## Journal of the Brazilian Society of Mechanical Sciences and Engineering

##
*Print version* ISSN 1678-5878*On-line version* ISSN 1806-3691

### J. Braz. Soc. Mech. Sci. & Eng. vol.25 no.2 Rio de Janeiro Apr./June 2003

#### http://dx.doi.org/10.1590/S1678-58782003000200003

**The elliptic-bi-parabolic planar transfer for artificial satellites**

**A. F. B. A. Prado**

Instituto Nacional de Pesquisas Espaciais, Depto. de Mecânica Espacial e Controle, Avenida dos Astronautas 1758, 12227-010 São José dos Campos, SP. Brazil. prado@dem.inpe.br, abertachini@terra.com.br

**ABSTRACT**

The "elliptic-bi-parabolic transfer" orbit for an artificial satellite is an extension of the bi-parabolic transfer that uses a Swing-By with a natural satellite of the main body to reduce the amount of fuel required by the maneuver. The objective is to find the minimum cost trajectory, in terms of fuel consumed, to transfer a spacecraft from a parking orbit around a planet to an orbit around a natural satellite of this planet or to a higher orbit around the planet. Graphics are built to show in more details the potential savings given by this technique. After that, the idea of using a natural satellite in the maneuver is applied to the problem of making a spacecraft to escape from the main planet to the interplanetary space with maximum velocity at infinity.

**Keywords:** Astrodynamics, orbital maneuvers, swing-by

**Introduction**

R. H. Goddard (1919) was one of the first researchers to work on the problem of optimal transfers of a spacecraft between two points. He proposed optimal approximate solutions for the problem of sending a rocket to high altitudes with minimum fuel consumption. The problem of optimal transfers (in the sense of reducing the fuel consumption) between two Keplerian coplanar orbits has been under investigation for more than 40 years. In particular, many papers solve this problem for an impulsive thrust system with a fixed number of impulses. The literature presents many solutions for particular cases, like the Hohmann (1925) and the bi-elliptic (Hoelker and Silber, 1959; Shternfeld, 1959) transfers between two circular orbits and their variants for ellipses in particular geometry.

The original Hohmann solution (see Fig. 1) was obtained for a bi-impulsive transfer between two circular and coplanar orbits. It is the most used result in orbital maneuvers and it is applied here to compare the results generated by the technique suggested in the present paper. This important transfer has the following steps:

a) In the initial orbit an impulse with magnitude

(where r_{0} (r_{f}) is the radius of the initial (final) orbit and V_{0} is the velocity of the spacecraft when in its initial orbit) is applied in the direction of the motion of the spacecraft. With this impulse the spacecraft is inserted into an elliptical orbit with periapsis r_{0} and apoapsis r_{f};

b) The second impulse is applied when the spacecraft is at the apoapsis. The magnitude is

and it circularizes the orbit. This result is largely used nowadays, as a first approximation of more complex models. Later, Hoelker and Silber (1959) (and others) showed that this transfer was not the best in all cases. A detailed study of those transfers can be found in Marec (1979). Next, the Hohmann transfer was generalized to the elliptic case (transfer between two coaxial elliptic orbits) by Marchal (1965). Smith (1959) shows results for some other special cases, like coaxial and quasi-coaxial elliptic orbits, circular-elliptic orbits, and two quasi-circular orbits.

The three-impulse concept is introduced in the literature by Shternfeld (1959) in Russia. He derived the bi-elliptic transfer (according to Edelbaum, 1967). This transfer was later independently derived by Hoelker and Silber (1959) and Edelbaum (1959). All those researches show that it is possible to find a bi-elliptical transfer between two circular orbits that has a DV lower than the one for the Hohmann transfer, when the ratio between the radius of the initial and the final orbits is greater than 11.93875. Following the idea of more than two impulses, there is also the paper by Broucke and Prado (1995) that uses three or four impulses passing through infinity. Two papers that document and summarize the knowledge about impulsive transfers are the ones written by Edelbaum (1967) and Gobetz and Doll (1969).

