Drag and the ballistic coefficient
Drag deceleration ad = ½ ρ v² CdA / m = ½ ρ v² / B with B = m / (Cd A). A heavier, more compact satellite (large B) is slowed less. Drag acts opposite to the velocity relative to the air; here the atmosphere is taken as static (no co-rotation, no winds).
Why a satellite speeds up as it loses energy
Orbital energy per unit mass is E = v²/2 − μ/r = −μ/(2a). Drag removes energy, so a shrinks and the circular speed v = √(μ/a) rises. Potential energy falls by twice as much as the total, so kinetic energy grows by the difference: the paradox is resolved. Period T = 2π√(a³/μ) shrinks.
Decay rate
Circular orbit: da/dt = −ρ (CdA/m) √(μa), or per revolution Δa = −2π a² ρ / B. For an eccentric orbit this program averages the Gauss variational equations da/dt = −(a²/μ) ρ v³/B, de/dt = −(e+cosν) ρ v / B over one revolution by numerical quadrature in eccentric anomaly (64 nodes). Drag is concentrated near perigee, so the apogee falls first and the orbit circularises, then the whole orbit sinks.
Thermosphere density (approximate empirical model)
Density comes from hydrostatic integration, d lnρ/dh = −mg/(kT) − (1/T) dT/dh, from 120 km, using a Jacchia-style temperature profile that rises exponentially to an exospheric temperature T∞ = 450 + 4.2 F10.7 + 5 Kp² K and a height-dependent mean molecular mass (28 to 2 u). It reproduces the broad behaviour of Jacchia-77 / NRLMSISE-00 (about 3×10-12 kg/m³ at 400 km for F10.7 = 150; a factor ~30 between solar minimum F10.7 = 70 and maximum F10.7 = 200 at 400 km) but is not a substitute for them: no diurnal bulge, latitude, season or composition detail, and the storm term is a simple heating of T∞.
Solar activity and storms
Extreme-ultraviolet heating tracks the 10.7 cm radio flux F10.7, which swings over the roughly 11-year solar cycle. Solar cycle 25 has been near its maximum in 2024–2026 (publicly reported by NOAA/NASA). Geomagnetic storms (Kp up to 9) dump energy at high latitude, heating and expanding the thermosphere within hours. Publicly reported examples: SpaceX said in February 2022 that a geomagnetic storm raised drag on a freshly launched batch of Starlink satellites deployed near 210 km and most re-entered; the strong storm of May 2024 raised drag on many low-orbit spacecraft (reported by NASA and ESA). The numbers here are illustrative, not a reconstruction of those events.
Station-keeping and disposal
Holding a circular orbit needs Δv/yr ≈ ½ ρ v² / B × 1 yr (the energy lost to drag divided by speed). The widely quoted 25-year post-mission disposal guideline comes from the Inter-Agency Space Debris Coordination Committee; the US FCC adopted a 5-year rule for US-licensed satellites (adopted 2022, applying to spacecraft launched from late 2024). The check in the readouts uses this program's own lifetime estimate and is for illustration only.
Re-entry footprint
An uncontrolled object can come down anywhere between latitudes ±i, so the map shades that band. Agencies typically quote re-entry time uncertainties of order ±10–20 % of the remaining life, which is why predictions only tighten in the final days. The footprint of surviving debris is an elongated strip of order a thousand km along the track.
Simplifications
- Spherical Earth (no J2 oblateness, no nodal regression), two-body motion, static atmosphere, constant ballistic coefficient, re-entry declared at 120 km perigee.
- The 3D drawing exaggerates altitudes (default ×3) and the decay ribbon magnifies radial change; the satellite model is far larger than true scale; the globe uses procedurally painted, stylised continents.
- At high time warp the on-screen orbital phase is only illustrative (it advances at most 400× real time); the physics uses exact elapsed time.
Sources
King-Hele, Satellite Orbits in an Atmosphere (1987); Vallado, Fundamentals of Astrodynamics and Applications; Jacchia (1977) thermospheric models; Picone et al. (2002) NRLMSISE-00; IADC Space Debris Mitigation Guidelines; FCC orbital debris rules (2022).