131 lines
5.7 KiB
Markdown
131 lines
5.7 KiB
Markdown
---
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title: Orbital perturbations
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summary: The forces that make an orbit deviate from an ellipse, and which of them each model includes.
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tags: orbit, perturbations, drag, oblateness, third-body
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---
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# Orbital perturbations
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A Keplerian orbit is an ellipse that repeats forever. A real orbit does not.
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The forces besides the point mass gravity of the Earth deform the ellipse
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and change its orientation. This document lists the forces and says which of
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them each model in the workspace includes.
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## Definition
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A perturbation is any acceleration on the satellite other than the central
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point mass gravity of the Earth. The main ones, in rough order of magnitude
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for a low Earth orbit, are:
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| Perturbation | Source | Typical size at low Earth orbit | Character |
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|--------------|--------|--------------------------------|-----------|
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| Oblateness, `J2` | The Earth is not a sphere | `1e-3` of central gravity | Secular in the node and perigee |
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| Atmospheric drag | Residual atmosphere | `1e-7` to `1e-4` of central gravity, depending on altitude and solar activity | Secular decay of the orbit |
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| Third body, Sun and Moon | Gravity of the Sun and Moon | `1e-6` at low Earth orbit, larger at high altitude | Long period |
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| Solar radiation pressure | Photon momentum | `1e-9` to `1e-8` of central gravity | Long period, depends on area and mass |
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| Higher zonals and tesserals | The Earth's gravity field | Diminishing series | Secular and resonant |
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| Relativity | General relativity | `1e-9` of central gravity at low Earth orbit | Secular in the node |
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| Tides and solid Earth tides | The Earth deforms | Below the above | Long period |
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The list is ordered by the size of the effect on a low Earth orbit. The
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order changes with altitude: above a few thousand kilometers the third body
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terms exceed drag, and the resonance with the Earth's gravity field becomes
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the dominant long period effect.
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## Mathematical meaning
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The central gravity is
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$$
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\mathbf{a}_{\text{central}} = -\frac{\mu \mathbf{r}}{\lvert \mathbf{r} \rvert^3}
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$$
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The `J2` term, which models the equatorial bulge, is
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$$
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\mathbf{a}_{J_2} = -\frac{3}{2} \frac{J_2 \mu R^2}{r^5}
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\begin{bmatrix} x \left( 1 - 5 z^2/r^2 \right) \\[2pt]
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y \left( 1 - 5 z^2/r^2 \right) \\[2pt]
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z \left( 3 - 5 z^2/r^2 \right) \end{bmatrix}
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$$
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with `R` the equatorial radius and `z` measured along the rotation axis. It
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has no component that changes the energy, so the size of the orbit is
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unaffected to first order; what it changes is the orientation. Its secular
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effects are
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$$
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\begin{aligned}
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\dot\Omega &= -\frac{3}{2} n J_2 \left( \frac{R}{a} \right)^2 \frac{\cos i}{\left( 1 - e^2 \right)^2} \\
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\dot\omega &= \frac{3}{4} n J_2 \left( \frac{R}{a} \right)^2 \frac{5 \cos^2 i - 1}{\left( 1 - e^2 \right)^2}
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\end{aligned}
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$$
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The first makes a low inclination orbit precess slowly westward and a polar
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orbit precess in right ascension. Both are large enough to matter within a
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day.
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Drag is not a conservative force. It removes energy and angular momentum,
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which lowers the semi-major axis and the eccentricity. Its size depends on
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the atmospheric density, which depends strongly on altitude and on the
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activity of the Sun. SGP4 models the effect with a single fitted number, the
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`BSTAR` term, rather than with a density model.
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Third body gravity and solar radiation pressure are first order effects on
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an orbit at high altitude, where they cannot be modelled by a single fitted
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number, and are treated by numerical integration in a force model propagator.
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## Unit convention
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- Accelerations are in **kilometers per second squared** where a force model
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is used, and are dimensionless ratios against the central gravity in this
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document for readability.
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- The zonal coefficients such as `J2` are **dimensionless**, and the
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reference radius they are scaled by is in **kilometers**.
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- The `BSTAR` term is in **inverse Earth radii**.
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## Applicability
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The choice of which perturbations to include follows from the altitude and
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the interval, not from a rule:
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- **A few hours, anywhere.** The central gravity and `J2` suffice for
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everything except a precise range.
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- **Days, low Earth orbit.** Drag must be included, and its uncertainty is
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the dominant error.
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- **Days, geostationary orbit.** Third body gravity and solar radiation
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pressure must be included; drag is negligible.
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- **Weeks or more.** The uncertainty of any model exceeds the effect it is
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trying to capture, and the answer should be treated as a prediction with
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an error that grows.
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SGP4 makes the choice for the caller by fitting the perturbations it models
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to the element set. That is its strength and its limit: it always agrees
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with the elements, and it cannot represent a force the fit did not include.
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## Comparison of what each model includes
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| Perturbation | Two-body | SGP4 | SDP4 | Force model integration |
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|--------------|----------|------|------|-------------------------|
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| Central gravity | Yes | Yes | Yes | Yes |
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| `J2` oblateness | No | Yes, secular | Yes, secular | Yes, full field |
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| Drag | No | Yes, one fitted term | Yes, one fitted term | Yes, with a density model |
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| Third body | No | No | Yes, long period | Yes |
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| Solar radiation pressure | No | No | No | Yes |
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| Resonance with the Earth's field | No | No | Yes | Yes, in the field |
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| Higher zonals and tesserals | No | No | Partial | Yes |
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| Relativity | No | No | No | Optional |
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## Source
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- The force models and their orders of magnitude follow Vallado,
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*Fundamentals of Astrodynamics and Applications*.
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- The SGP4 treatment of oblateness and drag follows Vallado et al.,
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*Revisiting Spacetrack Report #3*, AIAA 2006-6753, 2006.
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## See also
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- [[sgp4]] for the model that includes the first two
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- [[keplerian-elements]] for the orbit without any perturbation
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- [[propagator-comparison]] for the methods side by side
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