- Email: [email protected]
Free Return Trajectories in Lunar Missions
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
Chinese Astronomy and Astrophysics 37 (2013) 183–194
Free Return Trajectories in Lunar Missions† 1
2
HOU Xi-yun1,2
ZHAO Yu-hui1,2,3
LIU Lin1,2,3
School of Astronomy and Space Science, Nanjing University, Nanjing 210093 Institute of Space Environment and Astrodynamics, Nanjing University, Nanjing 210093 3 Beijing Aerospace Control Center, Beijing 100094
Abstract Four types of symmetric free return trajectories in the planar circular restricted three-body problem are computed and compared with each other. One of these four types is most applicable in practice. Concentrating ourselves on this special type of free return trajectory, the corresponding planar asymmetric cases are studied. Then the studies are generalized to the three-dimensional case. The restrictions on the inclination angle of the probe at the perilune are discussed. It is found that the maximum inclination at the perilune between the probe’s orbit plane and the Moon’s orbit plane is restricted by the heights of the perigee and the perilune of the free return trajectory. However, the inclination at the perigee is nearly not affected by them. At last, a strategy to design free return trajectories in the real Earth-Moon system is proposed. Some numerical simulations are done to show the feasibility of this orbit design strategy. The discussions from the planar case to the spatial case and then to the real force model can also be applied to the other three types of planar free return trajectories. Key words: celestial mechanics: three-body problem, restricted problem— methods: numerical
1. INTRODUCTION In recent years, we are witnessing an upsurge of lunar exploration across the world. After the successful progresses of “Chang-e” projects, the manned lunar mission must be put on the agenda of China’s space administration. Different from the normal lunar mission, †
Supported by National Natural Science Foundation Received 2011–11–22; revised version 2012–02–23 A translation of Acta Astron. Sin. Vol. 53, No. 4, pp. 308–318, 2012 [email protected]
0275-1062/13/$-see front matter © 2013 Elsevier All rights reserved. c 2013B.V. 0275-1062/01/$-see front matter Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2013.04.007 PII:
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the foremost consideration must be the astronauts’ safety in the manned spaceflight to the Moon. Therefore the safety factor of the mission must be guaranteed as much as possible in the mission design. As for the trajectory design, this means that the space vehicle should be able to go back to the Earth safely even when the mission goes wrong. The “free return trajectory” often used in the last century by USA and former USSR in the manned lunar missions is right the offspring of this safety requirement[1] . If the injection engine fails at the perilune, the space vehicle can return freely back to the Earth without another maneuver. Although such kind of free return trajectory was often mentioned in many pieces of literature, there is a lack of the details of its design. Thus we devote this paper to a detailed study on this type of transfer orbit, aiming to provide some reference for the lunar mission in China. For the sake of analyzing the transfer orbit in the Earth-Moon system, the circular restricted three-body problem (CRTBP) consisting of the Earth, the Moon and the space probe is an appropriate dynamical model. In this paper, we present first the 4 symmetrical free return trajectories in the plane of lunar orbit under the CRTBP model, then we extend one of them that is relatively applicable to the 3-dimensional case. We study the variations of inclination of such transfer orbit near the perilune and the perigee. Based on the results under the CRTBP model, we finally present a practical trajectory design and the numerical example for the realistic Earth-Moon system.
2
CIRCULAR RESTRICTED THREE-BODY PROBLEM
The space probe is mainly affected by the gravitations of the Earth and the Moon when it travels in the Earth-Moon space. Since the eccentricity of the lunar orbit is quite small, the CRTBP is a good approximation of the dynamical system. Generally, a rotating coordinate frame is adopted, in which the two primaries (the Earth and the Moon) situate on the abscissa and the origin coincides with their barycenter. Such a barycentric synodic coordinate system is illustrated in Fig. 1. Surely the x− y plane is the orbital plane of the two primaries and the z axis is set to make it a right-handed orthogonal coordinate system.
Fig. 1
An illustration picture of the barycentric synodic frame (for the Earth-Moon system)
Set the average distance between and the total mass of the Earth and the Moon to be the units of distance and mass, the equations of motion of the probe (assumed zero-mass)
Hou Xi-yun et al. / Chinese Astronomy and Astrophysics 37 (2013) 183–194
in this synodic coordinate system read[2] ¨r + 2(−y, ˙ x, ˙ 0) = ∂Ω/∂r , Ω = (μ(1 − μ) + x2 + y 2 )/2 + (1 − μ)/r1 + μ/r2
185
(1)
where μ = mMoon /(mMoon + mEarth ), mMoon and mEarth are the masses of the Moon and the Earth, r = (x, y, z) is the position vector of the probe with respect to the coordinate origin (the barycenter), and r1 , r2 are the distances of the probe from the Earth and the Moon. The Earth and the Moon locate at (−μ, 0, 0) and (1 − μ, 0, 0) in this coordinate system. Obviously, different perturbations exist in the realistic Earth-Moon system and will result in a deviation from the CRTBP model. We will analyze first the free return trajectory in the CRTBP and then extend our analysis to the realistic dynamical system.
