6. [PRACTICE] The gravitational field gait) at given point R (and a given time t) may be formally defined as the gravitational force per unit mass 111 that would be experienced by a point charge of mass m at position R at time I, in tlle limit as m a U. The gravitational field field g(R) from a given mass distribution p(R) satisfies Gauss's Law. in direct. analogy to the electrostatic field E(R) from a given charge distribution. (a) . Using Gauss's Law, show that the gravitational field g(R) of a spherically symmetric mass distri bution is the same as if all mass at radii mSlde the observation radius were concentrated at the center: while all mass at radii larger than the observation radius does not contribute. (b) . Usin f )art a and Newton's Third Law now: that the J'ravitational forces between two s )llericall F syrrnnetric mass distributions are exactly the same as if the mass of each system were concentrated at its center, as long as the systems do not collide or otherwise overlap. (C) . Use a symmetry argument to infer that the torque about a spherically symmetric object's own center of mass, induced by the gravitational effects of a remote spherically symmetric mass. necessarily vanish. (d) . Use superposition arguments to deduce that the dynamics of N spherically symmetric bodies are equivalent to those of N point masses, provided there are no collisions. This is the usual justification for replacing spherical stars and planets with point masses in astronomical simulations