Hohmann Transfer: LEO to Geostationary
The problem
A satellite is in a circular orbit 300 km above Earth's surface. Move it to geostationary orbit (42 164 km from Earth's centre) using the least fuel possible. Find the two burns required, the total Δv, and how long the trip takes.
This is the calculation behind every commercial satellite launch, and it needs exactly one equation you have not already met.
The one new tool: vis-viva
For any orbit around a body of gravitational parameter \(\mu = GM\):
It ties speed to position on any conic — circle, ellipse, parabola, hyperbola. Setting \(r = a\) gives the circular speed \(\sqrt{\mu/r}\); letting \(a\to\infty\) gives escape speed. Everything below is this equation applied four times.
For Earth, \(\mu = 398\,600\) km³/s².
The strategy
The Hohmann transfer is the minimum-energy route between two circular orbits: an ellipse tangent to the inner orbit at its perihelion and to the outer orbit at its aphelion. Two burns, both prograde, both at the ends of the major axis.
Burn 1: leaving LEO
Current circular speed:
Required speed at the perihelion of the transfer ellipse:
Burn 2: circularising at GEO
Coasting to aphelion, the satellite has slowed dramatically:
But a circular orbit at that radius requires
Total: \(\Delta v = 3.893\) km/s.
Without the second burn the satellite falls straight back to perigee — the transfer ellipse is a closed orbit, and arriving at the right altitude is not the same as staying there.
The trip time
Half the transfer ellipse's period:
Which is why a geostationary launch takes most of a day: the transfer coast is hours, and operators usually wait for the right perigee position first.
The counter-intuitive part
At aphelion the satellite is moving at 1.61 km/s and needs to reach 3.07 km/s — it must speed up to stop falling back. Meanwhile the whole manoeuvre took it from 7.73 km/s in LEO to 3.07 km/s in GEO: the final orbit is much slower than the initial one, yet both burns were prograde accelerations.
This is the standard confusion of orbital mechanics, and vis-viva explains it. Higher orbits are slower (\(v = \sqrt{\mu/r}\)), but reaching them costs energy, and energy is \(E = -\mu/2a\) — less negative for larger \(a\). You burn forwards to go higher and end up slower. Burning retrograde in LEO would lower the orbit and speed you up.
Sanity checks worth doing
- \(v_p > v_1\) and \(v_2 > v_a\): both burns are accelerations. ✓
- \(v_p/v_a = r_2/r_1\)? Angular momentum conservation on the ellipse says \(v_pr_1 = v_ar_2\): \(10.1516\times6678 = 67\,792\) and \(1.6078\times42\,164 = 67\,791\). ✓ This is Kepler's second law, and it is the cheapest check available.
- Escape speed from LEO is \(\sqrt2\times7.726 = 10.926\) km/s. Our transfer perigee speed of 10.152 km/s is just below it — as it must be, since the transfer orbit is bound. ✓
Common mistakes
- Forgetting the second burn. Reaching GEO altitude without circularising means falling back.
- Using circular-orbit speed on the transfer ellipse. The ellipse's speed varies; that is what vis-viva is for.
- Assuming a higher orbit needs a faster satellite. It needs more energy and results in a slower satellite.
- Ignoring that Hohmann is optimal only for \(r_2/r_1 < 11.94\). Beyond that ratio a three-burn bi-elliptic transfer uses less Δv — a genuinely surprising result, and GEO's ratio of 6.3 is comfortably inside the Hohmann regime.
Related
Kepler's laws & orbits · Gravitation · Newton's law of gravitation (eq.) · Conservation of energy · Circular motion · Orbit sandbox — vis-viva is checked live in the readout