Introduction
We generate a series of elliptical transfer orbits numerically, where these orbits exist. Each transfer is tangent to the paths of the orbits. Plotting the energy of each transfer allows us to identify the transfer of least energy – which is also the analytic transfer ellipse developed in the argument “General Hohmann Transfer”. Also, this numeric procedure offers an alternative method to calculate orbital transfers and rendezvous.
Numeric Transfer Method
The goal is to find all transfer ellipses between two elliptical orbits, tangent to both orbits at all points of contacts where the transfer ellipse exists. An ideal example of a transfer ellipse in tangent contact with both orbits is between circular orbits. However, in this method, the two orbits will generally be non-coaxial.
We begin by fixing a point at some angle (true anomaly) on the initial orbit, which will make constant tangent contact with our transfer ellipse. The transfer ellipse is created by a series of tangent burns at this point. Each burn creates a new ellipse, intercepting the target orbit at different points. We know there is a tangent point between the points of interception due to the Mean Value Theorem (see [2, sec. 6.4], for example). When the intercepting points meet, the transfer ellipse will be tangent to the target orbit.
Calculation of Transfer Ellipse
The new transfer ellipse is calculated from the parameters of the initial orbit at a particular angle (true anomaly) after each \Delta v as follows:
\begin{equation}
v_{new} = v_{old} + \Delta v = v
\end{equation}\begin{equation}
h = r v \cos(\gamma)
\end{equation}\begin{equation}
a = \mu (v^2/2 - \mu/r)/2
\end{equation}\begin{equation}
e = \sqrt{1 - h^2/\mu/a}
\end{equation}\begin{equation}
\phi = \theta - \arccos{((h^2/\mu/r - 1)/e)}
\end{equation}then
\begin{equation}
r(\theta) = h^2/\mu/(1 + e\cos(\theta + \phi))
\end{equation}where
- h = specific angular momentum (km^{2}/sec)
- e = eccentricity
- \mu = {G}{M} (km^{3}/sec^{2}). The standard gravitational parameter.
- \theta = true anomaly
- \phi = apse line rotation
- \gamma = flight path angle
- a = semimajor axis
The Points of Intersection
The points of intersection between two orbits are needed for the numerical routine. The points of intersection between any two orbits with a common focus can be found analytically by equating the radial equations of each ellipse as follows:
\begin{equation}
r_{initial}(\theta) = \frac{h_{1}^{2}}{\mu (1 + e_{1} \cos(\theta))} = \frac{h_{2}^{2}}{\mu (1 + e_{2} \cos(\theta - \theta_{0}))} = r_{target}(\theta - \theta_{0})
\end{equation}Resulting in
\begin{equation}
a \cos(\theta) + b \sin(\theta) = c
\end{equation}where
- h = specific angular momentum (km^{2}/sec)
- e = eccentricity
- \mu = {G}{M} (km^{3}/sec^{2}). The standard gravitational parameter.
- \theta_{0} = rotation of target apsis relative to the initial apsis
- a = e_{1} h_{2}^{2} - e_{2} h_{1}^{2} \cos(\theta_{0})
- b = - e_{2} h_{1}^{2}\sin(\theta_{0})
- c = h_{1}^{2} - h_{2}^{2}
and we finally solve for the angles where the orbits intersect
\begin{equation}
\theta = \tan^{-1}(b/a)\;\;\pm\;\;\cos^{-1}(c \: \cos(\tan^{-1}(b/a))/a)
\end{equation}The Transfer Orbit
Using the orbit data of Fig. 4 in “General Hohmann Transfer” ( or see Fig. 3 below) as a reference, we follow this general outline to generate the transfer ellipses numerically:
- Use one point of the initial orbit as the start for the transfer orbit. Both orbits are tangent at this point. This point will remain constant and is very similar to start of a Hohmann transfer.
- Vary the velocity at this point to create the transfer ellipse. Increasing/decreasing the velocity will expand/contract the ellipse.
- The points of intersections will move toward or away from each other as the transfer ellipse is varied. When the angle between the points of intersection becomes small (near zero), the target orbit is tangent to the transfer ellipse. Note: if there are no points of intersection, vary the ellipse in a direction so they will appear. Then start the procedure.
- Calculate the change in velocities at these points.
The General Transfer
We can generate a transfer ellipse at most points on the initial orbit that will be tangent to the target orbit. Although cluttered, a series of transfer orbits generated at 10 deg intervals is shown in Fig. 1 as dashed lines.
Fig. 2 shows the change in velocity needed per transfer ellipse per angle (that is, at the angle where the transfer orbit is generated on the initial orbit). Note some orbits near the points of intersection were excluded due to excessive energy or because the conic section generated became hyperbolic.
A search for the optimal transfer, from Fig. 2, shows the lowest energy transfer at 77 deg on the initial orbit. This transfer orbit is shown in Fig. 3 and for all intents and purposes identical with Fig. 4 in “General Hohmann Transfer”. And in Table 1 we determine the total change in velocity of 0.72 km/sec needed to effect this transfer.
| \Delta v_1 (Initial Orbit at 77^o) | \Delta v_2 (Target Orbit at 224^o) | Total\Delta v |
| 0.4627321 | 0.2564719 | 0.7192040 |
Rendezvous
When performing a rendezvous, you do not want the transfer ellipse to contact the target orbit or the target. The transfer ellipse should be 100 – 1000 meters away from the target orbit or whatever distance makes sense to dock safely. Transfer orbits generated for rendezvous at 10 degree intervals, 500 Km away from the target for clarity, are shown in Fig. 4 and Fig. 5. Compare with Fig. 3 where the transfer makes contact with the target orbit.
Each generated transfer ellipse, tangent to the target orbit at some point, are guaranteed by the Mean Value Theorem. We calculate the radial distance between the target and transfer orbits at these points until the correct distance is attained.
The above method makes a rendezvous more reliable, since tangent transfers require no changes in direction. The rendezvous is more safe, since the transfer ellipse never crosses the target orbit.
Conclusion
We numerically generate transfer ellipses at all points on the initial orbit, where these transfer orbits exist. We plot the total velocity (energy) needed for each transfer. We find the most efficient numeric transfer matches the analytic method. We conclude the analytic method generates the most fuel efficient tangent transfer.
The methods of orbital transfer presented are novel and efficient, but not optimal. And current methods of orbital transfer are most certainly optimized for velocity (fuel). An optimized transfer was calculated for the orbits presented – but not included – and the analytic transfer was within one percent. Close, but still higher than an optimized transfer.
If tangent transfers are less complex and safer than optimized transfers, this may more than compensate for the slight fuel advantage an optimal transfer offers. As space flight becomes common or popular, i.e. space tourism, fuel may not be the only consideration.
References
- Walter Hohmann, Die Erreichbarheit der Himmelskorper, Oldenbourg, Munich 1925
- Battin, R., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education Series, AIAA, New York, 1987.
- Howard D Curtis, Orbital Mechanics for Engineering Students, Elsevier, 2005.