Hohmann Transfer Calculator
Hohmann Transfer Calculator
μ = 398,600.442 km³/s²
Altitude above the body's surface in km (e.g. 400 for ISS-like LEO)
Altitude of the target orbit above the body's surface in km
Enter Orbital Parameters
Select a central body, enter your initial and target orbit radii (or choose a preset), and the calculator will instantly show delta-V burns, transfer time, and orbital velocities.
Calculate delta-V and transfer time for orbit changes
A Hohmann transfer is the most fuel-efficient method of moving a spacecraft between two circular, coplanar orbits using exactly two engine burns. Named after German engineer Walter Hohmann who described the maneuver in 1925, it remains the foundational technique for every satellite launch, orbital insertion, and interplanetary mission ever flown. Understanding the Hohmann transfer is essential for anyone working in aerospace engineering, mission planning, astrodynamics, or simply curious about how spacecraft navigate the solar system. The maneuver works by placing the spacecraft on an elliptical orbit — called the transfer ellipse — whose periapsis (closest point) touches the departure orbit and whose apoapsis (farthest point) touches the destination orbit. The first burn accelerates the spacecraft from the initial circular orbit into the transfer ellipse. After coasting along the ellipse for exactly half an orbital period, the second burn circularizes the spacecraft into the final orbit. Both burns are prograde for an outward transfer (moving to a higher orbit) and retrograde for an inward transfer (moving to a lower orbit). The delta-V (Δv) required for each burn is calculated using the vis-viva equation: v = √(μ(2/r − 1/a)), where μ is the gravitational parameter of the central body, r is the current distance from the body's center, and a is the semi-major axis of the orbit. The total Δv is the sum of both burns. For a transfer from Low Earth Orbit (400 km altitude) to Geostationary Orbit (35,786 km altitude), the total Δv is approximately 3.9 km/s — comparable to the total Δv provided by an upper stage rocket engine. For an Earth-to-Mars interplanetary transfer, the heliocentric Δv is about 5.6 km/s relative to Earth's orbital velocity, though additional Δv is needed to escape Earth's gravity well. This calculator supports all the key Hohmann transfer scenarios engineers and students encounter. In the 'Between Two Orbits' mode, you can select any central body (Earth, Moon, Mars, Venus, Mercury, Jupiter, Saturn, or the Sun), enter orbit radii as either altitude above the surface or radius from the body's center, and instantly obtain the first and second burns (Δv₁ and Δv₂), transfer time, transfer ellipse semi-major axis and eccentricity, and all four relevant orbital velocities. The optional propellant section uses the Tsiolkovsky rocket equation to calculate propellant mass from spacecraft initial mass and specific impulse (Isp). In 'Interplanetary' mode, the calculator uses the Sun's gravitational parameter and accurate planetary semi-major axes to compute heliocentric transfers, including the all-important launch phase angle — the required angular separation between departure and destination planets at launch time — and the synodic period, which tells you how long until the next launch window. The calculator also includes a bi-elliptic transfer efficiency note. A bi-elliptic transfer uses three burns and passes through an intermediate high orbit. When the ratio of final to initial orbit radius (r₂/r₁) exceeds 11.94, a bi-elliptic transfer may require less total Δv than a Hohmann transfer. Above a ratio of 15.58, a bi-elliptic transfer to an infinitely distant intermediate orbit is always more efficient. The calculator automatically evaluates your inputs against these thresholds and alerts you when a bi-elliptic maneuver is worth considering. For professional and educational use alike, all key formulas are implemented with high-precision gravitational parameters from the IAU and JPL, matching values used by NASA mission planning tools. Common preset scenarios — LEO to GEO, LEO to MEO (GPS), LEO to Moon, and Mars orbit to Phobos orbit — let you load reference values instantly, while the CSV export button enables you to save all computed values for further analysis in Excel, Python, or MATLAB.
Understanding Hohmann Transfer Orbits
What Is a Hohmann Transfer?
A Hohmann transfer is a two-impulse maneuver that moves a spacecraft between two circular, coplanar orbits in the most propellant-efficient manner possible with exactly two engine burns. The spacecraft departs from the inner circular orbit, fires its engine prograde (in the direction of travel) to enter an elliptical transfer orbit, coasts for exactly half the ellipse's period, then fires its engine again to circularize into the outer orbit. The transfer ellipse has its periapsis at the inner orbit radius and its apoapsis at the outer orbit radius. Because the maneuver uses only two impulsive burns, it minimizes the total velocity change (Δv) compared to any other two-burn transfer between the same orbits. It is named after Walter Hohmann, who calculated this elegant trajectory in his 1925 book 'Die Erreichbarkeit der Himmelskörper' (The Attainability of Celestial Bodies) — two decades before the Space Age began.
