Final answer:
The periastron distance can be found using the equation for angular momentum, while the speed at periastron can be calculated using the conservation of angular momentum. The gravitational potential energy at periastron can be determined using the formula for gravitational potential energy.
Step-by-step explanation:
a) The distance of closest approach (periastron) can be found using the equation for the angular momentum of the comet, which is given by l = mvr, where m is the mass of the comet, v is its velocity, and r is the distance from the star. Setting l equal to mvr, we can solve for r to find the periastron distance.
b) The speed of the comet at periastron can be found by using the conservation of angular momentum. Since angular momentum is conserved, we can set ml = m'v'r', where m' is the mass of the comet at periastron, v' is its speed at periastron, and r' is the periastron distance. Solving for v' gives us the speed at periastron.
c) The gravitational potential energy of the comet at periastron can be found by using the formula U = -GMm/r, where G is the gravitational constant, M is the mass of the star, m is the mass of the comet, and r is the distance between them. Plugging in the values for G, M, m, and r, we can calculate the gravitational potential energy at periastron.