Lyddane (1963): Singularity Fix

A concise 4-page paper that solves an essential problem: Brouwer’s 1959 formulas blow up for near-circular and near-equatorial orbits, which happen to describe most operational satellites. Lyddane’s fix is directly implemented in the SGP4 subroutine.

The Problem

Brouwer’s analytical solution expresses results in Delaunay variables $(L, G, H, l, g, h)$ . The resulting perturbation formulas contain two singularities:

Zero eccentricity ( $e \to 0$ ): Terms like $\delta l / e$ and $\delta g / e$ diverge.
Zero inclination ( $I \to 0$ ): Terms like $\delta h / \sin I$ diverge.

These are not physical singularities — the satellite coordinates remain perfectly well-defined. The singularities are artifacts of the coordinate system: at zero eccentricity, the argument of perigee $g$ becomes undefined; at zero inclination, the ascending node $h$ becomes undefined.

The Solution

Lyddane reformulates the problem using Poincare canonical variables instead of Delaunay variables:

\begin{aligned} x_1 &= L, & y_1 &= l + g + h, \\ x_2 &= [2(L-G)]^{1/2} \cos(g+h), & y_2 &= -[2(L-G)]^{1/2} \sin(g+h), \\ x_3 &= [2(G-H)]^{1/2} \cos h, & y_3 &= -[2(G-H)]^{1/2} \sin h. \end{aligned}

The perturbing potential is a power series in $x_2, x_3, y_2, y_3$ that remains regular throughout their range. The key insight is that the determining function $\tilde{S}_1$ and its derivatives are also regular, so the entire von Zeipel transformation can be carried out without encountering singular terms.

Rather than re-derive all of Brouwer’s work, Lyddane shows that the transformed Hamilton-Jacobi equations in Poincare variables reduce to exactly Brouwer’s equations in Delaunay variables. This means Brouwer’s solutions for $F^{**}$ , $S_1$ , and $S_1^*$ apply unchanged — only the final step of computing osculating elements from the transformed variables needs to change.

Lyddane Variables (Equation 8)

The output variables that replace the singular Delaunay elements:

Lyddane computes	Instead of	Singular when
$e \cos l$ , $e \sin l$	$e$ , $l$ separately	$e \to 0$
$\sin(I/2)\cos h$ , $\sin(I/2)\sin h$	$I$ , $h$ separately	$I \to 0$
$l + g + h$	$l$ , $g$ , $h$ separately	$e \to 0$ or $I \to 0$

By computing these combinations directly, the divisions by $e$ and $\sin I$ never occur. These are the “Lyddane variables” that appear in the SGP4 FORTRAN code.

Numerical Validation

Lyddane compared against Cowell numerical integration (10th-order, 60-second step, 12-hour spans, harmonics through $J_5$ ):

$e''$	Brouwer error	Lyddane error
0	undefined	15 m
0.008	1100 m	15 m
0.016	500 m	15 m
0.032	300 m	15 m

The residual 15 m error is the expected magnitude of second-order short-period terms neglected by both theories. Brouwer’s error grows as $\sim 1/e$ while Lyddane’s remains constant.

Correction to Smith (1961)

Lyddane identifies an error in O. K. Smith’s (1961) attempt to patch Brouwer’s formulas for small eccentricity. Smith’s constructed eccentricity $e_s = [e''(e'' + \delta e)]^{1/2}$ should be $e_s = e'' + \delta e$ . With this correction, Smith’s formulas agree with Lyddane’s.

Connection to SGP4

The Lyddane modification is directly implemented in the SGP4 subroutine from Spacetrack Report No. 3 (Hoots and Roehrich, 1980). After computing the Brouwer mean element perturbations, the osculating elements are recovered using Lyddane’s Equation 8 rather than Brouwer’s original formulas.

In the Vallado et al. (2006) Rev-1 paper, 5 of the 29 test satellites specifically exercise the Lyddane singularity fix:

04632 — Low inclination
14128 — Low inclination
20413 — Low inclination
23177 — Low inclination
23599 — Near-circular, low inclination

The Rev-1 paper also documents a bug in the original STR#3 FORTRAN: the choice of using the original vs. perturbed inclination for the Lyddane threshold switch affected results.

Connections

Document	Relationship
Brouwer 1959	The parent theory. Lyddane’s modification is a change of output variables, not a change to the theory itself.
STR#3	Implements Lyddane’s Eq. 8 in the SGP4 subroutine.
Hoots 1981	Exhibits the singular long-period terms ( $\delta l/e$ , $\delta h/\sin I$ ) explicitly — exactly what Lyddane’s change of variables eliminates.
Vallado Rev-1	Documents Lyddane-related bugs fixed in the corrected SGP4, provides 5 test satellites that exercise the fix.

Brouwer (1959): Mean Element Theory The parent theory with the singularities Lyddane removes

STR#3: Introduction The implementation document where Lyddane's fix lives

Vallado Rev-1 Corrects Lyddane-related bugs in the original FORTRAN

Hoots (1981): Brouwer Reformulation Makes the singular terms explicit