Geostationary Station Keeping

Geostationary Station Keeping (space-systems/orbit-mechanics/geostationary-station-keeping)

Use when the task is building the geostationary station-keeping plan quantities for a GEO satellite at the conceptual level: turning the sidereal day into the geosynchronous radius and speed, the inclination drift amplitude into the annual north-south delta-v with the two-burn-per-year model, the thrust and mass into burn time, the specific impulse into the annual propellant, and the longitude acceleration with the east-west deadband box into the drift-cycle period and maneuver cadence. This leaf implements the standard GEO station-keeping model (two north-south burns per year, box-limited east-west cycling) in pure Python, stdlib only. It pairs with space-systems/mission-design/mission-delta-v-budget, which sums the delta-v budget and takes the annual station-keeping contribution as an input line item, and with space-systems/orbit-mechanics/orbital- perturbations for the perturbation environment that drives the drift. It does NOT do total mission delta-v summation, three-body equilibrium orbit maintenance, constellation phasing, or one-shot plane rotation; those belong to sibling leaves.

Domain quick reference

• Geosynchronous radius from the sidereal day: r = (MU * (T_sid / (2
  • pi))**2) ** (1/3), with MU = 398600.4418 km3/s2 and the sidereal day T_sid = 86164.0905 s, giving 42164.2 km. The solar day of 86400 s must not be used here.
• Geosynchronous orbital speed: v = 1000 * sqrt(MU / r) with the radius in km, giving 3074.7 m/s (3.075 km/s).
• Annual north-south delta-v: the inclination drifts about one degree per year under lunisolar gravity, and two burns per year hold it in a band of half-width drift/2, so dv_annual = 2 * v * sin(radians(drift) / 2), 45.61 m/s at 0.85 deg/yr; the per-burn value is exactly half, 22.81 m/s.
• Burn time at constant thrust: t = m * delta_v / F, valid while the burn is short against the orbital period.
• Annual propellant by the rocket equation: m_prop = m * (1 - exp(- delta_v / (Isp * g0))) over the year, g0 = 9.80665 m/s2; for small delta_v this approaches m * delta_v / (Isp * g0).
• East-west deadband cycling: inside a longitude box of half-width h (deg) under a residual longitude acceleration a (deg/day2) the satellite drifts box edge to box edge in T = 2 * sqrt(2 * h / a) days, and the correction cadence is 365.25 / T maneuvers per year, 14.9 days and 24.5 per year at h = 0.05 deg, a = 0.0018 deg/day2.
• Uncontrolled drift: with station keeping off the inclination grows linearly, t = tolerance / drift, 0.1176 years at 0.1 deg tolerance and 0.85 deg/yr.
• Units are explicit per function: radius in km, speeds and delta-v in m/s, burn time in s, periods in days, drift time in years, mass in kg.
• ECSS frames the space systems and orbit environment context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Get the orbit geometry: geosynchronous_radius() and geo_speed() from the sidereal-day constants.
2. Fix the annual inclination drift amplitude in deg/yr from the lunisolar environment and compute ns_annual_delta_v; confirm the band model with ns_per_burn_delta_v, exactly half.
3. Turn the per-burn delta-v into a maneuver: burn_time with the thruster thrust and spacecraft mass.
4. Size the annual propellant: annual_propellant with the specific impulse over the annual delta-v.
5. Set the east-west control box: ew_cycle_period from the box half-width and the residual longitude acceleration, then ew_maneuvers_per_year for the cadence.
6. For the no-keeping case, run uncontrolled_drift_years to find how long the inclination tolerance holds before control is required.
7. Feed the annual delta-v and propellant as the station-keeping line item into a mission-delta-v-budget summation.
8. Confirm the deterministic checks with the contract test scripts/test_geostationary_station_keeping.py.

Worked example

A GEO satellite at 2000 kg wet mass with a 400 N apogee-class thruster at Isp = 280 s, an inclination drift of 0.85 deg/yr, and an east-west control box of 0.1 deg full width (0.05 deg half width) under a residual longitude acceleration of 0.0018 deg/day2.

