Force Model

The satkit numerical propagator integrates the forces acting on a satellite to produce a change in velocity, then integrates the velocity to produce a change in position:

\[ \vec{v}(t_1)~=~\vec{v}(t_0) + \int_{t_0}^{t_1}~\vec{a}\left ( t,~\vec{p}_{t},~\vec{v}_{t} \right ) ~dt \]

\[ \vec{p}(t_1)~=~\vec{p}(t_0) + \int_{t_0}^{t_1}~\vec{v}(t)~dt \]

This page describes each force in \(\vec{a}\left (t, \vec{p}, \vec{v}\right )\) that satkit models. The integration mechanics live in the ODE Integrators page; state vectors, the state transition matrix, and covariance propagation live in State Vectors, STM & Covariance.

The force model follows the treatment in O. Montenbruck and E. Gill, "Satellite Orbits: Models, Methods, Applications" — consult that book for more depth on any term.

Summary

Force	Default	Setting	Order of magnitude (LEO)
Earth gravity (spherical harmonics)	on	gravity_degree, gravity_order, gravity_model	\(10^0\) m/s²
Sun third-body	on	use_sun_gravity	\(10^{-6}\) m/s²
Moon third-body	on	use_moon_gravity	\(10^{-6}\) m/s²
Atmospheric drag (NRLMSISE-00)	when alt < 700 km	use_spaceweather, satproperties.cd_a_over_m	\(10^{-7}\) to \(10^{-3}\) m/s²
Solar radiation pressure	when craoverm > 0	satproperties.craoverm	\(10^{-8}\) to \(10^{-7}\) m/s²
Solid Earth tides (IERS 2010 §6.2.1 Step 1)	on	tide_model	\(10^{-7}\) m/s²
GR Schwarzschild (IERS 2010 §10.3)	on	use_relativistic_correction	\(10^{-9}\) m/s² (LEO)
Continuous thrust	when configured	satproperties.thrusts	user-specified

The Forces-vs-altitude plot at the bottom of this page shows how each contribution scales with orbital altitude.

Earth Gravity

Earth's gravity dominates by many orders of magnitude. The Earth is not a point mass; its non-spherical mass distribution is captured by an expansion in spherical harmonics with coefficients \(\bar{C}_{nm}, \bar{S}_{nm}\):

\[ V(r, \phi, \lambda) = \frac{GM_\oplus}{r}\sum_{n=0}^{N}\left(\frac{R_\oplus}{r}\right)^n \sum_{m=0}^{n} \bar{P}_{nm}(\sin\phi)\left[\bar{C}_{nm}\cos m\lambda + \bar{S}_{nm}\sin m\lambda\right] \]

The \(\bar{C}_{20}\) term (commonly called J2) captures Earth's equatorial bulge and is responsible for orbital precession.

Coefficient files come from ICGEM. satkit ships with four models, selectable via gravity_model:

Model	Description
egm96	Earth Gravitational Model 1996 (default)
jgm3	Joint Gravity Model 3
jgm2	Joint Gravity Model 2
itugrace16	ITU GRACE 2016

The gravity_degree and gravity_order parameters cap the expansion (default 4×4). For high-precision work, degree-8 to degree-20 is typical; gains beyond ~degree-20 are small for satellites above ~500 km. Order may be set lower than degree to zero out the longitudinal (tesseral) terms.

Third-Body Gravity (Sun, Moon)

The Sun and Moon each act as point-mass attractors. Their pull on the Earth must be subtracted to express acceleration in the geocentric frame:

\[ \vec{a}_\text{sun}~=~GM_\text{sun}\left[\frac{\vec{p}_\text{sun} - \vec{p}}{|\vec{p}_\text{sun}-\vec{p}|^3} - \frac{\vec{p}_\text{sun}}{|\vec{p}_\text{sun}|^3}\right] \]

The Moon expression has the same form. Body positions come from the JPL DE-series ephemerides (default DE440). Disable either with use_sun_gravity=False / use_moon_gravity=False.

Solid Earth Tides

The Sun and Moon deform the solid body of the Earth, which in turn perturbs the geopotential. satkit implements IERS 2010 §6.2.1 Step 1 (frequency-independent Love-number response) — about 99% of the total solid-tide signal at ~5% per-step CPU overhead. The correction modifies the degree-2, degree-3, and a small degree-4 set of Stokes coefficients \(\Delta\bar{C}_{nm}, \Delta\bar{S}_{nm}\) as a function of Sun and Moon ITRF positions and the IERS 2010 nominal Love numbers.

