Mathematical Modelling of Gravitational Control of Flights of Meteorites by Artificial Satellites

Transcription

1 Mathematical Modelling of Gravitational Control of Flights of Meteorites by Artificial Satellites A.S. Alekseev and Yu.A. Vedernikov Institute of Computational Mathematics and Mathematical Geophysics Novosibirsk, Russia Abstract Among the methods proposed for anti-asteroid protection of the Earth, gravitational control of the flights of meteorites by a group of artificial satellites is the most secure and environmentally safe. In this method, a threatening asteroid or meteorite is escorted by a system of artificial satellites controlled from the Earth. Calculations have shown that five such satellites forming a chain can shift considerably the center of the asteroid mass and thereby change its trajectory. For multiple corrections of the flight of an asteroid, several parades of its artificial satellites are necessary. Introduction We consider three techniques of a controlled deviation of the dangerous space body path from the original path. These techniques are environmentally clean and safe in contrast with the technique based on using the nuclear explosion devices (NEDs). Figure 1: The critical cone of the dangerous space body. Consider a dangerous meteorite moving towards the Earth (see Figure 1). Let us draw a line connecting the mass center of the meteorite with the center of the Earth. We now construct a right circular cone whose symmetry axis coincides with the above

2 line, the apex lies in the center of mass of the meteorite, and the base diameter coincides with the Earth diameter. Denote by α c the semi-apex angle of the cone. If the flight trajectory of the meteorite is inclined to the symmetry line of the cone by an angle α 0 < α c, then the meteorite will inevitably collide with the Earth. Thus, in order to avoid the meteorite impact with the Earth we can formulate the following problem: deviate the trajectory of the dangerous meteorite by an angle α > α c. As we will see below, there are at least three techniques for the solution of this task, which make no use of nuclear explosion devices. The satellite technique of asteroid deflection In this technique, several artificial satellites are delivered to the neighborhood of a dangerous space body. The x- and y-coordinates of the center of mass (inertia) of the mechanical system the space body + the satellites are determined by well-known formulas x c = 1 N s m k x k, y c = 1 N s m k y k, M Σ M Σ k=1 where N s is the number of artificial satellites, M Σ is the total mass of the system, m k is the mass of the kth satellite located at the distances x k and y k from the center of mass, see Figure 2. k=1 The artificial satellites y m 4 x m 3 m 2 m 1 The asteroid The center of mass Σ xc x1 x2 x3 x4 Figure 2: The satellite technique of the asteroid deflection. Let us now make some estimations involving the masses of the Earth, moon, and the asteroids. The moon mass amounts to 1/10 of the Earth mass, and 40 thousand of asteroids observed in telescopes amount to 1/1000 of the mass of our planet, which equals kg. We now take the same aspect ratio of the masses of the satellite and of the asteroid as for the asteroids belt and the Earth and calculate the deviation of the

3 center of mass of the linear system (see Figure 2) with ten satellites for the distances x k = 10kD, where D is the asteroid diameter. In this case the value of x c measured from the asteroid center equals 0.54D, that is, it goes beyond the asteroid perimeter. In the case of four satellites the position of the center of mass x c of the system equals approximately 0.1D. Retaining the ratio of the masses of satellites and the asteroid let us now halve the distances x k. As a result, the value of x c also nearly halves in the case of ten satellites. For four satellites, the position of the center of masses (inertia) becomes to be equal to 0.05D. If we follow the way of a ten-fold reduction of the satellite masses, then in the case of distances x k = 10kD the position of the center of masses of the system of ten satellites amounts to x c = 0.06D, and in the case of four satellites it amounts to x c 0.001D. With regard for the weight and dimensional constraints it appears to be hardly possible to deliver an artificial satellite with the mass of more than 1,000 kg. Then there remains the only possibility for realization of the given elegant method, which consists of the use of natural satellites near the dangerous asteroid. 10 % of small planets possess their satellites of natural origin [1]. It is advisable to augment right these natural satellites with a control system consisting of several artificial satellites. While distributing the parade of artificial satellites in the xoy plane one must take into account the shift of the center of masses (inertia) along the Oy axis and account for its averaged value. This relatively simple mathematical operation is advisable in the cases of multiple parades of the asteroid satellites, which are necessary for controlling its flight in the near-earth space. The third coordinate z is used in the regime of the verification of the process of the deviation of the dangerous asteroid from the Earth. Considerable efforts of the mathematicians will of course be needed to ensure the delivery of satellites to the space body to be deflected and to position the satellite orbits in the same plane. This is, however, already the scope of the conventional rocket dynamics, in which the Newton s law of universal gravitation is widely used [2, 3, 4]. Thus, besides the constraints for the weight of the relatively small asteroid to be deflected it is necessary to apply the gravitational flight control in the presence of natural satellites near the celestial body. Only in such case it is possible to guarantee the realization of environmental and technical advantages of the proposed gravitational method in the simplest case. Asteroid deflection with the aid of gravitational riders This technique for the deflection of dangerous asteroids was first proposed by [3]. It may be considered as a development of the foregoing technique and it eliminates some of the physical shortcomings of the above presented first technique.

