ALTERNATIVE PATHS TO EARTH-MOON TRANSFER

The planar, circular, restricted three-body problem predicts the existence of periodic orbits around the Lagrangian equilibrium point L1. Considering the Earth-lunar-probe system, some of these orbits pass very close to the surfaces of the Earth and the Moon. These characteristics make it possible for these orbits, in spite of their instability, to be used in transfer maneuvers between Earth and lunar parking orbits. The main goal of this paper is to explore this scenario, adopting a more complex and realistic dynamical system, the four-body problem Sun-Earth-Moon-probe. We defined and investigated a set of paths, derived from the orbits around L1, which are capable of achieving transfer between lowaltitude Earth (LEO) and lunar orbits, including high-inclination lunar orbits, at a low cost and with flight time between 13 and 15 days.


Introduction
Ever since the beginning of space exploration, the Moon has been the target of countless missions, and everything leads to believe that this will continue to be so.This certainty is focused on the recent discoveries of signs that point to the possible existence of ice at the lunar poles made by the American probes Clementine in 1994 [21] and Lunar Prospector in 1998 [10].The estimates indicate that there may be some 10 billion tons of ice at the lunar poles.Therefore, if the existence of this ice were confirmed, future lunar bases would have a water source capable of sustaining life.This ice could also serve as a source for rocket fuel, by separating the water into hydrogen and oxygen, thus making the Moon into a trampoline for future manned interplanetary missions.
In this context, a special type of path, capable of direct transfer between a low-altitude Earth orbit (LEO) and a low-altitude Moon orbit, with flight time between 13 and 15 days, could be of great interest for future lunar missions.This paper presents the results of our research on this type of path, and shows that these paths could be a good alternative for an Earth-Moon transfer, and also other kinds of transfer in Earth-Moon system.The origin of these paths is related to a family of generally unstable direct orbits around the Lagrangian equilibrium point L1, known as the Family G.These paths are predicted by the circular, planar, restricted three-body problem associated with the Earth-Moon system (Broucke [8]).This is why we initially considered the three-body Earth-Moonprobe problem to establish a set of initial conditions, close to the Earth, and final conditions, close to the Moon, in such a way that these conditions were related through an empirical mathematical expression.Next, we considered a more complex and realistic dynamical problem, the four-body problem Sun-Earth-Moon-probe, where we take into account the eccentricity of the Earth's orbit as well as the eccentricity and inclination of the Moon's orbit.An empirical expression related to the initial and final conditions of the paths continued to exist.On the other hand, the consideration of a more complex dynamical system opened up new lines of investigation with respect to the probe's inclination and speed on its final approach to the Moon.
This paper is structured as follows.In Section 2, to facilitate comparison, we have quickly described two of the main conventional methods of Earth-Moon transfer and the method of gravitational capture.In Section 3, we describe the two dynamical systems utilized in this study.In Section 4, we discuss the main properties of these orbits for the restricted three-body problem and the four-body problem.In Section 5, we present, through a curve and an empirical mathematical expression, the set of paths capable of making the maneuver.In Section 6, we present some examples of missions that could be made using these paths, and finally, in Section 7, we present our conclusions and the perspective for future researches.

Conventional methods of Earth-Moon transfers
The calculation of a transfer path between the Earth and the Moon can only be made through numerical integration of the equations of motion [1].These equations must take into account the Earth's, Moon's, and Sun's gravitational fields, and the mutual interactions between these bodies.In addition, given the complexity of the Moon's movement, a lunar mission must be planned and executed on an hour-by-hour and day-by-day basis.However, we can consider some approximations, based on the two-body and restricted three-body problem, to reach an initial estimate of the impulse needed to transfer a probe from the vicinity of the Earth to the vicinity of the Moon.
Next, we will describe two conventional methods used for preliminary analysis of the ΔV 's necessary for an Earth-Moon transfer and the gravitational capture method.Our goal is not to describe them rigorously, but rather to create a basis for comparison with the transfers that could be done by a set of paths derived from the unstable orbits around the Lagrangian equilibrium point L1; a set that we will present in Section 5.

