Semi analytical study of lunar transferences using impulsive maneuvers and gravitational capture

In this work we have studied transferences between the Earth and the Moon considering bi- and tri-impulsive conventional maneuvers (like Hohmann) in order to make the acquisition of trajectories which are captured by the Moon's gravitational field through the Lagrangian equilibrium point L1. Results show that these transfer models offer reduction in the ΔVTotal of the mission. However, they do not require long transfer times


Introduction
In the last twenty years, the activities related with the lunar exploration were intensified by the sending of several probes [1]. In this period we can highlight the detection of water in its solid state in the lunar poles [2] and the announcement of new lunar missions with several objectives, including studies leading to the establishment of permanent bases [1,3]. Therefore, the need for many unmanned and manned transfer missions between the Earth and the Moon can be foreseen. In this scenery, the control and the reduction of fuel consumption required for transfer maneuvers are needed in order to increase the spacecraft payload. Techniques to transfer spacecrafts between the Earth and the Moon using gravitational capture have been studied and applied since the 1980`s [4,5]. This work presents semi analytical studies for transferences which consider the gravitational capture phenomenon and bi-and tri-impulsive conventional maneuvers that do not require long transfer times.

Mathematical Models
The transfer maneuvers described here are divided into three parts. In the first part, a transfer from a circular parking orbit around the Earth with altitude h o up to a point in a capture trajectory is performed. This transference is done in two different ways: one based in a bi-impulsive Hohmann transfer and another one based in a tri-impulsive transfer [6]. The second part corresponds to the path up to the Moon by gravitational capture through the Lagrangian equilibrium point L1, and in the third part a maneuver to stabilize a spacecraft in an orbit around the Moon is performed. This latter maneuver may not be necessary for some missions. In the first and third parts the necessary ∆Vs can be analytically calculated considering the dynamics of the two-body problem. In the second part, the dynamics of the Restricted Three-body Problem is considered, and the ∆Vs are calculated taking into account numerical simulation results.
The Restricted Three-body Problem (R3BP) is well known in the literature [7] and the equations of motion of a particle (a spacecraft) in the xy plane of a rotating and barycentric coordinate system, also called synodic coordinate system, are given in components by µ 1 and µ 2 are the reduced masses of the Earth and the Moon, respectively, and their sum is 1, and the mean motion, n, of both around the common center of mass is also equal to 1. r 13 = [(x + μ 2 ) 2 + y 2 ] 1/2 and r 23 = [(xμ 1 ) 2 + y 2 ] 1/2 are the distances between the Earth and the particle, and between the Moon and the particle, respectively. In the R3BP, the angular momentum and the energy are not conserved. However, it admits an integral of motion called Jacobi constant, C j , given by The value of C j defines the surfaces of zero velocity. These structures delimit the boundaries of regions in the space where the particle cannot move. The equations 1.(a) and 1.(b) have five particular solutions which correspond to five points in the synodic coordinate system in which the velocity and acceleration of a particle are null. These points are called Lagrangian equilibrium points (L1, L2, L3, L4 and L5). When 3.17216 ≤ Cj < 3.18834 a passage is opened around L1, allowing a particle to move between the Earth and the Moon. This is the case studied in this work ( Fig. 1.(a)). And if Cj < 3.17216 another passage is opened around L2 ( Fig. 1.(b)), and the particle can also move beyond the Earth-Moon system. Being C j (L1) = 3.18834 and C j (L2) = 3.17216 [7,8].
The phenomena of escape and capture are temporary in the R3BP dynamics [4,7]. Therefore, we have considered this characteristic and also a family of periodic orbits around the Moon called H2 family [8]. The initial conditions of this family are given by [6,8] is the semimajor axis and e 0 is the eccentricity of the particle's osculating initial orbit round the Moon. The energy of two-bodies does not remain constant in the R3BP. However, its monitoring gives us an idea about the effects of the Earth and the Moon's gravitational fields on the motion of particles [9]. Thus, for each trajectory simulated and for each integration step, the Moon-particle two-body energy has been measured. If this energy becomes positive during the integration, the trajectory is considered an escape trajectory and it is classified taking into account the time at which the Moon-particle energy remains negative, or simply by its capture time around the Moon, and also if this escape occurs through L1 or L2. In order to analyze the capture through the L1, we have adopted the integration for negative times. It is possible to accomplish this study for positive times, but each initial condition has a time in which it remains in orbit around the Earth, so each trajectory is captured by the Moon at different times. Because of this, the work would become extenuating.
From the initial conditions defined by equations (4) and (5) it is possible to generate trajectories and to investigate them before the capture by the Moon. Figure 2 shows one of these escape trajectories with C j = 3.1111. While it remains captured, it reaches 118 km above the Moon's surface. In this point, the osculating lunar orbital elements are: a 0 = 23,200 km, e 0 = 0.920, ω 0 (argument of periapsis) = 270 o , Ω 0 (longitude of the ascending node) = 270º e i 0 (inclination) = 0. During its motion around the Earth, the same trajectory reaches a minimum distance of 144,795.55 km from the Earth and a maximum distance of 294,032.93 km.

