Theory
How SDN-AP Predicts Orbits
SDN-AP is a general-perturbations propagator: it describes each orbit by slowly changing mean elements and moves them forward with averaged forces, so a whole catalog can be propagated in one call.
Method
Mean Elements And Averaging
An orbit's shape and orientation change all the time. Most of that change is a fast wobble that repeats every revolution; underneath it, the forces on the satellite turn and stretch the orbit slowly. A general-perturbations theory separates the two.
| Part | What it is | How SDN-AP treats it |
|---|---|---|
| Mean elements | The orbit with the once-per-revolution motion averaged away | Carried by the element set; the epoch state of the theory |
| Secular and long-period rates | The slow drift of the mean elements under each force | Integrated on the theory's own fixed grid, with checkpoints, so any time costs at most one checkpoint interval of work |
| Short-period terms | The fast wobble around the mean orbit | Added back at output time to give the osculating position and velocity |
Because the slow part is smooth, results do not depend on the order in which times are requested, and each object's memory is fixed when its element set is loaded.
Theories
Three Theories In One Module
The theory travels with each element set, so one catalog may mix all three. The configuration message names a theory with THEORY; UNSPECIFIED means each set's own.
| Theory | Element sets | Basis | Forces |
|---|---|---|---|
| SGP4 | Ephemeris type 0, drag term B* | Spacetrack Report No. 3 (Hoots and Roehrich 1980), as revisited by Vallado et al. (2006) | Zonal harmonics, power-law atmospheric drag; lunar and solar terms and resonance for deep-space orbits |
| SGP4-XP | Ephemeris type 4: ballistic coefficient BTERM and solar radiation coefficient AGOM, in m²/kg | The SGP4-XP theory as published in the 2022 astrodynamics conference literature, built from a clean-room mathematical specification | Zonals J2 to J5 (EGM-96), Jacchia-70 drag with a generic flux model, solar radiation pressure with shadow, lunar and solar terms, resonances, short-period terms |
| SDN-AP | SDN's own sets (mean-element theory SDNAP) | SDN's advanced mean-element theory on the same averaged structure, labeled as its own | Everything in SGP4-XP, plus the additions below |
What the SDN-AP theory adds
| Term | Model |
|---|---|
| Atmospheric density | NRLMSISE-00 (Picone et al. 2002) times the open density correction exp(s(t)), averaged over each revolution: a trapezoid in eccentric anomaly for near-circular orbits, a perigee band otherwise. A shared density grid (79 heights, 13 latitudes, 16 local times, every 3 h) serves the whole catalog. |
| Drag coefficient | A signed effective ballistic coefficient Beff, free at any value, with the drag-makeup fraction carried. |
| Empirical accelerations | Transverse accelerations τ0, τc, τs fitted with the orbit. Beff and τ0 act in the drag regime (perigee below 2,500 km), where τc and τs decay after the fit arc; above it, τc and τs carry the once-per-revolution part of solar radiation pressure and are held. |
| Tesseral gravity | EGM-96 to degree and order 70: the m-daily short-period terms at output time, and the shallow resonances (odd degree to 31) integrated with the rates. |
| Zonal tail | J6 to J20: secular and long-period terms in the rates. |
| Resonance | The 12-hour, 1-day, 8-hour and 16-hour classes with the exact Hansen coefficients, valid at every eccentricity. Truncated power series in eccentricity diverge above the Laplace limit, e = 0.6627; Molniya orbits sit near e = 0.72. |
| Beyond Earth orbit | Segment sets about the Earth, the Moon or the Sun, described under regimes below. |
Regimes
One Module For Every Regime
The forces that matter change with altitude and shape. SDN-AP selects its terms per object, from orbits that graze the atmosphere to interplanetary cruise.
| Regime | Typical objects | What dominates | How SDN-AP handles it |
|---|---|---|---|
| Low Earth orbit | Earth observation, large constellations, debris | Atmospheric drag, J2 | Drag regime below a 2,500 km perigee: NRLMSISE-00 with the open density correction, Beff and the empirical accelerations |
| Medium Earth orbit | Navigation satellites (12-hour orbits) | 12-hour resonance, Sun and Moon, sunlight pressure | Resonance terms, solar radiation pressure with shadow, held transverse terms |
| Geostationary orbit | Communications and weather satellites | 1-day resonance, Sun and Moon, sunlight pressure | The 1-day resonance class, lunar and solar terms, solar radiation pressure |
| Highly elliptical orbit | Molniya and transfer orbits | Resonance at high eccentricity, drag near perigee | Exact Hansen coefficients at any eccentricity; perigee-band drag |
| Cislunar space | Earth-centered orbits reaching past 250,000 km | Earth, Moon and Sun together | Segment sets in ICRF axes, fitted and propagated about the Earth |
| Lunar orbit | Lunar orbiters | The Moon's uneven gravity, Earth and Sun | Moon-centered mean equinoctial elements in the DE440 principal-axis frame; GRGM660PRIM lunar gravity (rates to degree 70); Earth and Sun third-body terms; sunlight pressure with the Moon's shadow |
| Interplanetary cruise | Spacecraft between the planets | The Sun, the planets, sunlight pressure, thrust | A Sun-centered state with the Sun's gravity reduced by sunlight pressure, the DE440 planets (Venus, the Earth-Moon barycenter, Mars, Jupiter, Saturn) and a radial, transverse and normal acceleration a0 + a1t; arcs cut at sphere-of-influence passages |
Methods
What The Module Does
Each method takes Space Data Standards messages and, where it needs one, a single configuration message.