The present paper studies the problem of orbital maneuvers where a celestial body (a natural satellite of the main body considered) is used to decrease the DV (fuel consumed) required to complete the specified maneuver. It is called the "elliptic-bi-parabolic transfer" maneuver. A transfer from the Earth to the Moon and a transfer between two orbits around the Earth are used as examples, but the results are valid for any system of primaries.

**The Elliptic-Bi-Parabolic Transfer**

The elliptic-bi-parabolic transfer (Fig. 2) is an extension of the bi-parabolic transfer, that is the limit case of the bi-elliptic transfer invented by Shternfeld (1959) and Hoelker and Silber (1959). In their original version, they show how to make a transfer between two circular and coplanar orbits in three impulses. The sequence is: i) The first impulse is applied to send the spacecraft from its initial orbit to an elliptic orbit with large periapsis distance (parabolic in the limit case); ii) The second impulse is applied at the apoapsis of this first transfer orbit and it puts the spacecraft in a second elliptic (parabolic in the limit case) transfer orbit with periapsis tangential with its final orbit; iii) The third impulse is applied in the periapsis of this second transfer orbit and it completes the capture of the satellite in its final orbit. This transfer is not optimal, but it is shown (Marec, 1979) that this transfer is more economical (in terms of DV) than the Hohmann transfer when r_{f}/r_{0} > 11.93875.

The elliptic-bi-parabolic transfer takes advantage of an intermediate swing-by with the secondary body to reduce the amount of fuel required for the maneuver. It is useful to transfer a spacecraft from a low orbit around a central body to another body (a natural satellite) in orbit around this same central body or to transfer the spacecraft between two orbits around the central body. A useful application is the transfer of a space vehicle from LEO (Low Earth Orbit) to the Moon. There are many papers in the literature that describe the swing-by maneuver and applications in details. Some of them are: D'Amario, Byrnes and Stanford (1981), Battin (1987), Broucke (1988), Weinstein (1992), Broucke and Prado (1993), Prado and Broucke (1994). To develop the equations involved in this transfer it is assumed that: i) The initial LEO is circular with radius r_{0}; ii) The space vehicle is in a Keplerian orbit around the central body, except for the duration of the swing-by at the target body; iii) The swing-by at the target body can be modeled by the two-body scattering (Prado, 1993 and 1995); iv) The propulsion system is the usual impulsive system, able to delivery an instantaneous increment of velocity DV; v) The second body (the natural satellite) is in a circular orbit with radius r_{B}, coplanar with the initial orbit of the spacecraft; vi) The final orbit desired for the spacecraft is a circular orbit with radius r_{f} around the primary or the secondary body (so, r_{f} is measured in two different forms, depending on the version of the maneuver considered, but its exact meaning is always clear in the context).

**First Version: Transfer to a Satellite Body**

With those hypotheses, the complete transfer (for the case where the final orbit is around the natural satellite body) follows the steps:

i) From the initial circular parking orbit an impulse is applied to send the spacecraft to an elliptic Hohmann transfer to the target body. This impulse is tangential to the initial orbit, and the magnitude is given by Eq. (3). This magnitude is the same one obtained in the Hohmann transfer, because the intermediate transfer orbits are the same in both cases. So, Eq. (3) is equivalent to Eq. (1), but reorganized and using the variables that will be used in the sequence of the present paper.

