Alternative Transfers to the NEOs 99942 Apophis, 1994 WR12, and 2007 UW1 via Derived Trajectories from Periodic Orbits of Family G

Abstract

Swing-by techniques are extensively used in interplanetary missions to minimize fuel consumption and to raise payloads of spacecrafts. The effectiveness of this type of maneuver has been proven since the beginning of space exploration. According to this premise, we have explored the existence of a natural and direct link between low Earth orbits and the lunar sphere of influence to get low-energy transfer trajectories to the Near Earth Objects (NEOs) 99942 Apophis, 1994 WR12, and 2007 UW1 through swing-bys with the Moon. The existence of this link is related to a family of retrograde periodic orbits around the Lagrangian equilibrium point L1 predicted for the circular, planar, restricted three-body Earth-Moon-particle problem. The trajectories in this link are sensitive to small disturbances. This enables them to be conveniently diverted reducing so the cost of the swing-by maneuver. These maneuvers allow a gain in energy sufficient for the trajectories to escape from the Earth-Moon system and to stabilize in heliocentric orbits between the Earth and Venus or Earth and Mars. Therefore, the trajectories have sufficient reach to intercept the NEOs' orbits.

1. Introduction

The dynamics of the circular restricted three-body Earth-Moon-particle problem predicts the existence of the retrograde periodic orbits around the Lagrangian equilibrium point L1. Such orbits belong to the so-called Family G [1], and starting from them it is possible to define a set of trajectories that form a natural round trip link between the Earth and Moon and a link between the Earth, the Moon and the distant space of the Earth sphere of influence. These links occur even for more complex dynamical systems as the complete Sun-Earth-Moon-particle problem. In the last dynamical system, many kinds of transfers can be exploited. For example, the round trip link is useful for transfers between Low Earth Orbits (LEOs) and Low Lunar Orbits (LLOs), including polar LLOs, since some of its trajectories pass a few hundreds of kilometers from the Earth’s surface and a few dozens of kilometers from the Moon’s surface (Figure 1) [2]. Moreover, the round trip link is also useful for transfer between two LEOs with different altitudes and inclinations [3]. In these cases the Moon’s gravitational field, which is near the Earth, works as an extra impulse and provides an efficient method to minimize the fuel consumption to be used in plane change maneuvers. This is possible, because the trajectories of the links can lead spacecrafts in a natural way to accomplish swing-bys with the Moon. In this paper, we use the link between the Earth, the Moon and the distant space of the Earth sphere of influence and the energy gain of the swing-by with the Moon to get escape trajectories and to design missions to NEOs.

Figure 1 

Locations of Lagrangian equilibrium points and a typical periodic orbit of Family G in synodic frame (x, y).

In general, a swing-by, or a flyby, consists of the alteration of small celestial bodies orbits (comets, asteroids, or spacecrafts) when they have a close approach with a planet or a moon. Literature presents many studies about this subject with general descriptions, applications in interplanetary missions and in the Earth-Moon System. For example, Lawden [4], Minovitch [5], and Broucke [6] present general descriptions of the mechanics of gravity-assisted maneuvers. Successful examples in interplanetary missions were the Voyager 1 and 2 which accomplished several swing-bys with the visited planets to gain energy [7], and Ulysses solar probe which accomplished a swing-by with Jupiter to get a perpendicular orbit to the ecliptic and observe the Sun’s Poles [8]. Among the many studies on swing-bys in the Earth-Moon system, we can highlight Dunham and Davis [9], who designed missions with multiple swing-bys with the Moon; Uphoff [10], who gave a description about lunar gravity-assisted maneuvers, and Prado [11], who presented a study to use flyby with the Moon to accomplish transfers between Earth orbits with the same altitudes and different inclinations. Among the missions that accomplished swing-bys with the Moon, we should mention the ISEE-3 which after completing its mission in 1982 was renamed ICE (International Cometary Explorer) and accomplished several lunar encounters to gain energy and intercept the comets Giacobini-Zinner in September 1985 and Halley, in March 1986 [12]. On the other hand, also in general, the literature does not present works focused on getting the trajectory that will accomplish the swing-by. In this work we do not only use the swing-by to obtain transfer trajectories, but we have included a study that involves the use of trajectories derived from periodic orbits to accomplish the swing-by in an efficient way. This allows the reduction of the swing-by maneuver costs.

Our results, considering trajectories derived from periodic orbits of Family G and the quasiperiodic orbits that oscillate around them, have allowed us to present the parameters to design Earth-NEOs transfers. The trajectories studied in this work can offer a saving up to 4% in relation to the conventional methods used to send spacecrafts to these asteroids.

