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title	summary	tags
Orbital perturbations	The forces that make an orbit deviate from an ellipse, and which of them each model includes.	orbit, perturbations, drag, oblateness, third-body

Orbital perturbations

A Keplerian orbit is an ellipse that repeats forever. A real orbit does not. The forces besides the point mass gravity of the Earth deform the ellipse and change its orientation. This document lists the forces and says which of them each model in the workspace includes.

Definition

A perturbation is any acceleration on the satellite other than the central point mass gravity of the Earth. The main ones, in rough order of magnitude for a low Earth orbit, are:

Perturbation	Source	Typical size at low Earth orbit	Character
Oblateness, `J2`	The Earth is not a sphere	`1e-3` of central gravity	Secular in the node and perigee
Atmospheric drag	Residual atmosphere	`1e-7` to `1e-4` of central gravity, depending on altitude and solar activity	Secular decay of the orbit
Third body, Sun and Moon	Gravity of the Sun and Moon	`1e-6` at low Earth orbit, larger at high altitude	Long period
Solar radiation pressure	Photon momentum	`1e-9` to `1e-8` of central gravity	Long period, depends on area and mass
Higher zonals and tesserals	The Earth's gravity field	Diminishing series	Secular and resonant
Relativity	General relativity	`1e-9` of central gravity at low Earth orbit	Secular in the node
Tides and solid Earth tides	The Earth deforms	Below the above	Long period

The list is ordered by the size of the effect on a low Earth orbit. The order changes with altitude: above a few thousand kilometers the third body terms exceed drag, and the resonance with the Earth's gravity field becomes the dominant long period effect.

Mathematical meaning

The central gravity is


\mathbf{a}_{\text{central}} = -\frac{\mu \mathbf{r}}{\lvert \mathbf{r} \rvert^3}

The J2 term, which models the equatorial bulge, is


\mathbf{a}_{J_2} = -\frac{3}{2} \frac{J_2 \mu R^2}{r^5}
  \begin{bmatrix} x \left( 1 - 5 z^2/r^2 \right) \\[2pt]
                  y \left( 1 - 5 z^2/r^2 \right) \\[2pt]
                  z \left( 3 - 5 z^2/r^2 \right) \end{bmatrix}

with R the equatorial radius and z measured along the rotation axis. It has no component that changes the energy, so the size of the orbit is unaffected to first order; what it changes is the orientation. Its secular effects are


\begin{aligned}
\dot\Omega &= -\frac{3}{2} n J_2 \left( \frac{R}{a} \right)^2 \frac{\cos i}{\left( 1 - e^2 \right)^2} \\
\dot\omega &=  \frac{3}{4} n J_2 \left( \frac{R}{a} \right)^2 \frac{5 \cos^2 i - 1}{\left( 1 - e^2 \right)^2}
\end{aligned}

The first makes a low inclination orbit precess slowly westward and a polar orbit precess in right ascension. Both are large enough to matter within a day.

Drag is not a conservative force. It removes energy and angular momentum, which lowers the semi-major axis and the eccentricity. Its size depends on the atmospheric density, which depends strongly on altitude and on the activity of the Sun. SGP4 models the effect with a single fitted number, the BSTAR term, rather than with a density model.

Third body gravity and solar radiation pressure are first order effects on an orbit at high altitude, where they cannot be modelled by a single fitted number, and are treated by numerical integration in a force model propagator.

Unit convention

Accelerations are in kilometers per second squared where a force model is used, and are dimensionless ratios against the central gravity in this document for readability.
The zonal coefficients such as J2 are dimensionless, and the reference radius they are scaled by is in kilometers.
The BSTAR term is in inverse Earth radii.

Applicability

The choice of which perturbations to include follows from the altitude and the interval, not from a rule:

A few hours, anywhere. The central gravity and J2 suffice for everything except a precise range.
Days, low Earth orbit. Drag must be included, and its uncertainty is the dominant error.
Days, geostationary orbit. Third body gravity and solar radiation pressure must be included; drag is negligible.
Weeks or more. The uncertainty of any model exceeds the effect it is trying to capture, and the answer should be treated as a prediction with an error that grows.

SGP4 makes the choice for the caller by fitting the perturbations it models to the element set. That is its strength and its limit: it always agrees with the elements, and it cannot represent a force the fit did not include.

Comparison of what each model includes

Perturbation	Two-body	SGP4	SDP4	Force model integration
Central gravity	Yes	Yes	Yes	Yes
`J2` oblateness	No	Yes, secular	Yes, secular	Yes, full field
Drag	No	Yes, one fitted term	Yes, one fitted term	Yes, with a density model
Third body	No	No	Yes, long period	Yes
Solar radiation pressure	No	No	No	Yes
Resonance with the Earth's field	No	No	Yes	Yes, in the field
Higher zonals and tesserals	No	No	Partial	Yes
Relativity	No	No	No	Optional

Source

The force models and their orders of magnitude follow Vallado, Fundamentals of Astrodynamics and Applications.
The SGP4 treatment of oblateness and drag follows Vallado et al., Revisiting Spacetrack Report #3, AIAA 2006-6753, 2006.

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