Hohmann Transfer Calculator
Compute the two burns, total delta-v and transfer time of a Hohmann transfer between two coplanar circular orbits. Enter the gravitational parameter and the two orbit radii.
Find the Delta-V and Transfer Time Between Two Circular Orbits
Hohmann Transfer Formula
Where:
- \(\mu\) — gravitational parameter of the central body, \(\mu = GM\) (km³/s²)
- \(r_1\) — radius of the initial circular orbit, measured from the centre of the body (km)
- \(r_2\) — radius of the target circular orbit (km)
- \(\Delta v_1, \Delta v_2\) — velocity changes of the first and second burn (km/s)
- \(a\) — semi-major axis of the transfer ellipse (km)
- \(t\) — transfer time, i.e. half the orbital period of the transfer ellipse (s)
The first expression gives the burn needed at \(r_1\) to enter an ellipse that touches both circular orbits; the second gives the burn half a revolution later at \(r_2\) to circularise. The total delta-v is the sum of the two magnitudes, \(\Delta v = |\Delta v_1| + |\Delta v_2|\).
Hohmann Transfer – Calculation Example
Example: a spacecraft in a 300 km low Earth orbit (radius \(r_1 = 6678\) km) is transferred to geostationary orbit (radius \(r_2 = 42{,}164\) km). For Earth, \(\mu = 398{,}600.44\) km³/s².
\[ \Delta v_1 = \sqrt{\frac{398600.44}{6678}} \left( \sqrt{\frac{2 \times 42164}{6678 + 42164}} - 1 \right) = 7.726 \times 0.3140 = 2.426 \text{ km/s} \]
\[ \Delta v_2 = \sqrt{\frac{398600.44}{42164}} \left( 1 - \sqrt{\frac{2 \times 6678}{48842}} \right) = 3.075 \times 0.4771 = 1.467 \text{ km/s} \]
\[ \Delta v = 2.426 + 1.467 = 3.893 \text{ km/s} \]
\[ t = \pi \sqrt{\frac{24421^3}{398600.44}} \approx 18{,}986 \text{ s} \approx 5.27 \text{ h} \]
Result: the spacecraft burns 2.426 km/s to enter the transfer ellipse, another 1.467 km/s to circularise at GEO — 3.893 km/s in total — and the transfer itself takes about 5.27 hours (half of the transfer ellipse's 10.55-hour period).
How to use the calculator
- Enter the Gravitational parameter (μ) of the central body and pick the unit (km³/s² or m³/s²). For Earth use 398,600.44 km³/s²; the field descriptions list values for the Moon, Sun and Mars.
- Enter the Initial orbit radius (r₁). This is the distance from the centre of the body, not the altitude. If you only know the altitude, add the body's radius (Earth ≈ 6,371 km) first.
- Enter the Target orbit radius (r₂) the same way, and choose km, m or miles for both radii.
- Read the results: First velocity change (Δv₁), Second velocity change (Δv₂), Total delta-v, Transfer time, plus a details table with the circular and transfer-orbit speeds, the semi-major axis of the transfer ellipse and both orbit periods.
What is a Hohmann transfer?
A Hohmann transfer moves a spacecraft between two circular orbits that share the same plane using exactly two burns. The spacecraft first fires its engine prograde (or retrograde, when moving to a lower orbit), which raises — or lowers — the opposite side of its orbit so that it forms an ellipse tangent to both circles. At the far end of this ellipse, half a revolution later, a second burn adjusts the speed to match the target circular orbit. Because the ellipse touches both orbits exactly once, the geometry is perfectly symmetrical and the burn sizes are as small as geometry allows.
Walter Hohmann described this manoeuvre in 1925, and it was later proven to be the most propellant-efficient transfer between coplanar circular orbits that uses only two impulses. That efficiency is why it became the standard for orbit raising: communications satellites reaching geostationary orbit, and interplanetary probes travelling between planetary orbits around the Sun, all follow Hohmann-type trajectories. The classic Earth-to-Mars transfer, for example, is a Hohmann ellipse around the Sun taking about 8.5 months.
The total delta-v is the real “fuel cost” of the manoeuvre. Combined with the rocket equation, it determines how much propellant the spacecraft must carry, so mission planners compare transfers almost exclusively in km/s. The transfer time — half the period of the ellipse — is the price you pay for that efficiency: a Hohmann transfer is the slowest reasonable two-burn route. Faster transfers exist, but they need larger burns and a different trajectory shape.
Note the direction of the manoeuvre when interpreting the results. Going outward (raising the orbit), both burns are accelerations. Going inward, both burns are braking manoeuvres — the calculator shows their magnitudes, since a braking burn still spends the same amount of propellant.
Assumptions, limitations and practical tips
- Coplanar circular orbits. If the orbits are inclined relative to each other, plane-change burns are needed on top of the values shown here, and the optimal split of the burns changes.