3
SYMMETRICAL FREE RETURN TRAJECTORIES IN THE MOON’S ORBIT PLANE
Take a transfer orbit with a perilune height of 100 km and the perigee heights of 200 km at both the departure and the return ends, it is symmetrical with respect to the x−axis in the synodic coordinate system since the perigee heights at both ends are the same. This symmetry implies that the velocity vector of the space vehicle at the perilune of the free return trajectory is perpendicular to the x−axis. Denote the state vector of the space vehicle at perilune (that is the intersection of the trajectory and the x−axis) in the synodic coordinate system by X 0 X 0 = (r0 , r˙ 0 ) = (x0 , 0, 0, y˙ 0 ) . (2) Since the position vector is perpendicular to the velocity vector, X 0 is the perilune of the transfer trajectory. Thus x0 = 100 km is defined by the perilune height, and the adjustable variables are the velocity component y˙ 0 and the one-way transfer time T . Here by “one-way transfer time” we mean the time consumed by the probe from the departure point to the perilune or the time from the perilune to the return point. They are equal to each other due to the symmetry of the trajectory. Denoting the geocentric position and velocity vectors of the probe by r and r˙ , the constraints on the transfer trajectory at both ends read r · r˙ = 0,
r = aEarth + 200 km ,
(3)
with aEarth being the Earth radius. It is worth noting that the same constraints in Eq. (3) are hold for both ends of the trajectory because of the symmetry. The constraints can be fulfilled by adjusting y˙0 and T through the solution finding method for the boundary value problem of the ordinary differential equations[3] . As an example, we illustrate in Fig. 2 the result of such a solution, in which the probe enters the trajectory from the far side of the Moon and the transfer time is T = 2.8634 d. In the left panel, the trajectory is shown in the synodic coordinate system, while in the right panel it is in the geocentric inertial coordinate (this is the same for Figs. 3 ∼ 6). The dashed curves represent the launch trajectory and the solid curves indicate the returning trajectory. The “EM” indicates that the distance unit is the mean Earth-Moon distance [L] = 3.84747981 × 108 m. For such free return trajectory illustrated in Fig. 2, the one-way transfer time T increases with the perigee height and also
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along with the perilune height[4] . Lastly, the trajectory is a prograde orbit in the vicinity of the Earth. A retrograde orbit exists as well, with T = 2.8256 d, as shown in Fig. 3. Similar to the one in Fig. 2, the one-way transfer time T increases with the perigee and perilune heights[4] .
Fig. 2 A free return trajectory departs from the Earth in a prograde orbit and reaches the far side of the Moon.
Fig. 3 A free return trajectory departs from the Earth in a retrograde orbit and reaches the far side of the Moon.
If the injection is on the near side of the Moon, the transfer time of the free return trajectory will increase dramatically. We plot such trajectories for the prograde (in Fig. 4) and retrograde departures from the Earth (in Fig. 5). The one-way transfer time T for these two trajectories are 13.7657 d and 15.0158 d , respectively. For the free return trajectories shown in Fig.4∼5, T increases with the perigee height, however, it decreases with the perilune height[4] . All these 4 types of trajectories shown in Fig. 4 and Fig. 5 confine themselves in the lunar orbit plane and are symmetric with respect to the x−axis. These trajectories are called “Type I symmetrical free return trajectories” and have been investigated in Reference [4].
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We will not discuss them in this paper, for more details please refer to Reference [4]. It is noteworthy that the trajectories illustrated in Figs. 2 ∼ 5 are not closed in the vicinity of the Earth, therefore they are not periodic trajectories.
Fig. 4 A free return trajectory departs from the Earth in a prograde orbit and reaches the near side of the Moon.
Fig. 5
A free return trajectory departs from the Earth in a retrograde orbit and reaches the near side of the Moon.
4. NONSYMMETRICAL FREE RETURN TRAJECTORIES IN THE LUNAR ORBIT PLANE Since to launch a space probe from the Earth to a retrograde orbit needs a big amount of energy, we may abandon such kind of retrograde trajectory. Moreover, a trajectory with the injection point on the near side of the Moon means a much longer transfer time, therefore such trajectory is not a good choice for manned space vehicle. Only the prograde free return trajectory reaching the far side of the Moon as illustrated in Fig. 2 has the merits of practicality and economy. We will focus on this kind of orbit in the following. But the methods we will use can be applied to other types of return trajectories.
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For a returnable lunar probe, the perigee height of the Earth-Moon transfer orbit is determined by the height of the launch vehicle at the end of its sliding motion, while the perigee height of the Moon-Earth transfer orbit is defined by the height of the probe when it reenters into the atmosphere. Since generally these two perigee heights are not necessary to be the same, the free return trajectory is not necessarily symmetric about the x−axis. Denote the height of perigee of the return trajectory by hF , and of the launch trajectory by hB . The state vector of the probe at the perilune is denoted by X 0 = (r0 , r˙ 0 ) = (x0 , y0 , x˙ 0 , y˙ 0 ) .
(4)
Because the trajectory is not symmetric any more, we have y0 = 0, x˙ 0 = 0. Denoting the position and velocity vectors of the probe at the departure time by rB and r˙ B respectively, and denoting the corresponding position and velocity vectors at the returning time by rF and r˙ F , the constraints now read ⎧ rB = aEarth + hB ⎨ rB · r˙ B = 0, (5) rF · r˙ F = 0, rF = aEarth + hF , ⎩ ˙ r0 · r0 = 0, r0 = aMoon + hM where aMoon and hM are the radius of the Moon and the perilune height. Now the adjustable variables include x0 , y0 , x˙ 0 , y˙ 0 , the forward integration time TF and the backward integration time TB (the trajectory is not symmetric about x−axis, now TB = TF ). As in the previous case the constraints in Eq. (5) can be satisfied by adjusting x0 , y0 , x˙ 0 , y˙ 0 , TB and TF through shooting algorithm. Our calculations show that the trajectory is nearly symmetric when hB and hF do not differ from each other too much, and the nonsymmetric property of the trajectory can be evident only when hB and hF are quite different from each other. As an example, we plot in Fig. 6 a trajectory with hB = 3.6 × 104 km, hF = 200 km and hM = 100 km. The time consumption from the departure point to the perilune is 2.9765 d, while the returning from the perilune to the Earth consumes 3.1844 d.
