STR#2 — Lane & Hoots (1979): Near-Earth Theory

Source

Lane, M.H. and Hoots, F.R., “General Perturbations Theories Derived from the 1965 Lane Drag Theory,” Project Spacetrack Report No. 2, Aerospace Defense Command, Peterson AFB, CO, December 1979. DTIC: ADA081264.

The second Spacetrack Report provides the theoretical derivation behind the SGP4 subroutine — combining Lane’s 1965 atmospheric drag model with Brouwer’s gravitational perturbation theory for near-earth orbits (period less than 225 minutes).

Evolution of Lane’s Drag Theory

1965: The Original

Lane presented a power-law atmospheric density model:

$$\rho = \rho_0 \left(\frac{q_0 - s}{r - s}\right)^4$$

where $q_0 = a(1-e)$ was the perigee height of each satellite. This produced a unique atmosphere per satellite — useful for limited-arc fitting, but suboptimal for catalog maintenance.

1965—1969: SGP

Lane derived the remaining element rates and computed equinoctial elements to generate the first operational theory, SGP. This was drag-only, with no geopotential.

Late 1960s: GP4

Two key improvements:

• Fixed atmosphere model: $q_0 = 120\text{ km} + a_E$ and $s = 78\text{ km} + a_E$ for all satellites, eliminating the unique-atmosphere-per-satellite problem
• Brouwer geopotential: Combined with the complete Brouwer theory ($J_2$ through $J_5$) to produce GP4

1979: AFGP4 and SGP4

Lane and Hoots formalized two variants:

• AFGP4 (Air Force General Perturbations 4): Complete Brouwer geopotential with four zonal harmonics, second-order secular rates from $J_2^2$ and $J_4$, plus Lane drag
• SGP4 (Simplified General Perturbations 4): First-order $J_2$ secular rates only, eccentric-anomaly approximations for short-period terms, plus Lane drag

The simplification from AFGP4 to SGP4 proceeds through an intermediate theory (IGP4) and a geopotential simplification. The key insight: replacing $u = \omega + f$ with $U = \omega + E$ in the short-period trigonometric arguments introduces errors of order $e \cdot (k_2/p^2)$, negligible for typical LEO orbits.

Equation-to-FORTRAN Mapping

1. Initialization (computed once per element set)

STR#2 Equation	Description	sgp4.f Variable
Eqs. 103—108	Kozai-to-Brouwer conversion ($a_1$, $\delta_1$, $a_0$, $\delta_0$, $n_0''$, $a_0''$)	A1, DEL1, AO, DELO, XNODP, AODP
Eq. 3	$\theta = \cos i_0$	COSIO, THETA2, THETA4
Eq. 4	Atmospheric parameter $\xi$	TSI
Eq. 5	$\beta_0 = (1-e_0^2)^{1/2}$	BETAO
Eq. 6	$\eta = a_0 e_0 \xi$	ETA, ETASQ
Eq. 111	Primary drag coefficient $C_2$	C2
Eq. 112	$C_1 = B^* C_2$	C1, C1SQ
Eq. 113	Drag contribution to $\omega$: $C_3$	C3
Eq. 114	Eccentricity drag coefficient $C_4$	C4
Eq. 115	Eccentricity oscillatory drag $C_5$	C5
Eqs. 116—118	Higher-order drag (low perigee): $D_2$, $D_3$, $D_4$	D2, D3, D4
Eq. 119	Secular mean anomaly rate $\dot{M}$	XMDOT
Eq. 120	Secular perigee rate $\dot{\omega}$	OMGDOT
Eq. 121	Secular node rate $\dot{\Omega}$	XNODOT

2. Secular Update (at each propagation time)

STR#2 Equation	Description	sgp4.f Variable
Eqs. 122—124	Updated $M$, $\omega$, $\Omega$	XMD, OMGASM, XNODES
Eq. 125	Updated semimajor axis $a$	A
Eq. 126	Updated eccentricity $e$	E
Eq. 127/135	Updated mean longitude $\mathbb{L}$	XL
Eq. 128	Low-perigee drag correction $\delta\omega$	OMGADF
Eq. 129	Low-perigee drag correction $\delta M$	XMDF

