RSS Cape Canaveral to equatorial plane change

[NFunky]

Hey guys, haven't posted in a long time cause I've been completely revamping how I play KSP.  Got RSS, Real Fuels and basically every realism mod except RO itself.

Anyway, I've got a contract to launch a satellite into equatorial orbit.  I've launched other satellites already, including a two polar sats, but this is the first equatorial sat I've tried to launch.  In this career game, I'm only launching from Cape Canaveral or Vandenberg, so I'm going to have to make a quite significant plane change maneuver.  Minimum inclination I can seem to get from the Cape is around 27 degrees, maybe 26.x if I'm lucky.

So here's my question.  At the contract orbit (~1,336 km), it takes about 3200-3300 dV to zero my inclination.  I know that it is sometimes more efficient to push my apoapsis on the ascending/descending node way out before doing the plane change, so I thought I'd give that a try.  I'm wondering what the appropriate AP would be though, or if it would even be worth this extra maneuver.  How big of a plane change do I need to be making to make this technique wothwhile?  Also, how do I go about calculating the ideal AP for a given plane change?

Thanks!

[Plusck]

Well...

To start with, escape velocity is sq.root(2) x your current orbital velocity. Therefore in a perfect universe, since you'd have to burn almost to escape and then burn back after the plane change (giving 2x 41.4% your current orbital velocity in burns), you'd want to burn right to the edge of the SOI if your delta-v to make the plane change is more than 83% of your current orbital velocity. Obviously this is less than 100% accurate because a plane change at the edge of the SOI will not be free, while at infinity in a perfect universe it would be, but still, it should hold true enough as long as your starting orbit is relatively low.

And if it is less than 83%?

Call your original orbital velocity v0. When you increase your Ap, your velocity at Ap will be v1. Call your initial radius r0 and let it be equal to 1, so your radius at Ap is  r1 and is therefore a multiple of r0. Dont forget this is radius, not altitude.

The delta-v cost of a plane change = 2 * v * sin(angle/2). If you know the angle, then you have your figure for delta-v cost for a plane change. And from that you can deduce that if your plane change exceeds 47°, then you really should go to the edge of the SOI to do it. A plane change of 60° costs 100% of your orbital velocity.

But you don't actually need to know the figure for the delta-v cost - just the figure for your Ap velocity (v1), because the gain you make will be simply = 1 - v1/v0.

This page http://www.braeunig.us/space/orbmech.htm gives lots of helpful equations because you want to know the delta-v to raise your Ap, and the value of v1 at that Ap.

The delta-v for the first part of a Hohmann transfer is: dv = sq.root(GM/r0) * (sq.root(2r1/(r1+r0)) -1).

However, sq.root(GM/r0) is our initial velocity, and r0 = 1, so this simplifies to dv = v0 * (sq.root(2r1/(r1+1)) -1). And we need to do this twice, so our Hohmann transfers will cost us:

= 2 * v0 ((sq.root(2r1/(r1+1)) - 1).   As r1 tends to infinity, r1/(r1+1) tends to 1 therefore the cost of attaining escape velocity and returning is 2* v0 ((sq.root(2))-1). 

And finally, what our Ap velocity will be:

Ap velocity = sq.root(GM(2/r1 - 1/atx)). Where atx is the semi-major axis of the transfer elipse, and atx = (r0+r1)/2.

Since v0 = sq.root(GM/1), this again simplifies down to:

v1 = v0 * sq.root(2/r1 - 2/(r1+1)).

So if you double your radius, your plane change will be made at 0.58 * v0 and you "save" 0.42x your initial plane change dv, but it will cost you 0.31x your initial velocity on the transfers. The point at which 0.42x dv =  0.31x initial velocity is for a plane change of about 43°.

 

Therefore (assuming I didn't miss something above), the gains are zero or largely irrelevant until you get past about 40° plane change.

[NFunky]

 

 

That was awesome! Thank you. That was very well explained.

[numerobis]

It's a bit more complicated: If you're going to boost your Ap, you will do that at the equator. In the same burn as you boost the Ap, you can burn a bit normal as well, improving your inclination. For instance, burn 30 degrees normal of prograde and you'll have 86% of your thrust boosting Ap, plus 50% fixing inclination. So now you have less inclination to fix.

You can have your launch profile take this into account: launch to a very low Ap over the equator. When you get there, circularize, boost Ap to the target orbit, and burn normal a bit -- all at once. Once you get to the target orbit Ap, boost Pe and burn antinormal the rest of the way. Three burns, two of them combine prograde and normal components.

[Plusck]

2 hours ago, numerobis said:

It's a bit more complicated: If you're going to boost your Ap, you will do that at the equator. In the same burn as you boost the Ap, you can burn a bit normal as well, improving your inclination. For instance, burn 30 degrees normal of prograde and you'll have 86% of your thrust boosting Ap, plus 50% fixing inclination. So now you have less inclination to fix.

You can have your launch profile take this into account: launch to a very low Ap over the equator. When you get there, circularize, boost Ap to the target orbit, and burn normal a bit -- all at once. Once you get to the target orbit Ap, boost Pe and burn antinormal the rest of the way. Three burns, two of them combine prograde and normal components.

Absolutely, yes. Would this not also advocate a steeper launch profile and a higher circularisation burn at Ap, while still at suborbital speeds? Though gravity losses will be mounting up at the same time...

Seeing what you wrote, I went back and read OP's question again, noticing a bit I missed last night. The equations that I gave are only for a simple case where you're already in orbit, at more or less the intended target orbit. Indeed, if you already have to raise your Ap significantly, then clearly you want to combine the manouvres. Also, as pointed out in that link that I gave (http://www.braeunig.us/space/orbmech.htm), IRL insertion burns can also be done in three steps to save fuel, going higher than the target orbit to reduce the plane-change cost.

So yes, the "only worth it if above about 40°" rule only really applies if you're already at the target level. If you have to raise your Ap anyway, and all the more so if you are not even off the ground yet, then there are benefits to going higher, even if the plane change is much less than 40°. Might take a while to work out in actual figures, though.

 

edit: There is no way I'm going to even attempt working out how to find the best orbital manouvres. I have forgotten so much maths... 

Since my calculus (and even my geometry...) now sucks, I just used Excel to get the figures for simple plane changes. The cheat's way out. However, it struck me that it might be useful to fiddle with the numbers so that you can see what pops out, so here it is. Should be self-explanatory. And no macros.

Edited January 12, 2016 by Plusck

[numerobis]

Hold on, I'm only seeing you needing 1226 m/s to zero out inclination at that altitude. I can't reproduce your 3200 m/s.

Let's say you're going from 70km to your target altitude, and doing the inclination burns. The Hohmann transfer costs 503 to raise Ap, 382 to circularize. You can burn normal as much as you want in the first burn, and in the second you'll finish changing inclination. Here's a graph of how much total it'll cost you to change your orbit (raise altitude and zero inclination) versus how much normal burn you do in the first burn:

So in your first burn, you should boost Ap and also burn about 150 m/s normal (which knocks off about 4 degrees of your inclination). The second burn finishes the job. Combining the two tasks saves you nearly 1000 m/s compared to doing them separately.

[Plusck]

He's using RSS, which changes orbital speeds significantly, though the principle will be identical.

[numerobis]

Oh, of course!

And RO has a wiki page on this very subject: https://github.com/KSP-RO/RealismOverhaul/wiki/Equatorial-Orbits-From-Launchsites-Off-The-Equator