A small proof mass and a larger satellite do not travel in exactly the same orbit. The difference is small, of the order of the size of the satellite times the ratio of satellite size to orbit radius. For example, a one meter satellite in orbit around the earth could have an orbit radius which differed from that of the proof mass by 1 meter x 1 / (7 x 10⁶) » 10^-7 meters; and a LISA satellite in orbit around the sun would have a difference of about 1 meter x 1 / (1.4 x 10¹¹) » 10^-11 meters. These differences are very small, but the expected translation control performance of modern high-precision Drag-Free satellites exceeds the earth-orbit difference by a factor of 100 or more, and the LISA specification is of the same order as the difference. The additional specific force due to gravity-gradient effects which must be applied to the proof mass or satellite to compensate for this, however, is completely negligible.

Orbit-attitude coupling is interesting in its own right; and in the case of pitch resonance, orbit eccentricity can have a big effect on the attitude of a gravity-gradient stabilized satellite.

The Uncoupled Case

If the satellite motion is linearized about a locally-level coordinate system in a circular orbit, the characteristic equation of the translational motion (Euler-Hill equations) is

Euler-Hill In-Plane equation:	s2(s2 + n2) = 0
Euler-Hill Out-of-Plane equation:	s2 + n2 = 0

The s² + n² term corresponds to a harmonic oscillator, i.e. this mode oscillates at the orbit frequency. The s² term which applies only to the along-track motion indicates that this mode can grow as t . The Euler-Hill equations and their solutions are discussed in detail in Chapter 4 of my Ph.D. thesis, The Control and Use of Drag-Free Satellites, SUDAER 194, June 1964, pp 109-128.

Background: Linear Attitude Motions with Gravity-Gradient Stabilization

If pitch, roll, and yaw are defined in the usual way, the characteristic equation of the linearized attitude motion including gravity gradient coupling is:

Lagrange's Roll-Yaw equation:	s4 + (1 - 3k2 - k1k2) s2n2 - 4k1k2n4 = 0
Lagrange's Pitch equation:	s2 + 3k3n2 = 0

The stability of a gravity-gradient attitude-control system is determined by choosing the parameters, k₁, k₂, and k₃ , in the usual manner. This is discussed in detail in the classic paper by Dan DeBra and Dick Delp, Rigid Body Attitude Stability and Natural Frequencies in a Circular Orbit, Journal of the Astronautical Sciences, III, No. 1, Spring 1961, pp. 14-17.

The Coupled Case: Equilibrium Orbit of a Rigid Body

Because of its finite extent, a rigid body does not quite follow the same circular orbit as a point mass. A body in equilibrium with zero roll, pitch, and yaw rotations is in equilibrium when the radius, R , from the center of a spherical central body to the center of mass of the body satisfies the condition

n² = (k / R³) (1 - 3/2 (2 G₁² - G₂² - G₃²) / R²)

where G₁, G₂, and G₃ are the radii of gyration of the orbiting rigid body and k is the gravitational constant of the central body. The subscripts 1, 2, and 3 refer to the axes parallel to the radius vector, parallel to along-track, and perpendicular to the orbit respectively.

The difference between the orbit of a point mass and one with finite radius of gyration is approximately

DR » - (2 G₁² - G₂² - G₃²) / 2 R

In addition to the decoupling, the root s² = 0 factors out of the In-Plane/Pitch characteristic equation, and the root s² + n² = 0 factors out of the Out-of-Plane/Roll-Yaw equation. Thus the problem reduces to examining the behavior of two uncoupled 4th order equations.

Since there is no damping in a rigid body, there are no terms in s³ or s, and the substitution s² / n² = p can be made. Thus the problem reduces to the simple case of calculating the roots of two uncoupled 2nd order characteristic equations.

Physical Explanation of Why Each of the Uncoupled Roots Also Factor Out of the Coupled Equations

s² = 0 factors out of the In-Plane/Pitch characteristic equation because motion in a higher or lower circular orbit with the corresponding pitch rotation is also a solution of the linear coupled equations.

s² + n² = 0 factors out of the Out-of-Plane/Roll-Yaw equation because if the satellite were moving exactly in the reference orbit without librating; and if one changed the reference orbit by rotating it about any line of nodes contained in the old reference orbit (but did not disturb the body), the body would now appear to have an Out-of-Plane/Roll-Yaw motion with respect to this new reference orbit. This would consist of an oscillating roll motion in phase with an oscillating translation perpendicular to the new reference plane and an oscillating yaw motion 90º out of phase with respect to roll and z.

Thus all of the interesting information about the stability and natural frequencies is contained in only two quadratic equations.

The Coupled Characteristic Equations

With the substitution s² / n² = p the coupled characteristic equations are:

In-Plane/Pitch

p ((p + 1)(p + g) + (3a + b) p + 3ag - 3 (1 - a)b) = 0

(p + 1)(p² + (1 - 3k₂ - k₁k₂)p - 4k₁k₂ + (3e k₂ - d)p + k₁k₂ (d + 3e)) = 0

where

a = (k / n²R³) (2 G₁² - G₂² - G₃²) / R², b = (3k / n²R³) (G₂² - G₁²) / R²,

g = 3k₃ k / n²R³, d = 3k₂ (k / n²R³) G₂² / R², and e = 1 - k / n²R³.

k / n²R³ is very close to one; a, b, d, and e are all very small; and g is very close to 3k₃. Thus with the exception of pitch resonance the coupled equations behave very much like the two uncoupled 6th order systems; and when a, b, d, and e are zero, the equations reduce to the uncoupled equations.

Pitch Resonance

|g - 1| > 4e / (q₃ - dy / R) _max