David Dunham(1,3), Natan Eismont(2,3), Michael Boyarsky
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- 2. Gravity assist maneuver as a tool to target asteroid-projectile to hazardous near Earth object.
- 3. Lambert problem as the principal tool to design trajectories design using gravity assist maneuvers.
- Figure 2. Trajectories of Earth, Apophis, asteroid-projectile 2011 UK and spacecraft transfer trajectory from Earth to 2011 UK
- Table 1. Results of the selected candidate asteroids and the orbit designs
- 4. Optimization of spacecraft transfer trajectory with start from low Earth orbit (LEO) and landing on asteroid-projectile surface.
- Table 2. Key orbital parameters for transfer mission of the spacecraft to asteroid-projectile
- Duration of transfer, days ∆V s , km/s ∆V t
- Figure 3. Optimal transfer trajectory to 2011 UK asteroid with minimum total delta-V
- Table 3. trajectory parameters corresponding minimum delta-V
- 6. Analysis of results for ideal cases
- 7. Resonance orbits as the basis for constructing a planetary defense system
| DEFLECTING HAZARDOUS ASTEROIDS FROM COLLISION WITH THE EARTH BY USING SMALL ASTEROIDS
IAA-PDC13-04-16P David Dunham(1,3), Natan Eismont(2,3), Michael Boyarsky(2,3), Anton Ledkov(2,3), Ravil Nazirov(2,3), Eugene Chumachenko(3,2) and Konstantin Fedyaev(2,3) (1)KinetX, Inc., 7913 Kara Ct, Greenbelt, MD 20770, USA, +1-301-526-5590, david.dunham@kinetx.com (2)Space Research Institute of Russian Academy of Science, 84/32 Profsoyuznaya Str., 117997, Moscow, Russia, +7 916 628 6139, NEismont@iki.rssi.ru (3) National research university ‘Higher school of economics’, per. Trekhsvyatitelskiy B., d. 3, 109028, Moscow, Russia, +7-495-917-07-50, kommek@miem.edu.ru Abstract: The main idea of the proposed approach consists of targeting a very small asteroid to impact a larger dangerous one. The minimum size of this small asteroid is determined by the ability to detect it and to determine its orbit. The small object may have a diameter of about 10 -15 meters. Asteroids are selected from the near-Earth class that have a fly-by distance from Earth of the order of hundreds of thousands of kilometers. According to current estimates, the number of near Earth asteroids with such sizes is high enough. So there is a possibility to find the required small asteroid. Further, the possibility is evaluated of changing the small asteroid’s orbit so that by application of a very limited delta-V impulse to the asteroid, the latter is transferred to a gravity assist maneuver (Earth swingby) that puts it on a collision course with a dangerous asteroid. It is obvious that in order to apply the required V pulse it is necessary to install on the small asteroid an appropriate propulsion system with required propellant mass. A control system similar to that used on a spacecraft is also necessary. The Keck study [1] has already shown the feasibility of performing the above steps to change the orbit of similar asteroids to target them to a lunar orbit; our idea is similar in orbital energetics to that, only for slightly larger objects and targeting them to an Earth B-plane point rather than the vicinity of the Moon. Of course, any real test of this (or of any other deflection) strategy should first be performed on a benign asteroid whose orbital parameters give the asteroid no significant chance of a natural impact with the Earth during at least the next million years, even in the case of small changes in its orbit.The main goal of the paper is to demonstrate that this concept is feasible. For this purpose, studies were performed to optimize the whole list of possible missions, considering all known NEA’s regardless of their mass that might be used to deflect a hazardous asteroid from collision with the Earth. An asteroid that is too large can be considered since just a boulder of the right size could be lifted from its surface and used. The criterion of optimization is the total mass of the spacecraft to be launched from low Earth orbit in order to deflect the dangerous asteroid from hitting the Earth. For this the candidate small asteroids were selected whose targeting to the near Earth gravity assist maneuver needs a very limited V impulse and where the total V to place the spacecraft with the required amount of propellant on the surface