Low Thrust Spiral

Low-Thrust Spiral Transfer (space-systems/orbit-mechanics/low-thrust-spiral)

Use when the task is sizing a continuous-thrust orbit transfer: a low-thrust electric propulsion trajectory that spirals slowly outward (or inward) between two circular orbits while the thrust also turns the orbital plane. This leaf implements the Edelbaum approximation, which closes the delta-v budget, propellant mass, and transfer time analytically without integrating the spiral. It pairs with the impulsive-transfer leaves for the low-thrust versus chemical comparison: hohmann-transfer for the two-impulse coplanar baseline and lambert-transfer for time-constrained impulsive arcs.

Domain quick reference

• Circular orbit velocity: v = sqrt(mu / r), with mu the gravitational parameter (3.986004418e14 m^3/s^2 for Earth) and r the orbit radius in meters. A low earth orbit at 6878 km radius flies at about 7613 m/s; a geostationary orbit at 42164 km radius flies at about 3075 m/s.
• Edelbaum delta-v with a total inclination change di in degrees: dv = sqrt(v_i^2 + v_f^2 - 2 * v_i * v_f * cos(pi * di_rad / 2)), with di_rad = di * pi / 180 and v_i, v_f the circular velocities at the inner and outer radii. The approximation assumes the thrust is continuously steered so the plane change is spread over the whole spiral.
• No-plane-change case: with di = 0 the Edelbaum expression reduces to |v_i - v_f|, the pure spiral budget, so the two code paths agree exactly. A low-earth to geostationary spiral without a plane change costs about 4538 m/s against about 3816 m/s for the impulsive Hohmann transfer of the same coplanar radii.
• Exhaust velocity: c = g0 * I_sp with g0 = 9.80665 m/s^2 and I_sp the specific impulse in seconds. A 3000 s ion thruster exhausts at about 29420 m/s.
• Rocket equation: m_final = m0 * exp(-dv / c), propellant mass m_prop = m0 - m_final. Continuous thrust buys high specific impulse, so the propellant mass is small but the burn is long.
• Transfer time at constant thrust T: t = m_prop * c / T. Thrust is low (0.1 to 1 N class), so low-thrust transfers take weeks to months, not hours: the worked example below takes about 245 days.
• Hohmann comparison: hohmann_delta_v(r_i, r_f) = v_i * (sqrt(2 * r_f / (r_i + r_f)) - 1) + v_f * (1 - sqrt(2 * r_i / (r_i + r_f))), the two-impulse budget of the same end radii without a plane change, returned as a positive magnitude for both transfer directions.
• Units are SI throughout: m, m/s, s, N, kg, and degrees for the inclination change.
• ECSS E-ST-10C frames the mission analysis that owns transfer sizing; the relations above are standard astrodynamics methodology, summary-only.

Workflow

1. Establish the circular orbit radii r_i and r_f in meters (radius, not altitude) and the total inclination change di in degrees; use the Earth gravitational parameter unless another central body is given.
2. Compute the circular velocities at both radii with circular_velocity to know the starting and target speeds.
3. Get the delta-v budget with edelbaum_delta_v(r_i, r_f, di): this is the continuous-thrust cost including the plane change. For a transfer without an inclination change, read the same answer from spiral_no_plane_change_delta_v and confirm the two agree.
4. Size the propulsion with transfer_mass_and_time(dv, m0, thrust, isp): propellant mass, final mass, and transfer time for the constant-thrust, constant-specific-impulse assumption.
5. Or run low_thrust_transfer_summary(r_i, r_f, di, thrust, isp, m0) once for the packed result: v_i, v_f, delta_v, m_prop, mf, and t_transfer.
6. Compare against the impulsive baseline with hohmann_delta_v(r_i, r_f): the spiral pays roughly 20% more delta-v than the coplanar Hohmann transfer for the same radii (about 4538 m/s versus 3816 m/s for low earth to geostationary), while the Hohmann path needs a chemical stage and pays its own plane change.
7. Sanity-check the result: the Edelbaum low-earth to geostationary budget with the 28.5 deg inclination change is about 5846 m/s, the transfer time for a 0.5 N, 3000 s ion thruster on a 2000 kg bus is about 245 days.

Worked example

Low earth orbit at r_i = 6878 km (500 km altitude, v_i = 7612.68 m/s) to geostationary orbit at r_f = 42164 km (v_f = 3074.67 m/s), with the 28.5 deg inclination change of the low earth orbit removed during the spiral. Bus mass m0 = 2000 kg, ion thruster T = 0.5 N at I_sp = 3000 s (c = 29419.95 m/s).