Frequency-dependent corrections (IERS 2010 §6.2.2 Step 2, 71 tidal constituents) are reserved for TideModel::SolidFull — currently a placeholder that falls back to Step 1. The Step 2 contribution is sub-mm class at LEO.

Set with tide_model:

Value	Effect
tidemodel.solid_step1 (default)	IERS §6.2.1 Step 1 — recommended
tidemodel.solid_full	Step 1 + Step 2 (Step 2 not yet implemented; behaves as Step 1)
tidemodel.none	Disable solid tides (use for reproducibility with pre-tide versions)

Atmospheric Drag

At altitudes below ~600 km the residual atmosphere imposes a drag force:

\[ \vec{a}_\text{drag}~=~-\frac{1}{2}\,C_d\frac{A}{m}\,\rho\,|\vec{v}_r|\vec{v}_r \]

with \(C_d\) the drag coefficient (typically 1.5–3), \(A/m\) the area-to-mass ratio, \(\rho\) the atmospheric density, and \(\vec{v}_r\) the satellite velocity relative to the co-rotating atmosphere (assumed at rest in the Earth-fixed frame).

Density comes from the NRLMSISE-00 thermosphere model (pure Rust implementation), which reads space-weather indices (F10.7, Ap) automatically. Disable space-weather lookup with use_spaceweather=False to fall back to fixed nominal indices.

Drag is skipped above 700 km regardless of settings.

Solar Radiation Pressure

Solar photons absorbed or scattered by the satellite transfer momentum, producing a force away from the Sun:

\[ \vec{a}_\text{SRP}~=~-P_\text{sun}\,C_R\frac{A}{m}\,\hat{p}_\text{sun} \cdot \nu(\vec{p}, \vec{p}_\text{sun}) \]

where \(P_\text{sun} \approx 4.56 \times 10^{-6}\) N/m² is the radiation pressure at 1 AU, \(C_R A/m\) is the satellite's radiation susceptibility (the user-supplied satproperties.craoverm), and \(\nu(\vec{p}, \vec{p}_\text{sun}) \in [0, 1]\) is a shadow function that vanishes when the satellite is in Earth's umbra.

satkit uses a cannonball model — the satellite's surface is treated as if its normal points toward the Sun. For high-fidelity work (precise SRP modeling for GNSS or active satellite operations), a box-wing model is needed; that is not currently provided.

General-Relativistic Correction (Schwarzschild)

In strong gravitational fields, the static post-Newtonian Schwarzschild correction perturbs the orbit:

\[ \vec{a}_\text{GR}~=~\frac{GM_\oplus}{c^2 r^3}\left[\left(\frac{4GM_\oplus}{r} - v^2\right)\vec{r} + 4(\vec{r}\cdot\vec{v})\vec{v}\right] \]

(IERS 2010 §10.3 Eq. 10.12 with PPN parameters \(\beta = \gamma = 1\).) At GPS altitude this contributes ~1 m/day of position drift if omitted; ~3 m/day at GEO. Lense-Thirring and de Sitter terms are not yet implemented (sub-cm-class at MEO/GEO).

Toggle with use_relativistic_correction (default True).

Continuous Thrust

Constant acceleration in a chosen frame (used to model low-thrust maneuvers or — in orbit-determination contexts — empirical accelerations: a fitted catch-all that absorbs un-modeled physics). See satkit.thrust and the GPS Example tutorial for usage.

Future Propagation

When propagating into the future beyond the date range of downloaded data files:

• Earth Orientation Parameters (\(\Delta UT1\), \(x_p\), \(y_p\)): the last available values are held constant. This is much more accurate than defaulting to zero since EOPs change slowly.
• Space Weather (F10.7 solar flux, Ap geomagnetic index): if historical data isn't available, the NOAA/SWPC solar cycle forecast supplies predicted F10.7 (out ~5 years). Ap defaults to 4. If neither source is available, F10.7 = 150 and Ap = 4 are used. Run satkit.utils.update_datafiles() to refresh both.

Forces vs Altitude

The plot below, modeled on a similar figure in Montenbruck and Gill, shows each force's order of magnitude vs orbital altitude:

See Also

• Tutorial: GPS Example — fits a GPS orbit against ESA SP3 truth and walks through the empirical-acceleration concept.
• Theory: ODE Integrators for the integration mechanics; State Vectors, STM & Covariance for state representation and covariance propagation.
• API: satkit.propagate, satkit.propsettings, satkit.satproperties.