Fig. 6 A free return trajectory with hB = 3.6 × 104 km, hF = 200 km, and hM = 100 km
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5. FREE RETURN TRAJECTORY IN 3-DIMENSIONAL SPACE All the free return trajectories introduced in previous sections offer the backup trajectories only in the lunar orbit plane, so that the landing point has to be within ±5◦ around the equator, and this puts an unfavorable constraint on the fulfilment of some scientific tasks. In fact, for the sake of astronauts’ safety, in the early stage of the Apollo missions (including Apollo 8, 10 and 11) the free return trajectories were seriously considered in the mission design to ensure that the spaceship can safely return and land on the predetermined location when any failure happened to the spaceship, and such return trajectory can be achieved only through a minor orbit adjustment performed by the Reaction Control System (RCS) equipped onboard. But due to the constraints arising from the launching, reentry and recovery, it is not always possible in practice to confine the Earth-Moon and the MoonEarth transfer orbits onto the lunar orbit plane. The study on the free return trajectory has to be extended from the planar case to the case of 3-dimensional space. Such extension has been applied since the Apollo 12 mission. In the 3-dimensional case, the state vector of the probe at the perilune is X 0 = (r0 , r˙ 0 ) = (x0 , y0 , z0 , x˙ 0 , y˙ 0 , z˙0 ) .
(6)
While the constraints are the same as in Eq. (5), with x0 , y0 , z0 , x˙ 0 , y˙ 0 , z˙0 , as well as the forward and backward one-way transfer times TB , TF , being the adjustable variables. Clearly, six equations of the constraints cannot determine uniquely eight adjustable variables. However, we may find a solution by fixing z0 , z˙0 first, and then vary z0 , z˙0 to find other free return trajectories. When the perigee heights of the trajectory at the departure end and arrival end are equal to each other hB = hF , there exist two special 3-dimensional periodic orbits. One is symmetric with respect to the x − z plane in the synodic coordinate system, as shown in Fig. 7, and another is anti-symmetric about the x − z plane as shown in Fig. 8. For the former trajectory, the z-direction velocity component at the perilune is zero z˙0 = 0, while in the latter case the perilune is on the x − y plane, i.e. z0 = 0. Moreover, for the trajectory in Fig. 7 z0 = 1.1 × 10−3 (dimensionless unit), and the one-way transfer time is 2.8728 d, while for the trajectory in Fig. 8, z˙0 = 0.45 (dimensionless unit) and the one-way transfer time is 2.8412 d.
Fig. 7 A three-dimensional free return trajectory. At the perilune, z0 = 1.1 × 10−3 , z˙0 = 0.
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Fig. 8
A three-dimensional free return trajectory. At the perilune, z0 = 0, z˙0 = 0.45.
6. INCLINATION CONSTRAINTS AT PERIGEE AND PERILUNE Generally, after hB , hF , hM have been given, the variations of z0 , z˙0 at the perilune are confined in a given range. For an example with hB = hF = 200 km and hM = 100 km, the upper left panel in Fig. 9 shows the variation ranges of z0 and z˙0 . In this picture, each point (z0 , z˙0 ) represents a free return trajectory, and the contour line indicates the inclination of the probe orbit at the perilune with respect to the lunar orbit plane. Since the trajectory enters the lunar gravisphere in the retrograde direction, the inclination is larger than 90◦ . Obviously, due to the confinement arising from the variation of z0 and z˙0 , the inclination of the trajectory at the perilune is confined in a specific range. As the variation ranges of z0 and z˙0 increase with the perilune height, the maximal value of the inclination increases as well as shown by the upper right panel of Fig. 9 for another example with hB = hF = 200 km and hM = 1000 km. Similarly, the variations of z0 and z˙0 increase with the perigee height, and thus the inclination of the trajectory at the perilune, as shown in the lower left panel in Fig. 9, while we show an example of trajectory with hB = hF = 104 km and hM = 1000 km. The “adim” labeling along the axes of the four panels in Fig. 9 indicates that the values are given in dimensionless unit, which are defined as follows: [L] = 384747981.000000000 m,
[T ] = 375699.843898365 s,
[V ] = [L]/[T ] .
(7)
In all these four panels, the variation regions of z0 and z˙0 are symmetric about the z0 − and z˙0 −axis, respectively. This symmetry arises because hB = hF . However, if the difference between hB and hF is big, the asymmetry of the free return trajectory will result in a distinguishable asymmetry in the shape of the feasible regions of z0 and z˙0 . Such an asymmetric region is shown in the lower right panel in Fig. 9, in which hB = 36000 km, hF = 100 km and hM = 100 km. For the trajectories shown in the upper left panel of Fig. 9, we illustrate in Fig. 10 the contour lines of their inclinations at the perigee. Thanks to the symmetry of the transfer trajectory, it is necessary to discuss only the inclination of the near-Earth parking orbit at the departure end. This picture reveals that the inclination at the near-Earth end of the transfer trajectory may vary in the range 0 ∼ 90◦ , without being affected by any constraints at the near Moon end.
