2016 D1 | Timing a binary millisecond pulsar

Through systematic searches over the past decades, astronomers have found many millisecond pulsars (spin period below 10 ms). Most of them are in binaries with nearly circular orbits.

For a pulsar in a binary orbit, the measured spin period $P$ and the measured line-of-sight acceleration $a$ both vary systematically because of the orbital motion. For a circular orbit, in terms of the orbital phase $\phi$ ($0\le\phi\le2\pi$),

where $P_B$ is the orbital period, $P_0$ the intrinsic spin period and $r$ the orbital radius. The table gives measurements at heliocentric epochs $T$ in truncated Modified Julian Days (tMJD, days since MJD 2 440 000).

Plotted against each other, $a(\phi)$ and $P(\phi)$ trace an ellipse in the period–acceleration plane.

(D1.1) Plot the data, including error bars, in the period–acceleration plane. (7)

(D1.2) Draw an ellipse that appears to be a best fit to the data. (2)

(D1.3) From the plot, estimate $P_0$, $P_t$ and $a_t$, including error margins. (7)

(D1.4) Write expressions for $P_B$ and $r$ in terms of $P_0$, $P_t$, $a_t$. (4)

(D1.5) Calculate approximate values of $P_B$ and $r$, with error margins. (6)

(D1.6) Calculate the orbital phase $\phi$ for data rows 1, 4, 6, 8 and 9. (4)

(D1.7) Refine $P_B$ as follows.

• (a) Find the initial epoch $T_0$, the nearest epoch of zero phase before the first observation. (2)
• (b) The expected time of each observation's phase is $T_{\rm calc} = T_0 + (n + \phi/360^\circ)P_B$, where $n$ is the number of whole orbits elapsed. Find $n$, $T_{\rm calc}$ and $T_{O-C} = T - T_{\rm calc}$ for the five observations of (D1.6). (7)
• (c) Plot $T_{O-C}$ against $n$. (4)
• (d) Determine the refined initial epoch $T_{0,r}$ and orbital period $P_{B,r}$. (7)