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Hohmann Transfer Worked Example: 7000 km Orbit to Geostationary Radius

Calculate circular speeds, two Hohmann burns, total delta-v, and transfer time from a 7000 km Earth-centered orbit to geostationary radius.

By Neutron Studios LLC9 minute readPublished 2026-08-11Reviewed 2026-08-11

Why this calculation matters

A Hohmann transfer is the minimum-energy two-impulse transfer between two coplanar circular orbits in the ideal two-body model. It is an excellent hand-check for orbital-mechanics software because the equations are closed form.

This example transfers from a 7000 km Earth-centered circular orbit to the geostationary orbital radius of 42,164 km. The result excludes plane change, launch-site geometry, finite burn duration, atmospheric effects, and perturbations.

What you will calculate

Calculate circular-orbit velocity.
Use the vis-viva equation on the transfer ellipse.
Compute both ideal impulsive delta-v burns.
Calculate half-period transfer time and understand the model limitations.

Given values

Earth gravitational parameter μ = 398600.44 km³/s²
Initial circular radius r1 = 7000 km
Final circular radius r2 = 42164 km
Coplanar, prograde, impulsive burns
Two-body point-mass gravity

Governing equations

Circular speed

vc = \sqrt{\frac{\mu }{r}}

Speed in an ideal circular orbit.

Transfer semimajor axis

at = \frac{r_{1} + r_{2}}{2}

The Hohmann ellipse is tangent to both circular orbits.

Vis-viva

v = \sqrt{\mu \left(\frac{2}{r} \frac{- 1}{a}\right)}

Gives transfer-ellipse speed at each burn radius.

Transfer time

t = \pi \sqrt{\frac{at^{3}}{\mu }}

Half the period of the transfer ellipse.

Worked solution

1. Calculate the two circular speeds

At 7000 km radius the circular speed is about 7.546 km/s. At geostationary radius it is about 3.075 km/s. Higher circular orbits move more slowly even though more energy is required to reach them.

\begin{gathered}v_{1} \approx \frac{7.546\; \mathrm{km}}{s}\\v_{2} \approx \frac{3.075\; \mathrm{km}}{s}\end{gathered}

2. Define the transfer ellipse

The transfer semimajor axis is the average of the two radii: 24,582 km. The spacecraft must accelerate at perigee to enter this ellipse.

at = \frac{7000 + 42164}{2} = 24582\; \mathrm{km}

3. Calculate the departure burn

Vis-viva gives a transfer-ellipse speed at r1 greater than the original circular speed. The difference is the first ideal burn.

\Delta v_{1} \approx \frac{2.337\; \mathrm{km}}{s}

4. Circularize at the high orbit

At apogee the transfer-ellipse speed is lower than the final circular speed, so a second prograde burn raises the speed to circular velocity.

\begin{gathered}\Delta v_{2} \approx \frac{1.434\; \mathrm{km}}{s}\\\text{total}\; \Delta v \approx \frac{3.771\; \mathrm{km}}{s}\end{gathered}

5. Calculate transfer time

Half of the transfer ellipse takes about 5.33 hours. This is coast time between ideal instantaneous burns, not total mission time.

t \approx \pi \sqrt{\frac{24582^{3}}{398600.44}} \approx 5.33\; h

Result

Engineering interpretation

Ideal Hohmann burn 1: approximately 2.337 km/s; burn 2: approximately 1.434 km/s; total ideal delta-v: approximately 3.771 km/s; coast time: approximately 5.33 h.

A real geostationary mission usually has inclination, launch, finite-thrust, operational, and reserve requirements that make mission delta-v different from this idealized two-body transfer.

Sanity checks

The first burn for an outward Hohmann transfer should be prograde.
The transfer speed at apogee should be below the final circular speed, requiring a second prograde burn.
Swapping r1 and r2 should reverse the burn directions while preserving the ideal magnitude relationship.
Units must remain consistent: μ in km³/s² requires radii in km to obtain km/s.

Common mistakes

Using altitude above Earth instead of Earth-centered orbital radius.
Calling 42,164 km an altitude; it is the approximate geostationary radius from Earth center.
Ignoring plane-change cost when the initial orbit is inclined.
Using Hohmann equations for a transfer where burns are not approximately impulsive.

References and model boundaries

Two-body circular-orbit, vis-viva, and Hohmann-transfer equations from standard astrodynamics references.
Operational mission design requires perturbations, plane changes, finite burns, reserves, and navigation constraints beyond this example.

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