Hou Xi-yun et al. / Chinese Astronomy and Astrophysics 37 (2013) 183–194
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Fig. 9 The feasible regions of z0 , z˙ 0 , and the contour lines of the inclinations at the perilune for different perigee and perilune heights. Upper left: hB = hF = 200 km, hM = 100 km; Upper right: hB = hF = 200 km, hM = 103 km; Lower Left: hB = hF = 104 km, hM = 103 km; Lower right: hB = 36000 km, hF = 100 km, hM = 100 km.
Fig. 10
The feasible regions of z0 , z˙0 , and the contour lines of the inclinations at the perigee for hB = hF = 200 km, hM = 100 km.
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7. RESULTS IN REAL EARTH-MOON SYSTEM Different from the CRTBP model, there are different kinds of perturbations in the real Earth-Moon system. However, the perturbations from other planets in the solar system and the non-spherical gravitation perturbations of the Earth and the Moon are ignorable, especially compared to the perturbations from the Sun and the eccentricity of the lunar orbit. Therefore, the real system in this paper consists of the gravitations from the Sun, the Earth and the Moon as mass points. Their initial positions and velocities are taken from the DE405 Ephemeris. The results for such a real system can be obtained by making proper corrections to the results for the CRTBP model introduced in previous sections. Denote the position and velocity of the probe in the luncentric synodic coordinate system by r and r˙ , and the position and velocity in the geocentric inertial coordinate system (we adopt here the International and R. ˙ The transformation between them can be Celestial Reference System (ICRS)) by R described by =R M + Cr, R
˙ = R ˙ M + C ˙ r + Cr˙ , R
(8)
˙ M are the position and velocity of the Moon in the geocentric inertial M and R where R coordinate system. The rotation matrix C is defined as follow[5] : M ˙ M ) M × R R (R eˆ1 = , eˆ3 = (9) C = (eˆ1 , eˆ2 , eˆ3 ); , eˆ2 = eˆ3 × eˆ1 . RM M × R ˙ M || ||R The state vector of the perilune in the luncentric synodic coordinate system is denoted by X 0 = (r0 , r˙ 0 ) = (x0 , y0 , z0 , x˙ 0 , y˙ 0 , z˙0 )
(10)
If we set z0 = z˙0 = 0, the adjustable variables include x0 , y0 , x˙ 0 , y˙ 0 and the forward and backward integration times from the perilune TF and TB . The constraints are now ⎧ ˙ B = 0, B · R ⎪ RB = aEarth + hB ⎨R ˙ (11) RF · RF = 0, RF = aEarth + hF , ⎪ ⎩ ˙ r0 · r0 = 0, r0 = aMoon + hM B, R F, R ˙ B , R ˙ F are the position and velocity vectors at the departure In these equations, R and return times in the geocentric inertial coordinate system, and they can be computed via Eq. (8) from rB , rF , r˙ B and r˙ F . Similarly, the constraints (Eq. (11)) can be satisfied by adjusting x0 , y0 , x˙ 0 , y˙ 0 , TF and TB through the shooting method. It is notewothy that we have fixed z0 = z˙0 = 0 in our computation, but in practice the predetermined variables should be chosen according to the requirement of the real space mission. A real trajectory is computed and illustrated in Fig. 11. The origin of the Earth-Moon instantaneous synodic coordinates locates at the barycenter of the Earth-Moon system, and the distance unit is the instantaneous Earth-Moon distance. Thus, not like in Figs. 1∼ 4, the “EM” in Fig. 11 is the instantaneous Earth-Moon distance and it changes with time. The space probe spends 2.9329 d on the way from perigee to perilune, it reaches the perilune at epoch MJD57700.9, and it spends another 2.7081 d on its way back to perigee. The same trajectory is plotted in Fig. 12 in the geocentric inertial coordinate system, where “Ae” indicates the radius of the Earth equator aEarth .
Hou Xi-yun et al. / Chinese Astronomy and Astrophysics 37 (2013) 183–194
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Fig. 11 A free return trajectory in the real force model of the Earth-Moon system (drawn in the Earth-centered instantaneous synodic frame)
Fig. 12
The same trajectory as that in Fig. 11, but shown in the geocentric inertial coordinate system
8. DISCUSSION Not like the normal lunar explorations, the primary requirement for a manned lunar mission is the safety of astronauts onboard. The free return trajectory provides a backup trajectory for the lunar spaceship, ensuring that the spaceship can return to the Earth safely even when the orbit maneuver fails at the perilune. In this paper, based on the circular restricted three body problem, we systematically studied the different kinds of free return trajectories with different boundary constraints. First, adopting the planar circular restricted three body problem as the dynamical model, we calculated several symmetric free return trajectories in the lunar orbit plane, and the asymmetric trajectories were discussed as well. Then, considering the requirements of real space mission, in three dimensional space we described the constraints on the inclinations of the transfer trajectories in the luncentric and geocentric coordinate systems (in both cases, the reference plane is the lunar orbit plane). And finally, we presented the method for designing the free return trajectory in the real Earth-Moon system, and an example trajectory was given. The work presented in this paper may provide certain references for the orbit design of the future “Chang-e” projects that will perform the “collection - return” mission and even the manned lunar mission, and for the backuptrajectory design.
