ISSN 00109525, Cosmic Research, 2010, Vol. 48, No. 5 pp. 479–484. © Pleiades Publishing, Ltd., 2010. Original Russian Text © R.R. Nazirov, N.A. Eismont, 2010, published in Kosmicheskie Issledovaniya, 2010, Vol. 48, No. 5, pp. 491–496.

Gravitational Maneuvers as a Way to Direct Small Asteroids to Trajectory of a Rendezvous with Dangerous NearEarth Objects R. R. Nazirov and N. A. Eismont Space Research Institute, Russian Academy of Sciences, ul. Profsoyuznaya 84/32, Moscow, 117997 Russia Received February 24, 2010

Abstract—A possibility to prevent collisions with the Earth of dangerous celestial bodies by directing at them small asteroids is considered. It is proposed to solve this problem using a gravitational maneuver near the Earth. DOI: 10.1134/S0010952510050175

1. INTRODUCTION The discovery of the Apophis asteroid has shown that the hazard of a collision with the Earth of danger ous celestial bodies classified by parameters of their orbits as nearEarth ones is real. The bodies with dimensions exceeding 20–30 meters are referred to as dangerous. The abovementioned asteroid Apophis has a diameter of about 270 meters [1]. As a result of recognition of the asteroid hazard level, studies of this problem became of more system atic and purposeful character. The problem can be divided into two closely related parts: detection of dangerous celestial objects (they are mainly asteroids) with the following determination of parameters of their orbits and estimation of the probability of a colli sion with the Earth, and then determination of possi bilities and ways of preventing the collision. Currently many ways of changing the asteroid orbits with the aim of deflection of them from the tra jectory of encounter with the Earth are proposed. Among them one can mention such ways as direction to the asteroid of space vehicles [2] and a change of reflecting characteristics of asteroid surface [3]. In this paper we propose to direct to the dangerous asteroid another asteroid of sufficiently small dimen sion that within the available technical capabilities one can control parameters of its flight near the Earth in the regime of a gravitational maneuver. The aim of the latter is to change the trajectory of the small asteroid in such a way that the resulting orbit would pass through the dangerous asteroid. 2. AIMING OF A SPACECRAFT TO ASTEROID A substantial number of papers have been pub lished dedicated to the problem of aiming of a space craft to asteroid. The goal of these papers was, as a

rule, to choose under existing limitations such an orbit of the spacecraft hitting the asteroid which would lead to the maximal distance of the asteroid flyby from the Earth. In a substantial degree, it is equivalent to mini mization of the probability of encounter of the aster oid with the Earth after an impact of the spacecraft to the asteroid surface. However, such formulation of the problem meets considerable difficulties. In particular, the knowledge of physical characteristics of the aster oid, determining what will be the character of asteroid destruction and, accordingly, the velocity vector impulse obtained by the asteroid as a result of the impact is required. The latter depends also on the meeting angle of the spacecraft with the asteroid sur face. Under an assumption of completely inelastic impact without outburst of the substance formed at the collision, we have the simplest case when the asteroid obtains the momentum of the spacecraft calculated on the basis of its velocity relative to the asteroid. Note that the maximal velocity of the spacecraft rendezvous with the asteroid does not mean automat ically a maximal leading away of the asteroid from the Earth during the flight. The optimization process includes the choice of start parameters of the upper stage of the carrier from a low nearEarth orbit which at the hitting of the asteroid by the spacecraft would enhance maximally the flyby distance. Such an approach can be considered as a classical one. At the same time, other concepts are suggested in publica tions, for example, the use of small thrust engines at the flight phases following the spacecraft launch. In the paper by Dunham and Genova [4] published in this issue, a proposal is mentioned to use gravita tional maneuvers near the Earth for aiming spacecraft to dangerous asteroids. This idea was suggested as a part of the SHIELD program [5]. In the on duty regime, such spacecraft are located at highenergy heliocentric orbits and in case of a need the spacecraft

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Fig. 1. Collision of the striker of the NearShoemaker spacecraft with the nucleus of the 9P/Tempel comet.

