General Hohmann Transfer

Introduction

A profound pioneering effect for space travel came in 1925, when Walter Hohmann published his book, “The Attainability of Heavenly Bodies” [1]. The book describes many aspects of space travel including liftoffs and landings, passenger considerations and destinations. More importantly, he describes the route taken to these destinations. The route taken from Earth to Venus, say, is an elliptical path, connecting the orbit of Earth around the sun, to the orbit of Venus around the sun, i.e. connecting circular orbits. The points of connection require, in the words of Walter Hohmann, “changes in velocity, but no changes in direction”.

Since then, Hohmann transfers have been considered to apply only to circular orbits or coaxial elliptical orbits (see [2][chap. 6], for example). Here, we generalize Hohmann transfers between any two orbits in a plane, coaxial or not – offering an alternative procedure for orbital transfers and rendezvous. The advantages are similar to the standard Hohmann transfer – a less complex transfer, tangent to both orbits. As activities in space increase (space tourism, for instance), these transfers may be of interest.

The Hohmann Transfer

A typical Hohmann transfer is shown in Fig. 1. An elliptical orbit is generated connecting the initial and final orbits, first through an impulse velocity at point A. Then, another circular orbit is generated through an impulse velocity at point B. At the apse points of the transfer ellipse, the velocities are tangent to the paths of the orbits.

In contrast, two same plane non-coaxial orbits are shown in Fig. 2. The usual Hohmann transfer will not work, since the apse points do not line up and any generated intermediate ellipse will not be tangent to the point on the target orbit that crosses the apse line.

In Table 1 we define parameters for the initial and target orbits, designated by the subscripts 1 and 2 respectively, used to generate the plot in Fig. 2. The angle, \theta_{\circ}, is the rotation of the target orbit apse line relative to the initial orbit apse line. The data will be used to demonstrate the procedure.

The General Transfer

There are at least two angles (or true anomalies) that designate points where the flight path angles of two orbits are equal, i.e. the orbits are tangent to each other along the radial line at these angles. If an (transfer) ellipse coincides with these points, then two orbital equations for the known distances and a third equation for the known flight angles would determine the three unknowns of the ellipse: the eccentricity (e), the angular momentum (h) and the degree of apse line rotation (\phi) relative to the initial orbit.

Find Two Transfer Points

Tangent points between any two orbits with a common focus can be found analytically by equating the flight path angles as follows:

\tan(\gamma_{1}) = \tan(\gamma_{2})

where the flight path angle, \gamma, is the angle between the orbiting body’s velocity vector (the vector tangent to the instantaneous orbit) and the local horizontal, the line perpendicular to radial line passing through the object in orbit, as shown in Fig. 3.

The equations for the flight path angles can be written

\begin{align} 
e_{1}  \frac{\sin(\theta)}{(1 +  e_{1}  \cos(\theta))} = e_{2}  \frac{\sin(\theta - \theta_{0})}{(1 +  e_{2}  \cos(\theta - \theta_{0}))}
\end{align}

resulting in

\begin{align} 
a  \cos(\theta) + b  \sin(\theta) = c
\end{align}

where

• e = eccentricity
• \theta = true anomaly
• \theta_{0} = rotation of target apses relative to the initial apsis
• a = e_{2} \sin(\theta_{0})
• b = e_{1} - e_{2} \cos(\theta_{0})
• c = e_{1} e_{2} \sin(\theta_{0})

and we finally solve for two angles where the two orbits are tangent (77^{\circ} and 224^{\circ} in Figure 4, below).

