Pulsar Orbiting another body, doppler shifted pulse rate problem

[Mechanic403]

Hey all, new to the forum. Had a tough question that I was trying to work out about an orbiting pulsar and the doppler shifted pulse period. So, if we have a pulsar orbiting some central object in circular orbit, with a constant orbital velocity $v$, and a pulse period of $P_0$, how can we write an expression for the observed pulse period as a function of its position in orbit? I am assuming that we are observing with a line of sight in the plane of orbit. Let's say that θ is the angle the pulsar makes as it goes around the central body, and that when θ=0, the central body is directly in between the observer and the pulsar, and when θ=$\pi$ the pulsar is directly in between the observer and the central body. Let's say that the pulsar rotates counterclockwise and orbits around the central body counter clockwise as well.
So far, I expect that from $0 \leq \theta \leq \pi$, the pulsar's beam's angular velocity will be increased, from the observers POV, as its velocity component towards the observer is added with the orbital velocity component towards the observer, and like wise, the opposite effect occurs from $\pi \leq \theta \leq 2\pi$.
Im not exactly sure how to incorporate the fixed orbital velocity $v$ into an expression using $\theta$ and $P_0$. So far I only have $\omega=\frac{2\pi}{P_0}, v_{obs}=\omega + vsin\theta$ for the first half of the orbit and $\omega=\frac{2\pi}{P_0}, v_{obs}=\omega - vsin\theta$ for the second half. Any help would be greatly appreciated. Thanks guys!

 

[Mechanic403]

Ah, I just realized i can't add angular velocity to regular velocity. I came up with this:
$P_{obs}=(\frac{c-vsin\theta}{c})\cdot P_0$ w/ c the speed of light in a vacuum.
Does that seem correct?