Minimal energy path.
It is possible to establish a sort of minimal energy path for a transfer between the Earth and the Moon based on the two-body Earth-probe and Moonprobe problems.This maneuver is divided into two parts.In the first part, the Moon's gravitational field is neglected, and in the second part, the Earth's gravitational field is

Moon
Moon's parking orbit

Moon's orbit
Transfer ellipse (minimal energy path) Moon at the instant of injection of probe in transfer ellipse (a)

Moon
Lunar parking orbit (c) neglected.This way, we can establish a simple procedure to estimate the impulses required for the maneuver based on Hohmann's transfer [1]. Figure 2.1 shows the minimal energy path, an ellipse, it is tangent to Earth's and Moon's parking orbits.The first part starts with the application of a ΔV 1 when the vehicle attains the perigee's transfer ellipse.From Figure 2.1, we obtain the semimajor axis of transfer ellipse, a min , by where R 1 is the perigee's radius of the transfer ellipse and also of the Earth's parking orbit radius in the point of tangency with the transfer ellipse, D = 384400 km is the Earth-Moon distance which we are considering constant, that is, we are considering the Moon to be in a circular orbit around the Earth, and R 2 is the Moon's parking orbit radius in the point of tangency between it and the transfer ellipse.The energy per unit of mass of this ellipse is where G = 6.67×10 −11 m 3 /s 2 kg and M Earth = 5.9742 × 10 24 kg are the universal gravitational constant and the mass of the Earth, respectively.Thus, we can obtain the perigee's speed of the transfer ellipse, V E1 , by and the apogee's speed of transfer ellipse, V E2 , by If we consider that the Earth's and Moon's parking orbits are circular with radius R 1 and R 2 , respectively, the first impulse required to insert the vehicle in the transfer ellipse, ΔV 1 , is given by As we can see, the first part of the maneuver is equal to Hohmann's transfer.However, the second part of the maneuver corresponds to the application of a second impulse to insert the vehicle in lunar orbit and not to put it in a circular orbit with radius D − R 2 .Therefore, the second part is different from Hohmann's transfer procedure, because the ΔV 2 required for this maneuver should be applied in the opposite direction of the vehicle's motion.In order to calculate the magnitude of ΔV 2 , we should consider the apogee's vehicle speed, V E2 , relative to the Moon.This speed we denote by V L2 , and its magnitude is given by where V Moon = 1.023 km/s is the Moon's orbital speed around the Earth (assuming circular orbit).The magnitude of ΔV 2 is immediately calculated by where M Moon = 7.3483 × 10 22 kg is the Moon's mass, and (2.8) Earth's parking orbit The flight time for this maneuver is simply half of the orbital period of the transfer ellipse, that is, (2.9) The procedure described in this section is the simplest method to estimate the magnitude ΔV Total for a direct maneuver transfer between the Earth and the Moon.But, it makes use of two-body dynamics, for this reason, the ΔV Total given by (2.8) is not sufficient to take the vehicle to the final lunar orbit desired, requiring additional ΔV 's to complete the mission [18].