is transferred to the Earth flyby trajectory with the help of a small velocity impulse. The choice of the flyby parameters is performed in such a way that the result ing trajectory of the spacecraft would pass through the asteroid. The realization of such an approach makes it pos sible with relatively small consumption of the propel lant to obtain much larger (in comparison with the classical case) velocities of the spacecraft impact to the asteroid surface varying from a few km/s to ten and more km/s. However, even at such high velocities of encounter, the change in the asteroid velocity is very small because of the huge difference in the masses of the spacecraft and asteroid. For example, in the case of Apophis its mass at the diameter of 270 m and density of rock it consists of is estimated by a value of about 25 million tons [1], while the spacecraft launched to the trajectory of asteroid interception can have the mass not exceeding 5 tons. This means that the mag nitude of the vector of asteroid velocity change as a result of the impact will be equal (at the impact veloc ity of 10 km/s) to 0.002 m/s. Ivashkin and Chernov [2] estimated the obtained deviations of an asteroid from the Earth due to the impact of a spacecraft accelerated by electrojet engine up to reaching the impact velocities exceeding 20 km/s for fairly broad list of asteroids by their dimensions close to Apophis. The mass of the space craft was assumed to be close to the one accepted in our estimates. The time of flight after the impact is assumed to be equal to one–two years. In this case, the obtained deviations of the asteroid do not exceed a few

thousand km (for inelastic impact), which cannot be considered as a sufficiently convincing argument in favor of the use of the described method of asteroid deflection in practice. At the same time, the methods of preventing colli sions of asteroids with the Earth proposed in the above mentioned papers are considered as practically realiz able within the frames of current technologies of space missions, including the case of targeting the spacecraft at asteroids with the use of gravitational maneuvers near the Earth. As a confirmation of this point of view, one can consider the European Space Agency project Rozetta [6] whose final aim is landing on the nucleus of the Churyumov–Gerasimenko comet in 2014. In the process of execution of this continuing mission, three gravitational maneuvers near the Earth and one maneuver near Mars were conducted. The maneuvers were planned in such a way as (besides the main goal: approaching the comet nucleus and landing on the nucleus surface) to provide for flybys near asteroids at a distance close enough for their observation. The NASA Deep Impact Project [7] is an example of successful solution of the problem of striking a small celestial body (a comet nucleus) with high relative velocity. As a part of this project, on January 12, 2005 a spacecraft was launched to the nucleus of the 9P/Tempel comet. The goal of the project was hitting the nucleus by a special copper striker with the aim of creation as a result of the impact of a gas cloud (see Fig. 1) and the following spectrometric measurements of the emission of this cloud by the devices onboard the flyby spacecraft. The spacecraft reached the comet nucleus on July 4, 2005. The nominal minimal dis tance of the flyby module was 500 km. The striker was separated from the flyby spacecraft one day before the rendezvous with the nucleus. Prior to the separation, pointing of the spacecraft to the comet nucleus was performed with an error of ±50 km. This was achieved due to correcting impulses, the sum of two final impulses being equal to 6 m/s. After the separation, the maneuver of leading away the flyby module from the trajectory of hitting the nucleus was executed. The spacecraftstriker has its own autonomous aiming sys tem, and due to three correcting impulses (see Fig. 2) determined and executed by this system hitting of the striker to the comet nucleus with dimensions 7.6 by 4.9 km was ensured. The meeting velocity was 10.3 km/s. 3. METHOD OF CONTROLLING A GRAVITATIONAL MANEUVER The idea of the maneuver is based on using gravita tional attraction of the planet at the flyby of the planet by the body participating in this operation. Most easily this maneuver can be represented assuming that on some sphere of the planet influence, the gravitational field of the planet near which the maneuver is realized becomes so dominating that one can neglect the grav itational force of the Sun. (Maneuvers near planets of COSMIC RESEARCH

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Comet Nucleus

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Fig. 2. The concluding operations on aiming the striker to the nucleus of the comet and on obtaining scientific information via the flyby spacecraft ((1) observations 30 min after the flyby point; (2) zone of the use of protection screen; (3) transmitting the flyby data (recording); (4) transmitting the flyby data (real time); (5) the flyby spacecraft at the moment of impact; (6) maneuver of leading away the flyby spacecraft; (7) striker maneuver 12.5 min before; (8) striker maneuver 35 min before; (9) striker maneu ver 90 min before; (10) beginning of the autonomous navigation 2 h before; and (11) separation of the striker 24 hours before the impact).

