On the Orbit of the Lunar Science Mission GRAIL

Marco Giancotti

Abstract Gravity Recovery and Interior Laboratory (GRAIL) is a NASA mission launched in September 2011 with the purpose of accurately measuring the gravitational field of the Moon. The twin spacecraft traveled from Earth to the Moon in 112 days using a type of orbit different from the traditional ones, exploiting the gravitational effect of the Sun. This work analyzes the properties of this orbit, its launch conditions and the reasons behind its choice.

月探査機 GRAIL の軌道について

Abstract Gravity Recovery and Interior Laboratory (GRAIL)、月の重力分布の 測定を目的とした NASA の月探査機である。２０１１９月に打ち上げら れ、２機の宇宙機が僅かに異なる軌道を辿りながら、１１２日間をかけ て月に到着した。その軌道の設計は太陽重力をも利用していて、本来の ものとは大きく違う原理や特徴を持っている。本研究では、 GRAIL の打 ち上げ期間の選択や、月までの軌道の特性を解析した。

1 Introduction

Gravity Recovery and Interior Laboratory (GRAIL) is a science mission to the moon developed by NASA and launched in September 2011 [1, 2]. Its sci- entific objective is the detailed determination of the gravity field of the moon by means of a ranging system. The mission uses the same technology that was demonstrated by GRACE, which mapped the gravity field of the earth. It comprises of two twin spacecraft at a distance of 60 to 225 km between each other on the same low lunar orbit (∼55 km), with no effort done to keep this length constant. The fine measurement of the variation of this distance allows for the extraction of gravitational information [3, 4]. As an additional instru- ment GRAIL also carries MoonKam, an optical camera with outreach purposes. At the time of this writing GRAIL has completed all its main objectives and has entered an extended mission phase at lower altitudes for a higher-precision mapping.

1 Figure 1: Gravity gradient of the sun in the proximity of the earth (at the origin). The direction of the sun is shown.

This study is concerned with the unusual transfer orbits that were used by GRAIL to reach the moon. Moreover, it explores the reasons for the very long period of launch opportunities lasting 21 days and nonetheless using the same baseline trajectory structure.

2 Low-Energy Transfers

Instead of a standard direct transfer taking less than 5 days, the spacecraft employed a so called “low-energy transfer”, which initially moves very far away from the earth, up to 1.5 × 106 km, only to fall back towards it and reach the moon with a low speed and a favorable angle. This kind of transfer allows for considerable propellant savings at arrival at the cost of much longer flight times. It has been demonstrated in space already by several missions like Hiten, SMART-1 and Genesis. Classical transfer methods like Hohmann transfers are usually designed with a patched-conics approach that takes into consideration only 2 bodies at a time. The computations in this case are straightforward and any other weaker effect can be added later as a perturbation to refine the trajectory. Low-energy orbits like the ones used by GRAIL, however, are possible thanks to the 4th-body dynamical effects caused by the sun, and cannot be explained in a 2-body or in a 3-body system.

2.1 Tidal forces The gradient of the solar gravitational field in the surroundings of the earth is expressed by the tidal force tensor [ ] GM Φ = sun 3Rˆ Rˆ − u r R3

2 Figure 2: Orbits of the two GRAIL spacecraft (red and blue) at the beginning and end of the period of possible launch (NASA/JPL). where R is the Sun-Earth distance vector, R^ R^ is the dyadic product (or outer product) of the unit vector in the R direction with itself, u the unit dyadic and r is the Earth-spacecraft distance vector. The above expression neglects terms above the second order in ra/R where ra is the apogee distance. The result of this gradient is a force that pushes the spacecraft away from the earth on the Sun-Earth line and towards the earth in the plane normal to that line, as can be seen in Figure 1. The intensity of the force also grows with the distance from the earth.

2.2 GRAIL’s Orbit Although solar tidal forces are usually negligible perturbations, low-energy transfers like the one used by GRAIL exploit them by reaching out far from the earth with an appropriate angle with respect to the sun [5]. The 2-body osculating orbital elements change throughout the flight and, upon reaching the moon, the spacecraft’s velocity has a magnitude and direction similar to it, leading to smaller ∆v requirements. The shapes of GRAIL’s orbits are shown in Figure 2 for the two spacecraft and at two different times, that is, the first and the last day of possible launch. The reference is rotating so that Sun and Earth are fixed to the same horizontal line. A special aspect of the design of GRAIL’s transfer to the moon is the fixed arrival date. Although the launch date could have been almost any day between September 8 and October 3 2011, the arrival of the first spacecraft was scheduled at December 31 2011 in any case, and the second one exactly one day later. The transfer could therefore take from 89 to 115 days. The launch period only had one gap of 5 days due to the passage of the moon very close to the orbit’s path. GRAIL was eventually launched on September 10 and successfully entered into Moon orbit on the designated dates.

3 3 GRAIL’s Orbit Design

The transfer orbit designed for the mission is peculiar not only because it uses a low-energy strategy, but also because of its unusually large launch period. Moreover, its ∆v is not necessarily the absolute minimum that can be achieved with that kind of trajectory. The choice of these design features was guided by the specific properties and requirements of the mission [6].

