Hohmann transfer: Δv, transfer time and departure window

Two circular orbit radii and the mass of the central body give the Hohmann ellipse tangent to both orbits, its two impulses, the total Δv, the flight duration, the phase angle at departure and the synodic period. A second mode compares this cost with the three-impulse bi-elliptic detour: beyond a certain ratio of radii, a distant enough detour costs less, for a far longer journey.

The transfer ellipse touches the departure and arrival orbits at its two apsides; the transfer speeds at those two points come from vis-viva.

Δv

the sum of the velocity changes of the manoeuvres, in metres per second, independent of the engine used.

Impulse

a velocity change assumed instantaneous, delivered at one point of the orbit.

Tangential impulse

an impulse collinear with the velocity, changing only its magnitude. Both Hohmann impulses are tangential, the ellipse touching each circle at an apsis.

Phase angle

the angle between the departure point and the target at the moment of the first impulse.

Synodic period

the interval between two identical configurations of the two orbits, hence between two departure windows: T_syn = 1/|1/T₁ − 1/T₂|.

Bi-elliptic transfer

a three-impulse transfer: one ellipse climbs to a distant apoapsis, a second comes back down to the target orbit, the third impulse circularises.

• Both orbits are circular and coplanar. An eccentricity would make the cost depend on where the manoeuvre happens; an inclination would add a plane change, sometimes costlier than the transfer.
• The manoeuvres are impulsive: thrust is instantaneous. Real thrust spreads over an arc and costs more than the Δv of this model.
• The propellant mass is not computed; it follows from Δv through the Tsiolkovsky equation and the exhaust velocity of the engine.
• Pure two-body problem: no oblateness perturbation, no third body, no atmosphere, no sphere of influence. An interplanetary injection from a parking orbit is not modelled.
• The phase angle and the synodic period are those of the circular model. On real orbits, a launch date drawn from them would be wrong by several weeks.
• The bi-elliptic threshold is solved by bisection, with a tolerance of 10⁻¹² on the ratio of the radii. The critical apoapsis is searched by geometric bisection up to 10¹² times the small radius; beyond that, it is declared not found.

• r2 ≥ 0model · refusal

  Both radii are strictly positive, measured from the centre of the central body, and different from each other.

  Instead · For a descent, enter an arrival radius smaller than the departure one: the total Δv is the same in both directions.

• rb ≥ 0model · refusal

  The apoapsis of the bi-elliptic detour is at least equal to the larger of the two radii entered.

  Instead · Equality with the larger radius is allowed; the detour is then a Hohmann transfer, at the same cost.

Among the ellipses joining the two circles, the Hohmann ellipse, the only one tangent to both, is the least energetic. It is the cheapest two-impulse transfer between coplanar circular orbits, not the fastest. A tauter ellipse cuts the circles, arrives earlier and costs more. The velocity profile runs from one circular speed to the other; its two jumps are the two impulses.

Beyond a certain ratio of radii, a three-impulse transfer through a very distant apoapsis costs less than Hohmann, because an impulse costs less where the vehicle is slow. The detour through an infinite apoapsis costs (√2 − 1)(1 + 1/√R) times the circular speed at departure; this bound crosses the Hohmann cost at R = 11.938 765. Below it, no detour beats Hohmann. Above it, the detour wins if its apoapsis exceeds a critical value, which diverges near the threshold: 816 times the small radius at R = 12, 48.9 at 13. Beyond R ≈ 15.58, any apoapsis will do.

The form proposes the nominal solar GM (IAU 2015 B3) divided by G (CODATA 2018), a departure radius of 1 au and an arrival radius of 1.523 679 au, the semi-major axis of Mars (Standish table, JPL, approximate Keplerian elements 1800-2050). The model gives a first impulse of 2.944 7 km/s, a second of 2.648 9 km/s, 5.593 6 km/s in total, a flight of 258.87 d, a phase angle of 44.34° and a synodic period of 779.95 d (2.135 years). The default detour, through 15 au, costs 19.97 km/s for a journey 32.7 times longer; at a ratio of radii of 1.52, below the threshold, it cannot beat Hohmann.

1. Hohmann, W., Die Erreichbarkeit der Himmelskoerper, 1925.

   Location: Oldenbourg, Munich 1925, title transliterated oe for the umlaut that the delivered font subset does not carry: the tangential two-impulse transfer between two circular coplanar orbits · accessed 2026-08-26

2. Sternfeld, A., Sur les trajectoires permettant d’approcher d’un corps attractif central à partir d’une orbite keplerienne donnée, 1934.

   Location: Comptes rendus de l’Académie des sciences, Paris, 198: the three-impulse bi-elliptic transfer and the regime where it costs less than the two-impulse one · accessed 2026-08-26

3. International Astronomical Union, Resolution B3 on recommended nominal conversion constants, 2015.

   Location: Nominal solar mass parameter GM = 1.3271244 x 10^20 m^3/s^2 exactly, as tabulated by Prsa et al. 2016, AJ 152:41, table 1 · accessed 2026-08-26

4. Standish, E. M. (Jet Propulsion Laboratory), Keplerian elements for approximate positions of the major planets, 1800-2050.

   Location: Mars semi-major axis 1.523 679 au, used as a labelled example only · accessed 2026-08-26

Provisional · Sources and assumptions are provided; the documentary review is incomplete.

Physical model. The result follows the significant figures of the least precise input.

revision 1 · reviewed on 26 August 2026

Revision log

1. revision 1 · 2026-08-26 · scope change

   Creation of the tool: Hohmann transfer between two coplanar circular orbits (two tangential impulses from the vis-viva of the shared Kepler core, total Δv, half transfer period), departure window (phase angle and synodic period), and comparison with the three-impulse bi-elliptic transfer, whose radius-ratio threshold is solved by bisection (11.938 765 47) and whose critical apoapsis is searched by geometric bisection up to 10¹² times the small radius.

   First published result of the tool, with no earlier value to compare. Check values computed independently before the tests were written, for two idealised orbits of 1 au and 1.523 679 au about a mass of nominal solar GM divided by G: Δv₁ = 2 944.689 3 m/s, Δv₂ = 2 648.895 2 m/s, total 5 593.584 5 m/s, duration 258.865 8 d, phase angle 44.344 2°, synodic period 779.949 4 d.