the Solar System are considered). Usually as a body participating in the maneuver a spacecraft is consid ered with the mass which can be neglected when its motion in the gravitational field is analyzed. In our case, this is an asteroid with a mass that is three orders of magnitude higher than any possible mass of the spacecraft. However, even in this case it is negligibly small in comparison with the planet mass, so that the approach to the maneuver analysis remains the same. Under the accepted assumption, we have a Keple rian motion in the frame of reference related to the Earth at a nonzero velocity on the influence sphere, that is, the motion along a hyperbola. The velocity vector of the motion at infinity relative to the planet is calculated as a difference in vectors of the spacecraft (asteroid) and planet. It is assumed to be equal to the velocity vector of the spacecraft at the point of meeting of the planet and spacecraft and is calculated assuming that the gravitational field of the planet does not influ ence the spacecraft. As a result of the planet flyby, the velocity vector of the spacecraft VA at infinity relative to the planet turns by some angle α depending on the radius r of the pericenter and magnitude of the velocity vector at infinity V = |VA|. This dependence is deter mined by formula (1) from [8]: sin(α/2) = 1/(1 + rV2/μ),

(1)

where μ is the gravitational constant of the planet (the Earth). For the accepted pericenter radius of the flyby tra jectory and known flight velocity at infinity, the dis tance b of the straight line going along this vector from the planet center is determined unambiguously. The set of such lines forms a circular cylinder with the axis passing through the planet center. We denote the radius of this cylinder as b. This quantity is easily COSMIC RESEARCH

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determined from relation (2) of conservation of the angular momentum: b VA = rVπ,

(2)

where Vπ is the velocity in the pericenter calculated from energy conservation relation (3):

V π2 = 2μ/r + V 2.

(3)

Thus, choosing the pericenter radius, we choose the rotation angle α of the relative velocity vector at infinity. Choosing the position of the vector on the cyl inder of velocities, we choose the plane of rotation of the relative velocity vector. For the chosen α, we obtain a set of possible vectors V of the relative velocity forming a cone with the axis along the initial (before the maneuver of the relative velocity vector) cone, the length of the generatrixes of this cone being equal to the magnitude of the relative velocity vector. If the α angle can be obtained within the limits from zero to 180°, then possible positions of the relative velocity vector cover the entire sphere with the radius equal to the magnitude of this vector. The sphere center is located at the tip of the planet velocity vector. Accord ingly, the tips of resulting vectors of the velocity of the spacecraft (controlled asteroid) after the gravitational maneuver in the heliocentric frame of reference are located on this sphere. Indeed, the rotation angle of the vector V is limited, because the radius of the peri center at the flyby of the planet is not equal to zero. If we accept this radius equal to 7500 km, then for the relative velocity at infinity equal to 3, 6, and 10 km/s the α angle is equal to 117.6°, 73.2°, and 40.6°, respec tively. These values limit the sphere of possible relative velocity vectors down to the spherical sectors of above indicated dimensions. Nevertheless, it is easy to calcu late that the maximal velocity impulses got in this case

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Sun Moon

Fig. 3. Trajectory of the close flyby near the Earth of the small asteroid 2010 AL30 on January 12–13, 2010. The tick step on the trajectory is equal to 2 hours.

reach fairly large values if one compares them with the classical values obtained with jet engines: 5.13, 7.08, and 6.94 km/s, respectively. 4. CHOICE OF ASTEROID AS A CONTROLLED MISSILE The choice of an asteroid as a platform for installa tion of the systems which would make it possible to aim it to the celestial object in order to deflect the lat ter from the trajectory of collision with the Earth is determined by the possibility to meet a series of requirements. On the one hand, the asteroid should have sufficient dimensions in order to be discovered by the available or technically realized observation means with the subsequent determination of parameters of its orbit with the accuracy needed for spacecraft landing on its surface. On the other hand, the apparatus should have sufficient mass for solving the main problem: suf ficient deflection of the target asteroid from its initial trajectory. In addition, the asteroid orbit should be such that the realization of landing of the spacecraft on its sur face would not require too high consumption of the propellant. In particular, it is assumed that the charac teristic velocity needed at the start from a low near Earth orbit will not exceed 3.5 km/s (approximately the value which would be needed for missions to Mars or Venus), and the velocity for realization of landing on the asteroid surface from the velocity of transfer from the Earth to the asteroid does not exceed 2 km/s. In this case, as is shown by approximate estimates based on the published characteristics of the ProtonM carrier with the Briz booster block and on the assump tion that the specific impulse of the flight spacecraft

