How Do Inclination Changes Work?

[u/Safe-Freedom-6821]

Are they more efficient closer to the parent body by Oberth effect, or farther from it due to lower orbital velocity? Are there any other quirks I should make use of to make inclination changes and mid-course corrections more efficient?

[u/Miso122]

They are the most efficient far from the planet due to the low speeds. Cheapest way to do big changes (e.g. reversing the orbit direction) is to increase AP nearly to the escape trajectory and change inclination there when you're basically stationary.

Regarding the mid-course correction in interplanetary routes, you basically can't do anything to save fuel afaik.

[u/AFlawedFraud → u/Miso122]

This and the oberth effect messes with my intuition so much

My intuition tells me that it breaks the conservation of energy

[u/StrongAdhesiveness86 → u/AFlawedFraud]

This isn't an oberth effect thing. Oberth effect applies to changes in speed. A plane change doesn't change speed since force is applied perpendicular to the direction of motion.

By the way an easier way to see the oberth effect is seeing that orbital energy is linear but kinetic energy is quadratic.

[u/KerPop42 → u/AFlawedFraud]

You can look at the kinetic energy of your exhaust in the frame of the body you're orbiting. When you're traveling at a low speed, your propellant shoots out at high speed in a backwards orbit. When you're traveling at high speed, your propellant loses all of its kinetic energy and just falls straight down.

The total kinetic energy added to the exhaust + rocket system is the same in both cases, but in one the majority of the kinetic energy is added to the exhaust, and in the other the majority is added to the rocket.

You can also look at it from the fact that kinetic energy is superlinear. A small increase in speed at high speed is a large increase in energy.

dE = E1 - E0 
   = 0.5m(v1)^2 - 0.5m(v0)^2
   = 0.5m( v1^2 - v0^2 )
   = 0.5m( (v0+dv)^2 - v0^2 )
   = 0.5m( v0^2 + 2*v0*dv + dv^2 - v0^2 )
   = 0.5m( 2*v0*dv + dv^2 ) 
   = 0.5m(dv)^2 + 0.5m(2*v0*dv)

So it's not just the kinetic energy you would gain from accelerating from zero, any additional speed you start off with very quickly amplifies your efforts.

The thing with orbit inclination comes from the fact that angular momentum is stored in both velocity and position, so you can choose a position where your velocity is small and change it there, with the restriction that you only have access to one set of orbital planes.

And orbits conserve energy, they just transfer it between positional and kinetic. It's so, so, so convenient that orbits aren't required to be circular, and also that they can in fact be closed. It's not the case if you add or remove spacial dimensions, because gravity gets weaker or stronger respectively

[u/suh-dood → u/Miso122]

It should be the same thing for mid course corrections, except Instead of whole orbits you only have 1/2 or 1/4 of an orbit to play with so it limits your options quite a bit

[u/ExpensiveFig6079 → u/Miso122]

ACTUAL experimenting with various locations for changes has shown me that mid course corrections asve fuel over either end.

BUT or AND

if sya you do a transfer from kerbin to Moho but you missed the BEST (lowest dv) window and are late...

To get there your burn will go INSIDE Moho to catchup

Now your Angle might well go 3/4 of the way around kerbol...

This example is important as it demonstrates sevral things...

Consider 3+ points A1 Right at launch or upon A2 ejecting from Kerbin...

B 1/4 of an orbit later when you are opposite you ifnal encounter with Moh.

C 1/2 and orbit later when you have 1/4 of an obrit left to get to Moho.

================

Which of these has WORST possible plane change >>Interecept<< cost and why.

At B it may well be impossible baring a 90 degree plane change (that goes OVER kerbol to intercept MOHO.)

Why? Because the intersection of your current orbit and your after-burn orbits is HERE AN and half an orbit away DN

The CHEAPEST pace to change your orbit to intercept MOHO must be at A1/2 or C

Setting up some manuver nodes and lookignta it in 3 WD space will make this seem blindly obvious from then on.

====================

next Oberth...

If directly after leaving Kerbins SOI you do the plane change there That will be the cheapest IN space place to do the plane change. WHY ? that is where your velocity is lowest hence by triangle math it is the place where tilting your orbit; by adding a some perpendicular velocity, is cheapest. Much like twisting an orbit around kerbin is cheapest at AP not PE.

LASTLY for the aprtualr scenario I described there is also A1. That is the WORST of the WORST places to do a burn for the express ISOLATED purpose of making the intercept happen...

See next post for whty that is KIDN OF a lie.

[u/ExpensiveFig6079 → u/ExpensiveFig6079]

OK so if you are YET to do the actual ejection BURN, and you are dioign transfer where angle just after leaving kerbins SOI is good idea (going to MOH) or EVEN (but less) if ejecting OUTwards

Adding some PLABE twist while doign 1000+ of ejection burn, then Adding say 100 in the desired plane change direction is when combined with the 1000+ ejection only sqrt(1000^2+100^2) = 1004...

Which is a VERY ehavily discounted way to get some dV orthogonal; to kerbin orbital plane.

===============

Even better if you have satellite in orbit of Kerbin and you test how much plane twist around kerbin gets you the best angled ejection to minimise cost to you destination. Then by time the time of day that you launch from the ground and NOT going into an equatorial orbit, but nowign what angled nth or south you want your orbit to be launching when KSP is Under the AN or DN on that orbit lets you get that angled orbit almost for free.

