= Binary-pulsar Roemer delay
{title2=$\Delta_R=z_1/c$}
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 obtain
$$
E-e\sin E=M,\qquad
\Delta_R=x\left[\sin\omega(\cos E-e)+\sqrt{1-e^2}\cos\omega\sin E\right].
$$
The 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 $x$.
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