2.2.
Patched-conic approach.Similar to the transfer via minimal energy ellipse, the patched-conic approach is also divided into two parts.The basic idea is to consider the probe as being under the influence of Earth's gravitational field only since the moment of launch in transfer path, called geocentric conic, until the instant it reaches the Moon's sphere of influence (first part).At this time, the probe is considered to be under the influence of the Moon's gravitational field only (second part).The transition from geocentric motion to selenocentric motion occurs naturally along a finite arch of the probe's path when it is in the region where the geocentric conic touches or intercepts the Moon's sphere of influence; whose radius is R S = 66300 km. Figure 2.2 illustrates the basic geometry of patched-conic transfer.In this figure, we can see the first part of the maneuver that is, leaving the Earth's parking orbit and the geocentric-conic tangency of the Moon's sphere of influence.Theoretically, the geocentric conic could be an ellipse, a parabola, or a hyperbola, but generally, ellipses require the smallest ΔV 's for the maneuver.Figure 2.2 also shows the second part of the maneuver in which the probe gets inside the lunar sphere of influence.
The transfer starts when a ΔV 1 is applied injecting the probe into geocentric conic, this occurs at point P (Figure 2.2).At the point at which the probe reaches the Moon's sphere of influence, point I (Figure 2.2), it begins to move under the influence of the Moon.When the probe reaches the periselenium of the selenocentric path, point M (Figure 2.2), a ΔV 2 must be applied to conclude the maneuver.
In order to calculate the elements of the geocentric and selenocentric paths and the ΔV 's required to transfer, we need to know at least four independent initial quantities, or three independent initial quantities and one intermediary (on the lunar sphere of influence), or final quantities.The search for these quantities is an iterative procedure.Particularly, a convenient set for these quantities is R 1 , V E1 , φ 1 , and λ I (see Figure 2.2), where R 1 is the radial distance at the point P, V E1 is the injection speed into geocentric conic, φ 1 is the flight-path angle, and λ I specifies the point at which the geocentric trajectory crosses the lunar sphere of influence.
Thus, if the values attributed to R 1 , V E1 , φ 1 , and λ I lead the probe to periselenium equal to the radius of the final lunar orbit planed, the procedure is complete, otherwise, a new search for other values for R 1 , V E1 , φ 1 , and λ I must be started until the selenocentric path reaches periselenium equal to the radius of the final lunar orbit desired.
The procedure described above is a good approximation for preliminary mission analysis, but, in practice, errors occur in the encounter with the Moon's sphere of influence due to the disturbance of the Sun's gravitational field, and also due to the Moon's gravitational field in the first part of the maneuver and the Earth's in the second part of the maneuver.The calculations needed to reach the values of the ΔV 's are extensive and for this reason, we will not demonstrate them here.For more information, please consult [1].

Methods of Earth-Moon transfer by gravitational capture.
The phenomenon of gravitational capture can be understood as the mechanism by which an object, subject only to gravitational forces, approaches a celestial body at a low speed, relative to this body, in such a way that the object can be captured, and then temporarily orbits around that body.For the capture to be permanent, a dissipative force must act upon the object, such as atmospheric drag, for example.This phenomenon has been considered by various researchers to explain the origin of planetary satellites, see Murison [15] and Brunini [9].
Gravitational capture is possible for the general three-body problem.Usually, the initial distance between the three bodies is infinite, and after the approach, the distance between two of them remains limited.For the restricted three-body problem, the dynamical system considered in the great majority of the studies found in the literature, the third body (particle) approaches one of the primary bodies from an infinite or finite distance.After the approach, the distance between the particle and one of the primary bodies varies between well-determined values, and the primary-particle orbital energy remains negative as long as the capture lasts, as this is a temporary phenomenon.
Transfer methods based on the phenomenon of gravitational capture cannot be called conventional, especially due to the wealth of information that they offer from the semianalytic point of view.But the mechanism can be conveniently exploited to place spacecraft in orbit around celestial bodies as a technique to reduce fuel consumption.Some of the first studies of this theme were conducted by Belbruno [2][3][4][5], Belbruno and Miller [6,7], Miller and Belbruno [14], Krish et al. [12].Other more recent interesting studies in this area are those of Yamakawa [23], Koon et al. [11], Ocampo [17], Prado [19,20], and Winter et al. [22].They all study the mechanism by which spacecraft can be inserted into orbit around the Moon.There are also studies related to gravitational capture applied to low-consumption transfers to the moons of Jupiter, Saturn, and Uranus, for example, Lo and Ross [13].
The mechanism of gravitational capture was applied in 1991 in the Japanese Hiten mission [6].

