84 lines
4.8 KiB
Markdown
84 lines
4.8 KiB
Markdown
---
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title: Choosing a propagator
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summary: A comparison of the propagation methods, with the cost and the accuracy each one buys.
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tags: propagation, comparison, sgp4, hpop
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---
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# Choosing a propagator
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More than one method answers the question "where is the satellite". They
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differ in what they require as input, in what they cost, and in how long the
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answer stays true. This document compares them and gives the decision rule
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that follows.
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## The comparison
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| Method | Accuracy | Computational cost | Input required | Reference | Status |
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|--------|----------|--------------------|----------------|-----------|-------------------------|
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| Two-body Keplerian | Exact for the model; error grows without bound against the real orbit | Very low: one evaluation of Kepler's equation | Osculating elements and a state | Kepler, standard treatment in Vallado | Widely used |
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| SGP4 and SDP4 | About 1 km over the fit interval, degrading with time and solar activity | Very low: closed form evaluation | Two-line element set | Vallado et al. 2006 | Widely deployed |
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| Numerical integration with `J2` only | Error grows without bound because drag and third body are absent | Moderate: an integration step | Initial state and a gravity model | Textbook models | Textbook |
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| HPOP, numerical integration of a high order field | Meters over days, decimeters with a good force model and a good initial state | High: an integration per step, several steps per minute of arc | Precise initial state, force model, and station data | Each operator's own model | Operator use |
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| Interpolated table from a propagator | Bounded by the interpolation error of the table | Low per query, high to build the table | A table already sampled | Standard interpolation | Standard practice |
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| Machine learning correction of SGP4 | Improved within the fit interval, unbounded outside it | Moderate per query, high to train | TLE plus a trained model and its training data | Current research | Research |
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## How to read the table
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The accuracy column is not comparable across rows without its interval. A
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method that is exact for its own assumptions can be far from the real orbit,
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and a method with a large numerical cost can be no better than its input.
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The honest comparison is **error against the real trajectory as a function
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of the interval from the reference epoch**, and that curve depends on the
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altitude and on the activity of the Sun.
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Two statements that hold generally:
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- SGP4 is the best available model for a two-line element set, because the
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elements are defined by a fit to it. A more expensive model given the same
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elements is **worse**, not better, because it answers a different question.
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- A high precision model is only as good as its initial state. Without a
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precise orbit determination from tracking data, it will not reach the
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accuracy its force model would allow.
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## The decision rule
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| Requirement | Use |
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|-------------|-----|
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| Reproduce a catalogue position | SGP4 with the matching gravity model |
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| Inspect one state or plan an impulsive maneuver | The two-body model on the osculating state |
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| Query a track many times over a short interval | Sample once, then interpolate; see [[ephemeris-interpolation]] |
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| Predict over weeks | SGP4, with a stated error that grows and a check against a fresher element set |
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| Sub-meter accuracy over days | A precise orbit determination and a force model propagator, which are outside this workspace |
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| A rough geometry check | The two-body model, with the understanding that it drifts |
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## Cost in practice
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The cost column in the table is relative between methods, not absolute. Some
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orders of magnitude, for a single evaluation on a modern processor:
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| Method | Cost per evaluation | Notes |
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|--------|---------------------|-------|
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| Two-body | Tens of floating point operations | One Newton iteration for Kepler's equation |
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| SGP4 | A few hundred floating point operations | Closed form, no iteration beyond the element preparation |
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| SDP4 | A few thousand | The resonance and luni-solar terms add work |
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| Numerical integration | Milliseconds | Thousands of evaluations of the force model per day of propagation |
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| Interpolation | Tens of operations | A cubic Hermite evaluation per component |
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The ratio between SGP4 and a numerical integration is several orders of
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magnitude, which is why a coverage or access search uses an interpolated
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table rather than the propagator directly.
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## Source
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- SGP4 accuracy and the model: Vallado, Crawford, Hujsak, and Kelso,
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*Revisiting Spacetrack Report #3*, AIAA 2006-6753, 2006.
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- Force models and their relative size: Vallado, *Fundamentals of
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Astrodynamics and Applications*.
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## See also
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- [[sgp4]] for the model the element sets are fitted to
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- [[perturbations]] for what each model includes
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- [[ephemeris-interpolation]] for the table approach
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- [[keplerian-elements]] for the two-body model
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