Hohmann transfer orbit

A Hohmann Transfer Orbit is an elliptical orbital manoeuvre used to transfer a spacecraft between two circular orbits of different radii around the same central body, such as the Earth or the Sun. It is the most energy-efficient two-impulse transfer method for moving between coplanar orbits, making it a fundamental concept in astrodynamics and orbital mechanics.
This transfer technique was first described in 1925 by the German engineer Walter Hohmann, whose analysis laid the foundation for modern space navigation and mission planning.

Concept and Definition

A Hohmann transfer involves two engine impulses (velocity changes or delta-vs):

1. The first impulse is applied to move the spacecraft from its initial circular orbit onto an elliptical transfer orbit.
2. The second impulse is applied at the opposite end of the ellipse (the destination orbit) to circularise the spacecraft’s path into the new orbit.

The elliptical transfer orbit has its periapsis (closest point to the central body) at the radius of the initial orbit and its apoapsis (farthest point) at the radius of the final orbit.
Because it requires the least propellant for coplanar transfers, the Hohmann transfer is the most fuel-efficient method when time is not a major constraint. However, it is relatively slow compared to other transfer techniques such as bi-elliptic or high-energy transfers.

Step-by-Step Process

1. Initial Circular Orbit: The spacecraft begins in a circular orbit of radius r1r_1r1​ around a central body (e.g., Earth).
2. First Impulse (Δv₁): A tangential thrust increases the spacecraft’s velocity, raising its apoapsis to the radius r2r_2r2​ of the target orbit. The spacecraft now moves along an elliptical transfer orbit.
3. Transfer Orbit: The spacecraft coasts along the elliptical path from the periapsis (at r1r_1r1​) to the apoapsis (at r2r_2r2​). This part of the trajectory is unpowered and governed solely by gravitational forces.
4. Second Impulse (Δv₂): Upon reaching r2r_2r2​, a second tangential burn is performed to increase (or decrease) the spacecraft’s velocity to match that required for a stable circular orbit at r2r_2r2​.

Mathematical Formulation

If μ\muμ is the standard gravitational parameter of the central body, and r1r_1r1​ and r2r_2r2​ are the radii of the initial and final circular orbits respectively, then:

• Velocities in circular orbits:

  v1=μr1andv2=μr2v_1 = \sqrt{\frac{\mu}{r_1}} \quad \text{and} \quad v_2 = \sqrt{\frac{\mu}{r_2}}v1​=r1​μ​​andv2​=r2​μ​​

• Velocity in transfer orbit:

  vt1=μ(2r1−1at),vt2=μ(2r2−1at)v_{t1} = \sqrt{\mu\left(\frac{2}{r_1} – \frac{1}{a_t}\right)} , \quad v_{t2} = \sqrt{\mu\left(\frac{2}{r_2} – \frac{1}{a_t}\right)}vt1​=μ(r1​2​−at​1​)​,vt2​=μ(r2​2​−at​1​)​
  where at=r1+r22a_t = \frac{r_1 + r_2}{2}at​=2r1​+r2​​ is the semi-major axis of the transfer ellipse.

• Delta-v requirements:

  Δv1=vt1−v1,Δv2=v2−vt2\Delta v_1 = v_{t1} – v_1 , \quad \Delta v_2 = v_2 – v_{t2}Δv1​=vt1​−v1​,Δv2​=v2​−vt2​
  The total delta-v for the transfer is:

  Δvtotal=Δv1+Δv2\Delta v_{total} = \Delta v_1 + \Delta v_2Δvtotal​=Δv1​+Δv2​

Transfer Time

The time taken to complete the transfer (half of the elliptical orbit) is given by Kepler’s third law:
t=πat3μt = \pi \sqrt{\frac{a_t^3}{\mu}}t=πμat3​​​
This represents half the period of the elliptical orbit. For example, a Hohmann transfer from Earth to Mars typically takes about 8–9 months.

Conditions for Applicability

• Both orbits must be coplanar (lie in the same plane).
• The orbits should be circular or nearly circular.
• The spacecraft should start and end in orbits around the same central body.
• It is most effective when the ratio r2/r1r_2 / r_1r2​/r1​ is less than about 11.94; beyond that, a bi-elliptic transfer may require less energy.

Advantages

• Fuel Efficiency: Minimises propellant consumption for transfers between circular orbits.
• Simplicity: Easy to calculate and implement with precise timing.
• Predictability: Trajectories are stable and analytically defined.

Limitations

• Long Transfer Time: The spacecraft moves slowly along the elliptical path, making it unsuitable for time-critical missions.
• Coplanarity Assumption: Inefficient for orbits with significant inclination differences.
• Not Suitable for Non-Keplerian Orbits: Perturbations such as atmospheric drag or gravitational effects from other bodies can reduce accuracy.

Applications in Space Missions

The Hohmann transfer has been used in numerous space missions, particularly for transferring spacecraft between planetary orbits or from Earth orbit to interplanetary trajectories.
Examples include:

• Earth-to-Mars or Earth-to-Venus missions: Interplanetary spacecraft like Mariner, Viking, and Mars Orbiter Mission (Mangalyaan) used Hohmann-like transfer paths.
• Satellite transfers: Moving communication or weather satellites from Low Earth Orbit (LEO) to Geostationary Orbit (GEO).
• Orbital insertion and corrections: Used for efficient orbital adjustments around planets or moons.