WGS-72: SGP4's Geodetic Constants

Source

Seppelin, Thomas O., The Department of Defense World Geodetic System 1972, Defense Mapping Agency (DMA), Headquarters, Washington, D.C., May 1974. DTIC Accession Number: ADA110165.

Historical Context

The Cold War Geodesy Problem

By the late 1960s, the Department of Defense maintained multiple geodetic datums — NAD 27 for North America, ED 50 for Europe, Tokyo Datum for the Pacific, and dozens of regional systems. Intercontinental ballistic missile targeting, satellite tracking, and global navigation all required a single, unified, geocentric reference frame. The World Geodetic System program was DMA’s answer.

System	Year	Equatorial radius (m)	Flattening (1/f)	Key data sources
WGS-60	1960	6,378,165	298.3	Astro-geodetic, early Vanguard satellite obs.
WGS-66	1966	6,378,145	298.25	Transit Doppler data
WGS-72	1972	6,378,135	298.26	Improved Doppler, Baker-Nunn, radar altimetry
WGS-84	1984	6,378,137	298.257223563	GPS, SLR, VLBI (current standard)

DMA and Its Successors

The Defense Mapping Agency (DMA) was established in 1972 by consolidating the mapping functions of the Army Map Service, the Naval Oceanographic Office, and the Air Force Aeronautical Chart and Information Center. DMA was renamed the National Imagery and Mapping Agency (NIMA) in 1996, and again renamed the National Geospatial-Intelligence Agency (NGA) in 2003. NGA continues to maintain WGS-84 today.

How Constants Flow into SGP4

The chain of custody from WGS-72 to every SGP4 implementation in the world:

graph TD
    A["Seppelin (1974)<br/>DMA report defining WGS-72<br/>4 defining parameters + zonal harmonics"] --> B["NORAD / AFSPC<br/>Adopted WGS-72 for satellite catalog<br/>Used J2, J3, J4 for perturbation models"]
    B --> C["Hoots & Roehrich (1980)<br/>Spacetrack Report No. 3<br/>Codified SGP4 in FORTRAN"]
    C --> D["FORTRAN driver.f DATA statement<br/>XJ2, XJ3, XJ4, XKE, XKMPER"]
    D --> E["Kelso (1988)<br/>Pascal and C translations<br/>Preserved exact constant values"]
    E --> F["Vallado et al. (2006)<br/>C++ / Python / MATLAB<br/>Added wgs72 and wgs84 constant modes"]
    F --> G["Modern implementations<br/>satellite.js, python-sgp4, sgp4-rs<br/>All carry the same WGS-72 constants"]

The Constant Lock-in

Never substitute WGS-84 constants into SGP4

NORAD generates TLE element sets by fitting observed satellite positions using SGP4 with WGS-72 constants. The “mean elements” in a TLE are not true Keplerian elements — they are parameters of the SGP4 model that, when propagated forward, reproduce the observed trajectory. The propagator and the element sets form a matched pair.

The consequences of this coupling:

1. TLE elements absorb WGS-72 biases. The fitted mean motion, eccentricity, and inclination compensate for the specific J2/J3/J4 values used during orbit determination.
2. WGS-84 constants produce systematic errors. Propagating a TLE with WGS-84 constants introduces position errors of typically 1—3 km over 24 hours.
3. Every TLE consumer must use WGS-72. This is why SGP4 has never been updated to WGS-84, even though WGS-84 has been the DoD geodetic standard since 1984.

WGS-72 vs. WGS-84 Comparison

This is the canonical comparison table for the SGP4 theory archive. The WGS-84 page discusses the modern datum but refers back here for the detailed constant comparison.

Parameter	WGS-72	WGS-84	Delta	Notes
$a$ (m)	6,378,135	6,378,137	+2 m
$1/f$	298.26	298.257223563	~0.003	WGS-84 uses derived $f$
$GM$ (m$^3$/s$^2$)	$3.986005 \times 10^{14}$	$3.986004418 \times 10^{14}$	$-5.82 \times 10^{8}$
$\omega$ (rad/s)	$7.292115147 \times 10^{-5}$	$7.292115 \times 10^{-5}$	$\sim 10^{-12}$
$J_2$	$1.0826158 \times 10^{-3}$	$1.08262982 \times 10^{-3}$	$+1.4 \times 10^{-7}$
$J_3$	$-2.53881 \times 10^{-6}$	$-2.53215 \times 10^{-6}$	$+6.66 \times 10^{-9}$
$J_4$	$-1.65597 \times 10^{-6}$	$-1.61999 \times 10^{-6}$	$+3.6 \times 10^{-8}$

The differences are small in absolute terms but significant when accumulated over multiple orbit revolutions in a precision propagator.

Vallado’s Three Constant Modes

The Vallado et al. (2006) SGP4 revision defined three gravity constant modes:

• wgs72 — Original STR#3 constants from Seppelin/WGS-72. This is the correct mode for all operational TLE processing.
• wgs84 — Updated constants from WGS-84. Exists solely for academic comparison and is not operationally valid with standard TLEs.
• wgs72old — Constants from early Kelso implementations with minor rounding differences. Preserved for backward compatibility testing.

Derivation: How GM Becomes XKE

SGP4 uses Earth radii (er) for distance and minutes for time. The gravitational parameter must be converted from SI units to this internal system.

Starting from the WGS-72 value:

$$GM = 3.986005 \times 10^{14} \;\text{m}^3/\text{s}^2$$

Convert meters to Earth radii ($1 \;\text{er} = 6{,}378{,}135 \;\text{m}$):

$$GM = \frac{3.986005 \times 10^{14}}{(6.378135 \times 10^6)^3} = \frac{3.986005 \times 10^{14}}{2.59273 \times 10^{20}} = 1.53768 \times 10^{-6} \;\text{er}^3/\text{s}^2$$

Convert seconds to minutes ($1 \;\text{min} = 60 \;\text{s}$):

$$GM = 1.53768 \times 10^{-6} \times 3600 = 5.53566 \times 10^{-3} \;\text{er}^3/\text{min}^2$$ $$XKE = \sqrt{GM} = \sqrt{5.53566 \times 10^{-3}} = 0.0743669161 \;\text{er}^{3/2}/\text{min}$$

This is the value hardcoded in the FORTRAN DATA statement as .743669161E-1.

Derivation: How $C_{2,0}$ Becomes $J_2$

The WGS-72 defining parameter is the normalized spherical harmonic coefficient:

$$\bar{C}_{2,0} = -484.166 \times 10^{-6}$$

The unnormalized zonal harmonic is obtained by applying the normalization factor:

$$J_2 = -\sqrt{2 \cdot 2 + 1} \;\cdot\; \bar{C}_{2,0} = -\sqrt{5} \times (-484.166 \times 10^{-6})$$ $$J_2 = 2.2360680 \times 484.166 \times 10^{-6} = 1.0826158 \times 10^{-3}$$

The STR#3 FORTRAN truncates this to 1.082616E-3 (7 significant figures), introducing a rounding difference at the 6th decimal place. The WGS-84 value of $1.08262982 \times 10^{-3}$ comes from the higher-resolution EGM96 gravitational model and differs by $1.4 \times 10^{-7}$ — small in isolation, but consequential when the propagator expects the WGS-72 value.

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WGS-84: Modern Geodetic Datum The current standard -- used for display coordinates, not SGP4 propagation

FORTRAN Source: driver.f Where these constants appear as DATA statements in the original code

Kelso (2007): SGP4 Validation Empirical validation of SGP4 accuracy using these constants