What is delta-v in orbital mechanics?

Delta-v (change in velocity) is the fundamental measure of the effort needed to perform an orbital maneuver. It represents the total change in velocity a spacecraft's engines must produce, measured in km/s or m/s. Higher delta-v requirements mean more fuel is needed.

How long does a Hohmann transfer take?

The transfer time equals half the orbital period of the transfer ellipse. For an Earth LEO to GEO transfer, it takes about 5.3 hours. For an Earth-to-Mars Hohmann transfer, it takes approximately 8.5 months.

What is the gravitational parameter mu?

The gravitational parameter (mu or GM) is the product of the gravitational constant G and the mass M of the central body. For Earth it is approximately 398,600 km cubed per second squared. Using mu avoids the need to know G and M individually, which are harder to measure precisely than their product.

Hohmann Transfer Orbit Calculator

Calculate the delta-v required for the most fuel-efficient transfer between two circular orbits.

Starting Orbit Radius (r₁)

LEO ~6,771 km (400 km altitude)

Target Orbit Radius (r₂)

GEO ~42,164 km

Central Body

Gravitational Parameter (μ)

km³/s²

Total Δv

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Perigee Burn Δv₁

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Apogee Burn Δv₂

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Transfer Time

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Orbit Ratio (r₂/r₁)

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How Hohmann Transfer Orbits Work

A Hohmann transfer orbit is the most fuel-efficient method to move a spacecraft between two circular orbits around the same central body. Proposed by German engineer Walter Hohmann in 1925, it uses an elliptical transfer orbit that is tangent to both the initial and final circular orbits, requiring exactly two engine burns.

The Two-Burn Maneuver

The first burn (perigee burn) at the lower orbit accelerates the spacecraft into an elliptical transfer orbit. The spacecraft then coasts along this ellipse for half an orbital period until it reaches the higher orbit altitude. A second burn (apogee burn) circularizes the orbit at the target altitude. The total velocity change, or delta-v, is the sum of both burns.

The Hohmann Transfer Formulas

The delta-v for each burn is calculated using the vis-viva equation and orbital mechanics:

Δv₁ = √(μ/r₁) × (√(2r₂/(r₁+r₂)) - 1)

Δv₂ = √(μ/r₂) × (1 - √(2r₁/(r₁+r₂)))

The transfer time is half the orbital period of the transfer ellipse:

T = π × √((r₁+r₂)³ / (8×μ))

Comparison to Direct Burns

A direct, impulsive transfer at an arbitrary point in the orbit would require significantly more delta-v because the spacecraft would need to change both speed and direction simultaneously. The Hohmann transfer takes advantage of orbital mechanics by performing burns only at the points where the transfer ellipse naturally aligns with the circular orbits, minimizing wasted energy.

Real-World Applications

Hohmann transfers are used extensively in spaceflight. The transfer from Low Earth Orbit (LEO) to Geostationary Earth Orbit (GEO) is one of the most common, taking about 5.3 hours and requiring approximately 3.94 km/s of delta-v. Mars transfer orbits from Earth use a similar principle around the Sun, with a transit time of about 8.5 months. The Apollo missions used modified Hohmann-like transfers to reach the Moon.

Limitations

Hohmann transfers assume circular, coplanar orbits and instantaneous impulse burns. Real orbits have eccentricity, inclination differences, and finite burn times that require corrections. For very large orbit ratios (greater than 11.94:1), a bi-elliptic transfer with three burns can actually be more fuel-efficient despite taking longer.

Frequently Asked Questions

A Hohmann transfer orbit is an elliptical orbit used to transfer a spacecraft between two circular orbits of different radii using two engine burns. It is the most fuel-efficient two-impulse maneuver for coplanar circular orbit transfers, named after German engineer Walter Hohmann who proposed it in 1925.

A Hohmann transfer is most efficient when the ratio of the outer orbit radius to the inner orbit radius is less than approximately 11.94. Beyond this ratio, a bi-elliptic transfer using three burns and an intermediate orbit can be more fuel-efficient, though it takes significantly longer to complete.

Delta-v (written as Δv) represents the total change in velocity a spacecraft's engines must produce. It is the fundamental metric for the "cost" of an orbital maneuver, measured in km/s or m/s. Higher delta-v means more propellant is required, making it the key constraint in mission design.

The transfer time equals half the period of the transfer ellipse. For LEO to GEO around Earth, it takes about 5.3 hours. For a transfer from Earth orbit to Mars orbit around the Sun, the journey takes approximately 8.5 months. Longer transfer times are the trade-off for fuel efficiency.

The gravitational parameter (mu or GM) is the product of the universal gravitational constant G and the mass M of the central body being orbited. For Earth, mu is approximately 398,600 km³/s². Astronomers use mu because its value can be measured far more precisely than G and M individually.