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Hohmann transfer orbit: delta-v and transfer time

Calculate the two engine burns and the total delta-v needed to move a spacecraft between two circular orbits around the Earth, the Moon or Mars, with the transfer time.

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Results

Total delta-v

3.854

Two burns, without any change of orbital plane

Transfer time (half an ellipse)

5.29

First burn (leaving the initial orbit)
2.397
Second burn (circularizing at the final orbit)
1.456
Speed on the initial orbit
7.669
Speed on the final orbit
3.075

Changing orbits with two burns

A satellite in a circular orbit cannot simply point toward a higher orbit and fly there. Raising an orbit means speeding up at one point so the opposite side of the orbit rises, then speeding up again when you get there. In 1925 the German engineer Walter Hohmann showed that, for two circular orbits in the same plane, the cheapest two-burn route is an ellipse that just touches both circles. It is still the starting point of mission planning.

The formulas

With μ the gravitational parameter of the central body (3.986 × 10¹⁴ m³/s² for the Earth) and r₁, r₂ the radii of the two orbits from the center of the body:

  • Speed on a circular orbit: v = √(μ ÷ r).
  • First burn: Δv₁ = v₁ × (√(2r₂ ÷ (r₁ + r₂)) − 1).
  • Second burn: Δv₂ = v₂ × (1 − √(2r₁ ÷ (r₁ + r₂))).
  • Transfer time, half of the ellipse: t = π × √(a³ ÷ μ), with a = (r₁ + r₂) ÷ 2.

The calculator adds the body’s radius to the altitudes you enter.

Worked example: low Earth orbit to geostationary

From a circular orbit at 400 km to the geostationary altitude of 35,786 km:

  • Speed in the initial orbit: 7.669 km/s. Speed in the final orbit: 3.075 km/s.
  • First burn: 2.397 km/s, at the low point of the transfer ellipse.
  • Second burn: 1.456 km/s, at the high point, to circularize.
  • Total delta-v: 3.854 km/s.
  • Transfer time: 5.29 hours.

Notice that the spacecraft ends up moving more slowly in the higher orbit, even though both burns speed it up: it trades speed for height as it climbs.

Other Earth transfers

From To Total Δv Transfer time
200 km Geostationary, 35,786 km 3.93 km/s 5.3 h
400 km GPS altitude, 20,200 km 3.42 km/s 3.0 h
400 km 1,000 km 0.32 km/s 0.8 h
400 km Moon’s distance 3.91 km/s about 5 days

What the model leaves out

  • Plane changes: launches from Cape Canaveral start in an orbit inclined about 28.5° to the equator. Removing that inclination for a geostationary satellite adds about 0.36 km/s when it is combined with the second burn at the high point. Equatorial launch sites such as Kourou need less.
  • Finite burns: real engines take minutes, not an instant.
  • Other bodies’ gravity: for a trip to the Moon, its gravity changes the arrival, which is why lunar missions use patched conics or numerical models.

Bi-elliptic transfers

When the final orbit is more than about 11.9 times larger than the initial one, a three-burn bi-elliptic transfer, which goes beyond the target and then falls back, can use less delta-v than a Hohmann transfer, at the cost of a much longer trip.

Frequently asked questions

What is a Hohmann transfer?

The most fuel-efficient way, using two engine burns, to move between two circular orbits in the same plane. The first burn puts the spacecraft on an elliptical orbit that touches both circles; the second burn, half an orbit later, circularizes it. Walter Hohmann described it in 1925.

How much delta-v does it take to reach geostationary orbit?

From a 400 km low Earth orbit to geostationary altitude, 35,786 km, about 2.40 km/s for the first burn and 1.46 km/s for the second, 3.85 km/s in total, with a transfer time of about 5.3 hours. Real launches also need to change the orbital plane, which adds more delta-v.

What is delta-v?

The total change of velocity a spacecraft must produce with its engines. It is the currency of space travel: combined with the rocket equation, it tells you how much propellant a mission needs.

Is a Hohmann transfer always the best?

It is the cheapest two-burn transfer between circular orbits, but it is slow. When the final orbit is more than about 11.9 times larger than the initial one, a three-burn bi-elliptic transfer can use less delta-v, at the cost of a much longer trip.

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Results are estimates provided for information purposes. They do not replace professional advice.

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