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.
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 = √(μ/r)Speed in an ideal circular orbit.
Transfer semimajor axis
at = (r1 + r2)/2The Hohmann ellipse is tangent to both circular orbits.
Vis-viva
v = √[μ(2/r − 1/a)]Gives transfer-ellipse speed at each burn radius.
Transfer time
t = π√(at³/μ)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.
v1 ≈ 7.546 km/s; v2 ≈ 3.075 km/s2. 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 = (7000 + 42164)/2 = 24582 km3. 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.
Δv1 ≈ 2.337 km/s4. 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.
Δv2 ≈ 1.434 km/s; total Δv ≈ 3.771 km/s5. 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 ≈ π√(24582³/398600.44) ≈ 5.33 hEngineering 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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