Hohmann Transfer

Hohmann Transfer (space-systems/orbit-mechanics/hohmann-transfer)

Use when the task is sizing a two-impulse Hohmann transfer between two coplanar circular orbits: transfer-orbit geometry, circular and transfer-ellipse velocities, the departure and arrival burn impulses, the total delta-v budget, the transfer time, and the rendezvous lead angle.

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.
• Transfer-orbit semimajor axis: a = (r1 + r2) / 2. The transfer ellipse is tangent to the inner circular orbit at periapsis and to the outer circular orbit at apoapsis, so the semimajor axis is the mean of the two radii.
• Transfer period: T = 2 * pi * sqrt(a^3 / mu). The Hohmann transfer covers half the ellipse, so the one-way transfer time is T / 2. A low-earth to geostationary transfer takes about 19107 s, roughly 5.31 hours.
• Vis-viva on the transfer ellipse: v = sqrt(mu * (2 / r - 1 / a)). At r1 this is the periapsis speed, at r2 the apoapsis speed.
• Departure burn impulse: dv1 = |v_transfer(r1) - v_circular(r1)|. The first impulse at periapsis raises the spacecraft from the inner circular orbit onto the transfer ellipse; for an outward transfer it is prograde, for an inward transfer retrograde.
• Arrival burn impulse: dv2 = |v_circular(r2) - v_transfer(r2)|. The second impulse at apoapsis circularizes the transfer ellipse into the outer circular orbit.
• Total delta-v budget: dv_total = dv1 + dv2. The low-earth to geostationary coplanar transfer totals about 3816 m/s, the classic reference of about 3.9 km/s before any plane change.
• Burn timing: the departure burn fires at periapsis of the transfer ellipse (tangent to the inner orbit) and the arrival burn fires at apoapsis (tangent to the outer orbit); both burns are impulsive and aligned with the velocity vector for a coplanar transfer.
• Rendezvous phase angle: the chaser covers 180 degrees of true anomaly during the transfer while the target in the outer circular orbit sweeps (t_transfer / T2) * 360 degrees, so the chaser must lead the target by lead = 180 - (t_transfer / T2) * 360 degrees at departure. For the low-earth to geostationary case the lead angle is about 100.2 degrees.
• Specific orbital energy: epsilon = -mu / (2 * a). The transfer ellipse energy sits between the energies of the inner and outer circular orbits, which is why both impulses add energy for an outward transfer.
• Circular orbit period: T = 2 * pi * sqrt(r^3 / mu). A geostationary orbit period is about 86164 s, one sidereal day.

Workflow

1. Establish the two circular orbit radii r1 and r2 in meters (radius, not altitude) and the gravitational parameter mu of the central body.
2. Compute the circular-orbit velocities at both radii with circular_velocity to know what the spacecraft has before and after the transfer.
3. Build the transfer ellipse with transfer_semimajor_axis, then the one-way transfer time with transfer_time (or the full period with transfer_period).
4. Evaluate the periapsis and apoapsis speeds on the transfer ellipse with vis_viva_velocity and confirm they bracket the circular speeds.
5. Compute the departure and arrival burns with departure_delta_v and arrival_delta_v, then the total with total_delta_v; check the budget against the propulsion capability.
6. For a rendezvous, compute the departure lead angle with rendezvous_phase_angle so the target is met at the outer orbit.
7. Sanity-check the result: the total delta-v for a low-earth to geostationary coplanar transfer is about 3.9 km/s and the transfer time about 5.3 hours.

Pitfalls

• Using altitude instead of radius: the formulas take the distance from the body center; a 500 km altitude orbit has radius 6878 km, and mixing the two changes every velocity and time.
• Summing the impulses with the wrong sign: dv1 and dv2 are both positive magnitudes; the burn directions differ (prograde at departure, prograde at arrival for an outward transfer, retrograde for an inward transfer), and the budget never subtracts them.
• Using the full transfer period as the transfer time: the Hohmann arc is half the ellipse, so the time is T / 2; using the full period doubles the coast and breaks the rendezvous phase angle.
• Burning at the wrong point: the departure burn must fire at periapsis of the transfer ellipse and the arrival burn at apoapsis; an off-tangent burn wastes delta-v and leaves the transfer ellipse misaligned.
• Ignoring the plane change: the numbers above are for coplanar transfers; an inclination change between the two orbits adds its own delta-v on top of the Hohmann budget.
• Forgetting the target motion in a rendezvous: the chaser must lead the target by the rendezvous phase angle at departure, or the target has moved past the meeting point when the chaser arrives.
• Confusing the Hohmann transfer with the rocket equation: the rocket equation converts a delta-v budget into propellant mass through the specific impulse; the Hohmann transfer sizes the delta-v itself, and the two models are separate steps in the maneuver design.
• Assuming equal radii are a transfer: r1 equal to r2 means the spacecraft is already on the target orbit and no Hohmann burn exists; the logic rejects that input.

Behavior contract (gate 3)

The Hohmann transfer math is exercised by the gate 3 contract test: scripts/test_hohmann_transfer.py against scripts/hohmann_transfer_logic.py (stdlib unittest, offline). Run: python3 scripts/test_hohmann_transfer.py

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 transfer geometry and vis-viva relationships above are common astrodynamics methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.