ARTICLE IN PRESS Acta Astronautica 66 (2010) 1506–1519 Contents lists available at ScienceDirect Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro Electric sail missions to potentially hazardous asteroids Alessandro A. Quarta, Giovanni Mengali à Dipartimento di Ingegneria Aerospaziale, University of Pisa, I-56122 Pisa, Italy article info abstract Article history: Missions towards potentially hazardous asteroids require considerable propellant-mass Received 18 May 2009 consumption and complex flybys maneuvers with conventional propulsion systems. A Received in revised form very promising option is offered by an electric sail, an innovative propulsion concept, 23 November 2009 that uses the solar-wind dynamic pressure for generating a continuous and nearly radial Accepted 29 November 2009 thrust without the need for reaction mass. The aim of this paper is to investigate the Available online 24 December 2009 performance of such a propulsion system for performing rendezvous missions towards Keywords: all the currently known potentially hazardous asteroids, a total of 1025 missions. The Electric sail problem is studied in an optimal framework by minimizing the total flight time. Trajectory optimization Assuming a canonical value of sail characteristic acceleration, we show that about 67% Potentially hazardous asteroids exploration of the potentially hazardous asteroids may be reached within one year of mission time, with 137 rendezvous in the first six months. A detailed study towards asteroid 99942 Apophis is reported, and a comparison with the corresponding performance achievable with a flat solar sail is discussed. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction the absolute magnitude depends on the asteroid’s albedo, and since the albedo for most asteroids is not known, an The Solar System contains a long-lived population of albedo range between 0.25 and 0.05 is usually assumed asteroids and comets, some fraction of which are [3], and this results in a range for the equivalent diameter perturbed into orbits that may cross the Earth’s orbit. of the asteroid whose minimum value is approximately The potential threat posed by these objects of colliding 150 m. Although the annual likelihood that a PHA collides with Earth may be so catastrophic that it is important to with Earth is extremely small [4], it is important to quantify the risk and prepare to deal with such a threat investigate mission scenarios whose purpose is to send a [1]. The first step in a program for the prevention or spacecraft near the asteroid [5,6] to leave it a transponder mitigation of impact catastrophes involves a search for (or a reflector). In fact, tagging the asteroid may be potentially hazardous asteroids (PHAs) and a detailed necessary to track it accurately enough to determine the analysis of their orbits. PHAs are classified on the base of probability of a collision with Earth, and thus help decide parameters that measure the asteroid’s potential to make whether to mount a deflection mission to alter its orbit threatening close approaches to the Earth [2]. In parti- [7,8]. In particular, in this paper we study the potential- cular, all asteroids within an Earth minimum orbit ities offered by an electric sail spacecraft to fulfill the intersection distance of 0.05 AU and an absolute visual rendezvous mission. magnitude of 22.0 or less are considered PHAs. Because The electric sail is an innovative propulsion concept that uses the solar wind dynamic pressure for generating a thrust without the need for reaction mass [9–11]. The spacecraft is spun around a symmetry axis and uses the à Corresponding author. Tel.: +39 050 2217220; centrifugal force to deploy and stretch out a number of fax: +39 050 2217244. E-mail addresses: [email protected] (A.A. Quarta), thin, long conducting tethers [12]. The latter are held at a [email protected] (G. Mengali). high positive (or negative [13]) potential through an 0094-5765/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.11.021 ARTICLE IN PRESS A.A. Quarta, G. Mengali / Acta Astronautica 66 (2010) 1506–1519 1507 Nomenclature DV total variation of velocity (two-impulse trans- fer) Symbols Z propulsive acceleration parameter ðZ ¼ 7=6Þ k adjoint vector A state matrix m Sun’s gravitational parameter Aij generic entry of A (with i ¼ 1; ...; 6 and n true anomaly j ¼ 1; 2; 3) t switching parameter ac characteristic acceleration T RTN radial-tangential-normal orbital reference ^ 9 J J as sailcraft propulsive acceleration ðas as= as Þ frame b vector T heliocentric-ecliptic inertial frame bi generic entry of b c , c , c auxiliary variables, see Eqs. (13)–(15) 1 2 3 Subscripts f, g, h, k modified equinoctial elements J performance index 0 initial H Hamiltonian 1 final H0 reduced Hamiltonian, see Eq. (9) % target