Hoots (1981): Brouwer Reformulation

Source

Hoots, F. R., “Reformulation of Brouwer’s Solution of the Artificial Satellite Problem in Terms More Suitable for Automatic Computation,” Celestial Mechanics, Vol. 23, 1981, pp. 307-315.

Felix Hoots was the lead author of Spacetrack Report No. 3 (1980), which documented the SGP4, SDP4, SGP8, and SDP8 propagation models. This companion paper provides the theoretical basis for SGP8/SDP8 by reformulating Brouwer’s 1959 analytical theory into a form with cleanly separated perturbation components.

The Reformulation

Brouwer’s original solution uses a von Zeipel transformation that expresses osculating elements as implicit functions of mean elements. The transformation is mathematically elegant but computationally impractical because secular and long-period perturbations are buried inside the transformation rather than appearing as explicit terms.

Hoots makes the perturbations explicit:

$$\alpha(t) = \alpha''(t) + \delta\alpha_{lp}(t) + \delta\alpha_{sp}(t)$$

where $\alpha$ is any orbital element, $\alpha''$ is the Brouwer mean element (with secular rates applied), $\delta\alpha_{lp}$ is the long-period perturbation, and $\delta\alpha_{sp}$ is the short-period perturbation.

This separation enables modular code where each perturbation type can be independently computed, tested, and selectively applied.

Secular Rates

The secular rates include the familiar $J_2$ effects:

Node regression:

$\dot{h}'' \propto -J_2 \cos I / \eta^4$

with second-order $J_2^2$ corrections. The ascending node drifts westward for prograde orbits, a dominant effect for any LEO satellite.

Apsidal advance:

$\dot{g}'' \propto J_2 (1 - \tfrac{5}{2}\sin^2 I) / \eta^4$

This rate vanishes at the critical inclination $I \approx 63.43\degree$, freezing the argument of perigee — the physical basis of Molniya orbit design.

Mean motion: Modified by $J_2$ and $J_2^2$ terms, producing the secular acceleration (or deceleration) of the satellite along its orbit.

Long-Period Perturbations

The long-period terms depend on $g''$ (argument of perigee) and vary on the timescale of apsidal precession — months to years:

• Eccentricity: $\delta e_{lp} \sim J_3 \sin g'' / e''$
• Inclination: $\delta I_{lp} \sim J_3 e'' \cos g'' / \cos I''$
• Mean anomaly: $\delta l_{lp} \sim J_3 \cos g'' / e''$
• Argument of perigee: $\delta g_{lp} \sim J_3 \cos g'' / e''$
• Node: $\delta h_{lp} \sim J_3 e'' \cos g'' / \sin I''$

Lyddane Connection

Every long-period term listed above shows singularities at $e = 0$ or $I = 0$. These are exactly the singularities that Lyddane (1963) removes by changing the output variables. The $\delta l/e''$ and $\delta h/\sin I''$ terms exhibited here are the specific terms that Lyddane’s Poincare variable substitution eliminates.

Short-Period Perturbations

The short-period terms oscillate with the orbital period and depend on the true anomaly $f$:

• Semi-major axis: $\delta a_{sp} \sim J_2 (a/r)^3 \cos 2(f + g)$
• Eccentricity, inclination, node, perigee: all $\sim J_2 (a/r)^3$ with various angular dependences

These terms average to zero over one orbit and represent the oscillatory difference between mean and osculating elements at any instant.

Key Findings

Explicit separation of secular, long-period, and short-period perturbations enables modular code that can be independently tested and selectively applied.

Drag coupling becomes natural. Atmospheric drag modifies only the secular rates of $a''$ and $e''$; the gravitational perturbation formulas are unchanged. This is the fundamental design principle behind SGP8.

SGP4 vs. SGP8 distinction. SGP4 applies drag to a simplified (Kozai) mean motion, then corrects with Brouwer gravitational terms. SGP8 applies drag directly to Brouwer mean elements, preserving the theoretical structure. The difference is subtle but affects accuracy for high-drag orbits.

SGP4 vs. SGP8

Property	SGP4	SGP8
Drag coupling	Indirect (Kozai-to-Brouwer conversion)	Direct (drag on Brouwer $a''$, $e''$)
Geopotential	Simplified first-order $J_2$	Full Brouwer $J_2$—$J_5$
Short-period terms	Eccentric anomaly approximation	True anomaly (exact)
Theoretical basis	STR#2 (Lane & Hoots)	This paper (Hoots 1981)
Operational status	Standard (used worldwide)	Available but not operational

In practice, SGP4 became the standard because:

• It was simpler to implement and faster to execute
• The TLE fitting process absorbed most theoretical differences into the mean elements
• The accuracy difference was small relative to other error sources (drag modeling, geopotential truncation)

Historical Note

This paper was published in 1981 but the work was done in 1979 (received November 1979), concurrent with the preparation of Spacetrack Report No. 3 (published 1980). The two documents are companion pieces: STR#3 is the implementation manual, this paper is the theoretical derivation. Hoots authored both, which is why STR#3 implements the perturbation formulas in the explicit separated form rather than in Brouwer’s original implicit form.

Connections

Document	Relationship
Brouwer 1959	The parent theory. All perturbation formulas derive from Brouwer’s von Zeipel solution.
Lyddane 1963	Removes the singularities in the long-period terms that Hoots exhibits explicitly.
STR#1 (Hujsak)	Documents the drag theory feeding into both SGP4 and SGP8.
STR#3	Implements SGP4/SDP4/SGP8/SDP8. The SGP8 model is the direct computational realization of this paper.
Vallado Rev-1	Corrects and standardizes the SGP4 implementation derived from this theory.

Brouwer (1959): Mean Element Theory The parent theory that Hoots reformulates

Lyddane (1963): Singularity Fix Removes the singularities in Hoots's long-period terms

STR#3: Introduction The implementation document for all Spacetrack models