¶Orbital Maneuvers: The Hohmann Transfer

In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits of different radii around a central body in the same plane. It was first described by Walter Hohmann in 1925.

¶The Vis-Viva Equation

The foundation for calculating required velocity changes ($\Delta v$) is the vis-viva equation, derived from the conservation of specific mechanical energy: $v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right)$ where $v$ is orbital velocity, $r$ is the current distance to the central body, $a$ is the semi-major axis, and $GM$ is the standard gravitational parameter ($\mu$).

¶Maneuver Execution

The Hohmann transfer requires two impulsive velocity changes (engine burns):

1. First Burn ($\Delta v_1$): Accelerates the spacecraft from the inner circular orbit (radius $r_1$) into an elliptical transfer orbit. The periapsis of this ellipse is $r_1$, and the apoapsis is $r_2$. $\Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2r_2}{r_1 + r_2}} - 1 \right)$
2. Second Burn ($\Delta v_2$): Executed half an orbit later at apoapsis ($r_2$), accelerating the spacecraft again to circularize the orbit at the new altitude. $\Delta v_2 = \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2r_1}{r_1 + r_2}} \right)$

The total $\Delta v$ budget for the mission is the sum of both impulses: $\Delta v_{\text{total}} = \Delta v_1 + \Delta v_2$. The Hohmann transfer is proven to be the most fuel-efficient two-impulse transfer between coplanar circular orbits when the ratio of the final to initial radius is less than roughly 11.94.

Reference: Bate, R. R., Mueller, D. D., & White, J. E. (1971). Fundamentals of Astrodynamics. Dover Publications.

¶References

• Fundamentals of Astrodynamics (Roger R. Bate, Donald D. Mueller, Jerry E. White)
• Orbital Mechanics for Engineering Students (Howard D. Curtis)
• NASA: Orbital Mechanics Educational Material