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SGP4 vs SDP4: Propagator Selection and Error Analysis

Quick Reference

Orbit Type Period Mean Motion Propagator Altitude
LEO < 225 min > 6.4 rev/day SGP4 < 5,878 km
MEO/GEO/HEO ≥ 225 min ≤ 6.4 rev/day SDP4 ≥ 5,878 km

Critical Rule: The threshold is 225 minutes orbital period (or 6.4 revolutions/day)

What Each Propagator Includes

SGP4 (Simplified General Perturbations 4)

For near-earth orbits (period < 225 min)

Includes: - ✅ J2 gravitational harmonic (Earth oblateness) - ✅ J3 gravitational harmonic (pear-shaped Earth) - ✅ J4 gravitational harmonic (higher-order oblateness) - ✅ Atmospheric drag (via B* drag coefficient) - ✅ Short-period perturbations

Does NOT include: - ❌ Lunar gravity perturbations - ❌ Solar gravity perturbations - ❌ Resonance effects - ❌ Long-period perturbation integrator

SDP4 (Simplified Deep-Space Perturbations 4)

For deep-space orbits (period ≥ 225 min)

Includes everything SGP4 has, PLUS: - ✅ Lunar gravity (secular and periodic terms) - ✅ Solar gravity (secular and periodic terms) - ✅ Resonance effects (12-hour and 24-hour orbits) - ✅ Deep-space integrator (720-minute time steps) - ✅ Lyddane coordinate system (better for high-altitude orbits)

Error from Using the Wrong Propagator

Using SGP4 at GEO Altitude ❌

Expected Position Errors:

Time Period Estimated Error Cause
1 day 5-10 km Missing lunar/solar daily effects
7 days 30-70 km Long-period perturbations accumulate
15 days 100-200 km Compare to ≤40 km with proper SDP4
30 days 200-500 km Resonance errors dominate
1 year 1000+ km Secular drift, inclination errors

Published Research Findings: - Proper SDP4 at GEO: ~1.9 km average accuracy - 15-day SDP4 predictions: ≤40 km error - Using wrong propagator: "significantly larger errors"

Why Errors Grow So Large

  1. Comparable Force Magnitudes
  2. At GEO altitude (~35,786 km), Earth's oblateness and lunisolar gravitational forces are comparable in magnitude
  3. SGP4 only models Earth's oblateness

  4. Missing 50% of the dominant forces!

  5. Lunar Effects (Moon at ~384,400 km from Earth)

  6. Causes large amplitude, long-period perturbations
  7. Periodic effects with ~27-day lunar month cycle

  8. Unmodeled → errors accumulate linearly

  9. Solar Effects (Sun at ~150M km from Earth)

  10. Causes secular drift in orbital elements
  11. Perigee height changes at ~1 km/day over months
  12. Inclination drifts ~0.03° after 11 years

  13. Unmodeled → quadratic error growth

  14. 24-Hour Resonance

  15. GEO satellites are in 1:1 resonance with Earth's rotation
  16. Resonance amplifies certain perturbations
  17. SDP4 includes specific resonance terms for 24-hour orbits
  18. Without these → rapid orbital element drift

Using SDP4 for LEO ✅ (Safe but Slower)

Using SDP4 for LEO orbits is safe but computationally expensive: - ✅ Still produces correct results (lunar/solar effects are small at LEO) - ❌ ~4x more computation than SGP4 - ❌ GPU performance: 5.8M props/sec (vs 439M props/sec with SGP4)

The keplemon implementation automatically chooses the correct propagator based on mean motion.

Physical Intuition

Why 225 minutes?

At orbital periods ≥ 225 minutes: - Satellite is far enough from Earth that lunar/solar gravity becomes significant - Earth's gravitational harmonics (J2, J3, J4) decrease with altitude - Third-body effects increase with altitude - The crossover point is where these effects become comparable

Force Scaling

Altitude J2 Effect Lunar/Solar Effect Ratio
LEO (400 km) Dominant Negligible 1000:1
MEO (20,000 km) Strong Weak 100:1
GEO (35,786 km) Moderate Moderate 1:1

Implementation in Keplemon

The library automatically selects the correct propagator:

use keplemon::elements::TLE;
use keplemon::bodies::Satellite;

// LEO satellite (ISS)
let leo_tle = TLE::from_lines(
    "1 25544U 98067A   25105.52083333  .00012345  00000+0  22013-3 0  9991",
    "2 25544  51.6456 339.5765 0003456  35.8734  85.9834 15.48919755123456",
    Some("ISS")
)?;
// Mean motion: 15.49 rev/day → SGP4 automatically selected ✓

// GEO satellite (TDRS 3)
let geo_tle = TLE::from_lines(
    "1 19548U 88091B   26008.31539492 -.00000299  00000+0  00000+0 0  9994",
    "2 19548  12.7229 342.0612 0044050 345.9566 204.1506  1.00262839123781",
    Some("TDRS 3")
)?;
// Mean motion: 1.00 rev/day → SDP4 automatically selected ✓

No manual selection needed - the library handles it correctly.

GPU Performance Impact

From benchmark results (docs/gpu-benchmark-results.md):

Propagator GPU Speedup GPU Throughput Complexity
SGP4 (LEO) 83x 439M props/sec Low
SDP4 (GEO) 1.6x 5.8M props/sec High

Why is SDP4 75x slower on GPU? - More complex calculations (~4x more operations) - Deep conditional branching (resonance detection, coordinate conversion) - Iterative solvers (Newton-Raphson) - Poor GPU parallelism (warp divergence)

References

  1. "Analysis on the Accuracy of the SGP4/SDP4 Model"

  2. Comprehensive accuracy analysis across orbit regimes

  3. "High-Integrity TLE Error Models for MEO and GEO Satellites" (Virginia Tech)

  4. Statistical characterization of 148,984 MEO TLEs and 5,813 GEO TLEs

  5. PDF available

  6. "Lunar and solar perturbations on geosynchronous satellite orbits"

  7. Analysis of third-body effects at GEO altitude

  8. Celestrak - "Revisiting Spacetrack Report #3" (AIAA 2006-6753-Rev2)

  9. Official SGP4/SDP4 specification and implementation guide

  10. Wikipedia: Simplified Perturbations Models

  11. Good overview of the model family