SGP4 Constants & Symbols

All constants and symbols on this page are extracted directly from Spacetrack Report No. 3, Chapter 12: Users Guide, Constants, and Symbols (pp. 79—81), by Hoots & Roehrich (1980), as compiled by T.S. Kelso (1988).

Earth model

All constants below use the WGS-72 reference ellipsoid. This is intentional — SGP4/SDP4 were fitted to WGS-72 observations, and using WGS-84 values introduces systematic error. The propagator’s accuracy depends on using exactly these constants.

Constants

Except for the deep-space models, all ephemeris models are independent of units. Units input or output as well as physical constants can be changed by making the appropriate changes in only the DRIVER program.

Derived geopotential constants

These constants fold the gravitational zonal harmonics ($J_2$, $J_3$, $J_4$) and Earth equatorial radius ($a_E$) into the forms used throughout the SGP4 equations.

Name	Value	Definition	Units
CK2	$5.413080 \times 10^{-4}$	$\frac{1}{2} J_2 {a_E}^2$	er$^2$
CK4	$6.2098875 \times 10^{-7}$	$-\frac{3}{8} J_4 {a_E}^4$	er$^4$
XJ3	$-2.53881 \times 10^{-6}$	$J_3$	dimensionless

Atmospheric drag parameters

Name	Value	Definition	Units
QOMS2T	$1.88027916 \times 10^{-9}$	$(q_o - s)^4$	er$^4$
S	$1.01222928$	$s$ (atmospheric parameter)	er

Gravitational parameter

Name	Value	Definition	Units
XKE	$7.43669161 \times 10^{-2}$	$k_e \left(\frac{\text{er}}{\text{min}}\right)^{3/2}$	er$^{3/2}$ / min

Unit conversion constants

Name	Value	Definition	Units
AE	$1.0$	distance units per Earth radius	du/er
XKMPER	$6378.135$	kilometers per Earth radius	km/er
XMNPDA	$1440.0$	time units (minutes) per day	min/day
DE2RA	$1.74532925 \times 10^{-2}$	radians per degree	rad/deg
E6A	$1.0 \times 10^{-6}$	$10^{-6}$	dimensionless

Mathematical constants

Name	Value	Definition
PI	$3.14159265$	$\pi$
PIO2	$1.57079633$	$\pi/2$
TWOPI	$6.2831853$	$2\pi$
X3PIO2	$4.71238898$	$3\pi/2$
TOTHRD	$0.66666667$	$2/3$

Precision matters

These constants are defined to the exact precision shown. Using higher-precision values of $\pi$ or $2/3$ will produce different results from the reference FORTRAN implementation and invalidate comparison against the verification test cases in Spacetrack Report No. 3.

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Symbols

The following symbols appear throughout the SGP4/SDP4 mathematical formulations. They are grouped by category for easier reference.

Orbital elements at epoch

These are the Keplerian “mean” elements extracted from a TLE. The word “mean” is in quotes because these are not osculating elements — they include the perturbation model’s biases and should only be used with the matching SGP4/SDP4 propagator.

Symbol	Description	Units
$n_o$	SGP type “mean” mean motion at epoch	rad/min
$e_o$	”Mean” eccentricity at epoch	dimensionless
$i_o$	”Mean” inclination at epoch	rad
$M_o$	”Mean” mean anomaly at epoch	rad
$\omega_o$	”Mean” argument of perigee at epoch	rad
$\Omega_o$	”Mean” longitude of ascending node at epoch	rad
$\dot{n}_o$	Time rate of change of “mean” mean motion at epoch	rad/min$^2$
$\ddot{n}_o$	Second time rate of change of “mean” mean motion at epoch	rad/min$^3$

TLE unit conversion

TLE files express mean motion in rev/day and angles in degrees. SGP4 internally converts to rad/min and radians before propagation. The conversion constants XMNPDA and DE2RA serve this purpose.

Drag coefficients

Symbol	Description	Units
$B^*$	SGP4 type drag coefficient	1/er
$B$	Ballistic coefficient for SGP8: $\frac{1}{2} C_D \frac{A}{m}$, where $C_D$ is a dimensionless drag coefficient and $A$ is the average cross-sectional area of the satellite of mass $m$	m$^2$/kg

Physical constants

Symbol	Description	Units
$k_e$	$\sqrt{GM}$ where $G$ is Newton’s universal gravitational constant and $M$ is the mass of the Earth	er$^{3/2}$ / min
$a_E$	Equatorial radius of the Earth	er (or km)

Gravitational zonal harmonics

Symbol	Description	Units
$J_2$	Second gravitational zonal harmonic of the Earth	dimensionless
$J_3$	Third gravitational zonal harmonic of the Earth	dimensionless
$J_4$	Fourth gravitational zonal harmonic of the Earth	dimensionless

Derived constants

Symbol	Definition	Units
$k_2$	$\frac{1}{2} J_2 {a_E}^2$	er$^2$
$k_4$	$-\frac{3}{8} J_4 {a_E}^4$	er$^4$
$A_{3,0}$	$-J_3 {a_E}^3$	er$^3$

Atmospheric parameters

Symbol	Description	Units
$q_o$	Parameter for the SGP4/SGP8 density function	er
$s$	Parameter for the SGP4/SGP8 density function	er

Time

Symbol	Description	Units
$(t - t_o)$	Time since epoch	min

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Unit system

SGP4 uses Earth radii (er) as the fundamental distance unit and minutes as the fundamental time unit. This is baked into the constants above and into the structure of the equations. The conversion factors are:

$$1 \;\text{er} = 6378.135 \;\text{km}$$ $$1 \;\text{day} = 1440 \;\text{min}$$

The gravitational parameter $k_e$ absorbs the Earth’s $GM$ value such that the equations work directly in these natural units without requiring explicit $GM$ references.