Bi Elliptic Transfer

Bi-Elliptic Transfer (space-systems/orbit-mechanics/bi-elliptic-transfer)

Use when the task is analyzing the bi-elliptic three-impulse transfer between two coplanar circular orbits: the perigee raise to an intermediate apogee radius r_b, the burn at r_b that raises the perigee to the target radius, the circularization at the target, the total delta-v, and the crossover comparison against the Hohmann two-impulse transfer for the same orbit pair. This leaf implements the standard three-burn model in pure Python, stdlib only. It pairs with space-systems/orbit-mechanics/hohmann-transfer, which owns the two-impulse Hohmann transfer itself; this leaf re-derives the Hohmann budget only as the internal comparison baseline that decides whether the three-burn strategy wins at large radius ratios.

Domain quick reference

• Circular orbit speed: v = sqrt(mu / r), with mu the gravitational parameter (3.986004418e14 m^3/s^2 for Earth) and r the orbit radius in meters.
• First burn (perigee raise): dv1 = sqrt(mu * (2/r1 - 2/(r1 + r_b))) - sqrt(mu / r1). At the inner circular orbit radius r1 the spacecraft accelerates onto a transfer ellipse whose apogee reaches the intermediate radius r_b above the target radius r2.
• Second burn (at the intermediate apogee): dv2 = sqrt(mu * (2/r_b - 2/(r_b + r2))) - sqrt(mu * (2/r_b - 2/(r1 + r_b))). At r_b the spacecraft changes from the first ellipse (perigee r1) to the second ellipse (perigee r2), so it arrives at the target radius from above.
• Third burn (circularization): dv3 = sqrt(mu * (2/r2 - 2/(r_b + r2))) - sqrt(mu / r2). The spacecraft meets r2 at the perigee of the second ellipse, faster than the circular speed, so the closing impulse is retrograde and dv3 is the positive magnitude of that speed gap.
• Bi-elliptic total: dv_bi = dv1 + dv2 + dv3. The three impulses are magnitudes; the arrival burn direction differs from the first two.
• Hohmann baseline: dv1_h = sqrt(mu * (2/r1 - 2/(r1 + r2))) - sqrt(mu/r1) and dv2_h = sqrt(mu/r2) - sqrt(mu * (2/r2 - 2/(r1 + r2))), summed for the two-impulse total of the same orbit pair.
• Crossover saving: saving = dv_hohmann - dv_bi. The bi-elliptic transfer is cheaper for large radius ratios (roughly r2/r1 above 11.94 at the best r_b); at small ratios the extra middle burn only adds cost. Ties go to hohmann, the simpler strategy.
• Transfer time: the sum of the two half-period coasts, t = pi * sqrt((r1 + r_b)^3 / (8 mu)) + pi * sqrt((r_b + r2)^3 / (8 mu)). The overshoot to r_b costs days: the worked example below coasts about 14.5 days.

Workflow

1. Establish the gravitational parameter mu and the two circular orbit radii r1 (start) and r2 (target, larger) in meters; radius, not altitude.
2. Choose the intermediate apogee radius r_b above r2. The classic choice r_b = 2 * r2 shows the crossover saving at large radius ratios; r_b close to r2 degenerates into the Hohmann budget.
3. Compute the Hohmann baseline for the orbit pair with hohmann_delta_v(mu, r1, r2) to know what the two-burn strategy costs.
4. Compute the three impulses with bi_elliptic_delta_v(mu, r1, r_b, r2); check that dv3 is small compared with dv1 when r_b is far above r2, as the closing burn happens near the target.
5. Compare strategies with transfer_comparison(mu, r1, r2, r_b) and read the saving and the verdict ("bi-elliptic" when the three-burn total is lower, "hohmann" otherwise).
6. When mission time matters, size the coast with transfer_time_bi_elliptic(mu, r1, r_b, r2) and convert to days; the bi-elliptic option trades a much longer transfer time for its delta-v saving.
7. Sanity-check the choice: an intermediate apogee at 2 * r2 with r2 = 30 * r1 saves about 113 m/s over Hohmann, while at r2 = 2 * r1 the three-burn option loses and the verdict is "hohmann".

