A Gravitational Analog: A comet of 1014 kg mass describes a very elliptical orbit about a star of mass 3 × 1030 kg, with its minimum orbit radius, known as perihelion, being 1011 m and its maximum, or aphelion, 100 times as far. When at these minimum and maximum radii, its radius is, of course, not changing, so its radial kinetic energy is 0, and its kinetic energy is entirely rotational. From classical mechanics, rotational energy is given by L2/2I, where I is the moment of inertia, which for a "point comet" is simply mr2. (a) The comet's speed at perihelion is 6.2945 × 104 m/s. Calculate its angular momentum, (b) Verify that the sum of the gravitational potential energy and rotational energy are equal at perihelion and aphelion. (Remember: Angular momentum is conserved.) (c) Calculate the sum of the gravitational potential energy and rotational energy when the orbit radius is 50 times perihelion. How do you reconcile your answer with energy conservation? (d) If the comet had the same total energy but described a circular orbit, at what radius would it orbit, and how would its angular momentum compare with the value of part (a)? (e) Relate your observations to the division of kinetic energy in hydrogen electron orbits of the same but different ℓ.
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