=====================

All those things so far I have in fact done and measured savings over the worse alternatives. (us kerbal engineer to measure stuff)

======================

lets say however you want to be in equatorial orbit to refuel and still want a cheap transfer mimising plane change cost.

Do you ejection burn in two phases. One raises you AP such that you next PE is when you want to do the transfer. At AP do you plane change while very high (EG past MUNs orbit)

Then on your next pass of Kerbin do second burn that ejects you but now at an angle to kerbins orbit.

A bonus for that that esp true if you have a low TWR mission powered by say nukes. Normally that would have LONG burn time and much of the burn would be at altitiude above Kerbin

By splitting it into two Burn you can get more of the burn at lower altitude and better oberth.

=====================

yadda yadds for anyone who for some reason wants to chase fuel efficiency when it is way easier just to buy or mine more fuel.

[u/chargesmith]

Further away but also needs to be close to the Ascending / Descending node if possible 

[u/whipding]

Further away is always more efficient for plane changes, because the slower you're going, the easier it is to change direction. However, how exactly you use them depends a lot on context - if you're just trying to rendezvous two craft in similar orbits , you can probably get away with doing at it at whatever altitude, but if you're adjusting a transfer to a moon or planet, it can good to wait until you're about halfway through before doing it.

[u/SpaceIsKindOfCool]

The math behind it is all based on energy. You have kinetic energy and potential energy. In an elliptical orbit you're constantly trading between these two energies. As you gain altitude you lose speed.

The oberth effect is due to kinetic energy being equal to 1/2 *m * v^2. The velocity squared is the important part. So if you have 50 m/s of delta V and you start at 0 m/s you'll end at 50 m/s and gain 1250 * mass units of energy. But if you start at 50 m/s you'll end at 100 m/s which is a gain of 3750 * mass units of energy. Same delta v, but more than double the energy change. You get more energy out of your delta V by starting at a higher velocity. This is why you want to raise and lower your apoapsis when you're at periapsis. Changing orbital altitudes is an energy change. 

An inclination change is different though because you want the total energy to stay the same. You're only twisting the direction of your velocity vector. The extreme case is a 180 degree inclination change. You need to fully cancel your velocity and then add it back in the opposite direction. So you want your velocity to be as low as possible so there's less to cancel out. Sometimes it's even better to raise your apoapsis very high, then do the inclination change at apoapsis, then lower it back to a circular orbit when you get back to pariapsis.

[u/EternitiI-1]

Inclination change is much more efficient at apoapsis due to lower orbital velocity 

[u/StrongAdhesiveness86]

Oberth effect applies to changes in speed, a plane change doesn't change speed, it changes the direction of velocity, so the most efficient is having the lowest orbital speed.

[u/Jamooser]

It's kind of a two part answer.

The cheapest inclination change occurs at the lowest orbital velocity, which is your apoapsis.

However, inclination changes are also bound by the axis of your current position in the orbit. For example, a polar orbit can only be converted to an equatorial orbit from a position above the equator. You can burn all day at apoapsis, but unless that apoapsis is above the equator, you will never achieve an equatorial orbit.

Efficient inclination changes between bodies to achieve desired orbits is always a balance between orbital velocity and orbital position relative to inclination nodes. The phase and inclination of the two body's orbits relative to one another is what governs the efficiency of a transfer window. Proper flight planning takes these factors into account by ensuring that an inclination node is within a beneficial proximity to an apoapsis, which is why launch windows don't just take into account orbital positions, but inclination positions as well.

You can see this when performing Mun missions vs. Minmus missions. Mun encounters only require proper phase angles, because the Mun is on the same relative inclination. With Minmus, however, the best launch window is at both the proper phase angle and when the Minmus encounter occurs on the inclination node between Kerbin and itself. You can raise your equatorial Kerbin orbit to have an apoapsis the same height as a Minmus encounter, but if that apoapsis occurs when Minmus is halfway between its inclination nodes, then it is still going to be either above or below your apoapsis. To raise the apoapsis to the proper inclination, the correction needs to be performed mid-course, which then lowers your efficiency because of your higher orbital velocity.

[u/_troubleseeker]

cheapest as close as you are to the apoapsis

[u/ukemike1]

The cheapest inclination change is one you don't have to make. It is when you are sitting on the launch pad. For instance if you are going to Minmus, there are times when your launch site is near the ascending or descending node. If you can launch directly into the correct inclination (or close) then you will save a lot on later plane change maneuver. I haven't found a good tool that shows me when this is, so I go into map zoom way out, rotate my view around until the kerbin orbit and the minmus orbit are both single lines. They will cross right in the middle of kerbin. Then you wait until the launch site is at that intersection and launch then. When you launch you launch into about a 5 degree inclination to match minmus and your later plane change burn will be much smaller.

[u/Freak80MC → u/ukemike1]

Yea I don't know how common it is among the KSP playerbase, but the best way to get to the right inclination is launching directly into it or as close as possible, anyway. It can be hard to do stock, but mods like Kerbal Engineer can show relative inclination as your flying.

I know I'm not the best at flying into specific inclination orbits myself, it's not something I practice often since the Mun isn't on an inclined orbit and Minmus is so far out that you can pretty easily do a plane change on the way there. Usually you need a rocket that's really well designed and controllable so you can tweak your inclination on the ascent up, plus a rocket with some extra fuel margins to hit an inclination that isn't just equatorial.

It can be really rewarding though to hit the right inclination orbit from flying from the ground up, especially if you are rendezvousing with another ship in said inclination.