• Circular velocities: v_i = sqrt(3.986004418e14 / 6878e3) = 7612.68 m/s, v_f = sqrt(3.986004418e14 / 42164e3) = 3074.67 m/s.
• Edelbaum delta-v: dv = sqrt(7612.68^2 + 3074.67^2 - 2 * 7612.68 * 3074.67 * cos(pi * 0.4974 / 2)) = 5845.58 m/s, about 5.85 km/s with the plane change folded in. With di = 0 the same call returns 4538.02 m/s, equal to |v_i - v_f|, the pure spiral budget.
• Propellant mass: m_prop = 2000 * (1 - exp(-5845.58 / 29419.95)) = 360.40 kg, about 18% of the initial mass; mf = 1639.60 kg.
• Transfer time: t = m_prop * c / T = 360.40 * 29419.95 / 0.5 = 2.1206e7 s, about 245.4 days of continuous thrust.
• Hohmann comparison: the impulsive coplanar budget of the same radii is hohmann_delta_v = 3816.09 m/s, so the low-thrust spiral pays 5845.58 / 3816.09 = 1.53 times the coplanar chemical delta-v, and the chemical path would still need its own 28.5 deg plane change. The ion option trades that extra delta-v against the much higher specific impulse, at the cost of an 8 month transfer.

Pitfalls

• Passing altitude instead of radius: the Edelbaum call takes circular orbit radii (6878e3 m for a 500 km LEO), not altitude, so a surface-radius slip shifts both velocities and the whole budget.
• Confusing the plane-change and no-plane-change budgets: the 28.5 deg LEO-to-GEO spiral costs 5845.58 m/s while the di = 0 case costs 4538.02 m/s; quote the one that matches the mission's inclination requirement.
• Sizing propellant off the impulsive Hohmann budget: the low-thrust spiral pays 1.53 times the coplanar Hohmann delta-v in the worked example, so the chemical baseline underestimates the ion propellant unless the comparison is labeled as such.
• Ignoring the burn duration: transfer time is m_prop * c / T and the 0.5 N class thrust yields months of thrusting (245 days in the worked example); a schedule that plans weeks misreads the Edelbaum time.
• Feeding an inclination outside its domain: the total inclination change must lie in [0, 180] degrees and every radius, thrust, initial mass, and specific impulse must be positive; the functions raise ValueError rather than silently extrapolate.
• Treating the approximation as an integration: Edelbaum closes the budget analytically under continuously steered thrust, so the result is the steering-optimal budget, not a thrust-profile trajectory; J2 and drag accumulated over months belong to the perturbations leaf.

Verification

• Confirm edelbaum_delta_v(6878e3, 42164e3, 28.5) returns 5845.58 m/s (within 1% of the 5840 m/s reference) and that the di = 0 call returns 4538.02 m/s, exactly |v_i - v_f|.
• Confirm low_thrust_transfer_summary(6878e3, 42164e3, 28.5, 0.5, 3000, 2000) returns delta_v 5845.58 m/s, m_prop 360.40 kg, mf 1639.60 kg, and t_transfer 2.1206e7 s (245.4 days).
• Confirm hohmann_delta_v(6878e3, 42164e3) returns 3816.09 m/s, the impulsive comparison, and that the spiral no-plane-change budget exceeds it.
• Confirm the rocket equation round trip: mf * exp(dv / c) recovers m0, and t_transfer equals m_prop * c / T.
• Confirm every non-positive radius, thrust, initial mass, or specific impulse, and every inclination outside [0, 180] degrees raises ValueError.
• Run the contract test offline: python3 scripts/test_low_thrust_spiral.py (35 tests, deterministic).

Related leaves

• space-systems/orbit-mechanics/hohmann-transfer: the two-impulse coplanar transfer that low-thrust spirals compete against.
• space-systems/orbit-mechanics/lambert-transfer: time-constrained impulsive transfers for the chemical comparison.
• space-systems/mission-design/mission-delta-v-budget: rolls the low-thrust leg into the full mission delta-v and propellant budget.
• space-systems/orbit-mechanics/orbital-perturbations: J2 and drag effects that a many-month spiral accumulates.
• space-systems/orbit-mechanics/satellite-coverage: the target orbit geometry that motivates the transfer.

Behavior contract (gate 3)

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

python3 scripts/test_low_thrust_spiral.py

The test covers the low-earth to geostationary sizing contract (Edelbaum delta-v 5845.58 m/s within 1% of 5840 m/s, the di = 0 identity |v_i - v_f| = 4538.02 m/s, propellant mass 360.40 kg, final mass 1639.60 kg, transfer time 245.4 days, Hohmann comparison 3816.09 m/s), circular velocity anchors and mu scaling, plane-change-only and equal-radius boundary cases, inclination range boundaries, the rocket equation round trip, time-from-propellant consistency, and ValueError rejection of non-positive radii, thrust, mass, specific impulse, and out-of-range inclination.

Compliance

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