lowest dV way to get to circular equatorial orbit from highly eliptical - non 0 APe

[KerikBalm]

I get the impression you're hung up on your APe because it happens that your roid's orbit has both its AN and DN close to your Pe, which is close to Kerbin, so doing plane change burns at either one would be expensive. But the thing is, this has nothing to do with your APe. It's instead a function of your Pe's altitude and your inclination.

It does have to do with the APe. If APe was 0, then the plane change should be done at apoapsis for minimum dV. In the case of APe=0 or 180 either the ascending, or the descending node will be at apoapsis. But with APe != 0, its more complicated. I guess I should still try to reduce the velocityof one/both of those nodes, and do the burn there.

APe is defined as the angle between LAN and Pe as measured in the plane of your orbit.

No, it is defined as the angle between the AN and Pe. The longitude of the AN is not relevant.

APe's purpose is to help describe the direction in which your oribt's axes are pointing relative to a fixed, celestial coordinate system.

Not really a complete coordiante system, just a reference plane, and its not an arbitrary plane, or at least I would argue the equatorial plane is not arbitrary or irrelevant, given the reduced launch costs and most other bodies orbit in that plane or nearly so.

This is why some folks have suggested you circularize at Ap. While this increases your major axis, it also increases your minor axis. Increasing the major axis moves the Pe further from your AN and DN. Increasing the minor axis moves the AN and DN further away from Kerbin. The combination of these effects makes it cheaper to do a plane change burn at AN or DN.

Well, as I see it, the point is to make the velocity at the AN/DN as low as possible, so its as cheap as possible to change.

The relevance of a non-zero/180 APe is that you do the circularization at apoapsis.

This step would be skipped for an APe of zero.

Thus the relevance of APe != 0 is the relevance of the circularization burn.

Btw, you earlier made mention of flintknapping - I used to do a little of that myself. May I assume you use "old school" tools such and deer antler?

Do you prefer obsidian or flint? I always had and easier time coming by obsidian where I'm from.

[KerikBalm]

So I just set up a craft into an orbit with a high APe (I was aiming for 45) and apoapsis beyond Minmus orbit.

Doing the plane change at the DN would cost 312 m/s, and then another 18 m/s to aerobrake

Total: 330 m/s

Circularizing at the Apoapsis would cost 169 m/s

Doing the plane change and dropping the PE down to aerobraking cost 220 m/s

Total for circularizing first and then plane change: 389.

I conclude that I should just pick the AN/DN furtherst from Kerbin, and do the plane change there, unless my APe is really high... as in more than 45 degrees

[Kasuha]

since it looks like one of your inclination points is already almost on Minmus level, probably simplest approach is to coast to that point, burn retrograde to stop all your orbital speed, then burn slightly east to raise your periapsis to just the right altitude in atmosphere suitable for aerobraking. It can be done in one burn but might be tricky to set up. And it should not need more than some 300 m/s.

I conclude that I should just pick the AN/DN furtherst from Kerbin, and do the plane change there, unless my APe is really high... as in more than 45 degrees

I think we agree on that

[KerikBalm]

"burn retrograde to stop all your orbital speed, then burn slightly east to raise your periapsis to just the right altitude in atmosphere suitable for aerobraking."

I don't think there is any need to "burn retrograde", I'm fine with having an AP 50% farther than minmus, I think just the normal/antinormal burn to match planes, and retro only as needed to prevent reaching escape velocity (shouldn't be needed at all, unless you are trying to increase APe), then once AP is reached, then drop the PE...

Sure, that takes a lot longer, but we're talking minimum dV, not fastest

[Yasmy]

How about you give us the orbital parameters to play with? Even if you are not yet docked to it, the asteroid's parameters are in your save files. Now, lets do some math...

Givens: mu and R for the parent body, Pe, Ap, APe (argument of periapsis) and i (inclination). Then

a = (Ap + Pe + 2R)/2

e = (Ap - Pe)/(Ap + Pe + 2R)

Let P2 be your target Periapsis.