of the asteroid is minimal. For our study, we have selected Apophis [2] as the target hazardous asteroid just as an example, since that asteroid is not now actually expected to pose a significant threat in the next decades. All operations are supposed to be completed before the end of 2035. During the studies aimed to choose the appropriate asteroid to be directed to the dangerous near Earth object, it was found that in the available catalogue, there are candidates satisfying our needs, including some that need a V pulse of less than 10 m/s to strike the hazardous object. One asteroid was found needing only 2.5 m/s in order to transfer it to the necessary Earth swingby B-plane point to impact Apophis. Previous studies showed that using a small asteroid as a projectile with an Earth gravity assist, as described more below, is about two orders of magnitude more efficient than a direct kinetic impact using a spacecraft [3]. Optimal transfer trajectories to Apophis were found for the asteroids in the list of candidates to be targeted to Apophis. The V required to land a spacecraft on the asteroid’s surface starting from an initial low Earth orbit was used as an optimization criterion. It was shown that its value does not exceed 6 km/s. The problem of trajectory correction maneuvers is explored in the paper; they are strongly dependent on the achievable accuracy of the orbit parameters determined from processing the tracking data. In order to reach the best possible accuracy in targeting a small asteroid to Apophis, a beacon spacecraft is used on the surface of Apophis. Different types of engine units that can deliver the required nominal and trajectory correction maneuvers of the small asteroid are compared, including standard chemical and electric propulsion engines, and also mass drivers that can accelerate material from the asteroid to generate thrust. Keywords: Near-Earth asteroids, planetary protection 1. Introduction We want to answer the question: Do small asteroids exist in the contemporary catalogue that can be used to transfer onto a trajectory of collision with Apophis before that asteroid approaches the Earth in April 2036, assuming that we can use only launch vehicles and technologies of spacecraft motion control that are available now. We mean realizing a scenario such that a spacecraft with propellant of enough mass is sent to chosen small asteroid-projectile, then lands on that asteroid and, after its attachment to the surface, the spacecraft engine gives to the asteroid a velocity pulse, which transfers the asteroid to the trajectory of its collision with Apophis using an Earth gravity assist maneuver. Results are given below to give some preliminary confirmation that the described method of deflecting Apophis from a trajectory of possible collision with the Earth is feasible. Taking into account the latest successes in discovering new near Earth asteroids, one may state that the the chances to find an asteroid-projectile needed to implement the proposed method of Earth defense will be improving. 2. Gravity assist maneuver as a tool to target asteroid-projectile to hazardous near Earth object. Asteroids which are considered as small have mass about 1500 tons so to control their motion in classical use of this word i.e. by applying pulse of rocket engine in order to change their velocity by several kilometers per second is hardly possible. But if a gravity assist maneuver is used as the tool to amplify changes of orbital parameters with a small velocity pulse (delta-V) allows changing the pericenter height of the controlled body (asteroid-projectile) when it flys by the planet (Earth) by a value sufficient to turn the relative velocity vector by dozens of degrees, then we have an extremely efficient tool of orbital control for the celestial body having so large a mass as compared with the usual spacecraft. By selection of a relative velocity position at infinity with respect to the fly-by planet (with fixed pericenter radius) we receive any required plane of relative orbit with appropriate direction of relative velocity vector after the fly-by. This is illustrated in Fig.1 where a cylinder of possible vectors of relative velocities (at infinity) at arrival, and the resulting post-fly-by cone of velocity vectors of departure are shown.  