4 Figure 3: The forces acting on the gravitational rider. Figure 4: The motion diagram of the satellite. Consider an artificial satellite in the form of Indian clubs (Figure 3) with two equal spheres each of which has the mass m/2. We will neglect the mass of the connecting rope (the bar). Let the rope be perpendicular to the line connecting the Earth center with the rope center, and let r be the distance between the center of satellites masses and the Earth center, l be the half-length of the Indian clubs. Then R = l 2 + r 2 is the distance of a sphere from the Earth. The Newton s force F N = Mm 2R 2 = U N R acts on each sphere, where U N is the integral force function of each sphere. Each of the forces F 1 and F 2 is directed to the Earth center. The total force is determined with the aid of the parallelogram of forces (Figure 3) and is equal to F = U r = Mm r 2 1 (1 + d 2 ) 3 /2. It is seen that the total force F is smaller than the Newton s force F N. The effect of the body extension introduces an additional pushing force. Although this force is nearly vanishing for small satellites, it is available. This effect was proposed to be the basis for a resonance maneuvering in space [3]. The motion takes place along a planar orbit and is determined by the integrals of areas and energy: r 2 dϕ dt = c, V 2 2 U(r) m = h. Introducing the osculatory focal parameter p(t) and the osculatory eccentricity of the orbit e(t) we can present the radius vector r and the velocity V of the osculatory orbit

5 as r = p 1 + ecosν, V = M p (1 + e2 + 2ecosν). Here ν is the true anomaly in perturbed motion. By virtue of the constancy of p the area and energy integrals may be written as P = P 0 ; e 2 + 2p [ M M r 1 ] m U(r) = h where h is a new constant quantity. With regard for the fact that the integral force function is equal to U = Mm e 2 + r = Mm 2 r 1 + α, 2 where α = e/r, the last expression may be rewritten as e 2 + 2p r α α α = h 2 The actual motion can, however occur in some part of the plane (e 2,r = r/p) rather than in the entire plane: where e 2 ( p r 1 ) 2 e 2 (r), e 2 (r) = ( p r 1 ) 2. The diagram of the actual motion of the pulsating Indian clubs is presented in Figure 4 for fixed p and h. The satellite moves in a bounded neighborhood of the Earth so that it does not move away from the Earth and retains a similarity with the Kepler s ellipse. When the satellite is at the least distance from the Earth its eccentricity is minimal, and at a point farthest from the Earth the eccentricity is maximum. These additions to the forces may be taken into account in a refined method of gravitational rider with inert spheres (the upper half of Figure 5). An incorporation of the formation of ball lightnings represents a very exotic variant of the gravitational technique of the asteroid flight control, see the lower half of Figure 5. These artificial ball lightnings can be generated as high-temperature plasmoids by a micro-plasma galvano setup Bayasite [5]. Under the appropriate voltage and current it is possible to generate a ball lightning including a group of micro-plasmoids from the aluminum cathodes. After an additional improvement the above experimental scheme is delivered to the orbit of a dangerous space object and forms several ball lightnings. This set of lightnings is shifted in several stages along spiral curves to the center of masses (inertia) of the mechanical system asteroid + satellites. Upon achieving a certain energy concentration the cloud of ball lightnings explodes and ensures a shock deflection of the asteroid. Thus, the controlling effect of the gravitational technique can be enhanced in this way. This may also lead to an alteration of the pressure center.