In those equations, r_{B} is the radius of the initial circular parking orbit of the spacecraft and m_{C} is the gravitational parameter of the central body. The time to apply this impulse is chosen such that the spacecraft reaches the apoapsis of its transfer orbit at the same time that the target body is passing by that point, to have a near-collision encounter;

ii) In this point, the spacecraft makes a swing-by with the target body to transform its elliptic orbit O_{1} around the central body to a parabolic orbit (P_{1}). In a typical planar swing-by, there are three independent free parameters that can be varied to achieve the purposes of the maneuver: V_{¥} (the velocity of the spacecraft relative to the natural satellite body, when it is entering its sphere of influence); r_{p} (the distance during the moment of the closest approach); the approach angle y (the angle between the velocity of the spacecraft during the moment of the closest approach and the velocity of the planet). See Fig. 3 for more details. In this particular case, the values for V_{¥} and y are not free, since it is decided to approach the target body from a Hohmann transfer (to achieve the minimum DV for the first impulse). What is left to choose is r_{p}, and it has to be chosen in such way that the orbit after the encounter is parabolic. From the condition of the orbit of the spacecraft before the encounter, the information available is:

where, V_{i} is the velocity of the spacecraft relative to the central body before the encounter (the velocity at the apoapsis of the transfer orbit in the Hohmann maneuver) and V_{¥} is the same velocity relative to the target body. Equation (5) is valid because the velocity of the spacecraft and the target body are aligned, at the near-collision point. From the condition for the desired orbit for the spacecraft after the encounter (it has to be parabolic) it is possible to say that:

where V_{par} is the velocity of the spacecraft relative to the central body after the encounter (parabolic escape velocity). Using the rules to add two vectors the value for d (the turn angle of the swing-by, see Fig. 3) is found to be:

where V_{2} is the velocity of the satellite body with respect to the central body. Now, with the value of d, the desired value of r_{p} is found, from the equation:

where m_{T} is the gravitational parameter of the target body. The approach of the spacecraft has to be calculated to obtain a close encounter with the natural satellite with this distance;

iii) Then, the same principle used in the bi-parabolic transfer is used here. Theoretically, it is necessary to wait until the spacecraft reaches the infinity to apply a zero impulse to transfer the spacecraft to a new parabolic orbit, that will meet the natural satellite with a periapsis distance equals to r_{f}. This maneuver has a zero DV (called DV_{i} in Fig. 4), because it is performed at infinity, where the gravitational force from the central body is zero;

iv) The last step is the insertion of the spacecraft in an orbit around the target body. The same principle from the bi-parabolic transfer is used again.

The V_{¥} (the velocity of the spacecraft relative to the target body when entering its sphere of influence) is calculated by Eq. (9); then a conic around the target body with periapsis at r_{f} is constructed and an impulse at the periapsis of this conic is applied, opposite to the motion of the spacecraft, to reduce its velocity to the circular velocity at r_{f}. The magnitude of this impulse is given by Eq. (10).

All those equations can be combined to offer an expression for the savings in DV between the standard Hohmann transfer and the elliptic-bi-parabolic transfer. The expression is:

It is important to note that the first impulse is the same for both maneuvers, so Eq. (11) represents the difference between the velocity of the spacecraft when approaching the satellite body from an elliptic orbit that belongs to the Hohmann transfer (the first term) and the velocity of the spacecraft when approaching the satellite body from a parabolic orbit that belongs to the elliptic-bi-parabolic transfer (the second term).

As an example, it is calculated the impulses involved to transfer a spacecraft from a circular low orbit around the Earth to a circular orbit around the Moon. The data used are: r_{0} = 6545 km; r_{B} = 384400 km; r_{f} = 1850 km; m_{C} = 398600.44 km^{3}/s^{2}; m_{T} = m_{C}/81.3, where m_{C} is the gravitational parameter of the main body and m_{T} is the gravitational parameter of the natural satellite. The results are: DV_{1} = 3.140 km/s; V_{i} = 0.1863 km/s; V_{¥} = 0.832 km/s; V_{par} = 1.440 km/s; d = 39.13º; r_{p} = 4139.0 km; DV_{2} = 0.713 km/s. The total DV involved in this maneuver is DV_{T} = DV_{1} + DV_{2} = 3.853 km/s. To give an idea of the savings obtained, Table 1 shows the standard results available, obtained from Sweetser (1991).