This article is laid out according to the following order: in Section 2, the dynamical systems and the orbits of Family G are described; in Section 3, the set of trajectories of interest are defined. In Section 4, the studies about the mechanism of the proposed transfers between Earth and NEOs are shown and, in Section 5, we present and discuss the results. The conclusions on the work are presented in Section 6.

2. Earth-Moon Link

The equations of motion of a small particle of negligible mass, m, moving under the gravitational influence of two bodies with preponderant masses, written in a normalized rotating coordinate system (x, y), also-called synodic system, are

𝑥̈−2𝑛𝑦̇=𝜕𝑈,̈̇𝜕𝑥𝑦+2𝑛𝑥=𝜕𝑈𝜕𝑦, (2.1) where 𝑛 is the mean motion and 𝑈 is a scalar function given by

𝑛𝑈=²2(𝑥²+𝑦²𝜇)+¹𝑟₁+𝜇₂𝑟₂, (2.2)𝜇₁ and 𝜇₂ are the reduced masses of the two preponderant bodies, called primaries. 𝑟²₁=(𝑥+𝜇₂)²+𝑦² and 𝑟²₂=(𝑥−𝜇₁)²+𝑦² are the distances between the primaries and 𝑚. Equation (2.1) define the well-known restricted three-body problem in which the primaries have circular orbits around their common centre of mass [1, 13].

Equation (2.1) presents a special number of solutions that can be achieved searching for points in synodic frame where 𝑥̈=𝑦̈=𝑥̇=𝑦̇=0. There are five points with these features and they are called of Lagrangiam equilibrium points, L_𝑖, 𝑖=1,…,5. The locations of these points are obtained solving the simultaneous nonlinear equations 𝜕𝑈/𝜕𝑥=𝜕𝑈/𝜕𝑦=0. Figure 1 shows the locations of these points for Earth-Moon-particle problem. In this case, 𝜇₁=𝜇_Earth=0.9878494, 𝜇₂=𝜇_Moon=0.0121506, and 𝑛=1.

In the Earth-Moon-particle problem, groups of periodic orbits around the Earth, the Moon and the L_𝑖 points are known. We have considered a particular family of periodic orbits around L1 known as Family G [1–3]. In the normalized synodic frame, one of the groups of initial conditions that allow us to obtain the Family G orbits has the form

(𝑥₀,𝑦₀,𝑧₀,𝑥̇₀,𝑦̇₀,𝑧̇₀)=(𝑥₀,0,0,0,𝑦̇₀,0), (2.3) where (−𝜇_Moon+𝑅^∗_𝐸)<𝑥₀<𝑥(L1), with 𝑅^∗_𝐸=𝑅_𝐸/384,400 km = 0.016572, 𝑅_𝐸=6,370 km is the average radius of the Earth, 𝑥(L1)=0.836893 is the abscissa of the Lagrangian equilibrium point L1, and −9.389476≤𝑦̇₀≤601.045381 [1], which corresponds to a variation between −9.607 km/s and 614.964 km/s. The average Earth-Moon distance (384,400 km) is chosen as the unit of length of the normalized system. According to (2.3), for any orbit of Family G, in 𝑡=0, the Earth, the particle and the Moon are aligned in this order, that is, in inferior conjunction. All orbits in Family G are unstable and this feature is very important for the transfers here exploited. Figure 1 also shows a typical orbit of Family G.

Periodic orbits as the one shown in Figure 1 and the quasiperiodic orbits that oscillate around them are tangents to the Earth and Moon low orbits. In other words, they define a link between the Earth and the lunar sphere of influence. This property allied with the instability of Family G orbits allows the design of controlled swing-bys with the Moon to gain sufficient energy to escape from the Earth-Moon system [2]. Moreover, we remark that these properties allow the reduction of the swing-by maneuver costs besides providing the necessary energy to win the Earth’s and the Moon’s gravitational fields [3, 14]. Controlled swing-bys also allow the use of this energy to obtain trajectories inclination changes [3].