6 The conventional gravitational riders The asteroid The gravitational riders with ball lightnings Figure 5: The gravitational technique of the asteroid deflection. The shift of the asteroid pressure center by sun sails The sun wind represents the stream of particles and fine dust. The solar radiation pressure has a vanishingly small value at the Earth s orbit: p = Pa [3]. If the spacecraft is, however, supplied with a sail, which is sufficiently light and has a large size, then the total force of the sun wind pressure on the sail may give the spacecraft an acceleration necessary for the maneuvering in the solar system. This can result in an alteration of the center of pressure developed by the sun parachute similar to the Ural prototype shown in Figure 6. The testing of sun sails intended for transport operations in the near-earth space and for interplanetary flights can be carried out both on a ballistic trajectory and in the satellite orbit (Figure 7). To shift the pressure center of the asteroid the parachutes manufactured of special texture are delivered together with the satellites by a single carrier rocket or by a single artillery missile, see Figure 8. While performing the experiment in a near-earth orbit in 1999 the carrier rocket Volna (the Wave ) has brought onto the ballistic flight trajectory an orbital acceleration booster with spacecraft, which accommodated a small-size rocket engine. The engine has come into operation at the time when the ballistic trajectory apogee was reached, thus providing entry onto a circular orbit. The sun sail was then set up, and a further flight of the spacecraft continued. Gritzner [6] has proposed to enhance the sun wind effect with the aid of hemispherical reflectors rotating near the controlled asteroid. These reflectors may be used to redirect the laser beams emitted from the earth.

7 Figure 6: Accommodation of sun sail spacecraft on the missile: 1 3rd stage of missile; 2 apogee rocket engine; 3 adapter with spinning system; 4 container; 5 spacecraft; 6 container cover; 7 sun sail in transportation state; 8 sun sail in flight configuration. Figure 7: Scheme showing the launch of sun sail spacecraft. References [1] Tarashchuk, V.P., V.V. Prokofieva and N.N. Gar kavy, 1995, The satellites of asteroids. Reviews of topical problems. Uspekhi Fizicheskih Nauk, 165, 661 (in

8 Figure 8: The asteroid with several parachutes mounted on it and several gravitational riders in the asteroid orbits. Russian). [2] Gorbatenko, C.A., E.M. Makashov, Yu.F. Kolushkin and L.V. Sheftel, 1969, Flight Mechanics. Engineers Handbook. Mashinostroenie, Moscow (in Russian). [3] Beletsky, 1977, Essays about the Motion of Space Bodies, Nauka, Moscow (in Russian). [4] Alekseev, A.S., and Vedernikov, Yu.A., Ricochet and shock-cumulative methods of protecting the Earth against asteroids, pp in Ecology, Planetary Man, Creative Power, Inst. Theor. Appl. Mech., USSR Acad. Sci. Siberian Branch, Novosibirsk, and the Simferopol State University, Novosibirsk, Simferopol, 1993 (in Russian) [5] Vedernikov, Yu.A., B.S. Gisatullin and Yu.S. Khudyakov, 2001, A historical experience and prospects for the application of the micro-plasmoid technique for hardening of metallic plates for armored vests, in: The Topical Problems of Protection and Safety, The SM Scientific and Production Union, St. Petersburg, 181 (in Russian). [6] Gritzner, Ch., 1996, Analysis of Alternative Systems for Orbit Alteration of Near-Earth Asteroids and Comets. Institut für Planetenerkundung der DLR, Berlin, Germany.