Hohmann Transfer — First Burn Δv Calculator
Find the velocity change needed to leave a circular parking orbit and enter a Hohmann transfer ellipse toward a higher or lower target orbit.
Schematic — relative orbit sizes and the transfer ellipse connecting them. The central body is not to scale.
How to Use This Calculator
- Pick a central body from the preset list, or choose "Custom μ" and enter your own gravitational parameter.
- Enter your starting orbit radius, r₁ — the distance from the center of the body, not its surface.
- Choose what to solve for — the first-burn Δv₁ for a known target orbit r₂, or the target radius r₂ reachable with a given Δv₁ budget.
- Switch to Advanced to see the full transfer: both burns, transfer time, and orbit velocities.
- Read the step-by-step solution to check or show your work.
What This Tool Offers
- ✓Presets for Earth, Moon, Mars, Venus, and the Sun, or a custom gravitational parameter.
- ✓Solve for Δv₁ or r₂ — plan the burn for a known target, or find what target a given Δv₁ can reach.
- ✓Handles both directions — raising to a higher orbit or dropping to a lower one.
- ✓Advanced mode adds the second burn, transfer time, eccentricity, and circular/transfer-orbit velocities.
- ✓km/s or m/s output units, with a full step-by-step solution.
What is a Hohmann transfer's first burn?
A Hohmann transfer moves a spacecraft between two circular, coplanar orbits using an elliptical transfer orbit tangent to both. The first burn, Δv₁, is the impulsive velocity change applied at the starting orbit that puts the spacecraft onto that transfer ellipse — it's the difference between the transfer orbit's speed at that point and the original circular orbit's speed.
Formula
√(μ/r₁) is the speed of the original circular orbit at r₁. The second factor compares that to the transfer ellipse's speed at its perigee (or apogee, if r₂ < r₁). When r₂ > r₁ the result is positive — a prograde, speed-raising burn; when r₂ < r₁ it's negative — a retrograde, speed-lowering burn.
| Symbol | Meaning |
|---|---|
| Δv₁ | First-burn velocity change, applied at the starting orbit |
| μ | Standard gravitational parameter of the central body (GM) |
| r₁ | Radius of the starting circular orbit, from the body's center |
| r₂ | Radius of the target circular orbit, from the body's center |
Reference: gravitational parameters (μ)
Standard gravitational parameter, μ = GM, for commonly used central bodies:
| Body | μ (km³/s²) |
|---|---|
| Mercury | 22,032 |
| Venus | 324,859 |
| Earth | 398,600 |
| Moon | 4,902.8 |
| Mars | 42,828 |
| Jupiter | 126,686,534 |
| Saturn | 37,931,187 |
| Sun | 132,712,000,000 |
Reference: common orbit radii (Earth)
Typical starting and target radii, measured from Earth's center (add Earth's mean radius, ~6,378 km, to an altitude to get this figure):
| Orbit | Radius (km) |
|---|---|
| Low Earth orbit (≈300 km alt.) | ≈6,678 |
| ISS altitude (≈420 km alt.) | ≈6,798 |
| Medium Earth orbit / GPS (≈20,200 km alt.) | ≈26,578 |
| Geostationary orbit (GEO) | ≈42,164 |
| Lunar orbit (Moon's mean distance) | ≈384,400 |
Frequently asked questions
Orbital mechanics formulas are built around the distance to the body's center of mass, since that's what gravity acts from — to convert from altitude, add the body's mean radius (e.g. ~6,378 km for Earth) to your altitude above the surface.
The same formula applies — it simply returns a negative Δv₁, meaning a retrograde burn that slows the spacecraft down onto a lower, faster-perigee transfer ellipse.
It's the minimum-Δv two-burn transfer between two circular orbits when the radius ratio isn't too extreme, but for very large radius ratios a bi-elliptic transfer (three burns) can use less total Δv, at the cost of a much longer transfer time.
Assumes impulsive burns, coplanar circular orbits, and two-body dynamics — real missions must also account for plane changes, finite burn time, and perturbations.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Δv1 | First burn delta-v | m/s |
| μ | Standard gravitational parameter | m3/s2 |
| r1 | Initial orbit radius | m |
| r2 | Final orbit radius | m |
What it means
The Hohmann transfer is the most fuel‑efficient method for transferring between two circular orbits around a central body. The first burn (Δv1) is applied at the initial orbit to raise the apocenter to the target radius. The formula is derived from the vis‑viva equation. Δv1 is the difference between the transfer orbit velocity at the inner radius and the circular velocity. This burn is typically performed at the initial orbit. The total Δv for the transfer includes a second burn at the target orbit to circularise. The Hohmann transfer is used for interplanetary missions, satellite repositioning, and rendezvous. Understanding Δv1 is essential for calculating propellant requirements and mission timing. The formula assumes coplanar orbits and no gravitational assists.
Worked example
Hohmann Transfer First Burn – Two Examples
Real‑World| Parameter | Value |
|---|---|
| r₁ | 6.678×10⁶ m |
| r₂ | 4.216×10⁷ m |
| Parameter | Value |
|---|---|
| r₁ | 6.6×10⁶ |
| r₂ | 2.0×10⁷ |
Common mistakes
- Hohmann transfer first burn Δv: Δv₁ = √(μ/r₁) · (√(2r₂/(r₁+r₂)) − 1).
- Gravitational parameter μ: For Earth, μ = 3.986×10¹⁴ m³/s².
- Radii r₁, r₂: From the centre of the central body – absolute (not altitude).
- Units: r in m, μ in m³/s² → Δv in m/s.
- Assumes coplanar, circular orbits.
Applications
The first burn delta‑V for a Hohmann transfer, Δv₁ = √(μ/r₁)·(√(2r₂/(r₁+r₂)) − 1), is used to transfer a spacecraft from a lower circular orbit to a higher one. This elliptical transfer is the most fuel‑efficient two‑impulse manoeuvre between coplanar circular orbits. Engineers use this formula to plan interplanetary trajectories, to design satellite orbit‑raising manoeuvres, and to compute Δv budgets. It is essential for mission design, as it allows spacecraft to reach geostationary orbit or other destinations. By applying Hohmann transfer calculations, aerospace engineers can determine the required propellant and engine performance for space missions, balancing efficiency with mission time.
- Geostationary orbit (GEO) insertion from low Earth orbit
- Interplanetary trajectory design (e.g., Mars transfers)
- Satellite orbit raising and deployment
- Mission delta‑v budget calculation
- Optimisation of transfer times and fuel consumption
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