Restricted three-body problem (R3BP).
This problem, well known in the literature-see, for example, Murray and Dermott, [16]-considers three bodies, m 1 , m 2 , and m 3 , with m 3 being a particle with negligible mass that does not influence the other two bodies, which have preponderant mass and are called primary.These, in turn, have circular, coplanar orbits around the center of mass that is common to both; they maintain a constant distance from each other, and also have the same angular velocity relative to this center of mass.Due to these characteristics, it is useful to study the R3BP adopting a system of references whose origin is fixed in the center of the mass common to the primary bodies, with axes x and y rotating with the same angular velocity as the first bodies.This system is called a barycentric rotating or synodic system, and in it, the bodies m 1 and m 2 remain fixed over the x-axis, while m 3 moves on the xy-plane.The system is normalized, considering its reduced mass, μ, as unitary mass, that is, where G is the universal gravitational constant.The constant distance between the masses m 1 and m 2 is also considered equal to 1. Thus, the coordinates of m 1 and m 2 are (−μ 2 ,0) and (μ 1 ,0), respectively.The equations of motion for the third body in the synodic system are with where, considering the Earth-Moon-probe system, μ 1 = μ Earth = 0.9878494 and μ 2 = μ Moon = 0.0121506 are the mass parameters for the Earth and the Moon, respectively, r 13 is the Earth-particle distance and r 23 is the Moon-particle distance.Equations (3.do not have an analytical solution, but they do have symmetry properties that guarantee the existence of periodic orbits in the synodic system (Broucke [8]; and Murray and Dermott [16]), such as the Family G orbits that we investigated in this study.
The R3BP also has another interesting property which is the existence of five equilibrium points called Lagrangian equilibrium points, symbolized by the letter L. When a particle is placed on these points with a null velocity relative to the origin of the synodic system, it remains there indefinitely.Figure 3.1 illustrates this synodic system and the relative location of the five Lagrangian equilibrium points for Earth-Moon system.

Four-body problem.
In our numerical simulations, we also considered as dynamical system the four-body problem in three-dimensional space.Thus, for a system of fixed Cartesian coordinates, in which the position of a certain body is given by the vector x ki = (x k1 ,x k2 ,x k3 ) ∈ R 3 , the equations of motion are given by ẍki where k = 1,2,3,4; is the distance between the kth and the jth bodies, μ j is the mass parameter for the jth body, and with i = 1,2,3 denoting the three coordinates of the fixed Cartesian system.Equation (3.3) represents 12 second-order differential equations and expresses the fact that acceleration of a given body is the result of the sum of the forces exercised by the other three bodies.This means that (3.3) takes into account the mutual interactions between the four bodies in the system.Now, if we associate indexes 1 to the Sun, 2 to the Earth, 3 to the Moon, and 4 to the probe, and still consider that the system is normalized based on the Earth's and Moon's mass, so that μ 2 + μ 3 = 1, then we will have μ 1 = μ Sun = 328904.4747,μ 2 = μ Earth = 0.987849396, and μ 3 = μ Moon = 0.012150604.Considering that a lunar probe's mass can vary between some hundreds of kilograms and some tons, its mass parameter, μ 4 , will be at order of 10 −23 or 10 −24 , and even with (3.3) taking into account the mutual interactions between the four bodies considered, the probe would not influence the motion of the other three.For this reason, the terms of (3.3), which contain μ 4 , were suppressed.This is the four-body Sun-Earth-Moon-probe problem, and the word restricted could precede it, without a loss of generality, given the order of the size of μ 4 .The aforementioned normalization is completed adopting the average distance between the Earth and the Moon, 384400 km, as a unit of measurement.
The eccentricity of the Earth's orbit, e 2 , the eccentricity, e 3 , and the inclination, i 3 , of the Moon's orbit were included in the system via initial conditions, thus bringing the system closer to reality.The values for these elements are e 2 = 0.0167, e 3 = 0.0549, and i 3 = 5.1454 • (relative to the ecliptic).