asteroid i^ unit vector È Earth L true longitude c coasting p semilatus rectum max maximum r sailcraft position vector ðr9JrJÞ min minimum t time ss solar sail v velocity vector t transfer orbit x state vector z auxiliary complex variable, see Eq. (16) a sail cone angle Superscripts a~ unconstrained optimum cone angle, see Eq. (18) Á time derivative d sail clock angle T transpose Dt flight time electron gun, whose electron beam is shot roughly along mini-magnetospheric plasma thruster [27–29]. The aim of the spin axis. The resulting static electric field of the this paper is to provide a thorough analysis of electric sail tethers perturbs the trajectories of the incident solar wind potentialities to perform rendezvous missions towards protons, thus producing a momentum transfer from the any of the current catalogued PHAs. In particular, solar wind plasma stream to the tethers. The propelling minimum time transfer trajectories are studied, aiming thrust is almost radially directed, although a circumfer- at emphasizing the relationships between the electric sail ential component can also be generated by inclining the performance (in terms of characteristic acceleration) and sail plane at an angle with respect to the nearly radial the flight time. Moreover, a mission towards aspecific solar wind flow. This is possible acting on tunable asteroid, 99942 Apophis, is analyzed in detail and a resistors, placed between the spacecraft and each tether, comparison is made with the results achievable using a which allow each tether to slightly vary its potential. flat solar sail. Because the thrust magnitude depends on the tether potential, the resistors provide a way to control the thrust 2. Problem statement experienced by each tether individually. As a result, the sail plane can be rotated by modulating the resistors Consider an electric sail whose state x, at a generic settings with a sinusoidal signal synchronized to the time instant t, is defined through the modified equinoctial spacecraft rotation. The electric sail thrust concept has orbital elements (MEOE) p, f, g, h, k, and L as [30,31] been used to calculate successful and efficient mission 9 T trajectories in the Solar System for realistic payloads x ½p; f ; g; h; k; L ð1Þ [12,14,15]. Missions towards PHAs represent an interest- The sailcraft equations of motion can be written as [32,33] ing option for an electric sail whose peculiar character- x_ ¼ Aa þb ð2Þ istics allows the spacecraft to fulfill transfers that s 6Â3 otherwise will need either a considerable propellant mass where as is the sailcraft propelling acceleration, A 2 R [16] or significant complications such as planetary flybys and b 2 R6Â1 are suitable matrices whose generic entries [17–19]. Moreover, unlike conventional chemical propul- will be referred to as Aij and bi, respectively. An explicit sion systems [20], an electric sail offers some flexibility in expression for Aij and bi as a function of the MEOE is given the selection of the launch window, a feature that may be in Appendix A. obtained also with a solar sail [21–25] and, to a lesser The introduction of the modified equinoctial elements extent, with an electric propulsion system [26] and a into the equations of motion allows one to significantly ARTICLE IN PRESS 1508 A.A. Quarta, G. Mengali / Acta Astronautica 66 (2010) 1506–1519 ˆ iR ˆ iT ˆ aˆ s iN ˆ iR iˆ z r ˆ electric iN sail iˆ ˆ T iy RTN Sun ˆ i x orbit Fig. 1. Reference frames and electric sail control angles a and d. reduce the computational time necessary for the sailcraft spacecraft distance only. The latter characteristic represents trajectory integration. Also, it is useful to introduce a the main difference between an electric sail and a solar sail ^ ^ ^ rotating radial-transverse-normal T RTNðiR; iT ; iNÞ orbital [34–36], whose propelling acceleration magnitude depends reference frame [32], whose unit vectors are on the sail nominal plane orientation through cosa. r r  v i^ ¼ ; i^ ¼ ; i^ ¼ i^  i^ ð3Þ R JrJ N Jr  vJ T N R 2.1. Trajectory optimization where r and v are the sailcraft position and velocity Assume that at the initial time instant t090 the vector. Let ac be the electric sail characteristic accelera- sailcraft is placed on an orbit around the Sun coincident tion, that is, the spacecraft maximum propelling accel- with the Earth’s heliocentric orbit. This situation is eration at a distance of 1 AU from the Sun. The direction of representative of an electric sail deployment on a the propelling acceleration is unambiguously defined parabolic Earth-escape trajectory: that is, with zero through two independent control angles a 2½0; amax and hyperbolic excess energy ðC ¼ 0km2=s2Þ. Let n 2½0; 2p d 2½0; 2p. With the aid of Fig. 1 one has [12,14] 3 0 be the sailcraft true anomaly at t0.