Hohmann Transfer Simulator — Minimum-Energy Orbital Transfer
Compute the two-impulse Hohmann transfer between two circular orbits about the same central body in real time. Returns delta-v1, delta-v2, total delta-v and the half-period transfer time, and visualises the transfer ellipse and the efficiency curve.
Parameters
Departure altitude h1
km
Target altitude h2
km
Central body GM
km^3/s^2
Central body radius R
km
Defaults: Earth (GM = 398600 km^3/s^2, R = 6378 km). h1 = 400 km is LEO and h2 = 35800 km is near GEO; total delta-v of about 3.85 km/s matches a typical commercial launch scenario.
Results
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Burn 1 delta-v1
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Burn 2 delta-v2
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Total delta-v
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Transfer time t_tx
Transfer-orbit geometry
Centre = central body. Blue circle = departure orbit r1. Green circle = target orbit r2. Yellow dashed = half-ellipse transfer orbit. Yellow dot = delta-v1 burn point. Green dot = delta-v2 burn point.
Hohmann efficiency curve
x-axis = radius ratio r2/r1 (1 to 100, log). y-axis = total delta-v / sqrt(mu/r1). The efficiency peaks near R = 15.6 at about 0.536. Yellow marker = current radius ratio.
Theory & Key Formulas
The Hohmann transfer is the minimum-energy two-impulse path between two circular orbits of radii $r_1$ and $r_2$ around the same central body. Both burns are tangent to the orbit and the connection is a half-ellipse.
First burn delta-v1 (departure circle to transfer-ellipse periapsis):
$\mu = GM$ is the gravitational parameter (km^3/s^2), $r_i = R + h_i$ are orbital radii (km), delta-v in km/s and transfer time in s. For Earth $\mu = 398600$ km^3/s^2 and $R = 6378$ km.
What is the Hohmann transfer simulator?
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Communication satellites go from the ground straight to GEO, right? How much fuel does that take?
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Good catch — they don't actually go straight there. The standard procedure puts the satellite in LEO (a parking orbit around 400 km altitude) first and then uses a two-burn Hohmann transfer to reach GEO. With this tool's defaults (h1 = 400 km, h2 = 35800 km) the total delta-v is about 3.85 km/s — and the rocket equation tells you that is a substantial number for any payload.
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Two burns? Why not one?
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A single burn would cost more delta-v in total. A small first burn delta-v1 (about 2.40 km/s) tangent to the departure orbit puts the satellite on a half-ellipse with periapsis at r1 and apoapsis at r2. Half a period later it reaches the apoapsis, and a second tangent burn delta-v2 (about 1.45 km/s) circularises the orbit at GEO. The two-step "circle - ellipse - circle" sequence is provably the minimum for radius ratios up to 11.94.
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What sets the transfer time?
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Kepler's third law. The transfer ellipse has semi-major axis a = (r1+r2)/2, the full period is T = 2 pi sqrt(a^3/mu), and the spacecraft travels half of it: t_tx = pi sqrt(a^3/mu). The defaults give about 5.30 hours for LEO-to-GEO. An Earth-to-Mars transfer takes roughly 8.5 months because a is much larger; a Sun-to-Neptune transfer would take decades. Increase h2 here and watch how the lower-right efficiency curve also responds.
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Why does the efficiency curve have a peak?
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A great observation. delta-v_total / sqrt(mu/r1) is not monotonically increasing in R = r2/r1 — it peaks at R about 15.58 and value 0.536, then slowly drops. This is connected to the famous result that for very large radius ratios a bi-elliptic transfer (three burns) becomes cheaper than Hohmann (the crossover starts at R about 11.94 depending on the intermediate-orbit choice). For Moon- or Mars-class targets Hohmann is fine, but for very distant destinations alternative strategies are worth considering.
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Can I use this for interplanetary trajectories with the Sun as the central body?
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Yes. Set GM to the Sun's value (1.327 x 10^11 km^3/s^2), set R to a placeholder larger than a planetary radius, set h1 to Earth's orbital radius (1 AU = 1.496 x 10^8 km) and h2 to Mars's 2.279 x 10^8 km, and you get the heliocentric Hohmann delta-v. Real interplanetary missions need additional delta-v for departure and arrival in the Earth and Mars gravity wells, so this is only the heliocentric portion. The tool caps GM at 1.5 x 10^6 km^3/s^2 (about four times Earth's, up to lunar class) and is optimised for Earth-orbit use cases.
FAQ
The ideal Hohmann transfer assumes instantaneous impulses (infinite thrust, zero burn duration). Real engines have finite burn times. For chemical propulsion delta-v of a few km/s usually completes in minutes — short compared to orbital periods of hours or days — so the impulsive approximation is fine, with small Hohmann-formula corrections. Electric propulsion with months of low thrust is a different problem: the spacecraft spirals continuously and you need a low-thrust optimisation, not the Hohmann formulas. This simulator is the standard chemical-propulsion / impulsive teaching baseline.