Worked example

Earth mu = 3.986004418e14 m^3/s^2. Start radius r1 = 6578 km (300 km circular orbit), target radius r2 = 30 * r1 = 197340 km, intermediate apogee r_b = 2 * r2 = 394680 km. Module outputs (deterministic):

• Hohmann baseline: dv1 = 3045.4 m/s, dv2 = 1060.2 m/s, total = 4105.6 m/s.
• Bi-elliptic: dv1 = 3133.8 m/s, dv2 = 638.6 m/s, dv3 = 219.9 m/s, total = 3992.2 m/s.
• Crossover: saving = 113.4 m/s, verdict "bi-elliptic": the three-burn transfer is cheaper at this radius ratio with r_b = 2 * r2.
• Transfer time: 1248552.7 s, about 14.45 days, versus the roughly 5.3-hour Hohmann coast for the same orbit pair.

Pitfalls

• Passing altitude instead of radius: all inputs are circular orbit radii in meters (r1, r2, r_b), so a 300 km orbit enters as 6578000 m, not 300000.
• Choosing r_b at or below the target: the intermediate apogee must sit above r2 (r_b <= r2 raises ValueError); r_b close to r2 just degenerates into the Hohmann budget and buys nothing.
• Reversing the transfer direction: an inward transfer (r2 <= r1) is rejected with ValueError - this leaf models outward raises only.
• Expecting the extra burn to always pay: the bi-elliptic total only beats Hohmann at large radius ratios (the worked example saves 113.4 m/s at r2 = 30 r1); at small ratios like r2 = 2 r1 the verdict is "hohmann" and ties go to the simpler two-burn strategy.
• Comparing strategies on delta-v alone: the bi-elliptic coast runs 14.45 days versus a 5.3-hour Hohmann coast in the worked example, so a delta-v saving can be a mission-time loss.
• Forgetting the burn directions: dv1 and dv2 are prograde raises, but dv3 is the retrograde circularization at the target perigee; the three values are magnitudes, and only their sum forms the total.

Verification

• Worked-example bounds (contract anchors): Hohmann total in 3900-4300 m/s, bi-elliptic total in 3850-4150 m/s, saving in 50-180 m/s, verdict "bi-elliptic", transfer time in 10-40 days. The module outputs above sit inside every bound.
• Sanity at a small ratio: with r2 = 2 * r1 and a moderate r_b the Hohmann total stays below the bi-elliptic total and the verdict is "hohmann".
• Degenerate limit: with r_b very close to r2 (r_b = 1.01 * r2) the bi-elliptic total approaches the Hohmann total from above (bi_total > hohmann_total - 1.0 m/s).
• Round-trip: every impulse equals the difference of the two vis-viva speeds it is defined from, and the total equals dv1 + dv2 + dv3 within 1e-6.
• Determinism: pure math on module constants, no random number generation anywhere; repeated calls return identical floats.
• ValueError rejection of every non-physical input: mu <= 0, r1 <= 0, r2 <= 0, r1 == r2, r_b <= r2, and an inward transfer (r2 <= r1).
• Run the contract test offline: python3 scripts/test_bi_elliptic_transfer.py (33 tests, deterministic).

Related leaves

• space-systems/orbit-mechanics/hohmann-transfer: owns the two-impulse Hohmann transfer that this leaf re-derives only as the comparison baseline.
• space-systems/orbit-mechanics/keplerian-elements: the two-body orbital elements that frame both circular end orbits.
• space-systems/orbit-mechanics/orbital-perturbations: J2 and drag effects that a many-day transfer arc accumulates.

Behavior contract (gate 3)

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

python3 scripts/test_bi_elliptic_transfer.py

The test covers the worked example (Hohmann and bi-elliptic anchors with their magnitude bounds, the 50-180 m/s saving window, the "bi-elliptic" verdict, the 10-40 day time bound), the circular-speed identity, the small-radius-ratio sanity check with a "hohmann" verdict, the degenerate r_b near r2 limit, vis-viva round trips for every impulse, exact convenience-dict keys, run-to-run determinism, and ValueError rejection of every non-physical input in the validation list.

Compliance

• Standards referenced, not reproduced: ECSS-E-ST-10C (systems engineering general requirements) frames mission analysis and orbit design within the ECSS lifecycle, and the three-burn transfer geometry and vis-viva relationships above are common astrodynamics methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.