Properties of the Family G orbits
Considering the R3BP, the Family G generally has short-range orbits around L1, longrange orbits that pass a few kilometers from the Earth's surface and a few dozen kilometers from the Moon's surface, and even orbits that present a loop.In the synodic system, the first two kinds of orbits have initial conditions of the following type: x 0 ,0,0, ẏ0 , ( while the third kind has initial conditions of the following type: x 0 ,0,0,− ẏ0 .(4.2) Figure 4.1 exhibits an example of each kind, seen in the synodic system.We can observe that for the first two kinds of orbits, point x 0 is between the Earth and L1; and point x 1 , which corresponds to the first passage by the particle along the x-axis of the coordinate system under consideration, is located between L1 and the Moon.For the third kind, x 0 is also between the Earth and L1, and x 1 is located to the left of the Earth, between the letter and L3.All three kinds of orbits are unstable.
The kinds of paths illustrated in Figure 4.1(b) are those paths capable of making a direct transfer between the Earth and the Moon. Figure 4.2 exhibits, for the restricted threebody problem, a path of this type, a quasiperiodic orbit.In Figure 4.2(a), it is seen in the synodic system, in Figure 4.2(b), in the geocentric system, together with the Moon's orbit, and in Figure 4.2(c), we have a zoom showing a loop given by the path of the Moon's orbit.
Figure 4.3 shows for the four-body problem the orbit obtained with the same initial conditions as in Figure 4.2.Note that with the final approach to the Moon, the probe leaves the Moon's orbital plane.This fact allows the probe to be inserted into a highly inclined lunar orbit.
Note that for the initial conditions (4.1) and (4.2), the particle is always on the xaxis, between the Earth and the Moon, at t = 0.In this manner, considering a probe in   a terrestrial parking orbit, the injection impulse to acquire a transfer path could only be applied when the Earth, probe, and Moon were all lined up, in this order.For example, for a terrestrial parking orbit with an altitude equal to 200 km, there would be a launching window open every 1.47 hours, a time frame that is not restrictive to using these orbits for transfer maneuvers between the Earth and the Moon.

