Plane Change Maneuver

Plane Change Maneuver (space-systems/orbit-mechanics/plane-change-maneuver)

Use when the task is sizing an impulsive orbital plane change maneuver and trading the pure inclination-change burn against the combined burn that folds the plane change into an orbit transfer impulse. An inclination change at circular speed costs 2 v sin(di/2), which is brutal in LEO (3.803 km/s for 28.5 deg at 300 km) and mild at GEO altitude, so mission design pushes the plane change onto the slow apoapsis of a transfer ellipse and often merges it with the circularization burn there. This leaf implements the circular-orbit speed, the pure plane change law, the vis-viva transfer speed at the maneuver point, and the law-of-cosines combined burn in pure Python, stdlib only. It pairs with space-systems/orbit-mechanics/hohmann-transfer for the coplanar transfer ellipse that sets up the apsidal plane change, and with space-systems/mission-design/mission-delta-v-budget where the resulting contribution is summed into the mission budget.

Domain quick reference

• Circular orbit speed: v_c = sqrt(mu / r), with mu = 398600.4418 km3/s2 and r in km. At 300 km circular altitude (r = 6678 km) v_c = 7.7258 km/s; at geostationary radius (r = 42164 km) v_c = 3.0747 km/s.
• Pure plane change: dv = 2 v sin(di/2) rotates the velocity vector by the inclination change di with no speed change. At the LEO circular speed a 28.5 deg change costs 3.803 km/s; at the GEO circular speed the same change costs only 1.514 km/s.
• Transfer speed at the maneuver point (vis-viva): v = sqrt(mu (2/r - 1/a)) holds at any point of the transfer ellipse, periapsis or apoapsis, and every ellipse point satisfies r < 2a. On the GTO ellipse of semimajor axis 24421 km the speed at the 42164 km apogee is 1.6078 km/s.
• Combined one-burn delta-v: dv = sqrt(v1^2 + v2^2 - 2 v1 v2 cos(di)) from the law of cosines over the angle between the velocity vectors before and after the burn, used when the plane change and the speed change are applied by one impulse.
• Separate total: (v_after - v_before) + 2 v_after sin(di/2), the two burns done independently at the same point.
• Maneuver verdict: 'combined-cheaper' when the combined burn is below the separate total by more than 1e-9, otherwise 'pure-cheaper-or- equal'; the pure-only case returns 'pure-only'.
• Units are km, km3/s2, km/s and degrees throughout.
• ECSS E-ST-32 frames spacecraft dynamics and maneuver analysis; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the maneuver geometry: the circular orbit radius (radius_km) and the inclination change di in degrees.
2. Get the reference speed at the maneuver point with circular_orbit_speed(mu, radius_km), then the pure burn cost with plane_change_dv(speed, di).
3. For an apsidal plane change at the end of a transfer, set the transfer semimajor axis a = (r1 + r2) / 2 (the coplanar Hohmann ellipse from hohmann-transfer) and the maneuver radius at the apoapsis end.
4. Evaluate the transfer speed at the apoapsis with transfer_speed_at_radius(mu, apogee_radius_km, a) and the target circular speed at the same radius.
5. Compare the one-burn combined_burn_dv(v_before, v_after, di) against the separate total (v_after - v_before) plus the pure change, and read the winner from maneuver_verdict.
6. Run analyze_plane_change(mu, radius_km, di, transfer_semimajor_axis_km, target_radius_km) for the summary dict, or with only radius_km and di for the pure-only case.
7. Confirm the deterministic anchors with the contract test scripts/test_plane_change_maneuver.py.

Worked example

Case A, pure plane change: 28.5 deg inclination change on a 300 km circular orbit (r = 6678 km).

• circular_orbit_speed(MU_EARTH, 6678) = 7.7258 km/s.
• plane_change_dv(7.7258, 28.5) = 2 * 7.7258 * sin(14.25 deg) = 3.803 km/s.
• At GEO radius the same change costs plane_change_dv(3.0747, 28.5) = 1.514 km/s, which is why plane changes prefer high orbits.

Case B, combined with the transfer: GTO from 300 km to GEO with the 28.5 deg plane change done at apogee together with the circularization burn. Transfer semimajor axis a = (6678 + 42164) / 2 = 24421 km.

