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⟳ Impulsive transfers

Two- and three-burn transfers between circular orbits — Hohmann vs bi-elliptic — with the Δv and time each costs, side by side. The impulsive counterpart to Spiral's continuous thrust.

Transfer

From the body's center — Earth surface ≈ 6,371 km (LEO ≈ 6,778).
The intermediate apogee for the bi-elliptic route — must be beyond both orbits. Higher = less Δv, more time.
Inclination difference between the initial and target orbits. Split optimally across the burns — most goes to the slowest one, so it's cheapest at a far apogee.

Trajectory

The transfer path over the initial and target orbits (dashed), with burn points marked. Press play to fly it — the vehicle moves on Kepler time, so it visibly crawls at apogee (which is why a plane change is cheap out there). Drag to rotate and tilt, scroll or pinch to zoom, double-click to reset.

t— r—

Comparison

Hohmann (2 burns)

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time —
burns —

Bi-elliptic (3 burns)

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time —
burns —

About impulsive transfers

An impulsive maneuver idealizes an engine burn as instantaneous — a step change in velocity at a point — which is a good approximation when the burn is short compared with the orbit. The Hohmann transfer, published by Walter Hohmann in 1925, connects two coplanar circular orbits with a single ellipse tangent to both: one burn to leave the inner orbit, a second to circularize on the outer. For most radius ratios it is the minimum-Δv two-impulse transfer, which is why it is the default mental model for what it costs to get from one orbit to another.

The bi-elliptic transfer trades a third burn for a detour: it flings the vehicle far past the target to a high apoapsis, changes orbits cheaply where it is moving slowly, then drops back. For coplanar transfers it only beats Hohmann when the ratio of final to initial radius exceeds about 11.94 (and always beats it above 15.58), and even then the saving is a percent or two bought with a large increase in flight time. Its real value appears once a plane change is involved: because a plane change costs 2·v·sin(Δi/2), doing it at a slow, far apoapsis is cheap, so bundling inclination change into a bi-elliptic can win at far lower radius ratios.

This tool splits any requested plane change optimally across the burns as a combined maneuver — vector-adding the speed change and the rotation rather than doing them in sequence — which is exactly the reasoning behind the “supersynchronous” transfer orbits launch providers use to remove the launch-site inclination on the way to GEO.

References