Case 1: Plane change at the farthest node, followed by lowering periapsis from apoapsis. We don't know or care if it is the AN or the DN, so call it the SN, for slow node.

r_SN = radius at the slow node = a (1-e²) / (1 + e cos(APe)). (Actually, if pi/2 <= APe < pi/2, replace APe with pi-APe.)

v_SN = speed at the slow node = sqrt(mu (2/r_SN - 1/a))

Burn 1 is the plane change: ÃŽâ€V₁² = 2 v_SN² (1-cos(i))

Burn 2 lowers Pe to P2: ÃŽâ€V₂ = sqrt(mu (2/(Ap+R) - 2/(Ap + Pe + 2R))) - sqrt(mu (2/(Ap+R) - 2/(Ap + P2 + 2R)))

ÃŽâ€V = ÃŽâ€V₁ + ÃŽâ€V₂

Case 2: Circularize, followed by combined plane change and periapsis lowering burn:

Circularize: ÃŽâ€V₁ = sqrt(mu/(Ap + R)) (1 - sqrt((Pe + R)/a))

Now we are at v₂ = sqrt(mu/(Ap + R)) at inclination i, and want to go to v₃ = sqrt(mu (2/(Ap + R) - 2/(Ap + P2 + 2R))) at zero inclination:

ÃŽâ€V₂ = sqrt(v₂² + v₃² - 2 v₂ v₃ cos(i))

ÃŽâ€V = ÃŽâ€V₁ + ÃŽâ€V₂

Of course there are many variations on these cases, but these are the simplest two. For example, you could burn at Pe to push your Ap out first.

Maybe someone will feel like making plots of the two cases: for a few different eccentricities, make 2D plots of the difference between case 1 and case 2 vs APe and i...

Edited September 14, 2014 by Yasmy
Changed rSN again.

[Kasuha]

It depends a lot on how you will execute the burn and what are exact orbital parameters of the asteroid which you did not tell us. Plane change is the cheaper the slower you're moving at the moment. And you'll be burning retrograde to lower your periapsis anyway. So maybe not completely stop, but fixing the periapsis first and inclination second, or doing the whole thing in single straight burn (which will have substantial retrograde component) is probably best.

[KerikBalm]

Plane change at AN/DN nodes is cheaper when you have less vertical velocity. Slowing the horizontal component won't help.

Your vertical component will be velocity * sin angle at the AN/DN

Slowing your total velocity (as with wider orbits) will reduce the vertical component

The vertical component is really all you need to change to match planes.

If APe is 90, then all your prograde velocity is also your vertical velocity. But at say... 45 degrees, a 10m/s burn only reduces your vertical component by ~7 m/s, at 30 degrees, only 5 m/s.

I don't understand why you'd want to reduce your horizontal component - seems like a waste of dV. Eliminate the vertical component, at apoapsis, drop the PE into the atmosphere... aerobrake a few passes, raise PE, done.

[Geschosskopf]

No, it is defined as the angle between the AN and Pe.

OK, I give up. You obviously like confusing yourself so I'll leave you to it.

[sonicsst]

OK, I give up. You obviously like confusing yourself so I'll leave you to it.

His definition of APe is correct.

Edited September 14, 2014 by sonicsst

[Red Iron Crown]

On this he's right and you're wrong.

They're both right. Geshosskopf's definition as "the angle between LAN and Pe as measured in the plane of your orbit" is correct, LAN's intersection with the orbital plane is the AN. KerikBalm's more commonly used definition of "the angle between the AN and Pe" says the exact same thing more succinctly. Some hair-splitting going on here.

[Kasuha]

Plane change at AN/DN nodes is cheaper when you have less vertical velocity. Slowing the horizontal component won't help.

Your vertical component will be velocity * sin angle at the AN/DN

Slowing your total velocity (as with wider orbits) will reduce the vertical component

The vertical component is really all you need to change to match planes.

If APe is 90, then all your prograde velocity is also your vertical velocity. But at say... 45 degrees, a 10m/s burn only reduces your vertical component by ~7 m/s, at 30 degrees, only 5 m/s.

I don't understand why you'd want to reduce your horizontal component - seems like a waste of dV. Eliminate the vertical component, at apoapsis, drop the PE into the atmosphere... aerobrake a few passes, raise PE, done.

Most importantly, I don't see the difference in literally few m/s between your secret orbit and my approximation of it as significant, especially since my result is lower. You just keep talking and theorising instead of simply trying it out and coming with numbers. I described the approach the way I described it because that's how I found it relatively comfortable to set up using a maneuver. Which is what you might want to do since turning a ship attached to an asteroid around during burn is usually not simple. But you're of course free to do it whatever way you please.

[metaphor]

In that orbital image it looks like one of the equatorial nodes is already pretty far out from Kerbin, so it might be best to just change inclination there.

There are other ways too, especially if both your AN and DN are really close to Kerbin. You can do an inclination burn near apoapsis to raise your inclination, so that the node on the other side of your orbit gets closer to apoapsis. Then it takes less delta-v to fix the inclination at the new node since it's farther away from the planet, even though the inclination angle change is higher. You can also use a bit of a prograde component as well at the first burn. This is similar to Kasuha's first suggestion except you don't have to fully circularize the orbit at apoapsis.