Figure 1. a) Cylinder of possible vectors of relative velocities at arrival, and resulting post-fly-by cone of velocity vectors of departure, b) Geometry of gravity assist maneuver in coordinate system connected with the Sun The geometry of a gravity assist maneuver is illustrated in Figure 1b. In Fig. 1b Vp is the velocity vector of the planet with a fly-by during a gravity assist maneuver (the Earth in our case) given in coordinate system connected with the Sun, Va is arrival velocity vector of asteroid in the same reference (coordinate) system and V0 is the velocity vector of the asteroid relative to the planet at the moment of its arrival at the Earth referred to infinity (if gravity of the Earth would be equal to zero this vector would be equal to the one we consider). After planet fly-by this vector is turned by an angle α (the bend angle) which can be calculated using the following formula [4]:  (1)
where:
r – pericenter radius, V – relative velocity of asteroid-projectile at infinity, μ – gravitational constant of the planet. As one can see from the formula, with decreasing the pericenter radius, the α angle of relative velocity vector turn due to the fly-by increases, reaching 180 degrees if the pericenter radius reaches zero. Thus if there are no limits constraining the lowest value of the pericenter radius, then the asteroid’s relative velocity vector can be reversed, to opposite its initial direction. The whole family of the departure relative velocity vectors 3. Lambert problem as the principal tool to design trajectories design using gravity assist maneuvers. A scenario of a mission to deflect a hazardous near Earth object from an Earth collision trajectory consists of the following phases illustrated in Figure 2: 
Figure 2. Trajectories of Earth, Apophis, asteroid-projectile 2011 UK and spacecraft transfer trajectory from Earth to 2011 UK It is obvious that during the whole mission all available tracking instruments and facilities are needed to solve the navigation tasks with maximum achievable accuracy. The importance of this part of mission operations is much higher than in case of the usual spacecraft motion control because correction maneuvers must be kept small, due to the large mass of the projectile-asteroid and the limited amount of fuel available to change its orbital elements. The task of the mission design at large consists in the choice of all available free parameters in such a way that, with the maximum payload possible using currently available launch vehicles, results in the maximum deviation of the velocity vector of the target asteroid (Apophis) after it is hit by the controllable asteroid-projectile. This task is multi parametrical so it is to be solved in several phases. The first step is to choose an optimal transfer of the spacecraft from low Earth orbit (LEO) to the asteroid chosen as the asteroid-projectile. For this the Lambert problem is to be solved with the goal to minimize the required delta-V to execute this mission. As it is well known, the Lambert problem [5, 6] consists in the choice of initial orbital parameters that during a given time, allows transfer of a zero-mass object along a Keplerian orbit from one specified point to another. Our goal is to choose from possible dates of launch and arrival, the optimal ones in terms of total delta-V. During this phase acceptable candidate asteroids are to be chosen as the dates of their departure and arrival. It should be mentioned that on this phase of calculations of asteroid motion the gravity field is centered at the Sun. The coordinates of Earth and coordinates of asteroids are taken from appropriate catalogues using the SPICE system [7]. The next step is to find a transfer orbit from the Earth to Apophis with date and time of start the same as the ones of arrival to the Earth of the asteroid-projectile. The magnitude of the velocity vector relative to Earth is chosen to be the same as the magnitude of the arriving velocity vector. As a result of this second (modified) Lambert problem solution, we receive the departure relative velocity vector and transfer trajectory for the asteroid-projectile at large. Practically it was confirmed that the standard algorithm of Lambert problem solution may be used for both parts of the trajectory: for the part approaching the Earth and for the part from the Earth to Apophis. For this it is enough to minimize the sum of the delta-V for transfer of the asteroid-projectile from its initial orbit to the orbit reaching the Earth, and the difference of magnitudes of the arriving and departing relative velocities at the Earth. As calculations showed the result of such minimization is zero difference of these magnitudes that exactly correspond to the demand of mutual matching of these trajectories (that is, an unpowered Earth swingby), so we receive one uninterrupted trajectory of asteroid-projectile from the first correction velocity pulse to the collision with asteroid Apophis. With the use of the described method and the catalogue of the solar system bodies orbital parameters contained in SPICE calculations, we have fulfilled the aim to select the candidate asteroids for targeting them to Apophis with the use of a gravity assist maneuver near Earth. The criterion of selection of these asteroids was the required delta-V to transfer the asteroid from its original orbit to the trajectory of collision with Apophis after the gravity assist maneuver near Earth. The sizes of the asteroids were also taken into account as the necessary delta-V to deliver the spacecraft onto the surface of the asteroid-projectile. The best five asteroids, satisfying the described requirements that were chosen by our studies, and their key characteristics, are given in Table 1. Table 1. Results of the selected candidate asteroids and the orbit designs
