Liu (1974): Satellite Motion about an Oblate Earth

Source

Liu, J.J.F., “Satellite Motion about an Oblate Earth,” AIAA Journal, Vol. 12, No. 11, November 1974, pp. 1511—1516. DOI: 10.2514/3.49537. Northrop Services, Inc., Huntsville, Alabama. Contract: NASA-MSFC NAS8-21810.

Download PDF · DOI: 10.2514/3.49537

Liu’s 1974 paper derives the same perturbation terms that SGP4 evaluates — secular rates, long-periodic variations, and short-periodic corrections — using the method of averaging rather than Brouwer’s von Zeipel canonical transformations or Kozai’s secular perturbation theory. The agreement of three independent mathematical techniques on the same results provides strong validation of the perturbation terms implemented in the operational code.

Why This Matters for SGP4

The SGP4 perturbation terms were originally derived by Brouwer (1959) using von Zeipel transformations in Delaunay canonical variables — a powerful but opaque technique that makes it difficult to verify intermediate steps. Liu arrives at the same final expressions through a conceptually simpler approach: average the equations of motion over the fast variables and collect what remains.

This independent verification matters because:

1. Cross-validation. Three different methods (von Zeipel, Kozai, method of averaging) produce identical secular rates and periodic corrections. Any error in one method would produce a discrepancy.

2. Extensibility to drag and third-body effects. The method of averaging does not require canonical transformations, which means it extends naturally to non-conservative forces like atmospheric drag. This is exactly the mathematical foundation that Hujsak (STR#1) used for the deep-space perturbation theory.

3. Explicit initialization. Liu provides the osculating-to-mean element conversion directly, which is the inverse of SGP4’s propagation step and is needed when fitting TLEs from observations.

The Three-Step Averaging Procedure

Liu’s approach works with conventional Keplerian elements $(a, e, i, \Omega, \omega, M)$ rather than Delaunay or Poincare variables. The gravitational potential includes $J_2$ (first order), $J_3$ and $J_4$ (second order).

Step 1: Eliminate the Mean Anomaly

Average the equations of motion over $M$ (the fast variable, completing one cycle per orbit). This removes all short-periodic terms and produces the averaged differential equations containing only secular and long-periodic contributions.

The short-periodic variations emerge as the difference between the original and averaged equations.

Step 2: Eliminate the Argument of Perigee

Average the result of Step 1 over $\omega$ (the slow variable, precessing over months to years). This removes the long-periodic terms and yields the doubly-averaged secular equations.

Step 3: Integrate the Secular Equations

The doubly-averaged equations are integrable in closed form, giving the secular evolution of all six elements.

Secular Rates

The first-order secular rates match Brouwer and Kozai:

Node regression:

$\dot{\Omega}_1 = -\frac{3}{2} n J_2 \left(\frac{R_E}{p}\right)^2 \cos i$

Apsidal advance:

$\dot{\omega}_1 = \frac{3}{4} n J_2 \left(\frac{R_E}{p}\right)^2 (4 - 5\sin^2 i)$

Mean anomaly correction:

$\dot{M}_1 = \frac{3}{4} n J_2 \left(\frac{R_E}{p}\right)^2 \eta\,(2 - 3\sin^2 i)$

where $p = a(1 - e^2)$ is the semi-latus rectum, $\eta = \sqrt{1 - e^2}$, and $R_E$ is the Earth’s equatorial radius.

The Critical Inclination

The apsidal rate $\dot{\omega}_1$ vanishes when $4 - 5\sin^2 i = 0$, i.e., $i \approx 63.4\degree$. This is the same critical inclination singularity that appears in Brouwer’s theory and must be handled with special-case logic in SGP4. Molniya orbits deliberately exploit this property to freeze their argument of perigee.

A key result confirmed by Liu: the semi-major axis, eccentricity, and inclination have no secular changes under zonal harmonics alone:

$\dot{a}_s = \dot{e}_s = \dot{i}_s = 0$

Only the angular elements ($\Omega$, $\omega$, $M$) undergo secular drift. This reflects the physical reality that zonal harmonics produce no net energy or angular momentum change over complete orbits.

Short-Periodic Variations

The short-periodic corrections (Eqs. 20—21 in the paper) oscillate with the orbital period and represent the difference between mean and osculating elements at any instant. These correspond directly to the short-period correction block in sgp4.f.

Semi-major axis:

$\delta a_{sp} = -\frac{a^2 J_2}{p^2}\left[(3\sin^2 i - 2)\left(\frac{a}{r}\right)^3 + \sin^2 i \cos 2(\omega + f)\left[\left(\frac{a}{r}\right)^3 + (1 - e^2)^{-3/2}\right]\right]$

The other five elements ($e$, $i$, $\Omega$, $\omega$, $M$) have analogous short-periodic expressions, all proportional to $J_2$ and oscillating with the true anomaly $f$ or argument of latitude $(\omega + f)$.

Long-Periodic Variations

The long-periodic terms depend on $\omega$ (argument of perigee) and vary on the timescale of apsidal precession. These arise from the odd harmonic $J_3$ and couple eccentricity to inclination:

Element	Dominant Term	Period
$e$	$\sim J_3 \sin\omega / e$	Apsidal precession (~months)
$i$	$\sim J_3 e \cos\omega / \cos i$	Apsidal precession
$\omega$	$\sim J_3 \cos\omega / e$	Apsidal precession
$\Omega$	$\sim J_3 e \cos\omega / \sin i$	Apsidal precession

Singularities in Long-Period Terms

The $1/e$ and $1/\sin i$ factors in the long-periodic variations are the same coordinate singularities that Lyddane (1963) removes by switching to Poincare-type variables. Liu’s formulation exhibits these singularities explicitly, confirming they are artifacts of the element choice rather than physical effects.

Method of Averaging vs. Von Zeipel

Property	Von Zeipel (Brouwer)	Method of Averaging (Liu)
Variables	Delaunay canonical $(l, g, h, L, G, H)$	Conventional Keplerian $(a, e, i, \Omega, \omega, M)$
Transformation	Canonical (preserves Hamiltonian structure)	Non-canonical (no such requirement)
Extension to drag	Requires special treatment	Natural — drag enters the equations of motion directly
Extension to third-body	Possible but cumbersome	Natural — Liu explicitly notes this advantage
Final results	Identical secular rates and periodic terms	Identical secular rates and periodic terms
Physical intuition	Obscured by canonical formalism	More transparent

Liu states the advantage directly: the method of averaging is “feasible not only because the transformations involved need not be canonical, but also because it provides a rigorous, systematic and straightforward procedure” for treating drag and third-body effects jointly with the geopotential.

Connection to the Lineage

Document	Relationship to Liu (1974)
Brouwer (1959)	Von Zeipel derivation that Liu independently verifies
Kozai (1959)	Secular perturbation theory — the third independent derivation
Lane & Cranford (1969)	Cited as Ref 3; the operational SGP4 development from Lane’s drag theory
STR#1 (Hujsak, 1979)	Uses the same method of averaging for deep-space luni-solar perturbations
Hoots (1981)	Reformulates the same Brouwer perturbation terms into explicit secular/periodic separation

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Brouwer (1959): Mean Element Theory The von Zeipel derivation that Liu independently confirms

STR#1: Deep-Space Theory Hujsak extends the method of averaging to luni-solar perturbations

Hoots (1981): Brouwer Reformulation Explicit perturbation separation of the same terms Liu derives

Lyddane (1963): Singularity Fix Removes the zero-eccentricity singularities Liu exhibits explicitly