A STUDY OF CLOSE ENCOUNTERS BETWEEN MARS AND ASTEROIDS FROM THE 3: RESONANCE Érica C. Nogueira, Othon C. Winter Grupo de Dinâmica Orbital e Planetologia UNESP -- Guaratinguetá -- Brazil Antonio F.B. de A. Prado Divisão de Mecânica Espacial e Controle INPE -- São José dos Campos -- Brazil ABSTRACT In the main asteroid belt, the Kirkwood gap associated to the 3: mean motion resonance with Jupiter has been largely studied during the lasts decades. The studies suggest that objects in such resonance would suffer very large variations in their eccentricities becoming inner planet crossers, and finally, being removed by close encounters with such planets. Following the ideas of celestial mechanics of gravity assist we present a simple analytical analysis of the variation of some orbital elements of fictitious asteroids in 3: resonance with Jupiter after having a close encounter with planet Mars. The results indicate that Mars is not much effective in removing these asteroids. Keywords: Close encounter, Kirkwood gap, 3: resonance, sphere of influence - INTRODUCTION The existence of the Kirkwood gaps in the main asteroid belt is associated to mean motion resonances with Jupiter. In the case of the gap associated to the 3: resonance (Figure ), Wisdom (98, 983, 987) showed that asteroids in resonance would have chaotic trajectories such that their orbital entricities could suffer a very large variation. The orbits could reach values of eccentricities such that the asteroids would become Mars' (Figure ) or even Earth's crossers. The mechanism to remove the asteroids, in order to create the gaps, would be the gravitational effect due to close encounters with these planets. More recently, Gladman et al. (997) performed numerical simulations of particles placed in the main asteroid belt and found that asteroids from the 3: resonance are in the majority
(about /3 of them) removed by the planets Earth and Venus, another part (about /3 of them) by the Sun, and just a small fraction (less than 5% of them) are removed by Mars. The main goal of the present work is to better understand the effect of the close encounter between these asteroids and the planet Mars. Figure. Asteroidal distribution in the zone where is located the 3: Kirkwood gap. The borders of the libration region are denoted by full lines (From Murray (993)). Figure. Temporal evolution of the orbital eccentricity of a fictitious asteroid in the 3: resonance with Jupiter. The asteroid becomes a Mars' crosser (From Murray (993)).
Following the ideas of celestial mechanics of gravity assist (Broucke, 988) we present a simple analytical analysis of the variation of some orbital elements of fictitious asteroids in 3: resonance with Jupiter after having a close encounter with the planet Mars. - THE CELESTIAL MECHANICS OF GRAVITY ASSIST The idea of the gravity assist technique is to use the so called patched conics approach. The whole problem is divided into several steps and the keplerian two-body