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Pulsar Timing

For a detailed look at pulsar timing, see the Handbook of Pulsar Astronomy by Duncan Lorimer and Michael Kramer.

Pulsars are intrinsically interesting and exotic objects, but much of the best science based on pulsar observations has come from their use as tools via pulsar timing.  Pulsar timing is the regular monitoring of the rotation of the neutron star by tracking (nearly exactly) the times of arrival (TOAs) of the radio pulses. The key point to remember is that pulsar timing unambiguously accounts for every single rotation of the neutron star over long periods (years to decades) of time. This unambiguous and very precise tracking of rotational phase allows pulsar astronomers to probe the interior physics of neutron stars, make extremely accurate astrometric measurements, and test gravitational theories in the strong-field regime in unique ways.
The rotational phase $\phi(t)$ of a pulsar as a function of time $t$ (referenced to the nearly inertial frame of the Solar System barycenter) can be represented as a Taylor series
$$\phi(t) = \phi_0 + f(t-t_0) + \frac{1}{2}\dot f(t-t_0)^2 + \dots,$$ where $\phi_0$ and $t_0$ are arbitrary reference phases and times for each pulsar, and $f = 1 / P$ is the rotational frequency of the pulsar. In order to measure $\phi(t)$ in this form, though, many corrections have to be applied to the observed TOAs.

If we measure a pulse at our observatory on Earth at topocentric time $t_{\rm t}$, we can correct this time to the time $t$ in the inertial Solar-system center of mass or barycentric frame, which we assume to be the same as the time in a frame comoving with the pulsar.  Note that the measured pulse rates will differ from the actual pulse rates in the pulsar frame by the unknown Doppler factor resulting from the unknown line-of-sight pulsar velocity.
$$t = t_{\rm t} - t_{t_0} + \Delta_{\rm clock} - \Delta_{\rm DM} + \Delta_{{\rm R\odot}} + \Delta_{{\rm E\odot}} + \Delta_{{\rm S\odot}} + \Delta_{\rm R} + \Delta_{\rm E} + \Delta_{\rm S} .$$
As before, $t_{t_0}$ is a reference epoch,  $\Delta_{\rm clock}$ represents clock correction that accounts for differences between the observatory clocks and terrestrial time standards, $\Delta_{\rm DM}$ is the dispersion delay caused by the ISM, and the other $\Delta$ terms are delays from within the Solar System or, if the pulsar is in a binary, from within its orbit. The Roemer delay $\Delta_{\rm R}$ accounts for the classical light travel times across the orbits (with a magnitude of $\sim500\cos\beta$ s, where $\beta$ is the ecliptic latitude of the pulsar), the Einstein delay $\Delta_{\rm E}$ accounts for the time dilation from the moving pulsar (and observatory) and the gravitational redshift caused by the Sun and planets or the binary companion, and the Shapiro delay $\Delta_{\rm S}$ is the extra time required by the pulses to travel through the curved space-time containing the Sun/planets/companions.

pulsar timing
Figure 1:  Establishing a timing solution for an isolated pulsar.  In panel (e), you identify closely spaced days with unambiguous phase connection and fit for spin frequency.  In panel (f), you extend that phase connection until either RA or Dec errors dominate and then fit for it.  In panel (g), you fit for the other position component.  Finally, in panel (h), you fit for frequency derivative, which completes the timing solution.
timing examples
Figure 2: Pulsar timing examples.  Panel (a) shows a "good" timing solution with no unmodeled effects.  The sinusoidal ripple in Panel (c) indicates an error in position.  Panel (b) shows an error in the frequency derivative ($f = d\phi/dt$ so $\dot f = d^2\phi/dt^2$).  Panel (d) shows unmodeled pulsar proper motion.

