Binary-pulsar Roemer delay (${\Delta }_R=z_1/c$)

The geometric light-travel contribution from the pulsar's position within the binary orbit. To construct an eccentric Kepler orbit, use eccentric anomaly $E$, orbital eccentricity $e$, mean anomaly $M=2\pi (t-T_0)/P_b$, and argument of periapsis $\omega$. Here $T_0$ is passage through periapsis. Coordinates in the orbital plane are $X=a_1(\cos E-e)$ and $Y=a_1\sqrt{1-e^2}\sin E$. Rotate through $\omega$ and project by $\sin i$ to obtainThe first equation is Kepler's equation: equal time intervals give equal increments of $M$, not of $E$. Since $1-e\cos E>0$ for $0\le e<1$, each $M$ has a unique $E$ on the corresponding unwrapped orbit. A constant delay is absorbed into the arbitrary arrival-time origin. The circular limit is ${\Delta }_R=x\sin (M+\omega )$, so its amplitude measures $x$. For $e>0$, the delay's offset and amplitude also depend on $\omega$; its maximum is not generally $x$.