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Given that the $\Delta V$ required to perform an orbital inclination change is proportional to velocity, I was wondering when the best time to perform such a maneuver would be.

Would it be best to start the plane change maneuver as soon as possible in the ascent trajectory so that you'll end up with the correct orbital velocity and inclination when your engines finally cut off (i.e. incorporate the plane change as part of the ascent itself), or would it be better to first get into an inclined circular orbit at a higher altitude than your planned orbit, perform a plane change at this higher orbit (due to its lower velocity) and then change your orbit to the lower desired altitude?

Finally, would the worst case be to actually perform the plane change at your desired altitude once you're in an inclined circular orbit? As you can see, I'm slightly confused as to the best approach to take, so any guidance would be great.

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  • $\begingroup$ Remember that the Earth is spinning, so launches that don't happen at the poles already have some velocity. $\endgroup$ – uhoh Nov 18 '17 at 23:24
  • $\begingroup$ In practice Shuttle did it as soon as possible during ascent, which makes me think that's optimal. $\endgroup$ – Organic Marble Nov 18 '17 at 23:38
  • $\begingroup$ Met information, that Shuttles had just enough fuel at main engine cutoff to correct inclination up to 2 degrees, which also included docking and deorbiting reserve. $\endgroup$ – ZuOverture Nov 19 '17 at 3:36
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If you're going to a higher inclination than your launch latitude (for example, going from Cape Canaveral at 28º to ISS at 51º), you'll want to do it as early as possible, while your horizontal velocity is minimal -- essentially immediately when you start your gravity turn. The velocity you start with from Earth's rotation is less than that of even a very high orbit, so the inclination change is nearly free.

If you're going to a high altitude, lower inclination orbit, for example to equatorial GSO, there are two options:

  • Option A: Combine the plane change with the circularization/"apogee kick" burn, which makes the total burn requirement equal to the hypotenuse of the triangle made by the normal (plane change) and prograde (circularization) components of the burn, potentially yielding large savings;

or

  • Option B: Launch into a orbit higher than the destination orbit and do the plane change at apogee when you're going very slow. This writeup claims (without citation or derivation) this is more efficient than the first option for plane changes of more than 45 degrees.

The tricky case is when you're trying to reach a low orbit at lower inclination than your launch latitude, like equatorial LEO. In this case, to minimize the size of the plane change, you want to launch into the minimum inclination reachable from the launch site (i.e. fly a due-East initial ascent, making your inclination equal to your launch latitude). Circularization for LEO has to be done some 10-15 minutes into flight, but the descending node where you need to make the plane change is a quarter-orbit (23-30 minutes) from the launch site, so the burns can't be combined for efficiency. By launching into a more steeply inclined orbit (e.g. flying southeast instead of east from Canaveral) you can arrange for the circularization and plane change to coincide, but the plane change will have to be greater; I'm not sure how it optimizes.

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  • $\begingroup$ In my particular case, I'm launching from a site that is inclined 23º from the equator and am trying to get into a 0º circular equatorial orbit, so would this mean that doing the plane change as part of the ascent is actually a bad idea? $\endgroup$ – InquisitiveInquirer Nov 19 '17 at 22:30
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    $\begingroup$ From 23º you have to first get to the equator before you can do the plane change. If you're going to a high orbit like geosynch, you definitely want to combine circularization with plane change as in option A; if you're going to a lower orbit you will likely have to circularize before you reach the equator, and I'm not sure how best to optimize that. $\endgroup$ – Russell Borogove Nov 19 '17 at 22:55
  • $\begingroup$ Aha, this finally made sense, thanks Russel! $\endgroup$ – InquisitiveInquirer Nov 23 '17 at 4:33

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