The Hohmann transfer calculator above works out the classic two-impulse manoeuvre between two circular orbits. You burn once to raise your apoapsis out to the target radius, coast halfway round an ellipse, then burn again to circularise. Walter Hohmann described it in 1925, and a century later it is still the reference against which every other transfer is measured.
Arb Digital publishes free physics calculators that each own one job. This page owns the manoeuvre geometry: where the burns go, how big each one is, how long the coast lasts and when you have to leave. The live delta-v calculator owns the propulsion side — it takes a delta-v requirement and the Tsiolkovsky rocket equation and tells you how much propellant that costs for a given engine. The two pages chain together: this one produces the number, that one prices it.
What This Hohmann Transfer Calculator Does
Given a central body and two circular orbit radii, it computes the velocity change at each end, the total, the coast time and the phase angle between the two bodies at departure. It works in both directions: raising an orbit and lowering one. Lowering simply reverses both burns, and the total delta-v is identical, which is one of the tidier symmetries in orbital mechanics.
The phase angle is the output people most often forget they need. Raising an orbit takes time, and the target keeps moving while you coast. You must depart when the target is at a specific angle ahead of you, so that it arrives at the rendezvous point exactly when you do. Miss the angle and you arrive at the right altitude with the target somewhere else entirely.
The tool also reports the transfer ellipse itself in the line under the total, since its periapsis is your starting radius and its apoapsis is your target radius by construction.
How to Use It
- Pick the central body. Everything is a two-body problem about that body, so an Earth-to-Mars transfer is computed around the Sun, not around Earth.
- Choose whether your radii are from the centre or above the surface. Mixing the two is the single most common error here, and for low Earth orbit it changes the answer by a factor of more than twenty.
- Enter both circular orbit radii. They may be entered in either order; going down costs the same as going up.
- Read both burns separately. They happen half an orbit apart and are sized by different engines in many designs, so the split matters as much as the total.
- Note the phase angle before you plan a departure. For interplanetary transfers it is what defines the launch window.
The Formula: How A Hohmann Transfer Is Calculated
Write μ for the standard gravitational parameter of the central body, r1 for the starting radius and r2 for the target. The circular speed at any radius is √(μ/r). The transfer ellipse has semi-major axis a = (r1 + r2) ÷ 2, and the two burns are
Δv1 = √(μ/r1) × [√(2r2 ÷ (r1+r2)) − 1]
Δv2 = √(μ/r2) × [1 − √(2r1 ÷ (r1+r2))]
The transfer time is half the period of the ellipse, t = π√(a³ ÷ μ), and the departure phase angle is π minus the angle the target sweeps during the coast, which is ω2t with ω2 = √(μ ÷ r2³). NASA's Basics of Space Flight, Chapter 4 on trajectories describes the manoeuvre for interplanetary use and explains why it minimises propellant, including the Type I and Type II variants and where gravity assists depart from it. The gravitational constant behind every μ here is 6.67430 × 10−11 m³/kg·s², from the NIST CODATA page for the Newtonian constant of gravitation.
Work the defaults by hand. Earth's μ is 3.986004418 × 1014 m³/s². From 6,671 km to 42,164 km, the circular speeds are √(μ/6.671×106) = 7,729.9 m/s and √(μ/4.2164×107) = 3,074.7 m/s. The transfer semi-major axis is 24,417.5 km. The first burn is 7,729.9 × (√1.72670 − 1) = 7,729.9 × 0.31404 = 2,427.4 m/s. The second is 3,074.7 × (1 − √0.27320) = 3,074.7 × 0.47731 = 1,467.6 m/s. The total is 3,895.0 m/s. The coast is π√((2.44175×107)³ ÷ μ) = 18,987 s, which is 5.27 hours, and the required phase angle is 180° minus 7.2921 × 10−5 × 18,987 radians, or 100.7 degrees.
Why The Cheapest Route Is Almost The Slowest
The Hohmann transfer is optimal for two impulses between circular coplanar orbits, and its optimality comes at a price in time. The coast is exactly half an orbital period of the transfer ellipse, and nothing about that can be hurried without spending more delta-v.
Any faster trajectory has to be a more energetic ellipse or a hyperbola, which means a bigger first burn and, worse, a much bigger arrival burn because you turn up at the target moving fast in the wrong direction. The extra cost is steeply nonlinear: cutting an Earth-to-Mars transfer from around 259 days to 180 days roughly doubles the departure energy, and the arrival penalty is larger still.
This is why crewed Mars mission studies argue about trajectory class rather than about engines. Radiation dose and consumables scale with time, propellant scales with delta-v, and the two pull in opposite directions. For uncrewed cargo the Hohmann answer usually wins outright, which is why so many missions fly something very close to it.
The Phase Angle Is What Creates Launch Windows
For an orbit raise the phase angle is a scheduling detail. For an interplanetary transfer it is the whole reason launch windows exist. Earth and Mars must be positioned so that Mars arrives at the far end of the transfer ellipse just as the spacecraft does, which for the Hohmann case means Mars needs to be about 44 degrees ahead of Earth at departure.
