Hohmann and bi-elliptic transfers

Draft

Moving between two circular coplanar orbits is the most common manoeuvre in spaceflight. For most radius ratios the cheapest two-impulse solution was published by Walter Hohmann in 1925, and it remains the reference against which everything else is measured.

The Hohmann transfer

Connect the two circles with an ellipse tangent to both: periapsis on the inner circle, apoapsis on the outer. Two burns, both prograde, both purely tangential.

The transfer ellipse has

a_t = \frac{r_1 + r_2}{2}

Applying vis-viva at each end and differencing against the local circular speed:

Hohmann first burn

\Delta v_1 = \sqrt{\frac{\mu}{r_1}}\left(\sqrt{\frac{2r_2}{r_1+r_2}} - 1\right)

Hohmann second burn

\Delta v_2 = \sqrt{\frac{\mu}{r_2}}\left(1 - \sqrt{\frac{2r_1}{r_1+r_2}}\right)

The transfer takes exactly half the ellipse period:

t_{\text{transfer}} = \pi\sqrt{\frac{a_t^3}{\mu}}

Both burns are tangential because a velocity change parallel to the existing velocity is the most efficient way to change energy. Any component perpendicular to $\mathbf{v}$ rotates the velocity vector without contributing to $\Delta\varepsilon$ , which is wasted for a coplanar transfer.

Worked example: LEO to GEO

From a 300 km circular orbit ( $r_1 = 6678$ km) to geostationary ( $r_2 = 42\,164$ km), so $a_t = 24\,421$ km.

Quantity	Value
Circular speed at $r_1$	7.726 km/s
Transfer perigee speed	10.152 km/s
$\Delta v_1$	2.426 km/s
Transfer apogee speed	1.608 km/s
Circular speed at $r_2$	3.075 km/s
$\Delta v_2$	1.467 km/s
Total	3.893 km/s
Transfer time	5.28 hours

Note that the second burn is smaller despite raising the orbit by far more altitude. Speed at apogee is low, so a modest increment there produces a large change in perigee. This asymmetry is the reason apogee is where plane changes and other expensive manoeuvres are performed. See Plane changes and inclination.

Phasing

A Hohmann transfer arrives at a fixed point in space at a fixed time, so the target must be there to meet it. For a transfer to a specific body or spacecraft, the departure has to be timed so the target’s lead angle is correct.

\phi = \pi - n_2\, t_{\text{transfer}}

where $n_2$ is the target’s mean motion. If the phase angle is wrong, the options are to wait for the geometry to recur at the synodic period, or to fly a non-Hohmann transfer that costs more delta-v and arrives when required. This is why interplanetary launch windows exist at all, and why the Mars window recurs every 780 days. See Trajectory optimization.

Bi-elliptic transfers

For large radius ratios, three burns can beat two. Raise apoapsis far beyond the target radius, perform a very cheap plane-and-energy adjustment out there where speed is low, then drop periapsis to the target.

The mechanism is the Oberth effect running in reverse. At a very high apoapsis the orbital speed approaches zero, so changing the shape of the orbit costs almost nothing. The penalty is the extra energy to get out there and back.

The break-even points are fixed numbers, independent of $\mu$ :

Radius ratio $r_2/r_1$	Result
Below 11.94	Hohmann always wins
11.94 to 15.58	Depends on the intermediate apoapsis radius
Above 15.58	Bi-elliptic always wins, given a large enough intermediate radius

The LEO-to-GEO ratio is 6.3, comfortably in Hohmann territory. LEO to lunar distance is a ratio of about 57, where bi-elliptic wins on paper.

In practice bi-elliptic transfers are rarely flown for their own sake, because the delta-v saving is small in the region where it exists and the transfer time is enormous. A bi-elliptic LEO-to-GEO transfer via a 400 000 km apoapsis saves a few tens of metres per second and takes weeks instead of five hours.

Where the three-burn structure genuinely pays is when a plane change is also required, because plane-change cost scales directly with speed. A GEO mission launched into a high-inclination orbit can be cheaper to fly via a supersynchronous apogee, and several commercial satellites have used exactly this.

Low-thrust spirals

Electric propulsion cannot produce an impulsive burn. Thrust is applied continuously over weeks or months, and the trajectory is a slow spiral rather than a transfer ellipse.

For a circular-to-circular spiral, the delta-v required is simply the difference in circular speeds:

Low-thrust circular spiral

\Delta v = \left|\sqrt{\frac{\mu}{r_1}} - \sqrt{\frac{\mu}{r_2}}\right|

For LEO 300 km to GEO that is $7.726 - 3.075 = 4.651$ km/s, about 20% more than the Hohmann figure of 3.893 km/s. The spiral is genuinely less efficient in delta-v, because thrust is never applied purely at the optimal point.

But delta-v is not the deliverable, propellant is. Comparing a 320 s chemical apogee motor against a 1800 s Hall thruster:

	Chemical Hohmann	Electric spiral
Delta-v	3.893 km/s	4.651 km/s
Specific impulse	320 s	1800 s
Propellant fraction $1 - e^{-\Delta v/I_{sp}g_0}$	71%	23%
Transfer time	5.3 hours	3 to 6 months

The electric option delivers roughly three times the payload for the same launch mass. The cost is months of transfer time, during which the satellite earns no revenue, and repeated passage through the radiation belts, which consumes solar array life. Nearly every commercial GEO operator now flies some version of this trade, and many use a hybrid: chemical to raise perigee out of the belts quickly, then electric to finish.

The corresponding result when a plane change is included is Edelbaum’s:

\Delta v = \sqrt{v_1^2 + v_2^2 - 2v_1v_2\cos\left(\frac{\pi}{2}\Delta i\right)}

which shows the same effect as the impulsive combined burn: doing the plane change continuously while raising the orbit is far cheaper than doing it separately.

References

Hohmann, W. Die Erreichbarkeit der Himmelskörper, R. Oldenbourg, 1925.
Bate, R. R., Mueller, D. D., White, J. E. and Saylor, W. W. Fundamentals of Astrodynamics, 2nd ed., Dover, 2020, chapter 3.
Edelbaum, T. N. “Propulsion Requirements for Controllable Satellites”, ARS Journal, 31:1079, 1961.
Vallado, D. A. Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press, 2022, chapter 6.