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 ΔV 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 ΔV 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, ΔVSPC must be added to the total ΔV required for the transfer.
If the plane change is combined with the conclusion of the Hohmann Transfer, the ΔV 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 ΔVPC and ΔVHT represent the plane change and second ΔV of the Hohmann Transfer performed separately; the dashed arrow ΔVPC+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
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