Force ModelsThe invisible hands on an orbit
A perfect ellipse is a useful shape, but unfortunately it is a fantasy. Real spacecraft fly through a restless gravity field, a dynamic atmosphere, and a wash of photons, while the Sun, Moon, tides, and spacetime subtly and continuously rewrite the path.
A numerical propagator asks a deceptively simple question over and over: what is the spacecraft's acceleration right now?
The two-body answer, Keplerian motion, is the backbone. A useful prediction adds only the perturbations that can move the answer beyond the mission's error budget. More physics does not automatically equate to more truth though; with every extra model comes new parameters, environmental data, and uncertainty.
A force model is a negotiated boundary between fidelity, knowledge, and compute.
Build an Earth-orbit force stack
Change the orbit and spacecraft. The ranking below shows approximate acceleration scales, not an operational prediction.
Earth's shape
Earth's equatorial bulge makes its gravity field depart from a point mass. J₂ is the largest correction.
Drag uses a rough reference density and can vary by orders of magnitude. Third-body values show a maximum tidal scale; direction and geometry change continuously.
The first correction is the planet itself
Earth's equatorial radius is about 21.4 km larger than its polar radius, so the diameter difference is roughly 42.8 km
In simple terms, the bulge can be thought of as pulling a tilted orbit back toward the equator. That intuition is tempting, but an axisymmetric J₂ perturbation does not create a secular inclination decay. Its signature is mostly nodal precession and rotation of the orbit's apsides. Sun-synchronous missions deliberately tune this drift.
The coefficient file and its normalization convention are part of the model. The 2022 table also needed one numerical correction: EGM2008 is complete through degree and order 2159, with extra coefficients up to degree 2190 and order 2159. Most spacecraft do not need all of them; higher altitude filters out fine spatial structure.
The tug that matters is the difference
In an Earth-centered frame, the Sun's or Moon's raw attraction is not the perturbation. Earth is falling too.
Near Earth
J₂ and drag usually dominate LEO perturbation budgets. Lunisolar gravity is smaller but accumulates over long arcs.
High Earth orbit
The differential pull grows with geocentric distance. Sun and Moon become first-class model terms for GEO, highly elliptical, and cislunar trajectories.
Planetary ephemerides supply the geometry. The original paper's derivation can therefore be reduced to one robust rule: use the body's position relative to the same origin and subtract the origin's acceleration consistently.
At the edge of air, uncertainty becomes a force
Drag is predictable in form and difficult in inputs. In LEO, the density you do not know can matter more than the integrator you carefully chose.
The relative velocity includes Earth's rotation and winds. The effective drag coefficient depends on atmospheric composition, surface temperature, material, attitude, and the rarefied-flow regime. Values near 2–2.3 are useful engineering priors for some convex spacecraft, not universal laws.
Solar and geomagnetic energy expand the upper atmosphere.
F10.7 solar radio flux and geomagnetic indices are commonly used drivers. Density errors are not a fixed “15%”; they vary with altitude, epoch, storm state, model, and calibration data.
NRLMSIS 2.0 supersedes the study's MSIS history table.
The modern model extends from the ground to the exobase and uses location, season, time, solar activity, and geomagnetic activity. Operational orbit prediction often estimates or calibrates a density scale factor because no climatology perfectly predicts tomorrow's thermosphere.
Sunlight pushes, even without a solar wind
Solar radiation pressure is the transfer of momentum from photons to spacecraft surfaces, producing a small but measurable non-gravitational acceleration.
At 1 AU, the nominal pressure is about 4.56 μN/m². In a simple cannonball model, Cᵣ typically lies between 1 and 2. A box-wing model uses spacecraft attitude, articulated arrays, and surface optical properties. The eclipse factor ν must also transition smoothly through penumbra for precision work.
Tides, relativity, Earth radiation, and thrust
Time-varying gravity
Solid Earth, ocean, and pole tides change low-degree geopotential coefficients. They are standard ingredients in satellite geodesy, not one universal 10⁻¹⁵–10⁻¹² km/s² afterthought.
Relativistic corrections
Earth's Schwarzschild radius is about 8.9 mm, but that is a length scale and not a direct orbit-error threshold. Include first post-Newtonian terms when the accuracy budget demands them.
Earth radiation
Reflected sunlight acts mainly from the dayside; thermal infrared is emitted globally. Both can matter for high area-to-mass or precision geodetic satellites.
Thrust events
Impulsive burns change velocity discontinuously. Finite burns change acceleration at start and stop. Propagators should split or event-handle those boundaries, not blindly step across them.
Choose the stack by regime and question
“Include everything” is not a model-selection strategy. Start from the tolerated position error and propagation duration, then test sensitivity.
What changed from the 2022 study?
The original report had the right force-model map. This reworked edition preserves that architecture while fixing quantities, sharpening physical interpretations, and replacing claims that aged poorly.
Corrected diameter difference from ~20 km to ~42.8 km.
Corrected maximum degree/order and clarified that truncation is orbit-dependent.
Removed a blanket 15–20% accuracy claim; added NRLMSIS 2.0 and calibration context.
Changed energetic particles to photon momentum and refined Cᵣ interpretation.
Removed a single magnitude range that did not fit tides and Earth radiation broadly.
Distinguished impulsive velocity jumps from finite-burn acceleration discontinuities.
Sources behind the revision
- National Geospatial-Intelligence Agency. WGS 84 and Earth Gravitational Model 2008.
- Petit, G. & Luzum, B. (eds.). IERS Conventions (2010), Technical Note 36.
- Emmert, J. T. et al. (2021). NRLMSIS 2.0: A Whole-Atmosphere Empirical Model.
- Hughes, S. (2007). General Mission Analysis Tool Mathematical Specifications.
- Montenbruck, O. & Gill, E. (2000). Satellite Orbits: Models, Methods and Applications.
- Knocke, P. C., Ries, J. C. & Tapley, B. D. (1988). Earth Radiation Pressure Effects on Satellites.