3. Long-Period Periodics

STR#2 Equation	Description	sgp4.f Variable
Eq. 138	$a_{xN} = e \cos \omega$	AXN
Eqs. 139—141	$a_{yN} = e \sin \omega$ + $J_3$ correction	AYN, AYNL
Eq. 142	Long-period corrected mean longitude $\mathbb{L}_T$	XLT

4. Kepler’s Equation

STR#2 Equation	Description	sgp4.f Variable
Eqs. 146—147	Newton-Raphson iteration for $(E+\omega)$	U, EPW
Eqs. 148—149	$e\cos E$, $e\sin E$	ECOSE, ESINE
Eqs. 152—154	$r$, $\dot{r}$, $r\dot{f}$ from Kepler solution	R, RDOT, RFDOT
Eqs. 155—157	$\cos u$, $\sin u$, $u$	COSU, SINU, U

5. Short-Period Periodics

STR#2 Equation	Description	sgp4.f Variable
Eq. 158	Short-period correction $\Delta r$	DR
Eq. 159	Short-period correction $\Delta u$	DU
Eq. 160	Short-period correction $\Delta\Omega$	DNODE
Eq. 161	Short-period correction $\Delta i$	DI
Eq. 162	Short-period correction $\Delta\dot{r}$	implicit
Eq. 163	Short-period correction $\Delta r\dot{f}$	implicit

6. Position and Velocity

STR#2 Equation	Description	sgp4.f Variable
Eqs. 164—169	Osculating $r_k$, $u_k$, $\Omega_k$, $i_k$, $\dot{r}_k$, $r\dot{f}_k$	RK, UK, XNODEK, XINCK, RDOTK, RFDOTK
Eqs. 170—173	Orientation unit vectors $\vec{M}$, $\vec{N}$, $\vec{U}$, $\vec{V}$	MX/MY/MZ, NX/NY/NZ, UX/UY/UZ, VX/VY/VZ
Eqs. 176—181	Cartesian position and velocity	X, Y, Z, XDOT, YDOT, ZDOT

Known Errata

Code vs. Report Discrepancies

Where the report and code disagree, the code is authoritative for reproducing NORAD results. The FORTRAN in STR#3 was verified against independent numerical integration; the report contains known typographical issues.

The $C_4$ coefficient. A footnote on page 32 states: “The $k_2$ terms of $C_4$ in the operational coding do not agree with this equation. The present equation has been verified as correct with the author.” The FORTRAN implements a slightly different $C_4$ than the derived equation. The discrepancy has been retained in operational code for catalog consistency.

The $(1-\eta^2)$ exponent in $C_2$. The report shows $(1-\eta^2)^{-9/2}$ as the leading factor, with an additional $(1-\eta^2)$ multiplying $a_0$ inside the braces, giving an effective $^{-7/2}$ for the first term and $^{-9/2}$ for the $k_2$ correction. Some secondary references incorrectly show $^{-7/2}$ throughout.

Chapter Structure

Chapter	Report Section	Content
01	Title page	Bibliographic information
02	Section 1	Historical context, theory relationships
03	Section 2	Lane atmospheric drag theory
04	Section 3	AFGP4: Complete Brouwer + drag
05	Section 4	IGP4: Intermediate theory (not implemented)
06	Section 5	Simplification of short-period terms
07	Section 6	SGP4: Complete computational algorithm
08	Section 7	Position and velocity computation
09	References	Bibliography
10	Appendix	Constants, units, TLE format

STR#1: Deep-Space Theory The companion deep-space theory for periods >= 225 min

Brouwer (1959): Mean Element Theory The gravitational perturbation theory that STR#2 builds upon

STR#3: Introduction The unified FORTRAN implementation

FORTRAN: SGP4 Subroutine The actual code implementing these equations