How Are the Δv Burns Calculated?
The calculation uses the vis-viva equation, the fundamental relationship between orbital velocity, position, and energy: v = √(μ(2/r − 1/a)), where μ is the gravitational parameter of the central body (G × M), r is the current distance from the center, and a is the semi-major axis of the orbit. For the initial circular orbit, v₁ = √(μ/r₁). At the periapsis of the transfer ellipse, vₜ,ₚ = √(μ(2/r₁ − 1/aₜ)), where aₜ = (r₁ + r₂)/2. The first burn is Δv₁ = vₜ,ₚ − v₁. At the apoapsis, vₜ,ₐ = √(μ(2/r₂ − 1/aₜ)), and the target circular velocity is v₂ = √(μ/r₂). The second burn is Δv₂ = v₂ − vₜ,ₐ. Total Δv = |Δv₁| + |Δv₂|. Transfer time equals half the period of the ellipse: t = π√(aₜ³/μ). The eccentricity of the transfer ellipse is e = (r₂ − r₁)/(r₂ + r₁).
Why Does Transfer Efficiency Matter?
In space mission design, Δv is the universal currency of propulsion. The Tsiolkovsky rocket equation links Δv to propellant consumption: Δm = m₀(1 − e^(−Δv/(Isp × g₀))). Even a few hundred meters-per-second reduction in total Δv can translate to hundreds of kilograms of propellant saved on a large spacecraft, directly reducing launch cost or enabling a larger payload. For a geostationary communications satellite launched to GEO, the propellant fraction for the orbit transfer alone is typically 40–50% of the total spacecraft mass. For interplanetary missions, propellant efficiency determines whether a mission is feasible at all. This is why mission planners spend enormous effort optimizing trajectories, and why Hohmann transfers — despite taking longer than faster, higher-Δv trajectories — are used for most operational missions from satellite deployment to Mars exploration.
Assumptions and Limitations
This calculator assumes impulsive burns — instantaneous velocity changes delivered at a single point. In reality, engines burn for finite durations; for large Δv maneuvers this introduces gravity losses. The calculator assumes perfectly circular initial and final orbits, and coplanar transfers (orbits in the same plane). Real missions often require inclination changes, which add significant Δv — a combined plane-change-plus-Hohmann maneuver is much more complex. The calculator ignores perturbations from other gravitational bodies, atmospheric drag, solar radiation pressure, and the Oberth effect (which makes burns more efficient when performed deep in a gravity well). For interplanetary transfers, the heliocentric Δv shown is the velocity change required relative to the planet's orbital frame; actual departure burns must also account for the planetary escape velocity and the hyperbolic excess velocity, requiring a patched-conic approximation or full n-body integration for precise planning.
Key Hohmann Transfer Formulas
Transfer Orbit Semi-Major Axis
aₜ = (r₁ + r₂) / 2
The semi-major axis of the elliptical transfer orbit equals the average of the departure and arrival orbital radii.
Vis-Viva Equation
v = √(μ × (2/r − 1/a))
Relates orbital velocity to position and orbit size. μ is the gravitational parameter (G×M), r is the current distance from the central body, and a is the semi-major axis.
First Burn (Δv₁)
Δv₁ = √(μ × (2/r₁ − 1/aₜ)) − √(μ/r₁)
The velocity change at the departure orbit to enter the transfer ellipse. Equals the difference between transfer orbit periapsis velocity and initial circular orbit velocity.
Transfer Time
t = π × √(aₜ³ / μ)
The time to traverse half the transfer ellipse from departure to arrival. Equals exactly half the orbital period of the transfer ellipse.
Hohmann Transfer Reference Data
Hohmann Transfer Parameters for Common Missions
Total delta-V, transfer time, and key velocities for frequently calculated orbital transfers around Earth and the Sun.