The bi-elliptic transfer uses three burns: departure orbit -> first ellipse with a very high apoapsis -> second ellipse with a low periapsis -> target orbit. At the very high apoapsis the orbital speed is small, so the change-of-orbit delta-v is small. Analysis shows that beyond a target-to-departure radius ratio of about 11.94 (asymptotic value with the intermediate apoapsis at infinity), the bi-elliptic total delta-v is smaller than Hohmann's. The cost is a much longer transfer time — typically more than double — so real missions trade these against their constraints.
The Hohmann transfer assumes the two circular orbits are coplanar. With different inclinations you need an additional plane-change delta-v, roughly 2 v sin(delta-i / 2) where delta-i is the inclination change and v the speed at the burn point. Combining the plane change with delta-v1 or delta-v2 in a single vector burn (a "combined manoeuvre") saves fuel. GEO satellite launches benefit from low-latitude launch sites (Kourou for ESA, Tanegashima for JAXA) because the inclination penalty is smaller. This simulator is restricted to coplanar transfers.
Rough numbers: Earth LEO to the Moon (geocentric, r2 about 384400 km) is about 5 days. Earth to Mars (heliocentric, r1 = 1.0 AU, r2 = 1.524 AU) is about 8.5 months, Earth to Jupiter about 2.7 years and Earth to Neptune about 30 years. This simulator is optimised for Earth orbit (GM = 398600 km^3/s^2). Heliocentric missions can be approximated either by manual calculation or by switching GM to the Sun value with radii in km converted from AU. Real missions often shorten the transfer with gravity assists (Voyager, Cassini are classic examples).
Real-world applications
Geostationary satellite launches (LEO-GTO-GEO): communications and weather satellites are inserted into GEO (35786 km altitude) via a Hohmann transfer. The launcher first injects them into LEO (a parking orbit around 400 km), then a delta-v1 of about 2.4 km/s puts them on the GTO (Geostationary Transfer Orbit) ellipse, and 5.3 hours later a delta-v2 of about 1.45 km/s at the apoapsis circularises into GEO. SpaceX Falcon 9, Ariane 5, H-IIA / H3, ULA Vulcan and almost every commercial launcher use this pattern.
Trajectory design for interplanetary probes: Earth-to-Mars, Earth-to-Venus and Earth-to-Jupiter missions are designed around heliocentric Hohmann half-ellipses. NASA Mars probes (Mariner through Perseverance), ESA Mars Express, ISRO Mangalyaan and JAXA Nozomi / Akatsuki / Hayabusa2 all start from a Hohmann backbone and then optimise launch windows and arrival geometry. Real missions add small mid-course corrections (TCMs), but the Hohmann result is the design reference.
ISS resupply and crewed rendezvous: SpaceX Dragon, Cygnus, Soyuz and Shenzhou all approach the ISS (about 400 km, near-circular) with a sequence of Hohmann-like manoeuvres. Phasing requires careful choice of the transfer time and relative position, and tools like this simulator are the starting point for the operational delta-v analysis.
Orbit changes for space telescopes and science satellites: the HST servicing missions, the JWST insertion into a Sun-Earth L2 halo orbit, and ESA Gaia at L2 all use Hohmann-like two-impulse manoeuvres between intermediate orbits. For semi-stable points like L2 the standard recipe is "Hohmann-like two-burn approximation plus Lyapunov-orbit approach plus fine adjustment", and this simulator is well suited to learning the first stage.
Common misconceptions and pitfalls
The most common misconception is to think that "the Hohmann transfer is always optimal". It is the minimum-delta-v two-impulse transfer between coplanar circular orbits, but for radius ratios above 11.94 the three-impulse bi-elliptic transfer is cheaper. Plane changes (different inclinations), launch-window constraints, gravity assists and engine-thrust limits also push real missions toward different strategies. Treat the result here as the idealised delta-v lower bound, not a final answer.
The next pitfall is assuming "impulsive approximation always holds". Chemical propulsion is fine — a few-km/s delta-v completes in minutes, well inside the orbital period. But electric propulsion (ion engines, Hall thrusters) at thrust around 100 mN takes months to deliver several km/s, and the spacecraft spirals continuously. The two-burn model breaks down for those missions (Hayabusa, Dawn, BepiColombo). Electric propulsion needs more delta-v than Hohmann predicts, but the very high specific impulse Isp keeps propellant mass low — a different trade-off entirely.
The last pitfall is treating "the central body as a perfect point mass". The Earth is not a perfect sphere; the equatorial bulge (J2 perturbation) precesses orbital planes over the long term, and solar radiation pressure, atmospheric drag and third-body gravity (Sun and Moon) contribute non-negligible corrections. GEO satellites burn about 50 m/s per year in station-keeping, separate from the insertion delta-v this simulator computes. Use this tool for early-design estimates; for detailed mission analysis use professional packages such as STK, GMAT or Orekit.