model is adopted in each one of the steps. The problem of the asteroid's orbital evolution can be divided in the following way: i) an elliptic two-body problem, Sun-asteroid, before and after a close encounter with the planet; ii) a hyperbolic two-body problem, planet-asteroid, during a close encounter with the planet, also known as swing-by. In our case, the orbits of the planet and the asteroid cross at two different points where a close encounter between these bodies is possible (Figure 3). Defining a planet's sphere of influence one can fix the limits where the asteroid's orbital evolution will be determined only by the planet's gravity and where it will be determined only by the Sun's gravity. Figure 3. Diagram showing the orbits of a planet (dashed line) and an asteroid (full line) around the Sun. The areas A and B denotes the regions where the asteroid's orbital evolution could be well determined by the planet alone. In the present work we are interested in the orbital change of an asteroid after having a close encounter with the planet. The dynamical system considered is the planar circular restricted three-body problem: the Sun (M ), a planet (M) and an asteroid of negligible mass. The coordinate system OXY is centered at the center of mass of the two main bodies and rotates uniformly such that it keeps these two bodies on the X axis. In Figure 4 is given the geometry of the close encounter between the planet and the asteroid, the main variables are: V r inf - and V r inf +, the velocity vectors of the asteroid relative to the planet before and after the encounter; V r, the inertial velocity vector of the planet; Ψ, argument of the pericentre of the 3
asteroid's hyperbolic orbit (or encounter angle); r p, pericentre distance of the asteroid's hyperbolic orbit; δ, half of the curvature angle (between V r inf - and V r inf+ ). Before and after the swing-by the intensity of the relative velocity vectors are the same, V inf - = Vinf + = Vinf, the velocity vectors just rotate by an angle equal δ, which is given through the theory of hyperbolic orbits by sin δ = r p V () inf + µ where µ = GM and G is the gravitational constant. Figure 4. Geometry of the close encounter between the planet and the asteroid. The variables are: V r inf - and V r inf +, the velocity vectors of the asteroid relative to the planet before and after the encounter; V r, the inertial velocity vector of the planet; Ψ, argument of the pericentre of the asteroid's hyperbolic orbit; r p, pericentre distance of the asteroid's hyperbolic orbit; δ, half of the curvature angle (between V r inf - and V r inf + ). Using vector diagrams it can be shown that r r r V = V = Vinf - - Vinf + = Vinf sinδ () and that V r o makes an angle Ψ +80 with respect to the Sun-planet line. Such geometry results in the following increments on the velocity components:. X = - V. Y = - V inf inf sin δ cos Ψ sin δ sin Ψ (3) (4) 4