 \theta = \tan^{-1}\Bigl(\frac{b}{a}\Bigr)\;\;\pm\;\;\cos^{-1}\Biggl(c \quad \frac{\cos\Bigl(\tan^{-1}\Bigl(\frac{b}{a}\Bigr)\Bigr)}{a}\Biggr)

Solve Transfer Equation

For convenience, we reference the initial, target and transfer orbits, orbits 1,2 and 3, respectively. We also reference the two angles (true anomalies) where the flight path angles are equal, \theta_{1} and \theta_{2}. The distances of orbits 1 and orbit 2 are known at these angles. Since orbit 3 is tangent and touching both of these points, we have r_{3}(\theta_{1}) = r_{1}(\theta_{1}) and r_{3}(\theta_{2}) = r_{2}(\theta_{2}). This gives us two of the three equations needed to solve for orbit 3:

r_{3}(\theta_{1}) = \frac{h_{3}^{2}}{\mu (1 + e_{3} \cos(\theta_{1} - \phi))}\tag{5}

and

r_{3}(\theta_{2}) = \frac{h_{3}^{2}}{\mu (1 + e_{3} \cos(\theta_{2} - \phi))}\tag{6}

The third equation is obtained by equating flight path angles. First, the flight path angles, \gamma‘s, at \theta_{1} are equal and known, and the flight path angles at \theta_{2} are equal and known. Second, a simplification can be obtained by recognizing that the flight path angles at \theta_{1} are equal and opposite to the flight path angles at \theta_{2}. For example, in Fig. 1, the flight path angles at \theta = 0^{\circ} are equal and opposite to the flight path angles at \theta = 180^{\circ}. This is not obvious between non-coaxial orbits. However, it can be demonstrated as a general property by rotating orbit 2 relative to orbit 1 and evaluating the flight angles at various angular displacements. This was done at 45^{\circ} intervals and tabulated in Table 2 from the data in Table 1.

The flight angle equation (see equation 1) for orbit 3, at the appropriate true anomalies, \theta_{1} and \theta_{2}, can then be placed in the following relation:

e_{3} \frac{\sin(\theta_{1} - \phi)}{(1 + e_{3} \cos(\theta_{1} - \phi))} = (-)e_{3} \frac{\sin(\theta_{2} - \phi)}{(1 + e_{3} \cos(\theta_{2} - \phi))}\tag{7}

Using equations 5 and 6 gives us:

e_{3} \frac{r_{3}(\theta_{1})\mu \sin(\theta_{1} - \phi)}{h_{3}^{2}} = (-)e_{3} \frac{r_{3}(\theta_{2})\mu \sin(\theta_{2} - \phi)}{h_{3}^{2}}\tag{8}

To solve for \phi, rearrange Eq. 8 using a trig identity and A = (-) r_{3}(\theta_{2})/r_{3}(\theta_{1}):

\tan(\phi) = \frac{A\sin(\theta_{2}) - \sin(\theta_{1})} {A\cos(\theta_{2}) - \cos(\theta_{1})}\tag{9}

Using equations 5 and 6 and the definition of A above:

e_{3} = \frac{(-)(A + 1)} {\cos(\theta_{1} - \phi) + A\cos(\theta_{2} - \phi)}\tag{10}

h_{3} = \sqrt{r_{3}(\theta_{1}) \mu (1 + e_{3}\cos(\theta_{1} - \phi))}\tag{11}

The transfer ellipse, plotted in Fig. 4, is now fully defined and represented by:

r_{3}(\theta) = \frac{h_{3}^{2}}{\mu (1 + e_{3} \cos(\theta - \phi))}\tag{12}

Impulse Velocities

We now calculate the impulse velocities to produce the transfer. First, calculate the orbital parameters for the transfer ellipse. From eq. 5 and eq. 6, calculate the distances to the transfer points, used to calculate A in eq. 9 and eq. 10:

• at 77^\circ:\,\,\, r_3 = r_1 = 9923 Km
• at 224^\circ: r_3 = r_2 = 19915 Km
• A = -(19915/9923) = -2.007

From eq’s 9, 10 and 11 and the data in Table

• \phi_3 = 19^\circ
• e_3 = 0.429
• h_3 = 69670 Km^2/sec

The tangent velocities at the transfer points are calculated as follows:

• v_1 = \mu(1 + e_1 \textrm{cos}(\theta))/h_1
• v_2 = \mu(1 + e_2 \textrm{cos}(\theta - \theta_\circ))/h_2
• v_3 = \mu(1 + e_3 \textrm{cos}(\theta - \phi))/h_3

The impulse velocities are calculated similar to the standard Hohmann transfer (see appendix 1):

• at 77^\circ:\,\,\,\Delta v_{1} = v_3 - v_1 = (7.02 - 6.57) \textrm{ Km/sec}
• at 224^\circ:\Delta v_{2} = v_2 - v_3 = (3.77 - 3.50) \, \textrm{Km/sec}
• \Delta v_{\textrm{Total}} = \Delta v_{1} + \Delta v_{2} = 0.72 \textrm{Km/sec}

Discussion

Implementation of the general Hohmann transfer is similar to the standard Hohmann transfer. The impulse is always in the direction of the orbits. This makes transfers less complicated. However, the standard Hohmann transfer is also an optimal transfer [3]. Optimal transfers conserve fuel. General Hohmann transfers are fuel efficient, but not optimal.

An optimal transfer is a numeric calculation that minimizes fuel (velocities). A comparison between an optimal and tangent transfer is shown in appendix 2. The optimal transfer, \Delta v_{\textrm{Total}} = 711 m/sec, is less than the tangent transfer, \Delta v_{\textrm{Total}} = 719 m/sec – a difference of about 1%.

This is a general result. However, the optimal transfer is not a tangent transfer. The impulse is not in the direction of the orbits. This makes transfers more complicated. The tangent transfer offers an alternative, at a minimal increase in fuel.

The tangent transfer is close to optimal. If there is no practical difference in velocity between the two transfers, the tangent transfer should be considered. Conserving fuel is important, but safety and reliability of transfer may be items of concern, particularly if space travel becomes popular.

Conclusion

This procedure generalizes the Hohmann transfer. These transfers are less complex and easier to implement than other transfer methods. They are not optimal concerning fuel, but they are close.

This is a new method of transfer. It should be evaluated in areas of transfer applications, particularly where optimal transfers are used. If this method makes a transfer or rendezvous more safe or reliable, it should be considered.

Similar tangent transfers can be generated numerically at most points on the initial orbit where they exist (for example, a transfer would not exist at points of intersection). Numerically we can demonstrate the analytic method shown here is the lowest energy transfer of these types.

Appendix – Optimal Transfer

A comparison of impulse velocities, between an optimal and tangent transfer, is shown below. Optimization minimizes an item of interest in a set of orbital equations, such as time or fuel.

The parameters in Table 1, used to generate the orbits in Fig. 2, were input into optimization software with the results listed below. The optimization routine is based off the papers by McCue [4], Lee [5] and Bender and McCue [6].

The \Delta v_{\textrm{Total}} = 710.9 m/sec for the optimal transfer. The \Delta v_{\textrm{Total}} = 719.2 m/sec for the tangent transfer. This is an increase in fuel (velocity) of about 1%.

Bibliography

1. Walter Hohmann, Die Erreichbarheit der Himmelskorper, Oldenbourg, Munich 1925
2. Howard D Curtis, Orbital Mechanics for Engineering Students, Elsevier, 2005.
3. J.E. Prussing, Simple proof of the global optimality of the Hohmann transfer, Vol. 15, No. 4 (1992), pp. 1037-1038.
4. G. McCue, Optimum Two-Impulse Orbital Transfer and Rendezvous between Inclined Elliptical Orbits, AIAA Journal, vol. 1, no. 8, pp. 1865-–1872, 1963.
5. G. Lee, An Analysis of Two-Impulse Orbital Transfer, AIAA Journal, vol. 2, no. 16, pp. 1767-–1773, 1964.
6. G. McCue and D. Bender, Numerical Investigation of Minimum Impulse Orbital Transfer, AIAA Journal, vol. 3, no. 12, pp. 2328-–2334, 1965.