Hohmann Transfer
Two Burns, One New Orbit

A Hohmann transfer connects two circular, coplanar orbits with an ellipse tangent to both. In the ideal model, a short tangential burn enters the ellipse, the spacecraft coasts for half an orbit, and another burn matches the destination circle. The FAA orbital-maneuver chapter develops this geometry.

For a transfer outward, both burns are prograde. The first increases apogee; the second, at apogee, raises perigee to make the final circle. If the second burn is omitted, the spacecraft remains on the transfer ellipse and comes back down. Simply reaching the destination altitude does not establish the destination orbit.

The diagrams use a spherical Earth radius of 6,378.137 km and gravitational parameter mu = 398,600.4418 km³/s². These are teaching calculations with instantaneous burns, no drag and no change of plane. Altitudes must first become centre-to-centre radii: r1 = 6,778.137 km and r2 = 16,378.137 km.

The transfer semi-major axis is a = (r1 + r2) / 2, or 11,578.137 km. At either end, calculate transfer speed with v = sqrt(mu (2/r - 1/a)). The matching circular speed is sqrt(mu/r).

At departure, speed rises from 7.669 to 9.121 km/s: Burn 1 supplies 1.452 km/s. At arrival, the spacecraft has slowed to 3.775 km/s, while the target circle requires 4.933 km/s: Burn 2 supplies 1.159 km/s. Calculating with unrounded values gives 2.611 km/s total delta-v.

Coast time is pi sqrt(a3/mu), half the transfer ellipse period. The result is 103.32 minutes, about 1 hour 43 minutes. This is not the duration of either burn and does not include launch from the ground.

The starting circle has a higher speed than the final circle. That does not contradict the two prograde burns. During the outward coast, gravitational potential energy increases while kinetic energy decreases; the total specific orbital energy remains constant between the impulses.

Watch four distinct speeds, not just the first and last. The second burn accelerates the spacecraft relative to its slow arrival on the ellipse, not relative to its original low-orbit speed. The energy diagram below separates the coast from the burns; its conservation statement applies to the unpowered two-body segment.

Going from the higher circle back to the lower one uses retrograde burns in this ideal model. First reduce speed to lower perigee. At the low point, the transfer speed exceeds the circular speed there, so a second retrograde burn circularizes.

For the same two radii and assumptions, the burn magnitudes swap order and the coast time is unchanged. Reducing speed at an impulse can lead to a faster subsequent passage through a lower altitude. An engine burn changes an orbit, not just a height readout.

A target spacecraft moves during the coast. It must arrive at the transfer endpoint at the same time as the interceptor. NASA's trajectories chapter explains why transfer timing depends on the destination's future position rather than where it is at departure.

A different orbital plane adds another requirement. Two satellites can share altitude and inclination but have different node orientations. Read the rendezvous guide for timing and the inclination guide for plane geometry. Neither problem is solved by adding the two tangential burn costs alone.

Open Jewawud Orbital Planner, choose Earth and the Manual Transfer mission profile, set the initial and target periapsis altitudes to 400 and 10,000 km, and keep both eccentricities and inclinations at zero. Compare the two burns and time of flight before changing another input.

The planner currently uses a 6,371 km Earth radius rather than the diagram's 6,378.137 km. With its radius, the same altitude inputs give about 2.613 km/s total and 103.23 minutes. This small difference is a documented model convention, not a new physical effect.

The simulation is an educational calculation, not an operational burn plan. Finite burn duration, navigation errors, perturbations and mission constraints require more detailed analysis. Hohmann is not universally the lowest-delta-v strategy: sufficiently large radius ratios can favour a three-burn bi-elliptic transfer, discussed in NASA's orbital-transfer treatment. Low-thrust spirals are a different case again.