To show better the possible savings in more generic cases, Fig. 5 shows contour-plots for the savings obtained. These results are shown in Prado (1996) and are repeated here to make the present study more complete. The canonical system of units is used in those graphs, what means that m_{C} = r_{0} = V_{0} = 1, where the unit for velocity is chosen to be V_{0} (the velocity of a spacecraft in a circular orbit with radius r_{0}). The values of m_{T} are 0.001, 0.01 and 0.1, respectively. The vertical axis is used for the variable r_{f} (the radius of the final circular orbit of the spacecraft around the natural satellite) and the horizontal axis is used for r_{B} (the radius of the circular orbit of the natural satellite around the planet).

Of course, this maneuver is not practical, since the time required for the complete transfer is infinity. It should be considered as a limiting case of a more practical maneuver that performs the third step in a finite time (as large as possible) with DV ¹ 0 (but still very small). This practical maneuver is not studied here in detail, because the main goal of the present paper is to show limits of the maneuver. It is possible, in some cases, that the target body is not able to give the necessary impulse for the spacecraft to achieve parabolic orbit. In this case the swing-by maneuver can be used to get the maximum impulse possible to send the spacecraft to an elliptic orbit with the semi-major axis as large as possible, and the principles of the bi-elliptic transfer (Shternfeld, 1959 and Hoelker and Silber, 1959) are used to complete the maneuver. Another possible application for this transfer is a transfer between two planets, like an Earth-Mars transfer, using a swing-by in the target planet.

**Second Version: Transfer Between Orbits Around the Primary Body**

Considering the case where the final orbit is around the primary body, the steps are (see Fig. 6):

i) From the initial circular parking orbit an impulse is applied to send the spacecraft to an elliptic Hohmann transfer to the target body. This step is the same one used in the first version, so the cost is given by Eq. (3), using r_{B} = r_{f}. The result is ;

ii) At this point, the spacecraft makes a swing-by with the target body to transform its elliptic orbit O_{1} around the central body to a parabolic orbit (P_{1}), in the same way performed in the first version;

iii) Then, the same principle used in the bi-parabolic transfer is used again. Theoretically, it is necessary to wait until the spacecraft reaches the infinity to apply a zero impulse to transfer the spacecraft to a new parabolic orbit that will take the spacecraft at an altitude r_{f}. This maneuver has a zero DV (called DV_{i} in Fig. 6), because it is performed at infinity, where the gravitational force from the central body is zero;

iv) The last step is to circularize the orbit. The Hohmann transfer cost is

and the elliptic-bi-parabolic transfer cost is:

All those equations can be combined to offer an expression for the savings in DV between the standard Hohmann transfer and the elliptic-bi-parabolic transfer. The expression is given by Eq. (13), where the first term represents the savings obtained in the second impulse and the second term represents the extra expenses of the first impulse.

To show better the possible savings in more generic cases, Fig. 7 shows contour plots for the savings obtained over the Hohmann transfer. The canonical system of units is used in those graphs, what means that m_{C} = r_{0} = V_{0} = 1, where the unit for velocity is chosen to be V_{0} (the velocity of a spacecraft in a circular orbit with radius r_{0}). The vertical axis is used for the variable r_{f} (the radius of the final circular orbit of the spacecraft around the natural satellite) and the horizontal axis is used for r_{B} (the radius of the circular orbit of the natural satellite around the planet). Only situations where r_{B} > r_{f} are shown in this plot. Positive numbers means a gain for the elliptic-bi-parabolic maneuver and a negative number means a gain for the Hohmann transfer.

As an example, it is calculated the impulses involved to transfer a spacecraft between two circular orbits around the Earth. The data used are: r_{0} = 6545 km; r_{B} = 384400 km; r_{f} in the range 13090 km to 654500 km; m_{C} = 398600.44 km^{3}/s^{2}, the gravitational parameter of the main body. The total DV involved in this maneuver is shown in Fig. 8, as a function of r_{f}, compared with the Hohmann and the bi-parabolic transfers. For the biparabolic transfer the impulses are given by and .