Definition of the set of transfer paths
In order to select the paths capable of achieving transfer between a terrestrial and lunar parking orbits, we consider that the probe departs always from a circular orbit around the Earth.We consider altitudes for this orbit, H T , varying between 160 and 20000 km.Then, we select only those paths that reach the periselenium with altitudes, H L , between 0 and 100 km.This procedure allows us to find a curve that demonstrates the relation between the injection speed for acquisition of a transfer path, V I , and the altitude of the terrestrial circular parking orbit, H T , for transfer path's periselenium altitude, H L , less than 100 km from Moon's surface, including collision paths.But if we consider 160 ≤ H T ≤ 700 km, an empirical mathematical expression that relates V I and H T for H L ≤ 100 km can be written.Considering the four-body problem Sun-Earth-Moon-probe, the mathematical expression is given by The interval 160 ≤ H T ≤ 700 km was chosen taking into account the capacity of present launch vehicles.The set of transfer paths can be seen in Figure 5.1(a), and the set defined by (5.1) in Figure 5.1(b), which is a zoom of Figure 5.1(a) for 160 ≤ H T ≤ 700 km.Note that for this interval, we have a black band defining the region and not a line, this fact justifies the last term, δ, in (5.1). Figure 5.2 shows a zoom of the diagram of Figure 5.1(b) for 240 ≤ H T ≤ 245 km.In this figure it is possible to verify the extreme sensitivity of periselenium altitude with injection speed in achieving the transfer path.This structure exists in all the intervals studied, and can be expressed taking into account small variations in δ.These variations are presented in Table 5.1.
The numerical results found for PR3C are quite similar to those of the four-body problem, in terms of the curve as well as the mathematical expression.That is, graphs V I versus 10 000 20 000 30 000 40 000 50 000 60 000 Altitude of terrestrial circular parking orbit, H T (km), for periselenium's altitude of the path, H L , less than 100 km H T for the PR3C are practically identical to those shown in Figures 5.1(a) and 5.1(b), and the mathematical expression V I = V I (H T ) is almost the same as the one in (5.1).For this reason, the results were not shown here.In fact, the differences in the values for V I between the two dynamical systems only begin with the third decimal position in km/s.Nonetheless, there is a significant difference between the defined paths for each dynamical system.This difference is in the position of the periselenium of the transfer path; for the PR3C, it is obviously located in the Moon's orbital plane.However, for the four-body problem, the paths defined graphically by Figure 5.1, and also by (5.1), leave the Moon's orbital plane in their final approximation-see path shown in Figure 4.3, for example.In this manner, the osculating lunar orbit that contains the periselenium of the transfer path has inclination relative to the Moon's orbital plane and longitude of the ascending node that is different from zero.The inclination of this orbit varies between 40 and 42 degrees for transfer paths that leave terrestrial parking orbits with 160 ≤ H T ≤ 5000 km; for 5000 < H T ≤ 20000 km, the inclination declines gently until it reaches 29 degrees.These inclination values do not take into account the fact that the Moon's equator has an inclination of 6.5 degrees relative to its orbital plane.The longitude of the osculating orbit's ascending node also varies little, between 115 and 117 degrees, for those paths that leave terrestrial parking orbits with 160 ≤ H T ≤ 5000 km, and decreases gently until 108 degrees for paths with 5000 ≤ H T ≤ 20000 km.
It is interesting to observe that, in spite of the instability of the paths observed for the two dynamical systems under consideration, they define what we can call "links" between the Earth and the Moon.In order to better understand this, we will consider two paths obtained for PR3C: the first having H T = 160 km, V I = 10.969km/s, that is, in the geocentric coordinates system, (x 0 ;0;0; ẏ0 ) = (6610km;0;0;10.969km/s),the altitude of the periselenium H L = 10 km and C J (Jacob's constant) = 1.7469; and the second having H T = 20000 km, V I = 3.261 km/s, (x 0 ;0;0; ẏ0 ) = (20000km;0;0;3.261km/s),H L = 96 km, and C J = 2.3702.Note that these paths correspond exactly to the extremes of the interval considered for H T .The "linking areas" in the synodic system, one on the departure and one on the return, defined by these two paths, are illustrated in Figure 5.3(a).If we also consider that those paths which possess 100 km < H L ≤ 20000 km as the internal limit of the return "linking area" can be amplified, as shown in Figure 5.3(b), this guarantees faster Moon-Earth transfers.By analogy, considering paths obtained for the four-body problem with the same initial conditions, that are, H T = 160 km, V I = 10.967km/s, H L = 10 km, i (inclination of the osculating lunar orbit that contains the periselenium) = 41.44 degrees, and Ω (longitude of the ascending node of the osculating lunar orbit) = 116.87degrees, and H T = 20000 km, V I = 3.259 km/s, H L = 98 km, i = 41.89degrees, and Ω = 116.96degrees, the "area" of the departure journey continues to exist in the Moon's orbital plane.Nonetheless, the exit of the transfer path from the Moon's orbital plane in its final approximation introduces a "break in symmetry," with relation to that observed for the PR3C.The "area" of the return journey is transformed into a three-dimensional "canal" outside the Moon's orbital plane and its external limit is different from the external area of the departure journey, as shown in Figures 5.4

