1. Starting with the general expression for total energy of any orbit:
E = 1/2 m(GMm / L)2 (e2 - 1)
show that this reduces to the simpler expression for an ellipse,
E = - GMm/2a
by inserting the value of L at an appropriate place in the elliptical orbit (either at perihelion or aphelion), and simplifying the resulting expression. You'll need the value of rperi or rap from the general ellipse equation, and the expression for vperi or vap, given in Lecture 6.
2. To send a space probe directly to the Sun, we would have to remove
the angular momentum from its orbital motion. Consider a space probe
of mass 2500 kg, in circular orbit at 1 AU from the Sun. The orbital
speed of such an orbit (the same as Earth's speed) is about 30 km/s.
(a) What is the probe's orbital angular momentum, in kg m2/s?
(b) What is the minimum kinetic energy needed to lose this angular momentum, in J?
(c) Show that this is the same as the kinetic energy that would have to be added in order for the probe initially at 1 AU to escape the solar system. [Hint: The energy needed to escape the solar system is that needed to change its total energy to zero.]