3.1 Why a Low-Energy Orbit A low-energy orbit was chosen as GRAIL’s means of reaching the moon for several reasons. This type of orbit has the drawback of long transfer times, which can be undesirable for the increased monitoring and operation costs. However there are the following advantages:

• Lower ∆v requirements: for the final injection, GRAIL used approxi- mately 180 m/s instead of the 310 m/s that would have been necessary with a high-energy transfer. • Higher flexibility: the launch date can be altered considerably while keep- ing the arrival date fixes, using only small variations in the mid-course maneuvers.

• Wider time margins: longer transfers result in more time to monitor the two spacecraft one at a time, to perform complete instrument checkout and stabilization and wait for outgassing [7]; the spacecraft are almost ready for their main science phases when they reach the moon.

These considerations have led to the choice of this kind of transfer in the case of GRAIL. Note that ∆v is not the only optimization parameter.

3.2 Choice of the Arrival Dates A lunar eclipse of some degree happens roughly once every six months and poses a potential hazard for GRAIL considering its payload and instrumenta- tion. Moreover, the ∆v of low-energy transfers to the moon oscillate with the lunar period, reaching a minimum roughly every month. For these reasons the dates for the lunar orbit insertion were selected to be December 31 2011 and January 1 2012, at the first ∆v minimum after one such eclipse. This choice granted GRAIL enough time to safely complete its main mission objectives be- fore the following eclipse.

3.3 Transfer Optimization Algorithm For each day in the launch period there were two instantaneous launch op- portunities, each leading to gradually different orbit designs in order to reach the moon at the desired date. The family of orbits covering the whole period

4 Figure 3: Example of a typical relation between transfer ∆v and launch date for a low-energy orbit. The dots are the optimal transfers, one for each day. was computed with the 2-step optimization procedure that follows, applied at each “launch date”–“moon arrival date” pair [6]:

1. Backward propagation from the desired conditions at the moon, tuning it in order to find a perigee date corresponding to the chosen launch date; at this stage the boundary conditions at Earth are still mostly incorrect. 2. Forward propagation from the desired launch conditions, optimizing the trajectory through the launch parameters and two mid-course maneuvers in order to obtain insertion into the final segment calculated in the previ- ous step.

By repeating this procedure for every day in a wide period, it is possible to construct a plot like the one in Figure 3. In general, the curve of the ∆v forms a “valley” with a minimum value on some day. The choice of some threshold cost ∆vt (dashed horizontal line in Figure 3) translates into a finite period of time in which the transfers require less than that value. The transit of the moon close to the path of the transfer may perturb it disrupting the launch period and making launch undesirable for a few days, like in the case of GRAIL. The launch period was established in a trade-off between the mission’s total ∆v budget and the duration of the launch period (and therefore the mission’s flexibility).

4 Conclusions

The properties of the lunar transfer orbit of NASA’s GRAIL mission were studied, and the reasons behind their choices were explained. A low-energy orbit was judged appropriate for this mission because of its low ∆v requirements and its high flexibility. In fact, these two features were the main trade-off parameters in the determination of the orbit’s details, including the launch period and the mid-course maneuvers.

5 References

[1] R. Roncoli, K. Fujii, Mission Design Overview for the Gravity Recovery and Interior Laboratory, in: AIAA/AAS Astrodynamics Conference, Washing- ton, DC, 2010. [2] G. Havens, J. Beerer, Designing Mission Operations for the Gravity Recovery and Interior Laboratory Mission, in: spaceops2012 proceedings, 2012. [3] T. L. Hoffman, GRAIL: Gravity mapping the moon, 2009 IEEE Aerospace conference. [4] S. Hatch, R. Roncoli, H. Sweetser, GRAIL Trajectory Design: Lunar Orbit Insertion through Science, in: AIAA/AAS Astrodynamics Specialist Con- ference, no. AIAA 2010-8384, AIAA/AAS, Toronto, Ontario, Canada, 2010.

[5] C. Circi, P. Teofilatto, On the Dynamics of Weak Stability Boundary Lunar Transfers, Cel. Mec. Dyn. Astr. 79 (2001) 41–72.

[6] M. Chung, S. Hatch, J. Kangas, S. Long, R. Roncoli, H. Sweetser, Trans- Lunar Cruise Trajectory Design of GRAIL (Gravity Recovery and Interior Laboratory) Mission, in: AIAA/AAS Astrodynamics Specialist Conference, no. AIAA 2010-8384, AIAA/AAS, Toronto, Ontario, Canada, no. August, 2010. [7] T.-H. You, P. Antreasian, S. Broschart, K. Criddle, E. Higa, D. Jeffer- son, E. Lau, S. Mohan, M. Ryne, M. Keck, Gravity Recovery and Interior Laboratory Mission (GRAIL) Orbit Determination, in: 23rd International Symposium on Space Flight Dynamics, no. 1, 2012.

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