engine is equal to 3000 m/s, an apparatus of a total mass of 2800 kg can be delivered to the asteroid sur face. If we assume that 2000 km out of this mass is the propellant for subsequent operations, we can estimate possibilities of maneuvers of such controlled asteroid assuming some minimally permissible (from the point of view of observation capability) estimate of its dimensions. As it is known [9], the Lincoln Laboratory of the Massachusetts Technological University discovered on January 10, 2010 the asteroid 2010 AL30. This asteroid has flied near the Earth at a minimal distance of 128 thousand km at 13:46 UT on January 13, 2010. The trajectory of its flight is shown in Fig. 3 [9]. The processing of the asteroid observations showed that its perigee was near the Venus orbit and the apogee was in the region of the Mars orbit, its orbital period being close to that of the Earth. The dimensions of this aster oid referred to the nearEarth class are 10–15 meters. The number of such asteroids in the nearEarth space is about 2 millions. One can assume [9] that an aster oid of such dimensions flies near the Earth at the dis tances within the Moon orbit approximately once a week on the average. This gives us grounds to believe that in the future new asteroids of such dimensions will be discovered, and for our estimates we can assume a possibility to use them for the aims discussed above. Assuming that the asteroid has a diameter of 12 meters and the density of the substance forming it is 2 g cm–3 (as an example, we mention the (4179) Tou tatis asteroid with a density of 2.1 g cm–3), we obtain its mass equal to 1810 tons. At deploying on the sur face of such an asteroid of rocket blocks delivered by two Proton carriers with the working medium mass of 4 tons, the reserve of the characteristic velocity for maneuvers of the controlled asteroid is 6.6 m/s. 5. CONSTRAINTS ON AIMING OF ASTEROID MISSILE TO ASTEROIDTARGET Let us now consider the problem of transferring the asteroid chosen for the role of a missile from the initial orbit to the orbit of the flyby near the Earth with the parameters corresponding to the requirements of the gravitational maneuver for hitting the asteroidtarget. For estimation of the possibilities of aiming with constraints on the available reserve of the characteris tic velocity, we introduce, as it is usually done at plan ning missions to planets, the concept of picture plane. This plane passes through the center of the celestial body (in our case it is the Earth) to which the space craft is directed and orthogonal to the spacecraft’s rel ative velocity vector at infinity reduced to the body center, as it has been described above. In this plane, we direct one axis (Y) along the line of intersection of the picture plane with the ecliptic, and let another axis (Z) be orthogonal to Y in the picture plane. We assume that the orbit of the asteroid chosen as a missile is close to COSMIC RESEARCH

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the Earth’s orbit. Then, at the angular distance of 90 degrees of the maneuver point of the picture plane at the moment of reaching it, the impulse of 4 m/s turns the asteroid orbit plane by the value correspond ing to the change in the asteroid coordinate Z at the flyby at 20000 km. The velocity impulse equal to 1 m/s shifts the other coordinate at the flyby by 88000 km, if it is imparted one orbit prior to reaching the picture plane. Taking into account the above estimate of the number of asteroids (2 million) suitable for our prob lem and flying by at distances less than 400 thousand km from the Earth, one can hope for a possibility to choose an asteroid suitable for the role of a missile for deflecting dangerous objects. However, one has to take into account that, in addition to consumption of the characteristic velocity needed for launching to a nom inal orbit of the gravitational maneuver, one should take into account consumptions to conducting orbit corrections both before the gravitational maneuver and after it. Fairly accurate estimate of the values of correcting impulses needs a more detailed and partic ular analysis. Under our assumptions, the net value of 1.6 m/s remains for the correction, which is not enough if one compares it, for example, with the Deep Impact mission. At the most optimistic estimates of the achievable accuracies of determination of motion parameters of both the controlled asteroid and the asteroidtarget, the correction can require 8–10 m/s. This means that about 8 tons of the working medium will be needed to be delivered to the asteroid of the assumed mass, which means 4 launches of carriers of the Proton class. The improvement of the accuracy of determination of the orbit parameters of the con trolled asteroid and target is an alternative to such solution. Roughly speaking, the consumption of the working medium is proportional to the error in deter mination of the orbit parameters. 6. A SCENARIO OF PREPARATION AND EXECUTION OF THE MISSION While preparing the mission scenario, two types of base concepts are possible: one concept is based on the universal approach to the choice of an asteroidtarget, when the controlled asteroid is chosen not for deflec tion of a particular dangerous asteroid, but for some class of celestial objects. In the second concept, the mission is planned for the problem of deflection of a particular asteroid considered to be hazardous. The second type seems to be more understandable for the analysis. Since practically no asteroids of the suitable dimension, in spite of their huge number, were detected (the authors know only one: 2010 AL30), in the first place one should realize a program of detect ing and cataloging such asteroids. Then, a choice of an appropriate candidate should be performed, as it has been described in the previous section. COSMIC RESEARCH