• transfer_speed_at_radius(MU_EARTH, 42164, 24421) = 1.6078 km/s (the exact model value; the classic quoted GTO apogee speed 1.6057 km/s corresponds to a 24411 km ellipse and sits within 1e-2 of it).
• circular_orbit_speed(MU_EARTH, 42164) = 3.0747 km/s.
• combined_burn_dv(1.6078, 3.0747, 28.5) = 1.8302 km/s; the classic 1.832 km/s quote follows from the rounded 1.6057 km/s apogee input, both round to the familiar ~1.83 km/s combined GTO-to-GEO burn.
• Separate total: (3.0747 - 1.6078) + 1.514 = 2.981 km/s (within 0.01 of the 2.983 km/s rounded reference).
• maneuver_verdict(2.981, 1.8302) = 'combined-cheaper': the combined burn saves about 1.15 km/s.

Pitfalls

• Doing the plane change in LEO out of habit: a 28.5 deg change at 300 km circular costs 3.803 km/s versus 1.514 km/s at GEO, so the maneuver radius dominates the cost and the burn belongs at the slow apoapsis end of the transfer.
• Quoting the separate burns when one burn can do both: the combined apogee burn is 1.8302 km/s against a 2.981 km/s separate total in the worked example; the verdict function exists precisely to catch a plan that pays the inclination change twice.
• Mixing the transfer ellipse with the target orbit: the apogee speed on the GTO ellipse (1.6078 km/s at 42164 km) is not the GEO circular speed (3.0747 km/s); the combined burn is the law-of- cosines gap between them.
• Forgetting every point of an ellipse satisfies r < 2a: a maneuver radius with 2 a <= r is off the ellipse and raises ValueError.
• Feeding an inclination change outside (-180, 180] or a negative speed: the law-of-cosines geometry and the pure-change formula both reject non-physical inputs instead of returning a number.
• Expecting the model to pick the plane: this leaf sizes a specified di; choosing the target inclination (sun-synchronous geometry, launch-site constraints) belongs to the sun-synchronous-inclination and launch-window-analysis leaves.

Verification

• Confirm circular_orbit_speed(MU_EARTH, 6678) returns 7.7258 km/s and circular_orbit_speed(MU_EARTH, 42164) returns 3.0747 km/s.
• Confirm plane_change_dv(7.7258, 28.5) returns 3.803 km/s and plane_change_dv(3.0747, 28.5) returns 1.514 km/s.
• Confirm transfer_speed_at_radius(MU_EARTH, 42164, 24421) returns 1.6078 km/s, above the GEO circular speed at periapsis and below it at apoapsis.
• Confirm combined_burn_dv(1.6057, 3.0747, 28.5) returns 1.832 km/s within 0.001 and that di = 0 reduces the combined burn to the pure speed difference v_after - v_before.
• Confirm di = 0 returns zero delta-v, di = 90 at the LEO circular speed returns v * sqrt(2) = 10.926 km/s within 0.001, and di = 180 returns 2 v.
• Confirm every radius and semimajor axis of 0 or less, every point off the ellipse with 2 a <= r, every negative speed, and every di outside (-180, 180] raises ValueError.
• Run the contract test offline: python3 scripts/test_plane_change_maneuver.py (35 tests, deterministic).

Related leaves

• space-systems/orbit-mechanics/hohmann-transfer: the coplanar two- impulse transfer ellipse this leaf extends with an inclination change; hohmann-transfer treats coplanar orbits and defers plane changes here.
• space-systems/orbit-mechanics/low-thrust-spiral: the continuous low-thrust plane change alternative over many revolutions, opposite of the impulsive burn here.
• space-systems/mission-design/mission-delta-v-budget: sums this leaf's maneuver contribution with margin and converts the budget to propellant.
• space-systems/mission-design/launch-window-analysis: the launch-side plane change geometry that sets the initial orbital plane.
• space-systems/orbit-mechanics/sun-synchronous-inclination: the inclination target set by sun-synchronous geometry.

Behavior contract (gate 3)

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

python3 scripts/test_plane_change_maneuver.py

The test covers the circular speed anchors at LEO and GEO, the pure plane change anchors 3.803 and 1.514 km/s with the 2 v sin(di/2) law, the vis-viva GTO apogee speed 1.6078 km/s and its identity, the combined-burn anchor 1.832 km/s from the spec inputs (1.6057, 3.0747, 28.5) with the law-of-cosines identity, the separate total 2.983 km/s within 0.01, the combined-cheaper verdict and the ~1.15 km/s saving, the di = 0 zero delta-v and speed-difference limits, the di = 90 sanity value 10.926 km/s and di = 180 full-retrograde cases, the analyze chain dicts for the pure-only and combined cases, and ValueError rejection of non-positive radii, semimajor axes and mu, points off the ellipse with 2 a <= r, negative speeds, and inclination changes outside (-180, 180].

Compliance

• Standards referenced, not reproduced: ECSS E-ST-32 is a free ESA download (ecss.nl/standards); the plane change relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.