For binary pulsars, the pulsar Roemer delays comprise up to five Keplerian parameters: the projected semi-major axis $x\equiv a_1\sin i/c$, the longitude of periastron $\omega$, the time of periastron passage $T_0$, the orbital period $P_b$, and the orbital eccentricity $e$. Relativistic binaries may allow the measurement of up to 5 post-Keplerian (PK) parameters: the rate of periastron advance $\dot{\omega}$, the orbital period decay $\dot{P_b}$, the so-called relativistic $\gamma$ (i.e. the Einstein term corresponding to time dilation and gravitational redshift), and the Shapiro delay terms $r$ (range) and $s$ (shape).

PSR J0437-4715 timing
Table 1:  Millisecond pulsar timing example.  A timing ephemeris for the nearby MSP J0437$-$4715 by van Straten et al. 2001.  This is one of the best "timing" pulsars known (post-fit RMS timing residuals of $\sim$100 ns), and this measurement is one of the most accurate astrometric measurements ever made.  In addition, the timing accuracy allowed a fundamentally new test of general relativity.

In any theory of gravity, the five PK parameters are functions only of the pulsar mass $m_1$,  the companion mass $m_2$, and the standard five Keplerian orbital parameters. For general relativity, the formulas are: $$\dot\omega = 3 \left(\frac{P_b}{2\pi}\right)^{-5/3} (T_\odot M)^{2/3}\,(1-e^2)^{-1}$$
$$\gamma = e \left(\frac{P_b}{2\pi}\right)^{1/3} T_\odot^{2/3}\,M^{-4/3}\,m_2\,(m_1+2m_2)$$
$$\dot P_b = -\,\frac{192\pi}{5} \left(\frac{P_b}{2\pi}\right)^{-5/3} \left(1 + \frac{73}{24} e^2 + \frac{37}{96} e^4 \right) (1-e^2)^{-7/2}\,T_\odot^{5/3}\, m_1\, m_2\, M^{-1/3}$$
$$r = T_\odot\, m_2$$
$$s = x \left(\frac{P_b}{2\pi}\right)^{-2/3} T_\odot^{-1/3}\,M^{2/3}\,m_2^{-1}.$$
In these equations, $T_\odot\equiv GM_\odot/c^3 = 4.925490947\,\mu$s is the solar mass in time units, $m_1$, $m_2$, and $M\equiv m_1+m_2$ are in solar masses, and $s\equiv\sin i$ (where $i$ is the orbital inclination). If any two of these PK parameters are measured, the masses of the pulsar and its companion can be determined. If more than two are measured, each additional PK parameter yields a different test of a gravitational theory.

For the famous case of the Hulse-Taylor binary pulsar B1913+16, high-precision measurements of $\dot\omega$ and $\gamma$ were first made to determine the masses of the two neutron stars accurately. The Nobel-prize-winning measurement came with the eventual detection of $\dot P_b$, which implied that the orbit was decaying in accordance with general relativity's predictions for the the emission of gravitational radiation. The recently discovered double-pulsar system J0737$-$3039 is in a more compact orbit (2.4 hrs compared to 7.7 hrs for PSR B1913+16), which allows the measurement of all five PK parameters as well as the mass ratio $R$, giving a total of four tests of general relativity.  Kramer et al. (2006) showed that GR is correct at the 0.05% level and measured the masses of the two neutron stars to better than 1 part in $10^4$.

PSR B1913+16 MvMPSR B1913+16 orbital decay
Figure 3. Timing results for the Hulse-Taylor binary pulsar B1913+16.  The left panel shows the mass vs. mass plot for the pulsar and its companion neutron star.  The three lines correspond to the three measured post-Keplerian parameters.  The right panel shows the periastron shift caused by the decay of the orbit via emission of gravitational radiation.  The detection of gravitational radiation resulted in a Nobel prize for Hulse and Taylor.  (Figure provided by J. Weisberg).
PSR J0737-3039 MvM
Figure 4:  PSR J0737$-$3039 mass vs. mass diagram.  As in Figure 3, the diagram shows lines corresponding to the post-Keplerian parameters measured for the system.  In this case, though, six parameters were measured, including the mass ratio R since both neutron stars are pulsar clocks.  These measurements have tested GR to ~0.05% (Kramer et al. 2006).