In order to exploit these properties and to design more efficient and realistic Earth-NEOs transfers, the Sun’s gravitational field must be introduced. That is, new equations of motion must be considered, this time, taking into account the Sun, the Earth and the Moon mutual gravitational attractions, besides the characteristics of the Earth’s orbit around the Sun (eccentricity) and the Moon’s orbit around the Earth (eccentricity and inclination). Thus, we define the full four-body Sun-Earth-Moon-particle problem, whose 12 equations of motion in a Cartesian coordinates system (X, Y, Z) with origin in a fixed point of the space are 𝐑̈_𝑖=⁴_{𝑗=1𝑗≠𝑖}𝜇_𝑗𝑅³_𝑗𝑖(𝐑_𝑗−𝐑_𝑖), (2.4) where 𝑅_𝑖𝑗=|𝐑_𝑗−𝐑_𝑖|=[(𝑋_𝑗−𝑋_𝑖)²+(𝑌_𝑗−𝑌_𝑖)²+(𝑍_𝑗−𝑍_𝑖)²]^1/2, with 𝑗≠𝑖, are the distances between ith and jth bodies. The eccentricity of the Earth’s orbit and the eccentricity and inclination of the Moon’s orbit are introduced through initial conditions.

The link between the Earth and the lunar sphere of influence continues existing, even when the dynamical system given by (2.4) is considered. This can be seen in Figure 2(a) that shows a trajectory found considering the full four-body problem. Figure 2(b) shows the inclination gain relative to the ecliptic after the swing-by with the Moon. Note that the trajectory inclination change is about 52 degrees. As a first conclusion, we can say that the Moon gravitational field acts like a natural propeller, without fuel consumption. In the next section, we are going to exploit this property in a controlled way. That is, taking advantage of the existing natural link between the Earth and the lunar sphere of influence to obtain a change in inclination and the energy gain needed for a determined sort of mission.

(a)

(b)

Figure 2 

(a) Derived trajectory of a Family G periodic orbit considering the full four-body problem in geocentric frame (𝑥′,𝑦′,𝑧′). (b) Trajectory inclination change as function of the time.

3. Escape Trajectories

The existence of the link between the Earth and the lunar sphere of influence for the full four-body problem is essential to the design of the Earth-Moon system escape trajectories. For this dynamical system, the initial conditions of an escape trajectory in a geocentric frame (𝑥^′,𝑦^′,𝑧^′) are [3]

(𝑥^′₀,𝑦^′₀,𝑧^′₀,𝑥̇^′₀,𝑦̇^′₀,𝑧̇^′₀)=(𝑥^′₀,0,𝑧^′₀,0,𝑦̇^′₀,0), (3.1) where

𝑥^′₀=(𝑅_𝐸+ℎ₀)cos(𝑖_Moon𝑦),_′0𝑧=0,_′0=(𝑅_𝐸+ℎ₀)sin(𝑖_Moon𝑥̇),_′0𝑦̇=0,_′0=𝑉_𝐼,𝑧̇^′₀=0. (3.2)

Quantity V_I is called injection velocity. Considering a circular LEO, 𝑉_𝐼=𝑉_CE+Δ𝑉₁, where 𝑉_CE is the LEO’s velocity, 𝑉_CE=[𝐺𝑀_Earth/(𝑅_𝐸+ℎ₀)]^1/2, G is the gravitational constant, 𝑀_Earth is the Earth’s mass, and ℎ₀ is the LEO’s altitude. 𝑖_Moon is the inclination of the Moon’s orbit. Δ𝑉₁ is the velocity increment that must be provided to the spacecraft. Δ𝑉₁ is a function of the ℎ₀ and 𝑉_CE and its value is given by

Δ𝑉₁=𝑉_𝐶𝐸2(−2.3340×10⁻⁶ℎ₀+0.8085±0.0001)+𝛿+𝜗. (3.3)

Quantity 𝛿 assumes values that will take into account the relative position among the Sun, the Earth, and the Moon. In other words, the values assumed by 𝛿 reflect the influence of the eccentricities of the Earth’s orbit and, mainly, of the Moon’s orbit in the distance between them. Equations to estimate the values assumed by 𝛿 as a function of the true anomalies of the Earth’s and Moon’s orbits are given by de Melo et al. [3].