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To launch a spacecraft and to land it on the asteroid can be realized using practically tested technology, as it has been done in the NEARShoemaker project at landing the apparatus onto the asteroid surface [10]. However, here new problems arise related to small dimensions of the asteroid and necessity to use an engine allowing one to execute orbital maneuvers of the asteroid. We do not consider methods of solving these problems, not suppusing them to be impractica ble. After deploying on the asteroid all necessary sys tems needed for controlling its motion, we come to the problems of targeting which have no fundamental dif ference with those for spacecraft. 7. CONCLUSIONS If a problem is posed to deflect a dangerous celes tial body from the trajectory of collision with the Earth by aiming at it a comparatively small asteroid which can be detected and whose orbit can be determined, then one can technically realize, most likely, the deployment on such asteroid of necessary systems of navigation and control as well as a propulsion system with subsequent execution of a gravitational maneuver near the Earth. Estimating the efficiency of such an approach, one can take as a comparison criterion the changes in the momentum of the dangerous body at the same meeting velocity and the same mass of the spacecraft launched to the flight trajectory. In the case when an asteroid with a mass of about 1800 tons and the meeting velocity of 10 km/s is chosen as a con trolled missile, the change in the asteroidtarget momentum is equal to 1.8 · 1010 kg m s–1. In this case it is quite realistic to solve the problem by injecting into an intermediate orbit four spacecraft with a mass of 5500 kg each. Their direct aiming to the dangerous asteroid will allow one to change its momentum by 2.2 · 108 kg m s–1. Thus, the use of the proposed method of deflecting the asteroid from the trajectory of collision with the Earth can be by a factor of 81.8 more effective. If a more detailed analysis shows that for realization of the described mission a larger reserve of the characteristic velocity is needed for correction of the motion parameters, the proposed method should gain in efficiency still more. It is obvious that this method is much more com plicated than that used for the comparison, and the most uncertain part of it is the estimate of possibility to catalog asteroids suitable for the use in the role of mis siles. However, the programs of search for and obser vation of dangerous asteroids being currently devel oped in the world make it possible to hope for over coming the existing difficulties in solving this problem. REFERENCES 1. JPL SmallBody Browser: 99942 Apophis (2004 MNMN4), http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=99942

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2. Ivashkin, V.V. and Chernov, A.V., Optimal Trajectories for Space Flight to NearEarth Asteroids Using Elec tric Jet Propulsion System, Proc. of 17th Intern. Sympo sium on Space Flight Dynamics, Keldysh Inst. of Applied Math., 2003, vol. 2, pp. 328–342. 3. Hyland, D.C., Altwaijry, H.A., Ge, S., et al., Perma nentlyActing NEA Mitigation Technique via the Yark ovsky Effect, Kosm. Issled., 2010, vol. 48, no. 5. 4. Dunham, D.W. and Genova, A.L., Using Venus for Locating Space Observatories to Discover Potentially Hazardous Asteroids, Kosm. Issled., 2010, vol. 48, no. 5, pp. 433–439 (Cosmic Research, pp. 424 429). 5. Gold, R.E., SHIELD: A Comprehensive EarthPro tection Architecture, Adv. Space Res., 2001, vol. 28, no. 8, pp. 1149–1158.

6. ESA Science and Technology: http://sci.esa.int/sci encee/www/area/index.cfm?fareaid=13 7. Lakdavala, E., Deep Impact Successfully Splits in Final Hours before Comet Encounter, July 3, 2005, Planetary News: Asteroids and Comets, 2005. http://www.plane tary.org/news/2005/0703_Deep_Impact_Successfully_ Splits_in.html 8. Spacecraft Attitude Determination and Control, Wertz, J.R., Ed., D. Reidel Publishing Company, 1985. p. 60. 9. Yeomans, D., Chodas, P., Chesley, S., and Giorgini, J., Small Asteroid 2010 AL30 Will Fly past the Earth, NASA/JPL NearEarth Object Program Office, January 12, 2010. http://neo.jpl.nasa.gov/news/news167.html 10. NASA Science Missions: http://nasascience.nasa. gov/missions/near

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