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References 1
Woods W. D., How Apollo Flew to the Moon, Berlin: Springer, 2008, 395-427
2
Szebehely V., Theory of Orbits, New York: Academic Press, 1967, 53-97
3
Stoer J., Bulirsch R., Introduction to Numerical Analysis, New York: Springer-Verlag, 1980, 539-577
4
Schwaniger A. J., Trajectories in the Earth-Moon Space with Symmetrical Free Return Properties,
5
Hou X. Y., Liu L., CMDA, 2011, 110, 71
NASA TN D-1833, 1963
ELSEVIER
Chinese Astronomy and Astrophysics 37 (2013) 183–194
Free Return Trajectories in Lunar Missions† 1
2
HOU Xi-yun1,2
ZHAO Yu-hui1,2,3
LIU Lin1,2,3
School of Astronomy and Space Science, Nanjing University, Nanjing 210093 Institute of Space Environment and Astrodynamics, Nanjing University, Nanjing 210093 3 Beijing Aerospace Control Center, Beijing 100094
Abstract Four types of symmetric free return trajectories in the planar circular restricted three-body problem are computed and compared with each other. One of these four types is most applicable in practice. Concentrating ourselves on this special type of free return trajectory, the corresponding planar asymmetric cases are studied. Then the studies are generalized to the three-dimensional case. The restrictions on the inclination angle of the probe at the perilune are discussed. It is found that the maximum inclination at the perilune between the probe’s orbit plane and the Moon’s orbit plane is restricted by the heights of the perigee and the perilune of the free return trajectory. However, the inclination at the perigee is nearly not affected by them. At last, a strategy to design free return trajectories in the real Earth-Moon system is proposed. Some numerical simulations are done to show the feasibility of this orbit design strategy. The discussions from the planar case to the spatial case and then to the real force model can also be applied to the other three types of planar free return trajectories. Key words: celestial mechanics: three-body problem, restricted problem— methods: numerical
1. INTRODUCTION In recent years, we are witnessing an upsurge of lunar exploration across the world. After the successful progresses of “Chang-e” projects, the manned lunar mission must be put on the agenda of China’s space administration. Different from the normal lunar mission, †
Supported by National Natural Science Foundation Received 2011–11–22; revised version 2012–02–23 A translation of Acta Astron. Sin. Vol. 53, No. 4, pp. 308–318, 2012 [email protected]
0275-1062/13/$-see front matter © 2013 Elsevier All rights reserved. c 2013B.V. 0275-1062/01/$-see front matter Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2013.04.007 PII:
184
Hou Xi-yun et al. / Chinese Astronomy and Astrophysics 37 (2013) 183–194
the foremost consideration must be the astronauts’ safety in the manned spaceflight to the Moon. Therefore the safety factor of the mission must be guaranteed as much as possible in the mission design. As for the trajectory design, this means that the space vehicle should be able to go back to the Earth safely even when the mission goes wrong. The “free return trajectory” often used in the last century by USA and former USSR in the manned lunar missions is right the offspring of this safety requirement[1] . If the injection engine fails at the perilune, the space vehicle can return freely back to the Earth without another maneuver. Although such kind of free return trajectory was often mentioned in many pieces of literature, there is a lack of the details of its design. Thus we devote this paper to a detailed study on this type of transfer orbit, aiming to provide some reference for the lunar mission in China. For the sake of analyzing the transfer orbit in the Earth-Moon system, the circular restricted three-body problem (CRTBP) consisting of the Earth, the Moon and the space probe is an appropriate dynamical model. In this paper, we present first the 4 symmetrical free return trajectories in the plane of lunar orbit under the CRTBP model, then we extend one of them that is relatively applicable to the 3-dimensional case. We study the variations of inclination of such transfer orbit near the perilune and the perigee. Based on the results under the CRTBP model, we finally present a practical trajectory design and the numerical example for the realistic Earth-Moon system.
2
CIRCULAR RESTRICTED THREE-BODY PROBLEM
The space probe is mainly affected by the gravitations of the Earth and the Moon when it travels in the Earth-Moon space. Since the eccentricity of the lunar orbit is quite small, the CRTBP is a good approximation of the dynamical system. Generally, a rotating coordinate frame is adopted, in which the two primaries (the Earth and the Moon) situate on the abscissa and the origin coincides with their barycenter. Such a barycentric synodic coordinate system is illustrated in Fig. 1. Surely the x− y plane is the orbital plane of the two primaries and the z axis is set to make it a right-handed orthogonal coordinate system.
Fig. 1
An illustration picture of the barycentric synodic frame (for the Earth-Moon system)
Set the average distance between and the total mass of the Earth and the Moon to be the units of distance and mass, the equations of motion of the probe (assumed zero-mass)
Hou Xi-yun et al. / Chinese Astronomy and Astrophysics 37 (2013) 183–194
in this synodic coordinate system read[2] ¨r + 2(−y, ˙ x, ˙ 0) = ∂Ω/∂r , Ω = (μ(1 − μ) + x2 + y 2 )/2 + (1 − μ)/r1 + μ/r2
185
(1)
where μ = mMoon /(mMoon + mEarth ), mMoon and mEarth are the masses of the Moon and the Earth, r = (x, y, z) is the position vector of the probe with respect to the coordinate origin (the barycenter), and r1 , r2 are the distances of the probe from the Earth and the Moon. The Earth and the Moon locate at (−μ, 0, 0) and (1 − μ, 0, 0) in this coordinate system. Obviously, different perturbations exist in the realistic Earth-Moon system and will result in a deviation from the CRTBP model. We will analyze first the free return trajectory in the CRTBP and then extend our analysis to the realistic dynamical system.