The next important quantity to be obtained is the variation of the angular momentum h. From its definition, h = X Y - Y X, we have...... h = X Y + Y X - Y X + X Y (5) Using the assumption that the encounter is instantaneous( X = Y = 0 ) and that at the instant t = 0 we have X = X and Y = 0 (coordinates of the planet), then equation (5) becomes h = X Y. Substituting equation (4) results - V V sin δsin Ψ h = inf (6) where n o is the angular velocity of the rotating system. The third quantity to be computed is the variation of the energy, which is made through direct subtraction of the energy after and before the encounter between the asteroid and the planet. These energies are given by and The variation of the energy is n o.... E + = + X + X Y + Y (7).. E - = X + Y (8).. E = Vinf sin δ Vinf sin δ - X cos Ψ + Y sin Ψ (9) or just E = - V Vinf sin δ sin Ψ (0) Comparing equations (6) and (0) we also have that E h = () The orbital elements of interest in our analysis are the semi-major axis, a, and the eccentricity, e. Following, we derive the variation of these orbital elements after a close encounter with a planet..- The Variation of the Semi-Major Axis From the two-body problem we have the semi-major axis and the total energy related by µ E = -, () a where µ = GM. From equation () we can derive the expression for the variation of the semimajor axis as n o. 5
a - 4a V V a = E = inf sin δ sin Ψ (3) µ µ.- The Variation of the Eccentricity Also from the two-body problem we have that the eccentricity is related to the angular momentum and semi-major axis by The variation of the eccentricity can be found using h e = - = F( h, a) (4) µ a F F e = h + a h a (5) Computing the partial derivatives and substituting on equation (5) results h E e = µ h µ a - µ an o µ a + h (6).3- The Importance of the Encounter Angle Ψ Therefore, the variations of the semi-major axis and eccentricity are proportional to the variation of the energy due to the swing-by. A closer look to equation (0) reveals that the sign of E is defined by o o sin Ψ. So, The maximum gain of energy occurs for Ψ = 70 and the maximum lost for Ψ = 90. The same happening for a. 3- RESULTS In order to generate numerical results for the effect of a close encounter between asteroids and planets in the case of asteroids from the 3: resonance (a =.5 A.U.) and the planet Mars we have adopted the following steps: determination of the minimum eccentricity of the asteroid in order to become a planet crosser (Table ); determination of the crossing points between the orbit of the asteroid and the orbit of the planet; determination of the true longitude of the first crossing point; 5 determination of the sphere of influence, using the radius given by the expression R = µ, where µ is the mass ratio of the planet with respect to the Sun (Roy, 988). We considered two cases: one and two times the radius R Roy ; Roy 6