Here, we are interested in Earth-Moon escape trajectories. Then, we can consider the Earth and the Moon in circular orbits, with the average radius of the Moon’s orbit equal to 384,400 km. This does not represent a significant loss of precision. This way, we can assume that 𝛿=0 in (3.3) and, then, we just analyze the quantity 𝜃 whose values will determine the spacecraft position relative to the Moon and the Earth during the passage by the lunar sphere of influence and the swing-by. For −1.50×10⁻⁴ km/s ≤𝜗≤1.50×10⁻⁴ km/s, we will have collision trajectories with the Moon, for 𝜗<−1.50×10⁻⁴ km/s, the trajectories will have periselenium in the region close to the Moon, between itself and the Earth (anterior region). For 𝜗>1.50×10⁻⁴ km/s, the periselenium will be in the posterior region to the Moon relative to the Earth. For 𝜗<−1.50×10⁻⁴ km/s, it is possible to design transfers between LEOs and LLOs of any altitudes and also between two Earth orbits of different inclinations and altitudes, both at a low cost [2, 3]. Our interest, however, is the trajectories generated for 𝜗>1.50×10⁻⁴ km/s. In those cases, the gain of energy with the swing-by guarantees the necessary velocity increment to win the Earth’s and the Moon’s gravitational fields.

The spacecraft must reach the velocity 𝐕_𝐼 when the Earth, the spacecraft and the Moon are aligned in this order. This is necessary because the trajectory is derived from a Family G orbit, and this is the only situation in which the orbits of Family G are tangent to the circular LEOs. Note that this condition does not impose significant restrictions in practical purpose. For instance, if a satellite is in a LEO with an altitude of 200 km, this condition will occur every 1.48 hour, approximately, which is the period of the LEO. Figure 3 shows a typical escape trajectory derived from Family G orbits. From now on, we will call them of escape trajectories G.

(a)

(b)

Figure 3 

(a) Typical escape trajectory from the Earth-Moon system. (b) Energy gain relative to the Earth as function of the time for the trajectory. Note that it is a typical swingy-by energy gain.

It is important to note that the trajectories evolution of the Earth-Moon and escape links shown here (Figures 2 and 3) do not suffer any intervention since they start from the LEO. That is, the inclination and the energy gain occur naturally during the passage through the lunar sphere of influence. However, it is possible to take advantage of the inherent instability of these trajectories to control the spacecraft passage through the lunar sphere of influence [2, 15]. This idea will be exploited in the next section to generate specific escape trajectories for determined missions.

4. Escape Trajectories to Design Earth-NEOs Transfers

The basic idea is to find transfer trajectories between the Earth and the NEOs starting from escape trajectories derived from Family G periodic orbits. This will be done to three NEOs: the 99942 Apophis, the 1994 WR12, and the 2007 UW1. All these are NEOs of the Aten asteroids class and they are defined by having semimajor axes of less than one astronomical unit (1 AU is equal to 149.6×10⁶ km). Some data about these objects are provided in Tables 1 and 2. Table 1 shows the orbital elements: semimajor axis (a), eccentricity (e), inclination (i), longitude of ascending node (Ω), argument of perihelion (w), and the orbital period (P). Table 2 brings other information as average orbital velocity (𝑉_ave), Escape velocity (𝑉_esc), Aphelion, Perihelion, Mass, and diameter. In Table 3, the three next Closest Point Approaches (CPA) with the Earth are presented for 99942 Apophis, the 1994 WR12, and the 2007 UW1.

Table 1 

NEOs orbital elements.

Table 2 

Some characteristics of the NEOs orbits.

Table 3 

Three next closest point approaches.

Once the mission target is determined, it is necessary to find the point in the asteroid’s orbit where the spacecraft will reach it. However, our goal here is not to find an optimal trajectory exactly. We intend to show that transfer trajectories derived from periodic orbits around L1, generated after swing-bys with the Moon, can reach an NEO with ΔVs smaller than the ones required by trajectories that leave from the Earth directly. In order to do so, we only considered trajectories that are tangents to the NEOs’s orbits. In order to find an escape trajectory that will reach an NEO’s orbit, for example, an algorithm integrates a number of them starting from the interception point backward in time until they reach the closest point approach with the Moon during the swing-bys. These integrations are supplied for initial conditions determined starting from previous numerical simulations of trajectories that escape naturally from the Earth-Moon system, as the one shown in Figure 3. In these simulations, a group of trajectories that have a point closer to the NEO’s orbit is selected. The positions and the velocities of these trajectories are analyzed and adjusted to form a group of initial conditions for the integrations beginning in the interception point of the NEO’s orbit. When the closest point approach with the Moon is found, the algorithm calculates a small ΔV to determine which escape trajectory G is more suitable for the transfer. In other words, it calculates the components values of the spacecraft’s initial condition in the LEO (3.1). The algorithm provides a group of solutions with Δ𝑉_Total, Δ𝑉₁, and Δ𝑉_intermediary, transfer time and other quantities of interest. Besides the Δ𝐕_intermediary applied in the closest point approach with the Moon, the algorithm also calculates another Δ𝐕_intermediary applied in the first apogee of trajectory G. This apogee is reached about eight days after the departure from the LEO and prior to the passage by the lunar sphere of influence (Figure 3). This is the best point to do velocity and inclination changes. In this point, the trajectories G velocities are very small; so small ΔVs can provide significant velocity and inclination changes.

