Question

1. Gravitational Focusing (ALL ing asteroid is equal to Rp/. Here, I'll walk you through it. ) Show that a planet's gravitationally enhanced cross-sectional area for an approach- (ALL) Determine the periapsis distance of a hyperbolic orbit. Okay, that sounds com- plicated. Assume that the asteroid's trajectory is such that it would miss the center of the planet by a distance b with no gravity. This is called the impact parameter. Then use conservation of angular momentum and conservation of energy to determine the periapsis distance. Hint: since the distance that you're solving for will give you a quadratic, the periapsis is the distance where the determinant of the quadratic is equal to zero 1.1.

Answer 1

Angular momentum of the asteroid L = mv_infinity b effectibve potential v_e = -GM_p m/r + L^2 /2mr^2 m = asteroid mass M_p = planet mass v_infinity = velocity of asteroid at large distance v_e = escape velocity of planet total energy of asteroid at large distance E = 1/2 mv^2_infinity At minimum distance from planet v_e = E -GM_p m/r + L^2 /2mr^2 = E L^2 /2m (1/r^2) -am_1 m(1/r) -E = 0 1/r = GM_p m plusminus squareroot G^2 M^2_p m^2 + 4 middot L^2 /2m E/2 middot L^2 /2m r = L^2 /m/GM_p m + squareroot G^2 M^2_p m^2 + 2L^2 /m E = L^2 /GM_p m^2 [1/1 + squareroot 1 + 2L^2 L^5 /G^2 M^2_p m^3] = m^2 v^2_infinity b^2 /GM_p m^2 [1/1 + squareroot 1 + 2m^2 v^2_infinity b^2 /G^2 M^2_p m^3 1/2 mv^2_infinity] = 2v^2_infinity b^2 /v^2_c R_b [1/1 + squareroot 1 + 46^2 v^4_infinity /v^4_c R^2_p] Asteroid will miss the planet if r = R_p 2v^2_infinity b^2 /v^2_e R_p [1/1