Three-Burn Solutions for Very Large Inclination Change

This is the second of two blog posts which introduce functionality from the Orbit Transfer Module of the Spacecraft Control Toolbox (SCT). The Orbit Transfer Module has lots of tools to allow you to numerically optimize the engine burns that a spacecraft needs to apply to go from some initial orbit to its target orbit, with a minimum of fuel used, time elapsed, or some other metric. You can do this to impulsive burns, continuous (low-thrust) burns, and a special model that splits large impulsive burns into many small impulsive burns.

In this blog post, I’d like to discuss an interesting thing that happens when a spacecraft aims to change its inclination by more than 37 degrees. In real life, of course, this would never happen. This is a truly impractical amount of inclination change. If you were truly faced with a mission that required coverage of orbits that were different by 37 degrees, you would simply launch two spacecraft.

But if you were to explore this hypothetical, you would find an interesting feature emerge. Here we will consider the case that a spacecraft starts in a circular, equatorial LEO and targets a circular LEO which is 49.5 degrees inclined to the equator. The well-known solution is to burn once, at the common node of the initial and target orbits, to change direction by the target inclination:

The 1-burn inclination change solution for 49.5 degrees. This requires a truly impractical amount of delta-V.

As we can see from the above figure, this inclination change requires 6.47 km/s of delta-V, almost the spacecraft’s entire orbital velocity!

But, as the Orbit Transfer Module’s OptimizeElementsImpulsiveSearch finds, there is actually another transfer which can marginally reduce the delta-V required for this inclination change. It finds that the following transfer saves delta-V:

  • Burn prograde to raise apoapsis (and a very slight inclination change)
  • Coast up to apoapsis, where inclination change is cheaper
  • Perform an inclination change
  • Coast down to periapsis
  • Burn retrograde to lower apoapsis (and a very slight inclination change)
The 3-burn inclination change solution for 49.5 degrees. Loops out to high altitude before changing inclination. This also requires an impractical amount of delta-V, but it is a smaller amount.

As we can see from the above figure, this transfer only expends 5.97 km/s of delta-V. While this is still impractical, it is interesting that there exists this other category of optimal inclination transfers which exists for high inclination changes.

It is interesting to note that, for 0 through 37 degree inclination changes, the direct approach is superior. Then, from 37 degrees to 60 degrees, this 3-burn solution produces a smaller required delta-V. Above 60 degrees, the intermediate, high-apoapsis orbit actually exceeds escape velocity so the transfer takes infinite time.

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