Solar System Formation & Dynamics

Nothing on the exam is closely related to the problems in homework #2. However, it may be useful to review some basic points touched on in questions (2) and (3), as described below…

Question 2: you were asked to estimate the rotation period of the Sun if the angular momentum in Jupiter’s orbit were suddenly given to the Sun. To do this, we first note that the angular momentum,

where v is the tangential (i.e. orbital) velocity. For a Keplerian orbit,

so for the angular momentum content in the Jovian orbit we have,

.

Let’s suppose that the Sun is not spinning at all (not true, but a reasonable approximation that we can justify after the fact once we show that adding Jupiter’s angular momentum spins up the Sun tremendously). How fast will it spin if we put of angular momentum into it? There are various ways to estimate this. One of the simplest is to note that for a rotating body, the angular momentum is,

where is the rotation frequency and the moment of inertia. A uniform density sphere (which is not a good approximation for the Sun, but suffices for a quick estimate) has,

.

Equating the angular momentum of Jupiter’s orbit with the expression for the angular momentum of a spinning solid sphere gives us an estimate for the spin frequency, which we can then convert into a spin period if we want via,

.

Substituting numbers you’ll get a spin period of less than a day.

Question 3: the most important parts of this question were parts (a) and (b), where you were asked to calculate how much energy would need to be added to the protoplanetary disk to unbind it. One first needs to calculate the binding energy per unit mass for gas in a circular orbit at radius r from the star. To do this, note that gas in orbit around a star has both gravitational potential energy and kinetic energy. Per unit mass,

.

For a circular orbit, , so we can simplify this to read,

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This is the binding energy (or just the total energy) per unit mass… i.e. it’s the energy of 1 gram (or kg) of gas at this specific radius. To work out the total binding energy, we consider an annulus in the disk between radii r and (r + dr). The mass in that annulus is,

and hence that gas has binding energy (mass x specific binding energy) that is equal to,

.

The total binding energy is then just the integral of the above over all radii,

.

For the specified surface density profile, the radial dependence of the integrand is , so the integral gives you a natural log (hence you do need limits on both inner and outer radii to avoid divergence).