Hohmann transfer calculator
Choose a departure planet and a destination. The calculator gives both burns, the travel time and the orbital speeds, using the maneuver that spends the least fuel between two circular orbits.
Key formula
Δv₁ = v₁ · (√(2r₂/(r₁+r₂)) − 1) · Δv₂ = v₂ · (1 − √(2r₁/(r₁+r₂))) · t = π · √(a³/μ)
Try it
How to use the simulator
To get from one circular orbit to another with the least fuel, you use a Hohmann transfer: a first burn puts you on an ellipse that touches both orbits, and a second burn on arrival leaves you in the destination orbit. The calculator gives both burns (Δv), the travel time, which is half a lap of the ellipse, and the orbital speeds at origin and destination.
Controls
- Origin and destination: two dropdowns with Solar System planets and two Tau Ceti planets.
Things to try
- Try Earth → Mars: about 5.6 km/s in total and roughly 259 days, very close to real missions.
- Try Earth → Jupiter and compare: more Δv and nearly three years of travel. Distances grow faster than the speed you can reach.
- Keep in mind the calculator uses the Sun's mass. Tau Ceti has about 78% of it, so real transfer times there would be somewhat longer (around 13%) and the Δv somewhat lower.
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How the maneuver works
In 1925 Walter Hohmann described the cheapest way to move from one circular orbit to another in the same plane. It takes two engine burns. The first speeds the craft up along its direction of motion and puts it on an ellipse whose closest point to the Sun touches the departure orbit and whose farthest point touches the destination orbit.
The craft coasts along half the ellipse without burning fuel. At the far end it is moving slower than the destination planet, so a second burn speeds it up to the final circular orbit.
Launch windows
The travel time is fixed (half the ellipse's period), so the destination planet has to be in the right place when you arrive. That only happens when the two planets are in a particular relative position. For Mars the window opens roughly every 26 months, which is why Mars missions launch in batches.
What the calculator simplifies
It assumes circular, coplanar orbits and only the Sun's gravity. It leaves out the cost of escaping the departure planet's gravity and of entering orbit at the destination. Real missions also use gravity assists and faster trajectories that cost somewhat more fuel.
Worked example
Example: Earth to Mars
- Earth orbits at 1 AU from the Sun at about 29.8 km/s. Mars orbits at 1.52 AU at about 24.1 km/s.
- The transfer ellipse has a semi-major axis of (1 + 1.52) / 2 ≈ 1.26 AU.
- First burn: 29.8 · (√(2 · 1.52 / 2.52) − 1) ≈ 2.9 km/s. Second burn on arrival: about 2.6 km/s. Total ≈ 5.6 km/s.
- That ellipse has a period of 1.26^1.5 ≈ 1.42 years, so half a lap takes about 259 days: a bit over eight months, like real missions.
In Project Hail Mary
Late in the novel, maneuvers between the planets of another star system have to be planned. The calculator includes two Tau Ceti planets so you can compare them with the Solar System.
Frequently asked questions
Is the Hohmann transfer the fastest?
No, it is the cheapest between two circular orbits in most cases. Burning more fuel lets you take a more direct path and arrive sooner.
Why does Mars cost less Δv than Jupiter yet still take so long?
Δv grows with distance, but travel time grows even faster: it scales with the semi-major axis to the power 1.5 (Kepler's third law). Earth to Jupiter takes almost three years.
What is an AU?
The astronomical unit, the mean distance between Earth and the Sun: about 149.6 million kilometers.
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