change, I have remaining in LKO as the most efficient maneuver. 2,048 m/sec inclination change.

For a 54° inclination change, I have a 3 burn strategy all the way out to the SoI as the most efficient maneuver. 938 m/sec prograde, inclination change of 185 m/sec, and 938 m/sec retrograde = 2,061 m/sec total DV.

My math says that there is no case where an intermediate apoapsis will be more efficient than one of these two... at least not when starting from LKO.

Thanks!

-Slashy

[OhioBob]

@GoSlash27, I get what looks like a exponential function that confirms what @Red Iron Crown posted.  (There's no point in extending the curve beyond a 60^o plane change because we've reached the SOI.)

I'm not seeing the abrupt change between 53^o and 54^o that you are.

I'm using the vis viva equation to compute the Δv for raising and lowering the apoapsis, and this formula for computing the plane change, Δv = 2*v*sin(θ/2).

 

Edited August 29, 2016 by OhioBob

[foamyesque]

That assumes a Kerbin plane change. If I'm goofing about in the Kerbin system I'd just as soon launch into to the correct plane.

[OhioBob]

2 minutes ago, foamyesque said:

That assumes a Kerbin plane change. If I'm goofing about in the Kerbin system I'd just as soon launch into to the correct plane.

Same here.  When I make plane changes around Kerbin they are almost always small, thus I don't mess around with raising my apoapsis.

[OhioBob]

3 hours ago, GoSlash27 said:

For a 54° inclination change, I have a 3 burn strategy all the way out to the SoI as the most efficient maneuver. 938 m/sec prograde, inclination change of 185 m/sec, and 938 m/sec retrograde = 2,061 m/sec total DV.

(emphasis mine)

The plane change is where we are so different.  I get 938 m/s to transfer from 70 km to SOI, so it looks like we're the same there.  But for a 54 degree plane change at apoapsis, I calculate a Δv of only 23.4 m/s.  I think you have an error in there somewhere.
 

Quote

For a 53° inclination change, I have remaining in LKO as the most efficient maneuver. 2,048 m/sec inclination change.

I agree with the 2048 m/s number as well, so there's something funny in either the way you are calculating the plane change at apoapsis, or in the way you are computing the velocity at apoapsis.

For a 70km x SOI orbit, I compute an apoapsis velocity of 25.7 m/s, so a 54^o plane change is, 2*25.7*sin(54/2) = 23.4 m/s.
 

Edited August 29, 2016 by OhioBob

[GoSlash27]

42 minutes ago, OhioBob said:

The plane change is where we are so different.  I get 938 m/s to transfer from 70 km to SOI, so it looks like we're the same there.  But for a 54 degree plane change at apoapsis, I calculate a Δv of only 23.4 m/s.  I think you have an error in there somewhere.

Hmm... Let me dig a little deeper into the numbers for that case...

Starting from 70km altitude, Vorb is 2295.8 m/sec. Pushing the Ap up to 84,159,286m, the DV prograde is 938.2 and the Vap is.... Oh! *facepalm* I see where I screwed up. I used the orbital velocity for a circular orbit at Ap instead of the velocity at transfer.

Okay, I'll take it from the top and use the right numbers this time

Thanks!

-Slashy

 

Edited August 29, 2016 by GoSlash27

[OhioBob]

1 minute ago, GoSlash27 said:

Pushing the Ap up to 84,159,286m

Also, if you haven't already accounted for it, don't forget that that's the radius of the SOI, not the altitude.

[Kerbart]

23 hours ago, Norcalplanner said:

Then again, OhioBob has spoken.  Two degrees is the magic number.

“Two degrees is the magic number, and the magic number is two degrees.

Three degrees thy shalt not incline, and neither one, unless it is to proceed to two. Four is right out”

[OhioBob]

1 minute ago, Kerbart said:

“Two degrees is the magic number, and the magic number is two degrees.

Three degrees thy shalt not incline, and neither one, unless it is to proceed to two. Four is right out”

[GoSlash27]

Okay, so now that I've turned down the suck on the mixer board, here's what I get:

It's not worth it to raise apoapsis for any plane change less than 38.3°.

From 38.3° to 49.5°, It's pretty linear from zero DV added to escape velocity (as a function of %Vorb DV prograde)

Above 49.5°, always climb out to SoI and do the plane change there.

Sound about right? Note: I ran my figures from 70km.

Best,
-Slashy

Edited August 29, 2016 by GoSlash27

[Norcalplanner]

Looks correct to me.

After reading the other thread linked by RIC, it may be worthwhile to clarify that this assumes no aerobraking.

[Red Iron Crown]

5 hours ago, foamyesque said:

That assumes a Kerbin plane change. If I'm goofing about in the Kerbin system I'd just as soon launch into to the correct plane.

Have a look at the thread I linked, the parent body doesn't matter.