Simple Spacecraft Orbital Motion
Ron Graham

Parameters
Parameter Description Units
F forcing vector force
fR radial force force
fT tangential force force
g gravitational acceleration distance/time2
m mass force-time2/distance
R motion wrt Earth (inertial) distance
r motion wrt Earth (body) distance
t elapsed time time
z relative altitude distance
x relative range distance
w0 nominal orbit rate time-1
w orbital rate vector time-1

If a spacecraft maintains a circular orbit, then its motion relative to the Earth is given in Kaplan [1],

orbital dynamics equation 1

...where the orbital rate is

orbital dynamics equation 2

..and R is in the x-z plane. A forced spacecraft (such as undergoing reboost) has as its forcing function

orbital dynamics equation 3

If we convert from an Earth-centered to a spacecraft-centered coordinate system, we have two equations of motion. In the radial direction,

orbital dynamics equation 4

..and tangential motion is described by

orbital dynamics equation 5

Equations (4) and (5) describe simple orbital forced motion, also referred to as the Hohmann transfer [2]. In general, equation (4) is pure circular motion if unforced (fR, fT = 0). The introduction of a force will decircularize the orbit. This is referred to as a "delta-v," because it represents a change in the magnitude and direction of spacecraft velocity needed to change to another (usually at a higher altitude) orbit. When the desired orbit is achieved, another delta-v is used to circularize the orbit at that altitude.

Relative Motion of Two Spacecraft

Equations (4) and (5) may be used to model the relative motion of two spacecraft, such as might be used for studies of tethering, Shuttle approach, etc. In general, such a relative-motion problem may be treated with the following simplifying assumptions: A local coordinate system with x- and z-axes respectively representing relative altitude and range between the two bodies gives the equations of approach between the centers of gravity of the two bodies:

orbital dynamics equation 6

...where the equation in x represents relative altitude, and has a stiffness term governed by centripetal acceleration; and the equation in z represents range, and depends on each body's control system for an orderly rendezvous. In the case of the two bodies each being a system of inter-connected rigid bodies, for which translation is coupled with other degrees of freedom, then (6) becomes

orbital dynamics equation 7

...and the motion between two berthing points, one on each body and separated from their respective CGs by a vector

orbital dynamics equation 8

References

[1] Kaplan, M. H. Modern Spacecraft Dynamics and Control. New York: John Wiley & Sons, 1976.
[2] Bate, R. R., D. D. Mueller and J. E. White. Fundamentals of Astrodynamics. New York: Dover Publications, 1971.

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