Orbital Perturbations

Orbital Perturbations (space-systems/orbit-mechanics/orbital-perturbations)

Use when the task is secular J2 perturbation analysis of an Earth orbit: the RAAN drift rate, the argument-of-perigee drift, the nodal period change, and how the perturbation magnitude falls with altitude. All numbers below were verified by running scripts/orbital_perturbations_logic.py and by a J2 numerical propagation of the circular orbit.

Domain quick reference

Constants: Re = 6371000 m, mu = 3.986004418e14 m^3/s^2, J2 = 1.08262668e-3. Semimajor axis a = Re + altitude_km * 1000 (m), mean motion n = sqrt(mu / a^3), Keplerian period T_K = 2 pi / n.

• RAAN drift rate: om_dot = -1.5 * n * J2 * (Re / a)^2 * cos(i) rad/s. At 500 km, i = 30 deg: -1.3403e-6 rad/s, which is -6.6352 deg/day. Zero at i = 90 deg, positive for retrograde orbits. At GEO (35786 km, i = 30 deg): -0.0116 deg/day, about 572x smaller because the rate scales as a^-3.5.
• Argument-of-perigee drift: w_dot = 0.75 * n * J2 * (Re / a)^2 * (5 cos^2(i) - 1) rad/s. At 500 km, i = 30 deg: +2.1281e-6 rad/s, which is +10.5347 deg/day. The drift is zero at the critical inclination 63.435 deg, positive below it, negative between 63.435 deg and 116.565 deg.
• Nodal period: T_n = 2 pi / (n + om_dot) s (the ground-track convention, consistent with the ground-track-repeat leaf). At 500 km, i = 30 deg: T_K = 5668.14 s, T_n = 5675.01 s, so the nodal period change dT = T_n - T_K = +6.86 s (prograde orbits lengthen). At i = 90 deg, dT = 0; at i = 97.4 deg (sun- synchronous retrograde), dT = -1.02 s (shortens).
• Draconitic period (exact ascending-node crossing interval): T_d = 2 pi / (M_dot + w_dot), with M_dot = n + 0.75 * n * J2 * (Re / a)^2 * (3 cos^2(i) - 1). At 500 km, i = 30 deg: 5652.36 s (about 16 s shorter than T_K); at i = 90 deg: 5676.07 s (longer).
• Perturbation magnitude versus altitude: the J2 acceleration ratio to two-body is (3/2) * J2 * (Re / a)^2. At 500 km: 1.3962e-3; at GEO: 3.7077e-5, about 38x smaller. Drift rates fall even faster, as a^-3.5, so LEO-sized drift does not exist at GEO.

Workflow

1. Take the altitude in km and the inclination in radians; compute a, n, and T_K with semimajor_axis, mean_motion, keplerian_period.
2. Compute the RAAN drift with raan_drift_rate and convert to degrees per day with rad_per_s_to_deg_per_day.
3. Compute the argument-of-perigee drift with arg_perigee_drift_rate and check the sign against the critical inclination (critical_inclination_rad).
4. Derive the nodal period and its change with nodal_period and nodal_period_change; use the draconitic period (draconitic_period) for the true node-crossing cadence.
5. Scale the perturbation with altitude using perturbation_magnitude_ratio; compare LEO and GEO magnitudes.
6. Pack the full solution with secular_drift_properties and report rates in degrees per day for mission planning.

Pitfalls

• Confusing this leaf with sun-synchronous-inclination: that leaf solves for the inclination that makes the RAAN drift match the sun; this leaf takes the inclination as given and quantifies the drifts, periods, and magnitudes. Do not call the sun-synchronous solver when the task is drift rates for an arbitrary given orbit.
• Confusing this leaf with hohmann-transfer: a Hohmann transfer changes the orbit radius with two impulsive burns; perturbations drift the elements continuously with no propulsion. A perigee drift is not a perigee-raising burn.
• Forgetting the RAAN drift sign: negative for prograde (i < 90 deg), zero at i = 90 deg, positive for retrograde; cos(i) flips the sign.
• Assuming the perigee always advances: w_dot is positive only below the critical inclination 63.435 deg, negative between 63.435 deg and 116.565 deg, and positive again above.
• Using the Keplerian period for node-based cadence: the nodal period 2 pi / (n + om_dot) differs from T_K by about 7 s per orbit at 500 km, i = 30 deg, and the difference accumulates over a mission.
• Mixing the two period conventions: the nodal period 2 pi / (n + om_dot) (ground-track convention) is not the draconitic period 2 pi / (M_dot + w_dot); they differ by about 23 s at 500 km, i = 30 deg.
• Expecting LEO-sized drift at high altitude: the magnitude ratio falls as (Re / a)^2 and the drift rates as a^-3.5; at GEO the RAAN drift is about 572x smaller than at 500 km.
• Feeding degrees into radian-based functions, or quoting rad/s rates as deg/day without conversion.

Behavior contract (gate 3)

The RAAN drift, argument-of-perigee drift, nodal period change, draconitic period, and magnitude logic is exercised by the gate 3 contract test: scripts/test_orbital_perturbations.py against scripts/orbital_perturbations_logic.py (stdlib unittest, offline, 27 test methods). Run: python3 scripts/test_orbital_perturbations.py

Compliance

• Standards referenced, not reproduced: the ECSS series text is copyright ESA; the J2 secular rates and the draconitic period are common astrodynamics, summary-only per standards-map.yaml (ecss is a free ESA download).
• compliance: STANDARDS-REF, gated: false.