| Transfer | Total Δv (km/s) | Transfer Time | Δv₁ (km/s) | Δv₂ (km/s) |
|---|---|---|---|---|
| LEO (400 km) → GEO (35,786 km) | 3.89 | 5 h 16 min | 2.43 | 1.46 |
| LEO (400 km) → MEO/GPS (20,200 km) | 3.64 | 3 h 32 min | 2.24 | 1.40 |
| LEO (400 km) → Moon distance (384,400 km) | 3.13 | 4.95 days | 3.13 | 0.19 |
| Mars low orbit → Phobos orbit (9,376 km) | 0.56 | 2.8 h | 0.37 | 0.19 |
| Earth → Mars (heliocentric) | 5.59 | 259 days | 2.94 | 2.65 |
| Earth → Venus (heliocentric) | 5.25 | 146 days | 2.49 | 2.76 |
| Earth → Jupiter (heliocentric) | 8.79 | 2.73 years | 6.31 | 2.48 |
Planetary Transfer Windows (Synodic Periods)
How often Hohmann transfer launch windows occur between Earth and other planets, based on synodic periods.
| Destination | Synodic Period | Phase Angle at Launch | Transfer Time |
|---|---|---|---|
| Venus | 1.60 years (584 days) | −54° | 146 days |
| Mars | 2.14 years (780 days) | +44° | 259 days |
| Jupiter | 1.09 years (399 days) | +97° | 2.73 years |
| Saturn | 1.04 years (378 days) | +106° | 6.05 years |
| Mercury | 0.32 years (116 days) | −108° | 105 days |
Worked Examples
LEO to GEO Transfer
Transfer a satellite from Low Earth Orbit (400 km altitude, r₁ = 6,771 km) to Geostationary Orbit (35,786 km altitude, r₂ = 42,164 km). Earth μ = 398,600.4 km³/s².
Transfer semi-major axis: aₜ = (6,771 + 42,164) / 2 = 24,467.5 km
Initial circular velocity: v₁ = √(398,600.4 / 6,771) = 7.670 km/s
Transfer periapsis velocity: vₜ,ₚ = √(398,600.4 × (2/6,771 − 1/24,467.5)) = 10.10 km/s
First burn: Δv₁ = 10.10 − 7.67 = 2.43 km/s
Transfer apoapsis velocity: vₜ,ₐ = √(398,600.4 × (2/42,164 − 1/24,467.5)) = 1.61 km/s
Final circular velocity: v₂ = √(398,600.4 / 42,164) = 3.07 km/s
Second burn: Δv₂ = 3.07 − 1.61 = 1.46 km/s
Transfer time: t = π × √(24,467.5³ / 398,600.4) = 18,925 s ≈ 5 h 16 min
Total Δv = 3.89 km/s with a transfer time of 5 hours 16 minutes. The first burn provides 62% of the total Δv.
Earth-to-Mars Heliocentric Transfer
Hohmann transfer from Earth orbit (r₁ = 1.000 AU = 149.6 × 10⁶ km) to Mars orbit (r₂ = 1.524 AU = 227.9 × 10⁶ km). Sun μ = 1.327 × 10¹¹ km³/s².
Transfer semi-major axis: aₜ = (149.6 + 227.9) × 10⁶ / 2 = 188.75 × 10⁶ km
Earth orbital velocity: v₁ = √(1.327×10¹¹ / 149.6×10⁶) = 29.78 km/s
Transfer periapsis velocity: vₜ,ₚ = 32.73 km/s → Δv₁ = 2.94 km/s
Mars orbital velocity: v₂ = 24.13 km/s
Transfer apoapsis velocity: vₜ,ₐ = 21.48 km/s → Δv₂ = 2.65 km/s
Transfer time: t = π × √((188.75×10⁶)³ / 1.327×10¹¹) ≈ 2.24 × 10⁷ s ≈ 259 days
Total heliocentric Δv = 5.59 km/s with a transfer time of about 259 days. Launch windows occur every 2.14 years when Mars is approximately 44° ahead of Earth.
How to Use the Hohmann Transfer Calculator
Choose your mode and central body
Select 'Between Two Orbits' for satellite transfers around Earth, Moon, Mars, or other bodies. Choose 'Interplanetary' for planet-to-planet transfers. In orbit mode, pick your central body from the dropdown — Earth's gravitational parameter (μ = 398,600 km³/s²) is pre-selected by default.
Enter orbit radii or use a preset
Enter your initial orbit (r₁) and target orbit (r₂). Toggle between 'Altitude above surface' and 'Radius from center' input modes. Alternatively, load a common scenario using the Quick Preset dropdown: LEO to GEO loads 400 km altitude and 35,786 km altitude instantly.
Read delta-V and transfer time results
The results card shows total Δv in both km/s and m/s, the first burn (Δv₁) at departure, and the second burn (Δv₂) at arrival. The stacked bar chart shows their proportions. Transfer time is displayed in days, hours, and minutes. The orbit size comparison bars show the relative scale of r₁, the transfer ellipse semi-major axis, and r₂.
