The Multiple Encounters Problem is described in the literature as the problem of finding trajectories for a spacecraft that leaves from a mother planet, describes a trajectory in the interplanetary space, and then goes back to the mother planet. The present paper extends the literature and the departure and arrival angles of the spacecraft are generalized to be nonsymmetrical. The solutions are shown in terms of the true (
The literature is extensive with respect to problems involving transfer orbits and optimal spacecraft maneuvers. Goddard [
Regarding the Multiple Encounters Problem, it was introduced in the literature by Hénon [
The problem studied here is the transfer of a spacecraft from one body back to the same body, with a possible swing-by in the return passage of the spacecraft. There are numerous applications of this problem, such as (i) to transfer a spacecraft from the Earth to an interplanetary trip and then back to the Earth, without the need of maneuvers during this process, implying the optimization of the fuel consumption; (ii) to transfer a spacecraft from the Moon, which may include a passage by the Earth, and then put it back on the Moon; (iii) rendezvous maneuvers, when one desires that the space vehicle stands alongside another spacecraft; (iv) to maneuver the spacecraft so that it leaves and return to the Earth to perform a swing-by to change its energy, and then go to the outer Solar System; (v) to move a spacecraft that is around a planet, like Jupiter, to another location in the Solar System or even out to the interstellar space.
For these last two applications, it is necessary to combine the Multiple Encounters Problem with the gravity assisted maneuvers. This type of maneuver is used very often in interplanetary trajectories. It uses a close approach with a celestial body to change the orbit of a spacecraft. References [
To describe the Multiple Encounters Problem, let us call
The spacecraft
The geometry of Hénon’s problem formulated as Lambert problem [
Lambert’s problem can be formulated as follows. “Find an unperturbed orbit under the mathematical model given by a law that works with the inverse square of the distance (Newtonian formulation), that connects two given points
(i) The position coordinates of
(ii) The position coordinates of
There are different possibilities for the orbit of
To solve the problem it is necessary to impose that (
This set of equations will be used to generate the solutions of the problem. Hénon’s problem [
(i) The position of
(ii) The position of
(iii) The total transfer time is given by
(iv) The total angle
First of all, it is necessary to consider two possible choices for the transfer; the first one uses the sense of the shortest possible angle between
After considering these two choices, it is also necessary to consider the possibilities of multirevolution transfers. In this case, the spacecraft leaves the point
Lambert’s problem solution is the Keplerian orbit that contains the points
Possible applications for this technique are interplanetary research in the Solar System, a basis for a transportation system between the Earth (
To get the velocity variation (
(1) Find the radial and the transverse components of the velocity vector of
(2) Solve Lambert’s Problem for a transfer between the points
(3) After obtaining the components of the velocity vector immediately before and after the impulses, it is possible to calculate the magnitude of the two impulses (
To make numerical simulations, the procedure is to vary the initial and final positions of
The procedure for the simulation is the following. Values of the eccentricities of the primaries, An initial value for the point where the spacecraft leaves the body The angle that determines the point of the return of the spacecraft to Using the time of the transfer and the initial and final points defined earlier, it is possible to solve the associated Lambert’s problem. This solution gives the transfer orbit and the total increment of velocity to perform the transfer. Considering the possibility of several revolutions for the transfer, a family of solutions appears for each pair of points. They are plotted in figures that show the complete set of solutions.
Let us assume the following parameters:
Solution in terms of the true anomaly (
The same solutions can be seen in Figure
Solution in terms of the eccentric anomaly (
Figure
Velocity variation (
Figure
Semimajor axis (
Several transfer times (
Similar simulations were performed for different values of the eccentricity. The results are similar, and so they are omitted here.
The results of those simulations with various eccentricities showed that a system of primaries with smaller eccentricities are better for the execution of Multiple Encounters Orbits, with respect to the fuel consumption, because they are smaller when compared to system of primaries with high eccentricities (
Table
Solutions with
Eccentricity |
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True anomaly | Eccentric |
Velocity |
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The dynamics of the two bodies is used in the present formulation. The system is considered to be formed by three bodies. It is possible to say that the body a body
This encounter changes the orbit of
Using the “patched conics” approximation, the equations that quantify those changes are available in the literature [
Swing-by maneuver.
Having those variables, it is possible to obtain
A complete description of this maneuver and the derivation of the equations can be found in Broucke [
To solve this problem, the following assumptions are made. The system is formed by two bodies in elliptic orbits around their mutual center of mass and a third massless body moving under the action of the gravitational forces. The origin of the system is placed in the center of mass. The horizontal axis is the line connecting the bodies The spacecraft leaves the point After the close approach, the spacecraft modifies the velocity, energy, and angular momentum ( We used the canonical system of units. This formulation implies that the unit of distance is the distance between
In this system of units, the gravitational parameter of Jupiter is
Numerical simulations were performed using Jupiter for the body from where the spacecraft escapes and performs the swing-by in the return trip. This situation can happen in a practical situation where it is desired to use the gravity of Jupiter to help to move a spacecraft that is around Jupiter to another location in the space, like another planet, asteroid, and so forth. The velocity (
Swing-by with
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Angle of approach | Variation of velocity | Variation of energy | ||
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rad | Degree | rad | Degree | ||
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The numerical results show that the largest gain for the velocity variation (
Variation of velocity (
Variation of velocity (
Figures
Total
Positive balance in
In the simulations performed in this study, some results available in the literature [ If the swing-by occurs behind the body If the swing-by occurs in front of the body
Next, Figures
An important point to remember is to look at the type of the orbit of the spacecraft before and after the swing-by (circular, elliptic, hyperbolic, and parabolic). From the two body-problem, it is known that when the energy of the spacecraft is negative, the orbit is closed (circular or elliptical) and the spacecraft always remains at a finite distance from the central body, which means that the body remains orbiting when the energy of the spacecraft is positive, the orbit is open (hyperbolic) and the distance to the central body tends to infinity with the time, and so the body does not remain orbiting
It is also important to remember that is the sum of the initial spacecraft energy plus the variation obtained from the swing-by (
Next, another important question is studied related to this problem, which are the effects of the eccentricity of the primaries in the result of the swing-by maneuver in the return passage by
Variation of energy (
Variation of energy (
Figure
Swing-by with
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Angle of approach | Variation of velocity | Variation of energy | ||
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rad | Degree | rad | Degree | ||
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Optimum space maneuvers were studied, which the goal of finding trajectories that reduces the fuel consumption for interplanetary missions. In particular, in situations where the spacecraft leaves and comes back to the same body (Multiple Encounters Problem), considering nonsymmetrical situations for the start and finish of the maneuver. Several simulations were performed and shown, and the solutions found low-cost maneuvers that can be used for the planning of real missions.
Then, the study included swing-by maneuvers in the return passage by the mother body as a form of gaining energy. The numerical simulations showed that the velocity (
It was also shown that the gravity assisted maneuvers made with the Jupiter Planet (JGA) can provide a considerable variation of velocity and energy for a spacecraft, thus reducing the costs of the mission. In the figures and tables shown in this work, it is verified that this maneuver is a powerful tool that can also be used in interplanetary missions that require that the spacecraft leaves a body and returns later to this same body. This is interesting for a maneuver that needs to move a spacecraft that is around Jupiter to send it to another location of the Solar System or beyond.
This work was accomplished with the support of São Paulo State Science Foundation (FAPESP) under Contracts 2009/16517-7, 2011/09310-7, and 2011/08171-3, CAPES, CNPq (contract 304700/2009-6), and INPE—National Institute for Space Research—Brazil.