A similar procedure is used to find the transfer trajectory between the Earth and the NEO for Patched-conic approximation. In this method, the transfers are designed seeking for a heliocentric conic arc that links an LEO to an orbit around the asteroid [16].

5. Results

The results of the intercept missions to the NEOs Apophis, 1999 WR12, and 2007 UW1 are in Tables 4 and 5. Table 4 presents the values found for the missions conceived by escape trajectories G, and Table 5 presents the correspondent values found for the missions conceived by Patched-conic approach. For each sort of mission, the ΔVs applied in the departure from the LEO, for occasional velocity and inclination changes, the spacecraft velocities relative to the asteroids in the closest point approach (CPA), and the transfer times were calculated. The altitude of the circular departure LEO was considered equal to 200 km, and the altitude of the CPA with the asteroids was considered equal to 1 km. For the transfers between the Earth and Apophis and the Earth and 1994 WR12, it was possible to find two CPAs between the spacecraft and the asteroids. In Tables 4 and 5, we call these times of T1 for the 1st CPA and T2 for the 2nd CPA. It is interesting to note that spacecraft velocities relative to the asteroids are different in each CPA, and, depending on the mission goals, the choice of one of them could represent an extra fuel saving in an occasional insertion maneuver of the spacecraft into an orbit around the asteroid.

Table 4 

Missions’ data of the Earth-NEOs transfer by escape trajectories G.

Table 5 

Missions’ data of the Earth-NEOs transfer by Patched-conic approach.

Figure 4(a) shows the Earth’s and the Apophis’s orbits and the heliocentric ellipse arc generated by the Patched-conic approach. The times of the 1st and 2nd CPA are 155 and 198 days after the departure from the LEO, respectively, and the distances from the Earth to these points are 11.2×10⁶ km and 113.6×10⁶ km, respectively. The values of 𝑇1 and 𝑇2 of the other transfers are in Tables 4 and 5.

(a)

(b)

Figure 4 

(a) Earth’s and Apophis’s orbits and the transfer trajectory found for Patched-conic transfer. (b) Earth’s and Apophis’s orbits and transfer trajectory G (heliocentric frame).

By analogy, Figure 4(b) shows the escape trajectory G found for the Earth-Apophis transfer. The 1st and the 2nd CPAs take place 86 and 225 days after the departure from the LEO, and the distances from the Earth to these points are 52.27×10⁶ km and 204.45×10⁶ km, respectively.

The analysis of Tables 4 and 5 shows that the procedure described in Section 4 to generate transfer trajectories between the Earth and NEOs, derived from periodic orbits of Family G, requires ΔVs of 2% up to 4% less than the transfers conceived by Patched-conic approach. Besides, the spacecraft velocities relative to the asteroids in the CPAs, for the trajectories G, are the smallest. This can also provide fuel saving in occasional insertion maneuvers into an orbit around the asteroids. However, the smallest relative velocities always correspond to the longest transfer times.

6. Conclusions

In this work, a procedure capable of generating transfer trajectories between the Earth and the NEOs has been presented. These trajectories are derived from the periodic orbits around the Lagrangian point L1 and escape from the Earth-Moon system after a swing-by with the Moon.

In terms of Δ𝑉_Total, those required by escape trajectories G are, in general, fewer than the ones required by conventional transfers (Patched-conic), between 2% up to 4%. Besides, the spacecraft velocities relative to the asteroids are also, in general, less than those found by the conventional methods.

With regard to the transfer time, we verify that in two cases (Apophis and 1994 WR12) it was possible to find two CPAs. The time longest always corresponds to the smallest relative velocity in CPA for trajectories G. We also verify that there are not any discrepancies between the transfer times of the two considered methods.

This way, we can conclude that the escape trajectories G presented in this work are a good alternative to design future missions destined to the NEOs.

Finally, future studies of techniques to accomplish a swing-by with the Earth and, thus, to gain more energy and a larger reach could provide the planning of transfers to more distant objects, including the internal planets.

Acknowledgments

The authors are grateful to the FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo), processes 2005/05169-7 and 2006/04997-6, and the CNPq (Conselho Nacional para Desenvolvimento Científico e Tecnológico) Brazil, for the financial support.