*for departure delta-V optimization 4. Optimization of spacecraft transfer trajectory with start from low Earth orbit (LEO) and landing on asteroid-projectile surface. A natural criterion of optimization for choice of the trajectory of spacecraft delivery to the asteroid-projectile surface is the maximum mass of the spacecraft after landing. For our studies it is enough to use instead of this criterion, one very close to it: The total ∆Vt required to start from LEO and to execute the maneuver to land the spacecraft on the asteroid’s surface. For calculating first constituent (∆Vs) of the delta-V we assume that the spacecraft starts from the circular LEO with 200 km height. The second constituent we estimate as the relative velocity of spacecraft with respect to asteroid when arriving at it (rendezvous delta-V). In order to minimize available software [8, 9, 10] modification, another approach for solving the problem was used: we minimized the function F = W1*C3+W2*∆Va, where C3 is the square of relative asymptotic velocity with respect to Earth of the trajectory departing to the asteroid, ∆Va is the asteroid rendezvous delta-V, and W1 and W2 are the weight factors for the procedure used to minimize the function. Several values of W2 were tested keeping W1 =1 for all cases. It is obvious that with increasing the W2 value the influence of arriving velocity on the solution is raised. In the presented table of received solutions that includes the data of start, data of arrival, square of asymptotic velocity at departure, delta-V required for departure, and asteroid rendezvous deltaV. Table 2 presents key orbital parameters for the transfer mission of the spacecraft to be landed on 2011 UK10 asteroid calculated by the method of optimization for different weight factors W1, W2. The search of optimal trajectory was done for the spacecraft departure inside an interval of dates beginning with departure not earlier than 2020-01-01 and ending by arrival not later than 2025-08-15. Table 2. Key orbital parameters for transfer mission of the spacecraft to asteroid-projectile
As one can see from the table the last column presents the optimal trajectory with a total delta-V of 5.744 km/s. The impulse required to rendezvous with the asteroid is only 0.5427 m/s. This trajectory is shown in Fig.3, an ecliptic projection with the orbits of inner planets and orbit of 2011 UK asteroid. Rough estimations show that a standard Proton launch vehicle with Breeze upper stage using optimal transfer trajectory can deliver to the surface of 2011 UK10 asteroid (after consuming a total delta-V of 5.744 km/s) a payload with 2300 kg mass.
Thus the data given in the above list of key trajectory parameters confirms that we can use conic Lambert problem solutions for accurate-enough calculations for calculating the possibilities of the use of small asteroids to deflect dangerous ones from their original trajectory that might collide with the Earth. 6. Analysis of results for ideal cases The figures given in Table 1 and presented by key parameters of trajectories found by our studies are for ideal cases, i.e. for the ones when the trajectory is nominal with no (zero) deviation of the real trajectory from the calculated trajectory. But for the considered missions the part of propellant to be consumed during the mission for correction maneuvers may be comparable with those for the nominal maneuvers. If one considers the mission effectiveness only in terms of the nominal trajectory, then from the 5 cases presented in Table 1, the most promising case is the mission using the asteroid 1994 GV. The main argument for such a choice is the estimated mass of the asteroids in Table 1. The asteroids closest in size are 2011 UK10 and 2006 XV4 are supposed (if their densities are the same) to have a mass by a factor 30 times higher. Assuming that mass of 1994 GV asteroid is 1350 tons (radius 6 meters, density 1.5 t/m3), in order to change its velocity by the needed 17.72 m/s using engine unit with specific impulse 3300 m/s, one needs to consume 7.23 tons of propellant. It means that 4 Proton launch vehicles are to be used in order to deliver the required amount of propellant to the asteroid surface. It is possible but not so simple. In addition some consumption of propellant is necessary for trajectory correction maneuvers. If one refers to the Deep Impact mission to estimate the delta-V needed for these maneuvers, then the expected figures may reach a few dozen meters per second [11]. It means that for implementation of the proposed technology of dangerous asteroid deflection we need to find more asteroids satisfying our demands in terms of their size and required delta-V to target them to dangerous objects like Apophis. For example if 2006 XV4 would have the same size as 1994 GV and similar required delta-V to reach it then the required mass of propellant to target it to Apophis would be only 0.98 tons if one would not take into account the propellant needed for correction maneuvers. That might be done in any case by using a suitably-sized boulder lifted from the surface of 2006 XV4, rather than trying to move the whole asteroid. We note that even with the use of 10 Proton launches to accomplish the proposed plan for this selected case, it is 27 times more effective than a direct kinetic impact of the spacecraft to Apophis. Correction maneuvers might be decreased significantly by the use of a transponder to be delivered to the surface of Apophis before execution of the deflection mission. 