partition of the arc of the orbit inside the sphere of influence into ten parts, in order to have different timing of encounters (we use n from -5 to 5 to indicate each of the starting condition of the encounter); determination of the starting conditions, position and velocity of the asteroid, in the rotating frame; determination of the pericentre distance of the hyperbolic orbit relative to the planet, r p ; determination of the encounter angle, Ψ, and the value of sin δ ; computation of the variations of the energy, of the semi-major axis and of the eccentricity. We have written a program in Mathematica in order to simulate all these steps. The results are summarized in the tables at the end of this text. Each table is labeled with the initial eccentricity, e, of the asteroid, and the size of the sphere of influence fixed. Inside the tables we have the resulting pericentric distance of the hyperbolic orbit relative to Mars, rp, the encounter angle, Ψ, the value of sin δ and the resulting variations of energy, E, semi-major axis, a, and eccentricity, e, for each value of the longitude of Mars, Θ considered. The values of a are given assuming the distance Sun- Planet equals unity. We are interested in the variations of a and e that are large enough to remove the asteroid from the resonance. From Figure, we have that in order to create the gap the orbital semi-major axis has to change from 0.3% to %, depending on its eccentricity. Our results reveal that e is very small, not being significant to the process considered. Analyzing these results in terms of a we estimate the size of Mars orbital arc, i.e., the maximum angular distance from the collision point that Mars can be and still remove the asteroid from the resonance. A compilation of these results is given on Table, which shows that the size of the sphere of influence do not change significantly the size of Mars orbital arc. Table. Summary of the results given on the tables at the end of this paper. The size of Mars orbital arc, Θ, for a given initial eccentricity and a fixed sphere of influence. e θ ( R Roy ) θ ( R Roy ) 0.4 0.60 0.60 0.5 0.0 0.06 0.6 0.05 0.06 0.7 0.04 0.0 0.8 0.0 0.0 0.9 0.0 0.0 Conclusion From our results becomes clear that the asteroid has to have a really close encounter in order to suffer a perturbation by Mars enough to be removed from the 3: ressonance with Jupiter. As expected, due to its low mass, Mars is not a great contributor to this mechanism of clearing the gap. 7
APPENDIX Tables. Each table is labeled with the initial eccentricity, e, of the asteroid, and the size of the sphere of influence fixed. Inside the tables we have the resulting pericentric distance of the hyperbolic orbit relative to Mars, r p, the encounter angle, Ψ, the value of sin δ and the resulting variations of energy, E, semi-major axis, a, and eccentricity, e, for each value of the longitude of Mars, Θ considered. The values of a are given assuming the distance Sun-Planet equals unity. e = 0.4 *R Roy θ r p Ψ senδ E a/a e -..05x0-89.8 7.5x0-4 -.95x0-4 -9.68x0-4 3.90x0-7 -.0 9.57x0-3 89.68 7.88x0-4 -3.5x0-4 -.07x0-3 3.90x0-7 -0.5 4.75x0-3 88.60.59x0-3 -6.54x0-4 -.5x0-3 3.90x0-7 -0.3.84x0-3 87.3.65x0-3 -.09x0-3 -3.58x0-3.90x0-7 0.3.8x0-3 -87.7.69x0-3.0x0-3 3.6x0-3 3.9x0-7 0.5 4.69x0-3 -88.75.6x0-3 6.63x0-4.7x0-3 3.9x0-7.0 9.3x0-3 -89.74 8.09x0-4 3.34x0-4.0x0-3 3.9x0-7..0x0 - -89.86 7.37x0-4 3.04x0-4 9.98x0-4 3.9x0-7 e = 0.4 *R Roy θ rp Ψ senδ E a/a e -..06x0-89.04 7.x0-4 -.93x0-4 -9.6x0-4 3.90x0-7 -.0 9.63x0-3 88.83 7.83x0-4 -3.3x0-4 -.06x0-3 3.90x0-7 -0.5 4.78x0-3 86.79.58x0-3 -6.49x0-4 -.3x0-3 3.90x0-7 -0.3.86x0-3 84.05.63x0-3 -.08x0-3 -3.55x0-3 3.88x0-7 0.3.84x0-3 -85.59.65x0-3.09x0-3 3.58x0-3 3.90x0-7 0.5 4.7x0-3 -87.37.59x0-3 6.57x0-4.6x0-3 3.9x0-7.0 9.38x0-3 -89.0 8.04x0-4 3.3x0-4.09x0-3 3.9x0-7..03x0 - -89.9 7.3x0-4 3.0x0-4 9.90x0-4 3.9x0-7 e = 0.5 *RRoy θ r p Ψ senδ E a/a e -0.30 5.06x0-3 80.4 3.69x0-4 -3.0x0-4 -9.90x0-4 3.84x0-7 -0.0 3.37x0-3 75.35 5.54x0-4 -4.44x0-4 -.46x0-3 3.77x0-7 -0.05 8.40x0-4 36.78.x0-3 -.0x0-3 -3.6x0-3.34x0-7 0.05 8.45x0-4 -5.3.x0-3.45x0-3 4.75x0-3 3.09x0-7 0.0 3.37x0-3 -77.4 5.54x0-4 4.48x0-4.47x0-3 3.8x0-7 0.30 5.05x0-3 -8.4 3.69x0-4 3.03x0-4 9.93x0-4 3.88x0-7 8
e = 05 *R Roy θ rp Ψ senδ E a/a e -0.30 5.06x0-3 70.37 3.69x0-4 -.88x0-4 -9.46x0-4 3.67x0-7 -0.0 3.37x0-3 60.56 5.54x0-4 -4.00x0-4 -.3x0-3 3.39x0-7 -0.04 6.66x0-4 7.88.79x0-3 -3.8x0-4 -.04x0-3 5.36x0-8 -0.03 4.97x0-4.0 3.74x0-3 -.4x0-4 -3.73x0-4.43x0-8 -0.0 3.8x0-4 -3.7 5.65x0-3 3.05x0-4 9.99x0-4 -.54x0-8 -0.0.59x0-4 -9.47.6x0 -.58x0-3 5.9x0-3 -6.43x0-8 0.0.75x0-4 -0.33.06x0-3.05x0-3.00x0 -.36x0-7 0.0 3.44x0-4 -5.8 5.40x0-3.9x0-3 6.8x0-3.67x0-7 0.03 5.x0-4 -9.86 3.63x0 -.50x0-3 4.9x0-3.95x0-7 0.04 6.8x0-4 -34.06.73x0-3.7x0-3 4.7x0-3.9x0-7 0.0 3.38x0-3 -66.63 5.0x0-4 4.0x0-4.38x0-3 3.60x0-7 0.30 5.07x0-3 -73.40 3.68x0-4.93x0-4 9.6x0-4 3.76x0-7 e = 0.6 *R Roy θ rp Ψ senδ E a/a e -0.0 3.49x0-3 70.3.70x0-4 -.97x0-4 -9.73x0-4 3.67x0-7 -0.0.74x0-3 53.64 5.40x0-4 -5.08x0 4 -.67x0-3 3.4x0-7 -0.05 8.68x0-4 33.36.08x0-3 -6.95x0-4 -.8x0-3.4x0-7 -0.0.7x0-4 5.67 5.48x0-3 -6.33x0-4 -.08x0-3 3.87x0-8 0.0.76x0-4 -0.0 5.3x0-3.0x0-3 3.6x0-3 6.93x0-8 0.05 8.74x0-4 -36.46.08x0-3 7.47x0-4.45x0-3 3.33x0-7 0.0.75x0-3 -55.0 5.39x0-4 5.7x0-4.70x0-3 3.x0-7 0.0 3.49x0-3 -70.65.70x0-4.97x0-4 9.75x0-4 3.70x0-7 e = 0.6 *R Roy θ r p Ψ senδ E a/a e -0.0 3.48x0-3 53.7.7x0-4 -.55x0-4 -8.36x0-4 3.4x0-7 -0.0.73x0-3 33.46 5.43x0-4 -3.50x0-4 -.5x0-3.5x0-7 -0.05 8.6x0-4 7.9.09x0-3 -3.79x0-4 -.4x0-3.6x0-7 -0.0.63x0-4.8 5.75x0-3 -.3x0-4 6.70x0-4.4x0-8 0.0.84x0-4 -6.6 5.08x0-3 6.37x0-4.09x0-3 4.0x0-8 0.05 8.8x0-4 -.8.07x0-3 4.50x0-4.48x0-3.4x0-7 0.0.75x0-3 -36.40 5.36x0-4 3.7x0-4.x0-3.33x0-7 0.0 3.50x0-3 -55..70x0-4.60x0-4 8.46x0-4 3.x0-7 9
