Plane changes and inclination

Under two-body motion the angular momentum vector is constant, so the orbital plane never changes on its own. Changing it requires propellant, and it requires a great deal of it.

The cost of a pure rotation

Rotating the velocity vector by an angle $\Delta i$ without changing its magnitude means closing an isoceles triangle:

Two features of this expression drive every design decision that follows.

Cost is proportional to orbital speed. Not to altitude, not to energy, but to the speed at which the burn is performed.

Cost grows almost linearly for small angles. For $\Delta i$ below about 40°, $2\sin(\Delta i/2) \approx \Delta i$ in radians, so a 1° plane change costs roughly 1.75% of orbital speed.

A 90° plane change in low Earth orbit costs more than reaching orbit in the first place. This is why a satellite in the wrong plane is, for most purposes, unrecoverable, and why launch azimuth is treated as a hard requirement rather than a preference.

Do it where you are slow

Since cost scales with $v$, and $v$ is lowest at apoapsis, plane changes belong at apoapsis. The GTO apogee row above is the practical expression of this: at 1.60 km/s, the same 28.5° rotation costs less than half what it would at GEO speed and less than a quarter of the LEO figure.

This is the mirror image of the Oberth effect. Energy changes want high speed; direction changes want low speed. A three-burn manoeuvre that raises apoapsis first, rotates cheaply out there and then lowers it again can beat a direct plane change, for exactly the reason a bi-elliptic transfer can beat a Hohmann. See Hohmann and bi-elliptic transfers.

Combining with an energy change

Almost no real manoeuvre is a pure rotation. Usually the speed must change too, and doing both in one burn is dramatically cheaper than doing them in sequence, because the vector triangle closes more efficiently than two separate legs.

For the standard GTO-to-GEO case from a 28.5° launch site, with $v_1 = 1.597$ km/s at GTO apogee and $v_2 = 3.075$ km/s for GEO:

The combined burn saves 1.15 km/s against the better of the two sequential orders. On a satellite with a 320 s apogee engine this is roughly a third of the wet mass, which is the difference between a competitive commercial platform and one nobody buys.

The optimum is not always to put the entire rotation at apogee. When the inclination change is large, splitting it between the perigee and apogee burns does slightly better, because the perigee burn can absorb a few degrees almost for free while it is already changing the velocity substantially. For the 28.5° case the optimum puts about 2° at perigee and the rest at apogee, saving a further 10 to 20 m/s. Commercial launch providers publish GTO insertions with a deliberately supersynchronous apogee for the same reason: the higher the apogee, the slower the satellite is when it rotates.

Getting it free

Three mechanisms change the orbital plane without propellant.

Launch site latitude

The cheapest inclination is the one you launch into. A due-east launch achieves an inclination equal to the launch site latitude, and reaching any lower inclination requires a plane change later.

Kourou’s latitude is worth roughly 1.2 km/s of plane change to a GEO customer compared with Baikonur, which is a substantial part of why it commands the price it does.

J2 nodal drift

The oblateness of the Earth precesses the line of nodes at a rate that depends on altitude and inclination:

Two satellites at slightly different altitudes precess at different rates, so their planes separate over time at no propellant cost. A constellation can be deployed from a single launch into one plane, then allowed to drift apart until the planes are correctly distributed, and finally circularised into position.

This is slow, taking months, and it only changes the node $\Omega$, never the inclination $i$. But it is free, and it is how many multi-plane smallsat constellations are deployed on a single rideshare.

Lunar gravity assist

A trajectory taken out to lunar distance can use the Moon’s gravity to rotate its plane, then return. The manoeuvre costs the delta-v to raise and lower apogee but not the rotation itself.

The best-known application was a rescue. In 1998 the AsiaSat 3 satellite was stranded in a badly inclined orbit by an upper-stage failure. Flown as HGS-1, it was sent on two lunar flybys that removed the inclination the onboard propellant could never have paid for, and it reached a usable geosynchronous orbit. The technique is now a standard option for high-energy missions, although the transfer takes weeks.

References

• Vallado, D. A. Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press, 2022, chapter 6.
• Bate, R. R., Mueller, D. D., White, J. E. and Saylor, W. W. Fundamentals of Astrodynamics, 2nd ed., Dover, 2020.
• Ocampo, C. and Saudemont, R. R. “Initial Trajectory Model for a Multi-Maneuver Moon-to-Earth Abort Sequence”, Journal of Guidance, Control and Dynamics, 33:1184, 2010.
• Salvatore, J. et al. “The HGS-1 Lunar Flyby Mission”, AAS 98-4285, 1998.