3
SYMMETRICAL FREE RETURN TRAJECTORIES IN THE MOON’S ORBIT PLANE
Take a transfer orbit with a perilune height of 100 km and the perigee heights of 200 km at both the departure and the return ends, it is symmetrical with respect to the x−axis in the synodic coordinate system since the perigee heights at both ends are the same. This symmetry implies that the velocity vector of the space vehicle at the perilune of the free return trajectory is perpendicular to the x−axis. Denote the state vector of the space vehicle at perilune (that is the intersection of the trajectory and the x−axis) in the synodic coordinate system by X 0 X 0 = (r0 , r˙ 0 ) = (x0 , 0, 0, y˙ 0 ) . (2) Since the position vector is perpendicular to the velocity vector, X 0 is the perilune of the transfer trajectory. Thus x0 = 100 km is defined by the perilune height, and the adjustable variables are the velocity component y˙ 0 and the one-way transfer time T . Here by “one-way transfer time” we mean the time consumed by the probe from the departure point to the perilune or the time from the perilune to the return point. They are equal to each other due to the symmetry of the trajectory. Denoting the geocentric position and velocity vectors of the probe by r and r˙ , the constraints on the transfer trajectory at both ends read r · r˙ = 0,
r = aEarth + 200 km ,
(3)
with aEarth being the Earth radius. It is worth noting that the same constraints in Eq. (3) are hold for both ends of the trajectory because of the symmetry. The constraints can be fulfilled by adjusting y˙0 and T through the solution finding method for the boundary value problem of the ordinary differential equations[3] . As an example, we illustrate in Fig. 2 the result of such a solution, in which the probe enters the trajectory from the far side of the Moon and the transfer time is T = 2.8634 d. In the left panel, the trajectory is shown in the synodic coordinate system, while in the right panel it is in the geocentric inertial coordinate (this is the same for Figs. 3 ∼ 6). The dashed curves represent the launch trajectory and the solid curves indicate the returning trajectory. The “EM” indicates that the distance unit is the mean Earth-Moon distance [L] = 3.84747981 × 108 m. For such free return trajectory illustrated in Fig. 2, the one-way transfer time T increases with the perigee height and also
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along with the perilune height[4] . Lastly, the trajectory is a prograde orbit in the vicinity of the Earth. A retrograde orbit exists as well, with T = 2.8256 d, as shown in Fig. 3. Similar to the one in Fig. 2, the one-way transfer time T increases with the perigee and perilune heights[4] .
Fig. 2 A free return trajectory departs from the Earth in a prograde orbit and reaches the far side of the Moon.
Fig. 3 A free return trajectory departs from the Earth in a retrograde orbit and reaches the far side of the Moon.
If the injection is on the near side of the Moon, the transfer time of the free return trajectory will increase dramatically. We plot such trajectories for the prograde (in Fig. 4) and retrograde departures from the Earth (in Fig. 5). The one-way transfer time T for these two trajectories are 13.7657 d and 15.0158 d , respectively. For the free return trajectories shown in Fig.4∼5, T increases with the perigee height, however, it decreases with the perilune height[4] . All these 4 types of trajectories shown in Fig. 4 and Fig. 5 confine themselves in the lunar orbit plane and are symmetric with respect to the x−axis. These trajectories are called “Type I symmetrical free return trajectories” and have been investigated in Reference [4].
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We will not discuss them in this paper, for more details please refer to Reference [4]. It is noteworthy that the trajectories illustrated in Figs. 2 ∼ 5 are not closed in the vicinity of the Earth, therefore they are not periodic trajectories.
Fig. 4 A free return trajectory departs from the Earth in a prograde orbit and reaches the near side of the Moon.
Fig. 5
A free return trajectory departs from the Earth in a retrograde orbit and reaches the near side of the Moon.
4. NONSYMMETRICAL FREE RETURN TRAJECTORIES IN THE LUNAR ORBIT PLANE Since to launch a space probe from the Earth to a retrograde orbit needs a big amount of energy, we may abandon such kind of retrograde trajectory. Moreover, a trajectory with the injection point on the near side of the Moon means a much longer transfer time, therefore such trajectory is not a good choice for manned space vehicle. Only the prograde free return trajectory reaching the far side of the Moon as illustrated in Fig. 2 has the merits of practicality and economy. We will focus on this kind of orbit in the following. But the methods we will use can be applied to other types of return trajectories.
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For a returnable lunar probe, the perigee height of the Earth-Moon transfer orbit is determined by the height of the launch vehicle at the end of its sliding motion, while the perigee height of the Moon-Earth transfer orbit is defined by the height of the probe when it reenters into the atmosphere. Since generally these two perigee heights are not necessary to be the same, the free return trajectory is not necessarily symmetric about the x−axis. Denote the height of perigee of the return trajectory by hF , and of the launch trajectory by hB . The state vector of the probe at the perilune is denoted by X 0 = (r0 , r˙ 0 ) = (x0 , y0 , x˙ 0 , y˙ 0 ) .