One canonical unit of velocity corresponds to 7.8039 km/s. The results show that the elliptic-bi-parabolic transfer is better than the bi-parabolic in all situations, and the Hohmann transfer is the most economical only when r_{f} < 10.37745, that is a critical value for this maneuver. After this value the elliptic-bi-parabolic has the lowest cost, showing a difference about 0.5 canonical units (about 400 m/s) for r_{f} > 40.

**The Use of a Swing-by to Achieve Escape Velocity from the Planet**

A natural sequence of the maneuvers previously described is to use the swing-by with the satellite to achieve a hyperbolic orbit, instead of a parabolic one. In this way the spacecraft leaves the low circular orbit with an impulse a little bit smaller than the one required by the standard maneuver (with no swing-by), and it uses the natural satellite of the main body as an accelerator to compensate for this deficit. As an example, it is calculated the savings involved in a Hohmann transfer from Earth to all the planets in the Solar System and to the interstellar space, using the Moon as an accelerator. The standard procedure of the patched conic transfer (Taff, 1985) is used and the results are shown in Table 2. The "patched-conics" is a maneuver that has three phases: i) In the first one the natural satellite is neglected and the motion of the spacecraft around the central body is considered a Keplerian orbit; ii) In the second phase, it is assumed that the spacecraft is entering the sphere of influence of the natural satellite and the effect of the central body is neglected. The motion of the spacecraft around the natural satellite is hyperbolic in the case of interest for the present research. In this hyperbolic orbit the spacecraft is scattered by the natural satellite and its velocity vector (with respect to the natural satellite) rotates by an angle 2d, but it keeps its magnitude constant. Then the spacecraft crosses again the sphere of influence of the natural satellite and leaves it, to return to a Keplerian orbit around the central body. Two values for the periapsis distance during the swing-by (r_{p}) are used: 1750 km and 1840 km. They can show the importance of this parameter. The minimum possible value (about 1750 km, barely above the surface of the Moon) can provide better savings, but r_{p} = 1840 km can provide almost the same savings, with a comfortable distance greater than 100 km from the surface of the Moon during the swing-by (to avoid the risk of crashing into the Moon).

**Conclusions**

The use of the satellite of a planet to reduce the costs of several types of missions is explained. A maneuver that uses a swing-by with a natural satellite of a planet to transfer a spacecraft to the natural satellite or to a higher orbit around the planet was described. Analytical equations were derived and showed a savings in the order of 100 m/s in a transfer from the Earth to the Moon and in the order of 400 m/s in transfers between two orbits around the Earth over the standart Hohmann transfer. It was shown that there is a critical value for r_{f} (10.37745) for the elliptic-bi-parabolic to be more economical. After that, the idea of making a swing-by with the natural satellite was used in maneuvers to send a spacecraft to the other planets of the Solar System and to the interstellar space. The results show that the Moon is not a great accelerator, but savings in the order of 100 m/s can be achieved. Better results can be found for other hypothetical and real systems, where the natural satellite is a better accelerator.

**References**

Battin, R. A., 1987, *Introduction to the Mathematics and Methods of Astrodynamics,* New York, NY, American Institute of Aeronautics and Astronautics. [ Links ]

Broucke, R.A. and Prado, A.F.B.A., 1993, "Jupiter Swing-By Trajectories Passing Near the Earth," Advances in the Astronautical Sciences, Vol. 82, Part II, pp. 1159-1176. [ Links ]

Broucke, R.A. and Prado, A.F.B.A., 1995, "Orbital Planar Maneuvers Using Two and Three-Four (Through Infinity) Impulses". *Journal of Guidance, Control and Dynamics*, Vol. 19, No. 2, pp. 274-282. [ Links ]

Broucke, R.A., 1988, "The Celestial Mechanics of Gravity Assist," AIAA paper 88-4220. In: AIAA/AAS Astrodynamics Conference, Minneapolis, MN, 15-17 Aug. 1988. [ Links ]

