Lets consider two observer, which are at a position $r=R$ around a spherical mass which is described by Schwarzschild geometry. At some point, observer $1$ make a full round around the object on a circular orbit (without an engine, "free falling"), while observer $2$ stays fixed. My goal is to calculate $\Delta\tau_{2}/\Delta\tau_{1}$, where $\tau$ denotes the proper time measured by observer 1 and 2, respectively.
The proper time of observer $1$ is straightfoward. I can use the Binet equation in Schwarzschild geometry, which reads
$$\frac{\mathrm{d}^{2}u}{\mathrm{d}\varphi^{2}}+u=\frac{GM}{h^{2}}+\frac{3GM}{c^{2}},$$
where $u=1/r$ and $h$ is a constant related to the angular momentum. Observer $1$ is moving on a circular orbit and hence $u=1/R=\text{const.}$ This yields the relation
$$h=\sqrt{\frac{\mu c^{2}R^{2}}{(R-3\mu)}}$$
with $\mu=GM/c^{2}$. Applying one of the geodesics equations for Schwarzschild geometry, namely $r^{2}\frac{\mathrm{d}\varphi}{\mathrm{d}\tau}=h$, one obtains
$$R^{2}\frac{\mathrm{d}\varphi}{\mathrm{d}\tau_{1}}=h=\sqrt{\frac{\mu c^{2}R^{2}}{(R-3\mu)}}$$
and therefore
$$\Delta\tau_{1}=\int_{0}^{2\pi}\frac{1}{R^{2}}\sqrt{\frac{\mu c^{2}R^{2}}{(R-3\mu)}}\,\mathrm{d}\varphi=2\pi\sqrt{\frac{R^{2}}{\mu c^{2}}(R-3\mu)}$$
But I don`t know how to calculate $\Delta\tau_{2}$... In the end I should get
$$\frac{\Delta\tau_{2}}{\Delta\tau_{1}}=\sqrt{\frac{R-2\mu}{R-3\mu}}.$$
EDIT: My idea is the following: Observer $2$ doesn`t move and therefore: $$c^{2}\mathrm{d}\tau_{2}^{2}=\mathrm{d}s^{2}=c^{2}\bigg (1-\frac{2\mu}{r}\bigg) \mathrm{d}t^{2}$$.
This yields $$\frac{\mathrm{d}\tau_{2}}{\mathrm{d}t}=\sqrt{\bigg (1-\frac{2\mu}{r}\bigg)} $$
When I am now able to find a relation between $\tau_{1}$ and the coordinate time $t$, I can relate $\tau_{2}$ to $\tau_{1}$ and therefore to the $\varphi$-coordinate of observer $1$, which can be used to integrate from 0 to $2\pi$.