Flexibility of missions
The properties of the set of paths found for the four-body Sun-Earth-Moon-probe problem can be conveniently exploited in at least three types of missions.The first type of mission corresponds to a direct transfer between a terrestrial parking orbit, with altitude varying between 160 and 20000 km, and a lunar parking orbit, with altitude varying between 20 and 20000 km, and inclination varying between 29 and 40 degrees, with application of only two impulses.The second type of mission utilizes the instability inherent to the paths and gain in inclination to insert them into lunar parking orbits with other inclinations, including polar orbits, at a low cost, applying a ΔV director at mid-journey to direct the path in such a way that the lunar osculating orbit which contains the periselenium has inclinations greater than 40 degrees, or lower than 29 degrees, depending on the mission, of course.Finally, the third type of mission takes advantage of the instability of the paths and the "break in symmetry" caused by their exit from the Moon's orbital plane during the final approximation.The idea to be exploited is to launch a probe or satellite, towards the Moon in a path defined by (5.1), from a low-altitude terrestrial orbit (LEO), with H T ≤ 700 km, for example, and to take advantage of the gain in inclination and the return linkage "canal" to place the satellite in high-altitude terrestrial orbits and high inclinations, including polar orbits, possibly reducing the cost of this maneuver.Such a maneuver also requires the application of a ΔV director during mid-flight to adequately direct the path.Following we will discuss an example of the first type of mission, that is, a direct transfer between a low-altitude terrestrial parking orbit (LEO) with H T = 320 km, V I = 10.834km/s, and a lunar parking orbit with H L = 84.7 km, i = 41.15 degrees, and Ω = 116.84degrees.The ΔV 1 required for the first impulse, the injection impulse that will place the probe into the transfer path, is the difference between V I and the velocity of the terrestrial parking orbit, if we consider the parking orbit as circular, we have where R 1 is the Earth's parking orbit radius in the point of tangency with transfer path R 1 = R T + H T , with R T being the Earth's average radius (6370 km).The ΔV 2 required for the second impulse, the insertion impulse into the lunar orbit, will correspond to the difference between the velocity of the probe in the periselenium of the transfer path, V P , and the velocity of the planed lunar orbit at the point of its application.Supposing that the plane around the Moon is circular, we have 2) The value for V P is found through numerical integration, for this example V P = 2.560 km/s, R P = R L + H L , with R L being the average radius of the Moon (1738 km).In the periselenium of the transfer path, the probe's velocity is always perpendicular to the Moon-probe position vector, which facilitates the insertion into a circular lunar orbit.The ΔV Total is The flight time is 13.92 days.The cost of the maneuver calculated by Hohmann is ΔV Total = 3.910 km/s and by patched conic is ΔV Total = 4.118 km/s, and the flight time is of the order of 5 days for both.As we can see, the ΔV Total via the transfer path as defined by (5.1) is greater than the ΔV Total for the same maneuver via Hohmann (around 3%) and less than the ΔV Total via patched conic (around 2.2%).However, as we saw in Section 2, these methods are based strictly upon the dynamics of the two-body Earthprobe and Moon-probe.Numerical simulations considering the PR3C or the four-body problem show that the ΔV Total via Hohmann and patched conic are not sufficient to conclude the maneuver, requiring corrective ΔV 's to compensate for the disturbance caused by the Sun's and Moon's gravitational fields (first part of the maneuver) and of the Sun and the Earth (second part).These corrective ΔV 's can increase the cost of the maneuver by 5%.On the other hand, the ΔV Total via the transfer path defined by (5.1) already takes these disturbances into account.

