STR#1 — Hujsak (1979): Deep-Space Theory

Source

Hujsak, R.S., “A Restricted Four Body Solution for Resonating Satellites Without Drag,” Spacetrack Report No. 1, Office of Astrodynamics, Aerospace Defense Command, Peterson AFB, CO, November 1979. DTIC: ADA081263.

The first of three Spacetrack Reports that together define the complete SGP4/SDP4 propagation system. STR#1 provides the deep-space theory — lunar-solar perturbations and tesseral harmonic resonance — implemented as the DEEP subroutine in STR#3.

The Spacetrack Report Series

Report	Author(s)	Date	Subject	DTIC
STR#1	R.S. Hujsak	Nov 1979	Deep-space (luni-solar, resonance)	ADA081263
STR#2	M.H. Lane, F.R. Hoots	Dec 1979	Near-earth (drag + geopotential)	ADA083578
STR#3	F.R. Hoots, R.L. Roehrich	Dec 1980	FORTRAN implementation of all models	ADA093554

TLE Routing

When a TLE arrives, the orbital period determines which propagation path to use:

graph TD
    A[TLE Input] --> B{Period >= 225 min?}
    B -->|No| C[SGP4 / STR#2]
    B -->|Yes| D[SDP4 / STR#1]
    C --> E[J2 secular + drag]
    C --> F[Short-period J2 corrections]
    E --> G[Position and Velocity]
    F --> G
    D --> H[DEEP subroutine]
    H --> I[Lunar-solar secular rates]
    H --> J[Lunar-solar periodic corrections]
    H --> K{12h or 24h orbit?}
    K -->|Yes| L[Resonance integrator]
    K -->|No| M[Skip resonance]
    I --> N[Position and Velocity]
    J --> N
    L --> N
    M --> N

Physical Model

The theory models satellite motion under the gravitational influence of four bodies:

1. Earth (oblate): $J_2$, $J_3$, $J_4$ zonal harmonics; $J_{22}$, $J_{32}$, $J_{33}$, $J_{44}$, $J_{54}$ tesseral harmonics (resonance only)
2. Moon: Point mass, assumed in the ecliptic plane, circular orbit
3. Sun: Point mass, in the ecliptic plane, slightly eccentric orbit
4. Satellite: Massless test particle, no drag (deep-space regime)

Mathematical Approach

The method of averaging (Von Zeipel/Hori) with canonical Lie transformations:

1. Express the Hamiltonian in Delaunay variables $(l, g, h, L, G, H)$
2. Decompose into secular ($\bar{F}$), periodic ($F'$), and resonance ($F_R$) parts
3. Secular rates from $\partial\bar{F}/\partial(\text{momenta})$
4. Generating function $S = \int F'/\dot{\bar{l}}\,dl$ relates mean to osculating elements
5. For resonant orbits, $F_R$ is added to the secular Hamiltonian and the resulting equations are integrated numerically

DEEP Subroutine Mapping

The DEEP subroutine has three entry points. Each maps to specific sections of the paper.

Initialization (DPINIT)

Paper Section	DEEP Implementation	Description
Appendix I	Solar/lunar longitude computation	Compute $\lambda_\odot$, $\lambda_L$ at epoch
Appendix H, Sec H.2—H.4	Direction cosine variables $Z_1\ldots Z_{31}$, $a_1\ldots a_{10}$, $X_1\ldots X_8$	Satellite-to-third-body geometry
Appendix H, Sec H.5	$F_2$, $F_3$, FAB computation	Direction cosine combinations for force function
Appendix D	Central body secular rates	$\dot{g}_{J_2}$, $\dot{h}_{J_2}$, $\dot{g}_{J_4}$, etc.
Appendix E	Third body secular rates	$\dot{e}_L$, $\dot{i}_L$, $\dot{g}_L$, $\dot{h}_L$ (+ solar)
Sec 5.5	Resonance detection	Check $n$ against 12h and 24h bands
Appendix F, Sec F.6	Resonance initialization	Set $\lambda_{r,0}$, $\dot{\lambda}_{r,0}$, harmonic coefficients
Appendix J	Inclination/eccentricity functions	$F_{220}$, $G_{200}$, etc. for resonance terms

