Calculation of a Hohmann transfer is straightforward.
In a circular orbit, the centrifugal force is equal to the gravitational force, and the velocity can be obtained from the relation
\begin{equation}
\frac{v^{2}}{r} = \frac{G M}{r^{2}} \quad \text{or} \quad v = \sqrt{\frac{G M}{r}}
\end{equation}where,
- r is the radius of the orbit from the center of the primary
- v is the velocity of the object in orbit
- G is the gravitational constant
- M is the mass of the primary
In an elliptical orbit, motion is governed by the equation
\begin{equation}
r(\theta) = \frac{h^{2}}{G M (1 + e \cos(\theta))}
\end{equation}where,
- \theta is the angle of the orbiting object from periapsis (true anomaly)
- e is the eccentricity, (r_{2}-r_{1})/(r_{2}+r_{1})
- h is the specific angular momentum
At periapsis, the angle is zero, and the velocity, with no radial component, can be determined by
\begin{equation}
v_{ellipse@A}= \sqrt{\frac{G M (1 + e)}{r_{1}}}
\end{equation}
At apoapsis, since the angular momentum of an orbit around a central force is a constant, the velocity can be determined by
\begin{equation}
v_{ellipse@B}= r_{1} \frac{v_{ellipse@A}}{r_{2}}
\end{equation}
The impulsive thrusts are then calculated as follows:
- \Delta v_{1} = v_{ellipse@A} - v_{1}
- \Delta v_{2} = v_{2} - v_{ellipse@B}
- \Delta v_{Total} = \Delta v_{1} + \Delta v_{2}
It is assumed – and currently taught – that Hohmann transfers apply only to circular orbits (or coaxial elliptical orbits). This is simply not true. Please see General Hohmann Transfer for transfers between non-coaxial elliptical orbits.
Feeling adventurous? It is assumed – and currently taught – that degenerate electrons support the mass in a white dwarf. This also is not true. Please see Degenerate Electrons for a description of how they really behave. Then read Supernovas to see how easy they are to explain as a consequence.
This misunderstanding has been going on for almost 100 years…