- Impulsive burns. Real engines fire for minutes, not instantaneously, which adds small gravity losses; low-thrust (electric) propulsion uses an entirely different, spiralling trajectory.
- Point-mass body, no perturbations. Atmospheric drag in very low orbits, the oblateness of the body and third-body gravity are ignored. In low Earth orbit, atmospheric drag continuously lowers the orbit and must be compensated for separately.
- Radii, not altitudes. The most common input mistake is entering the altitude. Radius = altitude + body radius; the default example (300 km altitude → 6,678 km radius) shows the conversion for Earth.
- Very large radius ratios. When \(r_2/r_1\) exceeds roughly 11.9, a bi-elliptic transfer (three burns through an intermediate very high orbit) can beat the Hohmann transfer on total delta-v, at the cost of a much longer flight time.
- Interplanetary transfers. Use the Sun's \(\mu\) (1.32712×10¹¹ km³/s²) and the orbital radii of the two planets, and remember that the planets must be correctly phased at departure.
For quick reference, common gravitational parameters: Earth 398,600.44 km³/s², the Moon 4,902.8 km³/s², Mars 42,828.4 km³/s², and the Sun 1.32712×10¹¹ km³/s². Useful Earth radii: LEO ≈ 6,678 km, GPS ≈ 26,560 km, GEO = 42,164 km, and the Moon's orbit ≈ 384,400 km — all measured from Earth's centre.
Frequently Asked Questions
Why does my hand calculation differ from the calculator's result?
The two most common causes are unit mismatches. If you entered \(\mu\) in m³/s² while the radii are in km, the velocities come out wrong by a factor of \(\sqrt{10^9} \approx 31{,}623\). Always pair km³/s² with km, or m³/s² with m. The second cause is using altitude instead of radius: remember that \(r = \text{altitude} + R_{body}\), so 300 km above Earth means \(r = 6{,}671 + 300 \approx 6{,}678\) km.
What are typical delta-v values for common Hohmann transfers?
Some reference numbers: LEO to GEO costs about 3.9 km/s, LEO to the Moon's orbit roughly 3.1 km/s, and the heliocentric Earth-to-Mars Hohmann transfer needs about 5.6 km/s of combined departure and arrival burns. Small orbit raises within LEO, say 400 km to 800 km, cost only tens of m/s. If your result is far outside these ballparks, double-check whether you used radii or altitudes.
Does the calculator include plane changes or inclination differences?
No. A pure Hohmann transfer assumes both orbits lie in the same plane. If the target orbit is inclined, you need an extra plane-change burn costing \(\Delta v = 2v\sin(\Delta i/2)\), where \(v\) is the orbital speed at the point of the manoeuvre. Plane changes are cheapest at high altitude, where speed is low, so mission designers often combine them with the circularisation burn at apogee rather than performing them separately.
How do I find the right launch window for an interplanetary Hohmann transfer?
The spacecraft's transfer time must match the target planet's motion. Compute the transfer time \(t\) with this calculator, then find how far the target planet travels during it: \(\theta = 360° \cdot t/T_{target}\), where \(T_{target}\) is the target planet's orbital period. The required phase angle between the two planets at departure is \(180° - \theta\). For Earth-to-Mars this works out to about 44°, which is why Mars launch windows recur roughly every 26 months.
Can I use this calculator for elliptical or eccentric starting orbits?
No, the formulas are only valid between circular orbits. If your starting orbit is elliptical, the burn sizes depend on where in the ellipse you fire, and a simple two-impulse tangent transfer generally doesn't apply. For eccentric orbits you need a more general Lambert-solver approach, which finds a trajectory connecting two arbitrary points in space. That said, mildly eccentric orbits (e < 0.01, like most operational orbits) give results close to the circular case.
How do I convert the total delta-v into the amount of propellant needed?
Use the Tsiolkovsky rocket equation \(\Delta v = v_e \ln(m_0/m_f)\), where \(v_e\) is the exhaust velocity and \(m_0\), \(m_f\) are the initial and final masses. Rearranged: \(m_0 = m_f \cdot e^{\Delta v / v_e}\). The propellant fraction grows very quickly with delta-v, which is why mission planners fight for every m/s. For example, 3.9 km/s of LEO-to-GEO transfer with a chemical engine (\(v_e \approx 3\) km/s) means more than 70% of the departure mass is propellant.
Is a Hohmann transfer always the best choice?
Not always. It is the most efficient two-burn transfer between coplanar circular orbits, but not necessarily the best overall. When the radius ratio \(r_2/r_1\) exceeds about 11.9, a bi-elliptic transfer with three burns can be cheaper. If time matters, faster transfers using larger burns exist. And spacecraft with electric propulsion use continuous low-thrust spirals that slowly raise the orbit, which cost somewhat more delta-v but far less propellant thanks to their high exhaust velocity.