That alignment recurs at the synodic period, not the orbital period. Earth and Mars line up correctly roughly every 26 months, which is why Mars launch campaigns cluster into short windows every two years and then go quiet. The synodic period is set by the difference in angular rates, so it stretches out as the two orbits get closer in size — Earth and Venus repeat every 19 months, Earth and Jupiter every 13.
Note that the phase angle can come out negative for a large enough radius ratio. That is not an error: it means the target sweeps more than 180 degrees during the coast, so it must start behind you rather than ahead. The calculator reports the sign. Use the Kepler's third law calculator to get the periods that set the synodic interval, and the orbital velocity calculator for circular speeds at any radius.
Where This Model Stops Describing Real Missions
Four assumptions are baked in, and every real mission violates at least one. Both orbits are circular; both are in the same plane; the burns are instantaneous; and there is only one gravitating body.
Plane changes are the expensive violation. Changing inclination costs delta-v proportional to the orbital speed at the point you do it, so an inclination change in low Earth orbit is brutally expensive and the same change at the top of a transfer ellipse, where you are moving slowly, is comparatively cheap. Real geostationary insertions therefore combine the plane change with the second burn rather than doing them separately, which saves a great deal. This page does not model that combined burn.
Finite burn time is the quiet violation. A real engine burns for minutes, not instantaneously, and the vehicle moves and rotates while it does. The resulting gravity loss adds a few per cent to the ideal delta-v for a high-thrust stage and much more for a low-thrust one, where the impulsive approximation collapses entirely and a slow spiral is the right model instead. Finally, for a departure from a planet the true cost is not the number here but a hyperbolic escape from a parking orbit, where the Oberth effect makes a burn deep in the gravity well far more effective than the same burn far out. The escape velocity calculator and the gravitational force calculator cover that side.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing altitude with orbital radius — a 300 km orbit has a radius of 6,671 km around Earth. Entering 300 as a radius produces an answer that is wrong by a factor of more than twenty.
- Using the wrong central body — an Earth-to-Mars transfer is a heliocentric manoeuvre. Computing it around Earth gives a number that means nothing.
- Ignoring the phase angle — arriving at the right radius is only half the problem. If the target is not where you arrive, the transfer has achieved nothing.
- Assuming a plane change is free — inclination changes cost delta-v proportional to your speed, and none of that appears in this coplanar calculation.
- Treating the result as a mission budget — gravity losses, finite burn times, escape hyperbolas and margins all sit on top of this ideal impulsive figure.
Related Free Tools From Arb Digital
To turn a delta-v figure into propellant mass, use the delta-v calculator, which runs the Tsiolkovsky rocket equation in both directions. The orbital velocity calculator gives circular speeds directly, the Kepler's third law calculator gives orbital periods and the synodic intervals that set launch windows, and the escape velocity calculator covers the boundary between bound and unbound trajectories. The gravitational force calculator and gravitational time dilation calculator cover the wider gravitational physics, and the astronomical distance converter handles the units. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is a two-burn manoeuvre between two circular coplanar orbits. The first burn puts you on an ellipse whose periapsis is the starting orbit and whose apoapsis is the target orbit, and the second burn at the far end circularises you there. It uses the least delta-v of any two-impulse transfer between such orbits.
From a 300 km circular parking orbit to geostationary orbit the two burns are about 2,427 m/s and 1,468 m/s, giving roughly 3,895 m/s in total, with a coast of about 5.27 hours. That figure assumes no plane change, which a real geostationary insertion from an inclined orbit would also have to pay for.
Because the target keeps moving while you coast. You have to leave when the target sits at a specific angle relative to you, so that it reaches the far end of the transfer ellipse at the same moment you do. Getting the altitude right without the phasing achieves nothing.
Yes, in this ideal model the total delta-v is identical and both burns simply reverse direction. In practice a descent into an atmosphere can use aerobraking to supply part of the change for free, which has no equivalent going up.
Because the required alignment between Earth and Mars recurs at their synodic period rather than either orbital period. The synodic period is set by the difference in their angular rates, and for Earth and Mars that works out at roughly 26 months.
It is the cheapest two-impulse transfer between circular coplanar orbits in almost all cases. For very large radius ratios, above about 11.94, a three-burn bi-elliptic transfer becomes cheaper still, at the cost of a very much longer flight time.
Only by spending more delta-v. Any quicker route is a more energetic trajectory, which means a larger departure burn and a larger arrival burn because you reach the target with more excess speed. The extra cost rises sharply as the flight time falls.
No. Everything here assumes both orbits lie in the same plane. Inclination changes cost delta-v proportional to orbital speed, and real missions combine the plane change with the second burn at high altitude, where the speed is low and the change is cheapest.
This tool is provided for educational and study use. It implements the ideal impulsive two-body Hohmann transfer between circular coplanar orbits and does not account for plane changes, finite burn times, gravity losses, escape hyperbolas, atmospheric drag or third-body perturbations, so treat its output as a physics result rather than a mission-design figure.