7. Resonance orbits as the basis for constructing a planetary defense system. The proposed concept of using small asteroids to deflect hazardous objects from the trajectory of collision with Earth may be developed further. The idea is to transfer small asteroids onto Earth resonance orbits, for example with period of one year, using the Earth gravity assist method described above. Our preliminary studies have confirmed that it is possible to find 11 asteroids which are possible to transfer to such orbits with a delta-V not exceeding 20 m/s. Thus a system is constructed which allows sending asteroid-projectile to the hazardous object approximately each month during year. This is like a combination of two previous ideas, the use of nuclear “Soldier” spacecraft in Earth-return orbits as part of the SHIELD concept [12], and stationing small asteroids in quasi-periodic orbits about the collinear Sun-Earth Lagrange points for use as “David’s stones” against threatening “Goliath” asteroids [13]. 8. Conclusions The described method of deflecting dangerous asteroids from a collision trajectory with the Earth seems to be feasible, as shown using Apophis as an example. It was found that only a very small delta-V (2.38 m/s) may be needed to transfer a small asteroid to a trajectory with an Earth gravity assist maneuver, followed by collision of this asteroid with a hazardous object like Apophis. The proposed method allows a velocity change of a dangerous object by values greater than by any other contemporary technology. For practical implementation of the proposed approach, some further progress in broadening the catalogue of candidate asteroid-projectile is needed, especially for small asteroids. Also additional studies are needed to decrease correction maneuver delta-Vs. 9. References [1] Brophy, J., Culick, F., Dimotakis, P., and Friedman, L., “A Safe Stepping Stone Into the Solar System.” Paper IAC-12.A5.4.11, presented at the 63rd International Astronautical Congress, Naples, Italy, October 2012. [2] JPL Small-Body Browser: 99942 Apophis (2004 MN4), http:/ssd.jpl.nasa.gov/sbdb.cgi?sstr=99942 [3] Nazirov R., Eismont N. “Gravitational Maneuvers as a Way to Direct Small Asteroids to Trajectory of a Rendezvous with Dangerous Near-Earth Objects”. Cosmic Research, 2010, Vol.48, No.5, pp. 479-484 [4] Spacecraft Attitude determination and Control, Wertz, J.R., Ed., D.Reidel Publishing Company, 1985., p. 60. [5] Lancaster E.R., Blanchard R.C., “A unified form of Lambert's theorem.” NASA technical note TN D-5368,1969. [6] Gooding R.H., “A procedure for the solution of Lambert's orbital boundary-value problem.” Celestial Mechanics and Dynamical Astronomy (ISSN 0923-2958), 48, № 2:145-165, 1990. [7] “NAIF/SPICE ancillary information system”, http://naif.jpl.nasa.gov/naif/about.html [8] Rody P.S. Oldenhuis, “Robust solver for Lambert's orbital-boundary value problem”, http://www.mathworks.com/matlabcentral/fileexchange/26348-robust-solver-for-lamberts-orbital-boundary-value-problem [9] Izzo D., ESA Advanced Concepts team, http://www.esa.int/gsp/ACT/inf/op/globopt.htm [10] Adam Harden, “Trajectory Optimization Tool”, http://www.orbithangar.com/searchid.php?ID=5418 [11] Lakdavala, E, “Deep Impact Successfully Splits in Final Hours before Comet Encounter, July 3, 2005”, Planetary News: Asteroid and Comets, 2005. http://www.planetary.org/news/2005/0703_Deep-Impact_Successfully_Splits_in.html [12] Gold, R. E., “SHIELD: A Comprehensive Earth-Protection Architecture”, Adv. Space Res. Vol. 28, No. 8, pp. 1149-1158, 2001. [13] Massonnet, D. and Meyssignac, B., “A captured asteroid: Our David’s stone for shielding earth and providing the cheapest extraterrestrial material”, Acta Astronautica Vol. 59, pp. 77-83, 2006. 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forms in this case the sphere with V radius. The asteroid’s velocity vector in the heliocentric reference system is the sum of the planet’s velocity vector and the relative vector of the asteroid with respect to the planet (Earth) 
with initial point in the same point as the planet’s vector and any final point on the mentioned sphere. But this is valid only in case if the planet’s radius is zero. If one will take into account the radius of the planet, then only part of the surface of the sphere can be reached after the gravity assist maneuver. This part is constrained by the cone with the axis coinciding with the arriving vector of asteroid relative velocity V0. The semi-angle of this cone is αmax, calculated by use of the formula given above where the pericenter radius is equal to the minimum allowed one. With increasing possible radius of pericenter of the fly-by trajectory from zero to infinity, the area of achievable departure velocity vectors shrinks from the whole sphere to the one point which is the end point of the arriving velocity vector. Accordingly, the area of achievable velocity vectors of planet fly-by body (asteroid-projectile) is decreased.