(4)
Because the trajectory is not symmetric any more, we have y0 = 0, x˙ 0 = 0. Denoting the position and velocity vectors of the probe at the departure time by rB and r˙ B respectively, and denoting the corresponding position and velocity vectors at the returning time by rF and r˙ F , the constraints now read ⎧ rB = aEarth + hB ⎨ rB · r˙ B = 0, (5) rF · r˙ F = 0, rF = aEarth + hF , ⎩ ˙ r0 · r0 = 0, r0 = aMoon + hM where aMoon and hM are the radius of the Moon and the perilune height. Now the adjustable variables include x0 , y0 , x˙ 0 , y˙ 0 , the forward integration time TF and the backward integration time TB (the trajectory is not symmetric about x−axis, now TB = TF ). As in the previous case the constraints in Eq. (5) can be satisfied by adjusting x0 , y0 , x˙ 0 , y˙ 0 , TB and TF through shooting algorithm. Our calculations show that the trajectory is nearly symmetric when hB and hF do not differ from each other too much, and the nonsymmetric property of the trajectory can be evident only when hB and hF are quite different from each other. As an example, we plot in Fig. 6 a trajectory with hB = 3.6 × 104 km, hF = 200 km and hM = 100 km. The time consumption from the departure point to the perilune is 2.9765 d, while the returning from the perilune to the Earth consumes 3.1844 d.
Fig. 6 A free return trajectory with hB = 3.6 × 104 km, hF = 200 km, and hM = 100 km
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5. FREE RETURN TRAJECTORY IN 3-DIMENSIONAL SPACE All the free return trajectories introduced in previous sections offer the backup trajectories only in the lunar orbit plane, so that the landing point has to be within ±5◦ around the equator, and this puts an unfavorable constraint on the fulfilment of some scientific tasks. In fact, for the sake of astronauts’ safety, in the early stage of the Apollo missions (including Apollo 8, 10 and 11) the free return trajectories were seriously considered in the mission design to ensure that the spaceship can safely return and land on the predetermined location when any failure happened to the spaceship, and such return trajectory can be achieved only through a minor orbit adjustment performed by the Reaction Control System (RCS) equipped onboard. But due to the constraints arising from the launching, reentry and recovery, it is not always possible in practice to confine the Earth-Moon and the MoonEarth transfer orbits onto the lunar orbit plane. The study on the free return trajectory has to be extended from the planar case to the case of 3-dimensional space. Such extension has been applied since the Apollo 12 mission. In the 3-dimensional case, the state vector of the probe at the perilune is X 0 = (r0 , r˙ 0 ) = (x0 , y0 , z0 , x˙ 0 , y˙ 0 , z˙0 ) .
(6)
While the constraints are the same as in Eq. (5), with x0 , y0 , z0 , x˙ 0 , y˙ 0 , z˙0 , as well as the forward and backward one-way transfer times TB , TF , being the adjustable variables. Clearly, six equations of the constraints cannot determine uniquely eight adjustable variables. However, we may find a solution by fixing z0 , z˙0 first, and then vary z0 , z˙0 to find other free return trajectories. When the perigee heights of the trajectory at the departure end and arrival end are equal to each other hB = hF , there exist two special 3-dimensional periodic orbits. One is symmetric with respect to the x − z plane in the synodic coordinate system, as shown in Fig. 7, and another is anti-symmetric about the x − z plane as shown in Fig. 8. For the former trajectory, the z-direction velocity component at the perilune is zero z˙0 = 0, while in the latter case the perilune is on the x − y plane, i.e. z0 = 0. Moreover, for the trajectory in Fig. 7 z0 = 1.1 × 10−3 (dimensionless unit), and the one-way transfer time is 2.8728 d, while for the trajectory in Fig. 8, z˙0 = 0.45 (dimensionless unit) and the one-way transfer time is 2.8412 d.
Fig. 7 A three-dimensional free return trajectory. At the perilune, z0 = 1.1 × 10−3 , z˙0 = 0.
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Fig. 8
A three-dimensional free return trajectory. At the perilune, z0 = 0, z˙0 = 0.45.
6. INCLINATION CONSTRAINTS AT PERIGEE AND PERILUNE Generally, after hB , hF , hM have been given, the variations of z0 , z˙0 at the perilune are confined in a given range. For an example with hB = hF = 200 km and hM = 100 km, the upper left panel in Fig. 9 shows the variation ranges of z0 and z˙0 . In this picture, each point (z0 , z˙0 ) represents a free return trajectory, and the contour line indicates the inclination of the probe orbit at the perilune with respect to the lunar orbit plane. Since the trajectory enters the lunar gravisphere in the retrograde direction, the inclination is larger than 90◦ . Obviously, due to the confinement arising from the variation of z0 and z˙0 , the inclination of the trajectory at the perilune is confined in a specific range. As the variation ranges of z0 and z˙0 increase with the perilune height, the maximal value of the inclination increases as well as shown by the upper right panel of Fig. 9 for another example with hB = hF = 200 km and hM = 1000 km. Similarly, the variations of z0 and z˙0 increase with the perigee height, and thus the inclination of the trajectory at the perilune, as shown in the lower left panel in Fig. 9, while we show an example of trajectory with hB = hF = 104 km and hM = 1000 km. The “adim” labeling along the axes of the four panels in Fig. 9 indicates that the values are given in dimensionless unit, which are defined as follows: [L] = 384747981.000000000 m,
[T ] = 375699.843898365 s,
[V ] = [L]/[T ] .
(7)
In all these four panels, the variation regions of z0 and z˙0 are symmetric about the z0 − and z˙0 −axis, respectively. This symmetry arises because hB = hF . However, if the difference between hB and hF is big, the asymmetry of the free return trajectory will result in a distinguishable asymmetry in the shape of the feasible regions of z0 and z˙0 . Such an asymmetric region is shown in the lower right panel in Fig. 9, in which hB = 36000 km, hF = 100 km and hM = 100 km. For the trajectories shown in the upper left panel of Fig. 9, we illustrate in Fig. 10 the contour lines of their inclinations at the perigee. Thanks to the symmetry of the transfer trajectory, it is necessary to discuss only the inclination of the near-Earth parking orbit at the departure end. This picture reveals that the inclination at the near-Earth end of the transfer trajectory may vary in the range 0 ∼ 90◦ , without being affected by any constraints at the near Moon end.