Secular Update (DPSEC)

Paper Section	DEEP Implementation	Description
Appendix D, Sec D.4	Linear extrapolation of $g$, $h$	$\bar{g}(t) = \bar{g}_0 + \dot{g}\cdot t$
Appendix E, Sec E.8	Update $e$, $i$ from lunar-solar secular	$\bar{e}(t) = \bar{e}_0 + \dot{e}\cdot t$
Appendix I	Recompute $\lambda_\odot(t)$, $\lambda_L(t)$	Update third body positions
Appendix F, Sec F.5	Numerical integration step	For resonant orbits only

Periodic Correction (DPPER)

Paper Section	DEEP Implementation	Description
Appendix H	Recompute direction cosines at $t$	Updated $X_1\ldots X_8$, $F_2$, $F_3$, FAB
Appendix G, Sec G.2	$\delta e$ correction	Periodic eccentricity correction
Appendix G, Sec G.3	$\delta i$ correction	Periodic inclination correction
Appendix G, Sec G.4	$\delta g$ correction	Periodic perigee correction
Appendix G, Sec G.5	$\delta h$ correction	Periodic node correction
Appendix G, Sec G.6	$\delta l$ correction	Periodic mean anomaly correction

Key Variable Correspondence

Paper Symbol	DEEP Variable	Meaning
$l$	XLL	Mean anomaly
$g$	OMGASM	Argument of perigee
$h$	XNODES	Right ascension of ascending node
$e$	EM	Eccentricity
$i$	XINC	Inclination
$n$	XN	Mean motion
$\lambda_r$	XLI	Resonance angle
$\dot{\lambda}_r$	XNI	Resonance angle rate
$F_2$	F2	$\alpha^2 + \beta^2$ direction cosine combination
$F_3$	F3	$\alpha^2 - \beta^2$ direction cosine combination
$D$	FAB	$\alpha\beta$ direction cosine product
$\delta e$	PE	Periodic eccentricity correction
$\delta i$	PINC	Periodic inclination correction
$\delta g$	PGH	Periodic perigee correction
$\delta h$	PH	Periodic node correction
$\delta l$	PL	Periodic mean anomaly correction

Perturbation Hierarchy

Source	Order	Elements Affected
$J_2$ secular	$O(J_2)$	$\dot{l}$, $\dot{g}$, $\dot{h}$
$J_2^2$ secular	$O(J_2^2)$	$\dot{l}$, $\dot{g}$, $\dot{h}$
$J_4$ secular	$O(J_4)$	$\dot{g}$, $\dot{h}$
Lunar secular	$O(\mu_L/\mu)$	$\dot{e}$, $\dot{i}$, $\dot{g}$, $\dot{h}$
Solar secular	$O(\mu_S a^3/r_S^3)$	$\dot{e}$, $\dot{i}$, $\dot{g}$, $\dot{h}$
Lunar periodic	$O(\mu_L/\mu)$	$\delta e$, $\delta i$, $\delta g$, $\delta h$, $\delta l$, $\delta a$
Solar periodic	$O(\mu_S a^3/r_S^3)$	$\delta e$, $\delta i$, $\delta g$, $\delta h$, $\delta l$, $\delta a$
$J_{22}$ resonance	$O(J_{22})$	$\dot{n}$, $\dot{e}$, $\dot{i}$ (24-hour only)
$J_{32}$, $J_{33}$ resonance	$O(J_{3m})$	$\dot{n}$, $\dot{e}$, $\dot{i}$ (12-hour only)

Sign Convention Warning

Some sign conventions in the appendices are inconsistent between the theory development (Appendices A—B) and the implementation summaries (Appendices D—G). The FORTRAN code in STR#3 is the authoritative reference for sign conventions. When the paper and code disagree, trust the code.

STR#2: Near-Earth Theory The companion near-earth drag theory

STR#3: Introduction The unified implementation document

FORTRAN: DEEP Subroutine The actual FORTRAN code implementing this theory