D'Amario, L.A.; Byrnes, D.V.; Stanford, R.H., 1981, "A New Method for Optimizing Multiple-Flyby Trajectories," *Journal of Guidance, Control, and Dynamics*, Vol. 4, No 6, pp. 591-596. [ Links ]

Edelbaum, T.N., 1959, "Some Extensions of the Hohmann Transfer Maneuver". *ARS Journal*, Vol. 29, pp. 864-865. [ Links ]

Edelbaum, T.N., 1967, "How Many Impulses?".* Astronautics & Aeronautics*, Nov., pp. 64-69. [ Links ]

Gobetz, F.W, and Doll, J.R., 1969, "A Survey of Impulsive Transfers".* AIAA Journal*, Vol. 7, N° 5, pp. 801-834. [ Links ]

Goddard, R.H., 1919, "A Method of Reaching Extreme Altitudes". *Smithsonian Inst Publ Misc* *Collect*, Vol. 71, N° 2. [ Links ]

Hoelker, R.F., and Silber, R., 1959, "*The Bi-Elliptic Transfer Between Circular Co-Planar Orbits*," Alabama, Army Ballistic Missile Agency, Redstone Arsenal, (DA Tech Memo 2-59). [ Links ]

Hohmann, W., 1925, "*Die Erreichbarheit der Himmelskorper*".Oldenbourg, Munich. [ Links ]

Marchal, C., 1965, "Transferts Optimaux Entre Orbites Elliptiques Coplanaires (Durée Indifférente)". *Astronautica Acta*, Vol. 11, N° 6, pp. 432-445. [ Links ]

Marec, J.P., 1979, "*Optimal Space Trajectories*".New York, NY, Elsevier. [ Links ]

Prado, A.F.B.A., 1993, "*Optimal Transfer and Swing-By Orbits in the Two- and Three-Body Problems*". Ph.D. Dissertation, University of Texas, Austin, Texas, USA. [ Links ]

Prado, A.F.B.A., 1995, "The Dynamics of the Gravity-Assisted Maneuver". *Proceedings of the VI Symposium on Dynamic Problems of Mechanics (DINAME 95)*, pp. 256-259. Caxambú, Minas Gerais, March, 6-10. [ Links ]

Prado, A.F.B.A., 1996, "A Survey on Trajectories to the Moon". *SBA Controle & Automação*, Vol. 7, No. 2, pp. 91-95. [ Links ]

Prado, A.F.B.A.; Broucke, R.A., 1994, "A Study of the Effects of the Atmospheric Drag in Swing-By Trajectories," Journal of the Brazilian Society of Mechanical Sciences, Vol. XVI, pp. 537-544. [ Links ]

Shternfeld, A., 1959, "*Soviet Space Science*". Basic Books, Inc., New York, pp. 109-111. [ Links ]

Smith, G.C., 1959, "The Calculation of Minimal Orbits". *Astronautical Acta, *Vol. 5, N° 5, pp. 253-265. [ Links ]

Sweetser, T.H., 1991, "An Estimate of the Global Minimum DV Needed for Earth-Moon Transfer". AAS paper 91-101. *In: AAS/AIAA Spaceflight Mechanics Meeting*, Houston-TX, Feb. 11-13. [ Links ]

Taff, L.G., 1985, *Celestial Mechanics, A Computational Guide for the Practitioner,* New York, NY, John Wiley & Sons. [ Links ]

Weinstein, S.S., 1992, "Pluto Flyby Mission Design Concepts for Very Small and Moderate Spacecraft," AIAA paper 92-4372. In: AIAA/AAS Astrodynamics Conference, Hilton Head, SC, Aug. 10-12. [ Links ]

Paper accepted March, 2003

Technical Editor: Clóvis Raimundo Maliska

Presented at ICONNE-2000 - International Conference on Nonlinear Dynamics, Chaos, Control and Their Applications in Engineering Sciences, 31 July to- 04 Agost, 2000. Campos do Jordão, SP. Brazil