Formula & Calculator
Hohmann Transfer First Burn Delta-V
Velocity change required for the first (perigee) burn of a Hohmann transfer between two circular orbits.
Interpretation
Hohmann transfer first burn Δv1 = √(μ/r₁) · (√(2r₂/(r₁+r₂)) − 1). It is the velocity increment required to leave a circular orbit and enter an elliptical transfer orbit. Example: Earth to Mars (r₁=1 AU, r₂=1.524 AU) → Δv1 ≈ 2.94 km/s.
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
Frequently Asked Questions
It calculates the impulsive velocity change required at the perigee of the transfer ellipse to move from a lower circular orbit (radius r1) to a higher circular orbit (radius r2) via a Hohmann transfer.
μ = gravitational parameter of the central body (e.g., 3.986×10¹⁴ m³/s² for Earth)
r1 = radius of initial circular orbit (m)
r2 = radius of final circular orbit (m)
For a given pair of circular orbits, the Hohmann transfer (a half‑ellipse) requires the minimum total Δv because it is a two‑impulse transfer that uses the least propellant.
The first burn is applied in the direction of motion to increase the spacecraft’s speed from the circular velocity to the perigee velocity of the transfer ellipse.
- Mixing up r1 and r2 order (transfer up vs down changes the sign convention).
- Using orbital altitude instead of radius from the Earth’s centre (must add planetary radius).
- Forgetting that the transfer ellipse perigee is at r1 and apogee at r2.
Transfer from LEO (r1 = 6,378 km + 300 km = 6,678 km) to GEO (r2 = 42,164 km). μ = 3.986×10¹⁴. Compute vc1 = √(μ/r1) = √(3.986e14/6.678e6) ≈ 7723 m/s. Then Δv1 = √(μ/r1)·(√(2r2/(r1+r2)) − 1). Calculate √(2×42164/(6678+42164)) = √(84328/48842) = √1.726 = 1.314. Δv1 = 7723×(1.314−1) ≈ 7723×0.314 = 2425 m/s.
The second burn occurs at apogee to raise the perigee and circularise the orbit. It is given by Δv2 = √(μ/r2)·(1 − √(2r1/(r1+r2))).
A Hohmann transfer is the minimum two‑impulse transfer. A direct ascent (one burn) is not possible between circular orbits; the Hohmann transfer is the most fuel‑efficient.
The transfer time is half the period of the transfer ellipse: t = π·√(a3/μ), where a = (r1+r2)/2.
It assumes coplanar orbits (same inclination) and is only optimal for circular orbits. For high eccentricity or non‑coplanar orbits, other transfers (e.g., bi‑elliptic) may be more efficient.