Circular orbits in Schwarzschild geometry

[alc95]

Homework Statement

Hartle, Gravity, P9.8
A spaceship is moving without power in a circular orbit about a black hole of mass M, with Schwarzschild radius 7M.
(a) What is the period as measured by an observer at infinity?

(b) What is the period as measured by a clock on the spaceship?

Homework Equations

Eqn 9.46 from Hartle:
$$\omega =\sqrt{(M/r^3)}$$

Proper time:
$$d\tau^2=-ds^2$$
and the Schwarzschild metric.

The Attempt at a Solution

(a) This part is fine. Using T=2*pi/omega and substituting r=7M, T=14*pi*sqrt(7)*M.

However, I'm not sure how Eqn 9.46 is derived? Can it be derived using the Schwarzschild metric and noting that the coordinate r is constant?

(b) The clock in the spaceship measures proper time:
$$d\tau=\sqrt{(1-2M/r)dt^2+r^2d\phi^2}=\sqrt{(1-2M/r)+M/r}dt$$
Here, $$d\phi^2=M/r^3dt$$
$$T'=\sqrt{(1-2M/r)+M/r}T$$
Is my reasoning correct? Assuming this is correct, is all that remains is to substitute r=7M and the period T from part (a)?

 

[mfb]

A spaceship is moving without power in a circular orbit about a black hole of mass M, with Schwarzschild radius 7M.

Calling 7M "Schwarzschild radius" is confusing.

Can it be derived using the Schwarzschild metric and noting that the coordinate r is constant?

Sure (but don't ask me how exactly).

For (b), I would have expected an expression that becomes 0 at the photon sphere, but I'm not sure if that is right.