Conclusions
In this study, we have been able to verify the existence of a well-defined set of paths, derived from the direct periodic orbits around the Lagrangian equilibrium point L1 (Family G [8]), and capable of carrying out, among other things, a direct transfer maneuver between low-altitude terrestrial and lunar parking orbits.This set was graphically defined starting with injection velocity V I , or velocity required to acquire the transfer path, versus the altitude of the terrestrial parking orbit, H T , for 160 ≤ H T ≤ 20000 km and with H L (altitude of the lunar parking orbit) ≤ 100 km (Figures 5.1 and 5.2), and via an empirical mathematical expression for 160 ≤ H T ≤ 700 km (5.1).The dynamical systems considered were the PR3C and the four-body Sun-Earth-Moon-probe problems, to which the eccentricity of the Earth's orbit and the eccentricity and inclination of the Moon's orbit were taken into account.
In addition to the direct transfer between the Earth and the Moon, the paths defined by the graphs in Figures 5.1 and 5.2 can also be used to insert a probe into lunar orbits with high inclinations, including polar orbits, thanks to the instability and to the fact that they leave the Moon's orbital plane in their final approximation.It is also possible to exploit this gain of inclination to transfer, at a low cost, a probe or a satellite from a lowaltitude terrestrial orbit (LEO), to an orbit with a much higher altitude, with or without high inclinations.This maneuver is possible because the set of investigated paths defines "linking areas" and/or "canals" between the Earth and the Moon, as shown in Figures 4.  With respect to the cost of the direct transfer maneuver, when compared to the conventional methods of Hohmann and patched conic, we see that the paths studied have ΔV Total that are very close to those obtained by these methods.However, the ΔV 's found via Hohmann and patched conic are not sufficient to conclude the maneuver, because they do not take into account the disturbances of the gravitational fields of the Sun, Earth, and Moon during the entire maneuver.This makes it necessary to apply corrective ΔV 's to conclude the transfer, which can increase the cost of the maneuver by 5%.Nevertheless, the disturbances of the Sun's, Earth's, and Moon's gravitational fields are present in the investigations that led to the set of paths herein defined.With respect to the flight time, the conventional methods are unbeatable, but a flight time between 13 and 15 may be acceptable in logistical missions, transfer of automatic probes, and even in some manned missions given the presumed savings.
When compared to the traditional gravitational capture transfer methods, in general, the cost of a maneuver carried out by a path defined in (5.1), for example, is greater (around 5%).However, the flight times for gravitational capture transfers are very long, and are measured in months (7)(8)(9)(10)(11)(12)(13)(14)(15)(16), while flight times for the paths herein presented do not exceed two weeks.
Therefore, we can conclude that this study reveals a set of alternative paths for transfer missions in the Earth-Moon system that, if properly exploited, could reduce the costs of some important maneuvers in the present context and also in the future of space exploration.

Figure 2 . 1 .
Figure 2.1.(a) Minimal energy transfer ellipse and the Earth's and Moon's circular parking orbits in geocentric system.(b) Velocities of the Moon and of the probe (in its apogee's transfer ellipse) in geocentric system.(c) Velocities of the probe relative to the Moon at the moment of application of the ΔV 2 in selenocentric system.(Not to scale.)

Figure 2 . 2 .
Figure 2.2.Geocentric conic of transfer (ellipse) and the transfer geometry seen in the geocentric coordinates system (not to scale).

Figure 3 . 1 .
Figure 3.1.Synodic reference system and the relative location of the Lagrangian equilibrium points for Earth-Moon system.

Figure 4 . 1 .
Figure 4.1.Family G of periodic orbits obtained considering the R3BP: (a) short range and (b) long range, with x 0 between the Earth and L1 and with x 1 between L1 and the Moon.(c) Orbit that has loops with x 0 between the Earth and L1, and with x 1 between the Earth and L3.

Figure 5 . 2 .
Figure 5.2.Injection speed, V I , versus the altitude of the Earth's parking orbit, H T , for paths with various altitudes of periselenium, H L , indicated using a color code.

Figure 5 . 3 .
Figure 5.3."Linking areas" between the Earth and the Moon in the synodic system for the PR3C with (a) 160 ≤ H T ≤ 20000 km and H L ≤ 100 km and (b) 160 ≤ H T ≤ 20000 km and 100 < H L ≤ 20000 km.

Figure 5 . 4 .
Figure 5.4."Linking areas and canals" between the Earth and the Moon in the synodic system, obtained for the four-body problem with (a) 160 ≤ H T ≤ 20000 km and H L ≤ 100 km and (b) 160 ≤ H T ≤ 60000 km and 100 < H L ≤ 20000 km.