Transfer Methods and Results
Two points can be also seen in Figure 2. They are B and B'. Point B corresponds to the point with lower distance from the Earth, and B' to the maximum distance, both belonging to the trajectory whose initial conditions of the lunar osculating orbit are a 0 = 23.200 km, e 0 = 0,920, ω 0 = 270 o , Ω 0 = 270º and i 0 = 0. From an Earth circular parking orbit with a radius R C = h 0 + 6370 km, two transfers are designed up to the point B and other two up to the point B'. For each point (B and B') a bi-impulsive transfer (direct transfer) and a tri-impulsive transfer (bi elliptical transfer) are designed. Therefore, four cases are considered. In the first case, a transfer ellipse tangent to the Earth orbit at points P and to the capture trajectory at point B (Fig. 3(a)) is obtained, and the application of two ∆Vs is necessary. In the second case, two ellipses with the same apses line are obtained. The first one is tangent to the Earth orbit at P' and it has an apogee at point A. The second ellipse also has its apogee at point A and its perigee at point B in the capture trajectory ( Fig. 3(b)). In the third case (bi-impulsive transference), a transfer ellipse tangent to the Earth`s parking orbit at a point P and also tangent to the capture trajectory at the point B' is obtained. By analogy, in the fourth case (tri-impulsive transfer), two ellipses are obtained -the first ellipse is tangent to the Earth`s parking orbit at a point P' and its apogee is at point A'. The second ellipse also has its apogee at point A' and it is tangent to the capture trajectory at point B'. In all cases, it is necessary to know the capture trajectories velocities at points B and B'. And theses quantities are known by the numerical simulations of the capture trajectories. In the first and third cases (bi-impulsive transfers), ∆V 1 is analytically determined and the ∆V 2 is defined by the difference between the velocity of the capture trajectory at point B or B', and the apogee velocity of the transfer ellipse (Table 1). In second and fourth cases, that is, in the tri-impulsive transfers (bi-elliptic), ∆V 1 is also analytically determined. The determination of ellipses apogee radius, R A12 (Fig. 3.(b)), it is linked to the total time of transfer between the Earth and the Moon. In general, the farther from the Earth the apogee of an ellipse is, the lower its velocity. This means, in terms of orbital maneuvers, a smaller ∆V 2 to jump from the first transfer ellipse to the second one, since ∆V 2 is the difference between the apogee velocities of the two transfer ellipses (analytically calculated). Though, starting from certain values of R A12 the observed variation for ∆V 2 is small, however, the time of transfer becomes large. Still in the second and fourth cases, there is a ∆V 3 and it corresponds to the difference between the velocities of the capture trajectory in B or B' and the perigee of the second transfer ellipse. Figure 4.(a) shows the variation of ∆V Total (∆V Total = ∆V 1 + ∆V 2 + ∆V 3 ) for one bielliptical transfer as a function of the apogee distance from the Earth, R A12 , and in Figure 4.(b), ∆V Total is given as function of the time of transfer. The values are obtained for the trajectory shown in Figure  2 considering that the acquisition of the capture trajectory occurs at point B'. For times of transfers larger than 120 days, which corresponds to R A12 > 1.5x10 6 km, ∆V Total is at 3.785 km/s, a value having little variation. Thus, starting from a certain point, the time of transfer becomes more important in order to determine the transfer parameters than ∆V Total .  Tables 1 and 2 show the results for the capture trajectory in Figure 2. But the technique for calculating ∆V Total can be considered for any trajectories whose initial conditions satisfy eq. (4) and (5). Table 1 brings the values for the bi-impulsive transfer for cases 1 and 3 that correspond to a direct transfer between points P and B, and P and B'. Table 2 brings some values for the tri-impulsive transfer (bi-elliptical) for ∆V Total and makes the relationship between them and the time of transfer, and distance from the apogee to the Earth (R A12 ) for case 4. It is possible to observe that for ∆V Total = 3.800 km/s, the time of transfer is 82 days, and for ∆V Total = 3.785 km/s, 123 days are needed to complete the maneuver. In other words, there is an increase of 41 days for a reduction of only 15 m/s in the ∆V Total . For a ∆V Total = 3.900 km/s, the time of transfer is of 33.8 days. This value is practically similar to the one in the third case (direct transfer among the points P' and B). An analysis of the second case is similar to the fourth case, however, ∆V Total below 4.000 km/s is obtained only for time of transfers longer than 100 days. This makes any application of this maneuver unfeasible, since there are faster and more economical alternatives.
Finally, it remains to analyze the one which will happen to the spacecraft after it is captured by the Moon. Naturally, it will remain in an orbit around the Moon for some time until a new escape happens. This time may be enough for the development of a lot of missions. The trajectory shown in Figure 2, for instance, this time is 37.4 days. But, if this time is not enough to accomplish a mission. The spacecraft can be stabilized in a keplerian orbit around the Moon. To stabilize the spacecraft in an orbit around the Moon, it would be necessary to apply an extra ∆V in a point that corresponds to the larger distance from the Moon (in the Moon's sphere of influence), or in the point of smaller distance, both extra ∆V are analytically calculated. If this maneuver is necessary, the extra ∆V should be added to the ∆V Total of tables 1 and 2.

Conclusions
The transfers presented in this work are planed starting from the two-and three-body problem dynamics, and they correspond to alternative ways to study Earth-Moon transfers, and they are perfectly feasible. The results, analyzed in terms of ∆V Total and the time of transfer (here presented for the purpose of an example) are attractive; especially if they are compared to the Patched-conic approximation [10] used in the Apollo missions, for instance, whose ∆V Total ranged between 4.100 and 4.500 km/s. The choice, or study, of these alternative transfers necessarily involves the analysis of the time of transfers. In the cases of directed bi-impulsive transfers, for example, between points P and B', they present ∆V Total = 3.900 km/s and the time of transfer is not very long. In this case, 33.26 days. This is a long time for manned missions, but taking into account unmanned missions of many natures, as supply transport, given the ∆V Total reduction; these transferences represent consistent alternative.