High vs. Low-Altitude Plane Changes
As reflected in the equation for computing a satellite's velocity from its specific mechanical energy,
the velocity of a satellite decreases as the radius of its position vector increases, and vice versa. This relationship can be seen very clearly at the apsides: to derive a satellite's velocity at apoapsis from its specific angular momentum, we divide by the radius of apoapsis; to derive velocity at periapsis we divide by the radius of periapsis:
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The following figure shows a satellite's velocity before and after a plane change:
The dashed arrows represent the Delta-V required for an inclination change in a high-radius, low-velocity situation (left) and a low-radius, high-velocity situation (right). It can be seen graphically how much more energy is required for the latter than for the former.
Combined vs. Simple Plane Changes
The Delta-V needed for a simple plane change is given by:
where V is the velocity at the time of the plane change, and θ is the plane change angle. In a situation where the plane change is carried out separately from the Hohmann Transfer, Delta-VSPC must be added to the total Delta-V required for the transfer.
If the plane change is combined with the conclusion of the Hohmann Transfer, the Delta-V for that combined maneuver is given by:
where VT is the velocity of the satellite in the transfer orbit (at apogee), V2 is its velocity in the final orbit and θ is the plane change angle.
A picture is worth a thousand words (and at least two or three equations):
In the above figure, V1 is the velocity vector of the satellite at the apogee of the transfer orbit; V2 is the velocity vector after the plane change and the completion of the Hohmann Transfer. The dotted arrows Delta-VPC and Delta-VHT represent the plane change and second Delta-V of the Hohmann Transfer performed separately; the dashed arrow Delta-VPC+HT represents the combined maneuver. As the geometry plainly reveals, the combined maneuver is substantially more efficient.
These notes are based on an example discussed in Sellers, Jerry Jon, Understanding Space: An introduction to Astronautics, New York: McGraw-Hill (1994), pp. 191-192