e = 0.7 * R Roy θ rp Ψ senδ E a/a e -0.0 3.46x0-3 66.65.65x0-4 -.7x0-4 7.46x0-4 3.58x0-7 -0.0.73x0-3 50.5 3.3x0-4 -3.8x0-4 -.5x0-3 3.0x0-7 -0.03 5.6x0-4 4.00.x0-3 -6.75x0-4 -.x0-3.59x0-7 -0.0 3.43x0-4 8.56.67x0-3 -7.95x0-4 -.6x0-3.4x0-7 -0.0.69x0-4.74 3.37x0-3 -.x0-3 -3.65x0-3 8.63x0-8 0.0.76x0-4 0.40 3.4x0-3 -3.46x0-5 -.4x0-4 -.80x0-9 0.0 3.50x0-4 -5.84.63x0-3.49x0-4 8.8x0-4 3.99x0-8 0.03 5.3x0-4 -.96.09x0-3 3.39x0-4.x0-3 8.x0-8 0.0.74x0-3 -44.40 3.9x0-4 3.45x0-4.3x0-3.74x0-7 0.0 3.47x0-3 -64.33.65x0-4.3x0-4 7.30x0-4 3.54x0-7 e = 0.7 * R Roy θ r p Ψ senδ E a/a e -0.08.37x0-3 9.5 4.7x0-4 -3.04x0-4 -9.99x0-4.90x0-7 -0.07.0x0-3 6.69 4.77x0-4 -3.x0-4 -.05x0-3.76x0-7 -0.06.0x0-3 4.3 5.58x0-4 -3.4x0-4 -.x0-3.60x0-7 -0.03 5.05x0-4 5.8.3x0-3 -4.6x0-4 -.5x0-3.07x0-7 -0.0 3.3x0-4.87.7x0-3 -5.74x0-4 -.88x0-3 8.7x0-8 -0.0.59x0-4 9.85 3.59x0-3 -9.x0-4 -3.0x0-3 6.69x0-8 0.0.87x0-4 3.66 3.05x0-3 -.9x0-4 -9.59x0-4 -.50x0-8 0.0 3.60x0-4 0.5.59x0-3.8x0-5 7.4x0-5 -3.58x0-9 0.03 5.33x0-4 -.6.07x0-3 7.33x0-5.40x0-4.79x0-8 0.06.05x0-3 -.87 5.43x0-4.67x0-4 5.49x0-4 8.06x0-8 0.07.3x0-3 -4.84 4.66x0-4.79x0-4 5.87x0-4.00x0-7 0.08.40x0-3 -7.73 4.09x0-4.87x0-4 6.x0-4.9x0-7 e = 0.8 * R Roy θ rp Ψ senδ E a/a e -0.0.68x0-3 49.59.3x0-4 -3.4x0-3 -.03x0-3.97x0-7 -0.05 8.39x0-4 35.66 4.48x0-4 -4.8x0-4 -.58x0-3.8x0-7 -0.03 5.0x0-4 8.4 7.49x0-4 -6.53x0-4 -.4x0-3.84x0-7 -0.0 3.33x0-4 3.9.3x0-3 -8.45x0-4 -.77x0-3.59x0-7 -0.0.64x0-4 9.4.8x0-3 -.40x0-3 -4.60x0-3.30x0-7 0.0.73x0-4 9.64.7x0-3 -6.7x0-4 -.x0-3 -6.55x0-8 0.0 3.4x0-4 4.49.0x0-3 -.60x0-4 -5.3x0-4 -3.07x0-8 0.03 5.0x0-4 -0.7 7.37x0-4.7x0-5 5.60x0-5 4.9x0-9 0.04 6.79x0-4 -5.9 5.54x0-4.06x0-4 3.46x0-4 4.0x0-8 0.05 8.48x0-4 -.03 4.43x0-4.57x0-4 5.4x0-4 7.50x0-8 0.0.69x0-3 -33.07.x0-4.4x0-4 7.35x0-4.4x0-7 0
e = 0.8 *R Roy θ rp Ψ senδ E a/a e -.500.5x0-8.76.49x0-5 -.74x0-5 -8.98x0-5 3.73x0-7 -.00.85x0-79.85.04x0-5 3.70x0-5 -.x0-4 3.74x0-7 -.00.70x0-78.9.x0-5 -4.0x0-5 -.3x0-4 3.74x0-7 -.005.69x0-78.87.3x0-5 -4.04x0-5 -.33x0-4 3.74x0-7 -.00.68x0-78.8.4x0-5 -4.05x0-5 -.33x0-4 3.74x0-7 -.000.68x0-78.8.4x0-5 -4.06x0-5 -.33x0-4 3.74x0-7 0.500 8.46x0-3 -63.85 4.45x0-5 7.38x0-5.4x0-4 3.56x0-7.000.69x0 - -77.37.x0-5 4.0x0-5.3x0-4 3.9x0-7.00.86x0 - -78.64.0x0-5 3.66x0-5.0x0-4 3.94x0-7.500.54x0 - -8.07.48x0-5.7x0-5 8.90x0-5 4.03x0-7 e = 0.9 * R Roy θ r p Ψ senδ E a/a e -0.0.60x0-3 50.55.58x0-4 -.75x0-4 -9.0x0-4 3.0x0-7 -0.05 7.96x0-4 39.39 3.6x0-4 -4.53x0-4 -.49x0-3.48x0-7 -0.03 4.76x0-4 33.57 5.9x0-4 -6.6x0-4 -.7x0-3.6x0-7 -0.0.55x0-4 6.86.6x0-3 -.65x0-3 -5.43x0-3.77x0-7 0.0.65x0-4 9.5.5x0-3 -.3x0-3 -3.7x0-3 -.9x0-7 0.03 4.86x0-4 0.87 5.9x0-4 -.x0-4 -7.4x0-4 -7.39x0-8 0.05 8.07x0-4.99 3.x0-4 -.46x0-5 -8.07x0-5 -.37x0-8 0.0.6x0-3 -9.70.57x0-4.9x0-4 3.9x0-4.3x0-7 e = 0.9 *R Roy θ r p Ψ senδ E a/a e -0.0 3.8x0-3 50.69 7.93x0-5 -.39x0-4 -4.55x0-4 3.0x0-7 -0.0.58x0-3 39.5.60x0-4 -.9x0-4 7.5x0-4.48x0-7 0.07.0x0-3 35.4.9x0-4 -.99x0-4 -9.80x0-4.5x0-7 -0.06 9.39x0-4 33.70.68x0-4 3.36x0-4 -.0x0-3.7x0-7 -0.03 4.59x0-4 8.75 5.48x0-4 -5.95x0-4 -.95x0-3.88x0-7 -0.0.39x0-4 5.6.8x0-3 -.7x0-3 -5.69x0-3.66x0-7 0.0.80x0-4.35.40x0-3 -.5x0-3 -3.77x0-3 -.43x0-7 0.03 5.00x0-4 7.33 5.03x0-4 -3.39x0-4 -.x0-3 -.7x0-7 0.05 8.x0-4 3.3 3.07x0-4 -.57x0-4 -5.7x0-4 -8.9x0-8 0.0.6x0-3.08.55x0-4 -.8x0-5 -4.0x0-5 -.43x0-8 0.0 3.x0-3 -9.66 7.8x0-5 5.94x0-5.95x0-4.3x0-7
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