• Geosynchronous radius: geosynchronous_radius() = 42164.17 km, the 42164.2 km anchor; geo_speed() = 3074.66 m/s, the 3074.7 m/s anchor.
• Annual north-south delta-v at 0.85 deg/yr: ns_annual_delta_v(0.85) = 45.61 m/s; per burn, ns_per_burn_delta_v(0.85) = 22.81 m/s, and the identity dv_annual = 2 * dv_per_burn holds exactly.
• Burn time per N/S burn: burn_time(22.81, 400.0, 2000.0) = 114.05 s, the 114.0 s anchor within 0.1 s.
• Annual propellant: annual_propellant(45.61, 280.0, 2000.0) = 32.95 kg, the 33.0 kg anchor within 0.2 kg.
• East-west cycle: ew_cycle_period(0.05, 0.0018) = 14.907 days and ew_maneuvers_per_year(0.05, 0.0018) = 24.50 per year; the product period * cadence = 365.25 days exactly.
• Uncontrolled drift: uncontrolled_drift_years(0.1, 0.85) = 0.1176 years, so a 0.1 deg inclination tolerance is spent in about 43 days with station keeping off, which motivates the two-burn-per-year N/S control.

Verification

• Confirm geosynchronous_radius() returns 42164.2 km and geo_speed() 3074.7 m/s within tolerance, and that r * v**2 = MU * 1e6 closes the circular-orbit identity.
• Confirm ns_annual_delta_v(0.85) returns 45.61 m/s within 0.05 and that the annual value equals twice the per-burn value at any drift.
• Confirm ew_cycle_period(0.05, 0.0018) returns 14.907 days within 0.05 and the cadence 24.5 per year, with period times cadence equal to 365.25.
• Confirm the small-delta-v linearization of annual_propellant and the sqrt scaling of the deadband cycle with box half-width and acceleration.
• Confirm every non-physical input raises ValueError: negative inclination drift, thrust or mass or Isp at zero or below, negative delta-v, box half width or acceleration at zero or below, tolerance at zero or below, drift rate at zero or below.
• Confirm repeated calls are bit-identical (pure deterministic functions, no RNG).
• Run the contract test offline: python3 scripts/test_geostationary_station_keeping.py (33 tests, deterministic).

Related leaves

• space-systems/mission-design/mission-delta-v-budget: the mission delta-v summation that consumes the annual station-keeping delta-v from this leaf as an input line item.
• space-systems/orbit-mechanics/orbital-perturbations: the perturbation environment (lunisolar gravity, longitude acceleration) that drives the drift this leaf controls.
• space-systems/orbit-mechanics/three-body-libration: equilibrium orbit maintenance in the circular restricted three-body problem, a different regime from the GEO ring.
• space-systems/orbit-mechanics/walker-delta-constellation: constellation design and phasing, not per-satellite orbit keeping.
• space-systems/orbit-mechanics/plane-change-maneuver: a one-shot plane rotation between circular orbits, not the annual inclination-drift control loop.

Pitfalls

• Quoting the annual N/S delta-v as the per-burn value: the year is controlled with two burns, so the maneuver and its burn time run on 22.81 m/s, not 45.61 m/s, and the burn time anchor is 114 s, not 228 s.
• Sizing the radius with the solar day: the geosynchronous period is the sidereal day, 86164.0905 s, not 86400 s; the solar-day radius comes out about 70 km high.
• Mixing kilometers and meters: the speed is 3074.7 m/s or 3.075 km/s; feeding 3074.7 into an equation expecting km/s inflates the delta-v by a factor of 1000.
• Treating the E/W drift cycle as linear: under a constant residual longitude acceleration the longitude follows a parabola in the box, so the cycle period carries the sqrt(2 * h / a) dependence; a naive linear crossing estimate shortens the period and inflates the cadence.
• Reading the east-west box as full width: the cycle period uses the half width; a 0.1 deg box is 0.05 deg half width, which is why the worked example cycles in 14.9 days and not 21.1 days.
• Ignoring the uncontrolled-drift verdict: 0.1 deg of inclination tolerance is consumed in 0.1176 years (about 43 days) without N/S control, so a GEO payload that needs tight pointing cannot ride the natural drift.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_geostationary_station_keeping.py

The test covers the geosynchronous radius and speed anchors (42164.2 km, 3074.7 m/s) and the circular-orbit identity, the annual and per-burn N/S delta-v anchors at 0.85 deg/yr with the annual-equals- twice-per-burn identity and the small-drift linear approximation, the burn-time anchor and thrust scaling, the annual propellant anchor with the small-delta-v rocket-equation linearization, the east-west cycle and cadence anchors with the period-times-cadence identity of 365.25 and the sqrt scaling laws, the uncontrolled-drift anchor and its scaling, ValueError rejection of every non-physical input, and determinism of repeated calls.

Compliance

• Standards referenced, not reproduced: ECSS is the family spine for space systems and the orbit environment context (ecss.nl/standards); the station-keeping relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.