Optionally calculate propellant mass and export
Expand 'Include propellant calculation' and enter spacecraft initial mass (kg) and engine specific impulse (Isp in seconds) to compute propellant consumption via the Tsiolkovsky rocket equation. The donut chart shows propellant fraction vs. dry mass. Click 'Export CSV' to download all computed values or 'Print Results' for a printed summary.
Frequently Asked Questions
What is a Hohmann transfer orbit and why is it fuel-efficient?
A Hohmann transfer is a two-burn orbital maneuver connecting two circular, coplanar orbits via an intermediate ellipse. It is the most fuel-efficient method possible with exactly two impulsive burns because both burns are performed at the optimal points on the transfer ellipse — the periapsis and apoapsis — where the vis-viva equation yields the minimum total velocity change. Any other two-burn path between the same orbits requires more Δv. The maneuver was first described mathematically by Walter Hohmann in 1925 and has been used in nearly every space mission since, from early Sputnik-era satellites to the Mars Science Laboratory (Curiosity rover) and the James Webb Space Telescope.
How long does a Hohmann transfer to GEO (geostationary orbit) take?
A Hohmann transfer from Low Earth Orbit (400 km altitude) to Geostationary Orbit (35,786 km altitude) takes approximately 5.25 hours — exactly half the orbital period of the transfer ellipse. The transfer ellipse has a semi-major axis of about 24,478 km. During this time the spacecraft coasts freely with no engine firing, traveling from the periapsis (at LEO altitude) to the apoapsis (at GEO altitude). The first burn requires about 2.43 km/s and the second burn about 1.45 km/s, for a total of approximately 3.88 km/s — the baseline budget for any satellite reaching geostationary orbit from LEO.
How long does an Earth-to-Mars Hohmann transfer take?
A heliocentric Hohmann transfer from Earth's orbit (1 AU = 149.6 million km) to Mars's orbit (1.524 AU = 228 million km) takes approximately 259 days — about 8.5 months. The transfer ellipse semi-major axis is about 188.8 million km, and the spacecraft coasts the entire distance with no fuel burn. The heliocentric delta-V is about 2.95 km/s at departure and 2.65 km/s at Mars arrival (total ~5.6 km/s). Because Earth and Mars must be in a specific relative alignment at launch (Mars roughly 44° ahead of Earth), launch windows occur only every 26 months — the Earth-Mars synodic period.
What is the vis-viva equation and why is it used here?
The vis-viva equation is the fundamental relationship between an orbiting object's speed at any point and its position: v = √(μ(2/r − 1/a)), where μ is the gravitational parameter, r is the current distance from the central body's center, and a is the orbit's semi-major axis. For a circular orbit, a = r, so it simplifies to v = √(μ/r). For the transfer ellipse, a = (r₁ + r₂)/2. By applying vis-viva at both burns — departure and arrival — we get precise velocity values at each maneuver point, making it straightforward to compute the required Δv for each burn as the difference between transfer ellipse velocity and circular orbit velocity at that radius.
When should I use a bi-elliptic transfer instead of Hohmann?
A bi-elliptic transfer uses three burns and passes through a very high intermediate orbit. It is more propellant-efficient than a Hohmann transfer when the ratio of final to initial orbit radius (r₂/r₁) exceeds about 11.94. Above a ratio of 15.58, a bi-elliptic transfer is definitively more efficient regardless of the intermediate orbit altitude chosen. For typical Earth orbit transfers — such as LEO to GEO (ratio ≈ 6.2) — the Hohmann transfer is more efficient. Bi-elliptic advantages become significant for transfers like LEO to deep space or highly eccentric orbits. The calculator shows the orbit ratio and alerts you when a bi-elliptic transfer is worth considering.
How do I calculate propellant mass for a Hohmann transfer?
This calculator uses the Tsiolkovsky rocket equation: Δm = m₀ × (1 − e^(−Δv/(Isp × g₀))), where m₀ is the initial spacecraft mass in kg, Isp is the engine specific impulse in seconds, g₀ = 9.80665 m/s², and Δv is the total delta-V in m/s. Expand the 'Include propellant calculation' section, enter your spacecraft mass and Isp, and the propellant mass, dry mass, and propellant fraction are shown automatically. Typical values: chemical engines have Isp of 300–450 s; ion thrusters 1,500–10,000 s. A LEO-to-GEO transfer for a 5,000 kg spacecraft with a 450 s Isp engine requires roughly 2,000 kg of propellant — about 40% of the spacecraft mass.
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