The optimal sequence of burns really depends on your specific orbital parameters. It's a nonlinear optimization problem.

[KerikBalm]

Most importantly, I don't see the difference in literally few m/s between your secret orbit and my approximation of it as significant, especially since my result is lower.

Its not about the magnitude of the difference, as I realize doing a burn at minmus orbit, or beyond minmus orbit, will not have massive differences.

Its more about the theory, which is why I haven't given specific numbers, its not because an orbit is such a secret.

Here are the orbital parameters for the numbers I arrived at, if that will make you happier:

ORBIT

{

SMA = 11396924520.3127

ECC = 0.2170260877019

INC = 1.50113290597947

LPE = 338.895144038711

LAN = 137.362053173262

MNA = 3.79013058444902

EPH = 3617460.69961997

REF = 0

}

After having thought about it some more, what one really wants is to have the AN be as far out as possible, and the DN as close as possible (or vice versa) so that the velocity at the node that you'll do the plane change is as low as possible. Assuming the AN is farther than the DN, this means burning at the DN to raise the AN, or as close to the DN as is possible without reaching escape velocity... which if the AP is already near the edge of the SOI, is basically at the apoapsis.

But this is for large plane changes, I guess it would be nice to know the "break even point" going to the edge of the SOI may be most efficient for a 90 degree plane change, but not a 1 degree plane change.

[Kasuha]

Its more about the theory, which is why I haven't given specific numbers, its not because an orbit is such a secret.

Theory is not clear cut, though principles are simple:

- the slower you move at inclination point, the less dv it takes to change the inclination

- it is usually beneficial to join multiple simple burns into one, sometimes even if it means performing neither of these at the place where it would be best for the burn alone

Steps you need to perform for optimum approach are IMO:

simple burn 1: move the inclination point to a better suitable place

simple burn 2: fix inclination at the inclination point

simple burn 3: lower your periapsis

In general I believe you need just two burns to do it. First burn will do complete "sb1" task and may do part of "sb2" and "sb3" tasks. Second burn will then align the inclination and lower the periapsis. Or perhaps you may really want to lower the periapsis separately. The theory does not go that deep. You need to do the math to get optimal approach, or at least some experimenting to get close to it.

[Tw1]

There is some truth in what the OP says, if your apoapsis and periapsis do happen to line up with the AN and DN, then it's perfect for plane changes.

The problem is, it's often not the case.

APe can't be manipulated directly. You can muck around with prograde/retrograde and zenith/nadir burns, and move the AP and PE, and normal or antinormal burns can move nodes to an extent, but none of this going to help set up an efficient plane change.

Go with Kashua's plan. Burning at AP efficiently raises the PE, high orbits are slower, that makes plan changes easier.

[Red Iron Crown]

MSPaint diagrams are the best diagrams.

[MarvinKitFox]

The optimal sequence of burns really depends on your specific orbital parameters. It's a nonlinear optimization problem.

That's the essence of it.

Kasuha's plan is totally *not* the ultimate in efficiency, but it is a good first approximation of it. Even in worst case, it requires only less than 1.4x the ultimate optimal delta-v.

(I base this on simply comparing the energetics between the two orbits, and utterly ignoring technique. *nothing* can reduce that as a minimum, without a third-party influence such as a SOI change, gravity sling, etc.)

Compared to the first-attempt approach of correcting inclination at the An/Dn, his method saves you heaps and gobs of fuel.

[einsteiner]

There is some truth in what the OP says, if your apoapsis and periapsis do happen to line up with the AN and DN, then it's perfect for plane changes.

Using a setup that looks like the OP's pictures, it takes 52.5 m/s to make the orbit equatorial. If it were in the optimal position (ie. Ap is on AN, meaning Ape=0), it would take only 48.67 m/s. So not being right on the Ap for the equatorialization burn loses you less than 4 m/s. Include a burn to reduce your periapsis to aerobrake (about 100 m/s) and you'll never miss it.

Here are the orbital parameters for the numbers I arrived at, if that will make you happier:

ORBIT

{

SMA = 11396924520.3127

ECC = 0.2170260877019

INC = 1.50113290597947

LPE = 338.895144038711

LAN = 137.362053173262

MNA = 3.79013058444902

EPH = 3617460.69961997

REF = 0

}

This isn't the right asteroid, this one is in orbit around the sun (REF=0).

Edited September 15, 2014 by einsteiner