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Fig. 9 The feasible regions of z0 , z˙ 0 , and the contour lines of the inclinations at the perilune for different perigee and perilune heights. Upper left: hB = hF = 200 km, hM = 100 km; Upper right: hB = hF = 200 km, hM = 103 km; Lower Left: hB = hF = 104 km, hM = 103 km; Lower right: hB = 36000 km, hF = 100 km, hM = 100 km.
Fig. 10
The feasible regions of z0 , z˙0 , and the contour lines of the inclinations at the perigee for hB = hF = 200 km, hM = 100 km.
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7. RESULTS IN REAL EARTH-MOON SYSTEM Different from the CRTBP model, there are different kinds of perturbations in the real Earth-Moon system. However, the perturbations from other planets in the solar system and the non-spherical gravitation perturbations of the Earth and the Moon are ignorable, especially compared to the perturbations from the Sun and the eccentricity of the lunar orbit. Therefore, the real system in this paper consists of the gravitations from the Sun, the Earth and the Moon as mass points. Their initial positions and velocities are taken from the DE405 Ephemeris. The results for such a real system can be obtained by making proper corrections to the results for the CRTBP model introduced in previous sections. Denote the position and velocity of the probe in the luncentric synodic coordinate system by r and r˙ , and the position and velocity in the geocentric inertial coordinate system (we adopt here the International and R. ˙ The transformation between them can be Celestial Reference System (ICRS)) by R described by =R M + Cr, R
˙ = R ˙ M + C ˙ r + Cr˙ , R
(8)
˙ M are the position and velocity of the Moon in the geocentric inertial M and R where R coordinate system. The rotation matrix C is defined as follow[5] : M ˙ M ) M × R R (R eˆ1 = , eˆ3 = (9) C = (eˆ1 , eˆ2 , eˆ3 ); , eˆ2 = eˆ3 × eˆ1 . RM M × R ˙ M || ||R The state vector of the perilune in the luncentric synodic coordinate system is denoted by X 0 = (r0 , r˙ 0 ) = (x0 , y0 , z0 , x˙ 0 , y˙ 0 , z˙0 )
(10)
If we set z0 = z˙0 = 0, the adjustable variables include x0 , y0 , x˙ 0 , y˙ 0 and the forward and backward integration times from the perilune TF and TB . The constraints are now ⎧ ˙ B = 0, B · R ⎪ RB = aEarth + hB ⎨R ˙ (11) RF · RF = 0, RF = aEarth + hF , ⎪ ⎩ ˙ r0 · r0 = 0, r0 = aMoon + hM B, R F, R ˙ B , R ˙ F are the position and velocity vectors at the departure In these equations, R and return times in the geocentric inertial coordinate system, and they can be computed via Eq. (8) from rB , rF , r˙ B and r˙ F . Similarly, the constraints (Eq. (11)) can be satisfied by adjusting x0 , y0 , x˙ 0 , y˙ 0 , TF and TB through the shooting method. It is notewothy that we have fixed z0 = z˙0 = 0 in our computation, but in practice the predetermined variables should be chosen according to the requirement of the real space mission. A real trajectory is computed and illustrated in Fig. 11. The origin of the Earth-Moon instantaneous synodic coordinates locates at the barycenter of the Earth-Moon system, and the distance unit is the instantaneous Earth-Moon distance. Thus, not like in Figs. 1∼ 4, the “EM” in Fig. 11 is the instantaneous Earth-Moon distance and it changes with time. The space probe spends 2.9329 d on the way from perigee to perilune, it reaches the perilune at epoch MJD57700.9, and it spends another 2.7081 d on its way back to perigee. The same trajectory is plotted in Fig. 12 in the geocentric inertial coordinate system, where “Ae” indicates the radius of the Earth equator aEarth .
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Fig. 11 A free return trajectory in the real force model of the Earth-Moon system (drawn in the Earth-centered instantaneous synodic frame)
Fig. 12
The same trajectory as that in Fig. 11, but shown in the geocentric inertial coordinate system
8. DISCUSSION Not like the normal lunar explorations, the primary requirement for a manned lunar mission is the safety of astronauts onboard. The free return trajectory provides a backup trajectory for the lunar spaceship, ensuring that the spaceship can return to the Earth safely even when the orbit maneuver fails at the perilune. In this paper, based on the circular restricted three body problem, we systematically studied the different kinds of free return trajectories with different boundary constraints. First, adopting the planar circular restricted three body problem as the dynamical model, we calculated several symmetric free return trajectories in the lunar orbit plane, and the asymmetric trajectories were discussed as well. Then, considering the requirements of real space mission, in three dimensional space we described the constraints on the inclinations of the transfer trajectories in the luncentric and geocentric coordinate systems (in both cases, the reference plane is the lunar orbit plane). And finally, we presented the method for designing the free return trajectory in the real Earth-Moon system, and an example trajectory was given. The work presented in this paper may provide certain references for the orbit design of the future “Chang-e” projects that will perform the “collection - return” mission and even the manned lunar mission, and for the backuptrajectory design.
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NASA TN D-1833, 1963