| Method | In | Out |
|---|---|---|
ingest_omm | Element sets ($OMM) | The catalog the module holds |
propagate | Element sets ($OMM), the configuration, an optional grid template ($OEM), space weather ($SPW, $PRW) and the density correction ($DCL) | One ephemeris ($OEM) per element set, in TEME, GCRF or EME2000 |
fit | Ephemerides ($OEM), the configuration, space weather, the density correction and an optional prior set ($OMM) | Fitted SGP4-XP or SDN-AP element sets ($OMM) and an orbit-determination summary ($OBD) |
maneuver | An element set ($OMM), its maneuvers ($MNV), the configuration and an optional grid template | The element set after the burns ($OMM), the maneuvers ($MNV) and the ephemeris across them ($OEM) |
detect_maneuvers | Ephemerides ($OEM) and the configuration | Element sets ($OMM) and the detected maneuvers ($MNV) |
reentry | Element sets ($OMM) and the configuration | The reentry hand-off and window ($RDM) |
launch_set | A launch ($LAM) or burnout vector ($BOV), the configuration and an optional prior set | The first SDN-AP element set of each launched object ($OMM) |
launch_batch | A launch or burnout vector, ephemerides, the catalog and the configuration | Element sets for a batch released from one parent ($OMM) |
load_environment | Space weather ($SPW, $PRW) and the density correction ($DCL) | The environment the drag terms use |
Beside these, a structure-of-arrays entry takes plain element rows and writes six planes of positions and velocities for many objects at once; the demo uses it.
Runtime
Built To Run Anywhere
| Property | What it means |
|---|---|
| One WebAssembly module | The same file runs in a web browser and in WasmEdge on a server; the two give identical ephemeris bytes. |
| Threads without drift | Results are the same bytes at any thread count. |
| Bounded memory | Each object's memory is fixed when its element set is loaded; nothing allocates while propagating. |
| Refusals by name | A configuration value the module cannot apply is refused with the table and field it concerns. |
References
Literature
- Brouwer, D. (1959). Solution of the problem of artificial satellite theory without drag. AJ 64, 378–397.
- Lyddane, R. H. (1963). Small eccentricities or inclinations in the Brouwer theory of the artificial satellite. AJ 68, 555–558.
- Kozai, Y. (1959). On the effects of the Sun and the Moon upon the motion of a close Earth satellite. SAO Special Report 22.
- Kaula, W. M. (1966). Theory of Satellite Geodesy. Blaisdell.
- Jacchia, L. G. (1970). New static models of the thermosphere and exosphere with empirical temperature profiles. SAO Special Report 313.
- Broucke, R. A., Cefola, P. J. (1972). On the equinoctial orbit elements. Celestial Mechanics 5, 303–310.
- Hujsak, R. S. (1979). A restricted four body solution for resonating satellites without drag. Project Space Track Report No. 1.
- Hoots, F. R., Roehrich, R. L. (1980). Models for propagation of NORAD element sets. Spacetrack Report No. 3.
- Danielson, D. A., Sagovac, C. P., Neta, B., Early, L. W. (1995). Semianalytic satellite theory. NPS-MA-95-002.
- Lemoine, F. G. et al. (1998). The development of the joint NASA GSFC and NIMA geopotential model EGM96. NASA/TP-1998-206861.
- Montenbruck, O., Gill, E. (2000). Satellite Orbits. Springer.
- Picone, J. M., Hedin, A. E., Drob, D. P., Aikin, A. C. (2002). NRLMSISE-00 empirical model of the atmosphere. JGR 107(A12), 1468.
- Vallado, D. A., Crawford, P., Hujsak, R., Kelso, T. S. (2006). Revisiting Spacetrack Report #3. AIAA 2006-6753.
- Lemoine, F. G. et al. (2013). High-degree gravity models from GRAIL primary mission data. JGR Planets 118, 1676–1698.
- Park, R. S., Folkner, W. M., Williams, J. G., Boggs, D. H. (2021). The JPL planetary and lunar ephemerides DE440 and DE441. AJ 161, 105.
The mathematical specification and the module source: propagator/sdn-ap (SPEC.md, README.md).