Hohmann Transfer

Hohmann transfer is an orbital transfer maneuver performed between two coplanar orbits around the same central body, involving two impulses (two instantaneous changes in velocity). The transfer is accomplished via a single elliptical orbit tangent to both the initial and target orbits. This type of maneuver is defined as the solution that minimizes the total velocity change required among all two-impulse orbital transfers.

The Hohmann transfer is widely regarded as the fundamental reference maneuver for transitions between two circular orbits and is used as a benchmark in the analysis of more complex orbital optimization problems.

Basic Principle

The maneuver begins when the spacecraft performs its first velocity change while in the initial orbit, entering an elliptical transfer orbit. After traversing half of this elliptical orbit, the spacecraft executes a second velocity change to insert itself into the target orbit. The first impulse is typically applied at the periapsis of the initial orbit, and the second impulse is applied at the apoapsis of the transfer orbit.

The semi-major axis of the transfer orbit is determined as the arithmetic mean of the radii of the initial and target orbits. This geometric property defines the distinguishing feature of the Hohmann transfer.

Basic Velocity Change (Δv) Formula

The total velocity change required for a Hohmann transfer is defined as the sum of the two impulses and is expressed in terms of the orbital radii as follows:

^{Basic Formula}

In this formula:

• r1: radius of the initial circular orbit,
• r2: radius of the target circular orbit,
• μ: standard gravitational parameter of the central body.

The first term represents the velocity increase required to depart from the initial orbit into the transfer orbit, while the second term represents the velocity change needed to enter the target orbit from the transfer orbit.

Mathematical proofs have established that, under specific assumptions, the Hohmann transfer provides the globally minimum total Δv among all two-impulse transfers.

Energy and Optimality

The key characteristic of the Hohmann transfer is that it requires the globally minimum total velocity change among all coplanar two-impulse orbital transfers. Prussing’s work mathematically demonstrated that this optimality is not merely local but global.^【1】

This optimality depends not only on the magnitudes of the impulses but also on their directions and application points. Considering the total velocity change as the sum of the absolute values of the impulses, rather than their vector sum, ensures accurate representation of propellant consumption.

Generalized Elliptical Hohmann Transfer

In the generalized Hohmann transfer approach proposed by El Mabsout and colleagues, the initial and target orbits are not required to be circular; they may instead be elliptical and coplanar.^【2】 In this case, the transfer orbit is also elliptical, and the system remains defined by two impulses.

In this generalized problem, the semi-major axis and eccentricity of the initial orbit and the semi-major axis of the target orbit are given. The optimization process is carried out over the eccentricity of the transfer orbit. The total velocity change has been analyzed as a function of the transfer orbit’s eccentricity and shown to decrease monotonically within certain bounds.

The analysis reveals that as the eccentricity of the target orbit increases, the transfer requires less energy, and the energy requirement reaches its maximum when the target orbit is circular. This result demonstrates that the classical circular Hohmann transfer is a special case of the generalized elliptical transfer.

Transfer Duration

The duration of a Hohmann transfer equals half the orbital period of the transfer ellipse. This time depends on the semi-major axis of the transfer orbit and the gravitational parameter of the central body. Although the Hohmann transfer provides the most fuel-efficient solution, it is not always the shortest in terms of time.

Application Context

The Hohmann transfer is a fundamental concept in orbital mechanics, used in altitude change maneuvers for artificial satellites, mission design analyses, and theoretical studies of orbital transfer problems. It also serves as a reference solution for evaluating more complex transfer strategies, such as multi-impulse or plane-change transfers.