TECHNICAL NOTES · ORBITAL MECHANICS
Bi-elliptic or Hohmann? When the extra burn saves Δv
The Hohmann transfer is taught as the cheapest way between two circular, coplanar orbits. It usually is — but not always. Past a certain ratio of orbit radii, going further out first and coming back costs less Δv. Here are the thresholds, and the numbers.
The setup
Start in a 400 km circular orbit — r1 = 6 778.137 km, using Earth's equatorial radius of 6 378.137 km — and end in a circular, coplanar orbit of radius r2, with μ = 398 600.4418 km³/s². Burns are impulsive and the model is two-body only.
Hohmann: Δv₁ = √(μ/r₁)·(√(2r₂/(r₁+r₂)) − 1) · Δv₂ = √(μ/r₂)·(1 − √(2r₁/(r₁+r₂)))
The bi-elliptic transfer uses three burns: raise apoapsis to an intermediate radius rb beyond r2; at rb, raise periapsis to r2; at r2, burn retrograde to circularise. Each burn is a difference of vis-viva speeds:
v = √(μ · (2/r − 1/a))
Two ratios decide it
With R = r2/r1:
- R < 11.94 — Hohmann wins for every choice of rb.
- 11.94 < R < 15.58 — bi-elliptic wins only if rb is large enough.
- R > 15.58 — bi-elliptic wins for any rb beyond r2.
Underneath sits a curiosity: the Hohmann Δv itself peaks at R ≈ 15.58 — 4.112 km/s from this starting orbit — and then falls slowly as the target moves further out, because the arrival orbit becomes slower and cheaper to match.
Worked: R = 10 (r2 = 67 781 km)
| TRANSFER | TOTAL Δv | VS HOHMANN | FLIGHT TIME |
|---|---|---|---|
| Hohmann | 4.063 km/s | — | 0.42 d |
| Bi-elliptic, rb = 30 r1 | 4.180 km/s | +117 m/s | 4.84 d |
| Bi-elliptic, rb → ∞ | 4.181 km/s | +118 m/s | ∞ |
Below the first threshold, the extra burn only costs.
Worked: R = 20 (r2 = 135 563 km)
| TRANSFER | BURNS (km/s) | TOTAL Δv | VS HOHMANN | FLIGHT TIME |
|---|---|---|---|---|
| Hohmann | 2.915 + 1.186 | 4.101 km/s | — | 1.09 d |
| Bi-elliptic, rb = 40 r1 | 3.043 + 0.722 + 0.265 | 4.031 km/s | −70 m/s | 8.26 d |
| Bi-elliptic, rb = 60 r1 | 3.087 + 0.521 + 0.385 | 3.993 km/s | −107 m/s | 13.54 d |
| Bi-elliptic, rb → ∞ | — | 3.887 km/s | −214 m/s | ∞ |
The saving is real, and it grows with rb. But so does the honesty problem: rb = 60 r1 is 406 688 km — beyond the Moon's orbit, where lunar gravity would dominate the trajectory and the two-body result no longer holds.
What the savings cost
Time, first: the R = 20 Hohmann transfer takes about a day; the bi-elliptic options take one to two weeks, and the limiting case never arrives. Then the environment — longer radiation exposure, third-body perturbations, and Earth's sphere of influence at roughly 925 000 km. For most missions Hohmann, or a low-thrust spiral, remains the answer. The bi-elliptic transfer earns its place when Δv is the binding constraint and time is cheap — and it pairs well with plane changes, which cost far less at a high, slow apoapsis.
Check it yourself
Every value above comes from two equations and two constants, so it can be re-run in a few lines. ARITHMOS computes transfers like these with the equation, assumptions and validity range attached — including a warning when an intermediate apoapsis leaves the regime where two-body results hold.
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Published · 5 October 2026 · All technical notes