Pulsar timing

2.3 Pulsar timing

Once dispersion has been removed, the resultant time series is typically folded modulo the expected pulse period, in order to build up the signal strength over several minutes and to obtain a stable time-averaged profile. The pulse period may not be very easily predicted from the discovery period, especially if the pulsar happens to be in a binary system. The goal of pulsar timing is to develop a model of the pulse phase as a function of time, so that all future pulse arrival times can be predicted with a good degree of accuracy.

The profile accumulated over several minutes is compared by cross-correlation with the “standard profile” for the pulsar at that observing frequency. A particularly efficient version of the cross-correlation algorithm compares the two profiles in the frequency domain [129 ]. Once the phase shift of the observed profile relative to the standard profile is known, that offset is added to the start time of the observation in order to yield a “Time of Arrival” (TOA) that is representative of that few-minute integration. In practice, observers frequently use a time- and phase-stamp near the middle of the integration in order to minimize systematic errors due to a poorly known pulse period. As a rule, pulse timing precision is best for bright pulsars with short spin periods, narrow profiles with steep edges, and little if any profile corruption due to interstellar scattering.

With a collection of TOAs in hand, it becomes possible to fit a model of the pulsar’s timing behaviour, accounting for every rotation of the neutron star. Based on the magnetic dipole model [106 , 55 ], the pulsar is expected to lose rotational energy and thus “spin down”. The primary component of the timing model is therefore a Taylor expansion of the pulse phase $ϕ$ with time $t$ :

where $ϕ0$ and $t0$ are a reference phase and time, respectively, and the pulse frequency $ν$ is the time derivative of the pulse phase. Note that the fitted parameters $ν$ and $ν˙$ and the magnetic dipole model can be used to derive an estimate of the surface magnetic field $B sinα$ :

where $α$ is the inclination angle between the pulsar spin axis and the magnetic dipole axis, $R$ is the radius of the neutron star (about 10⁶ cm), and the moment of inertia is $45 2 I ≃ 10 g cm$ . In turn, integration of the energy loss, along with the assumption that the pulsar was born with infinite spin frequency, yields a “characteristic age” $τc$ for the pulsar:

2.3.1 Basic transformation

Equation (1 ) refers to pulse frequencies and times in a reference frame that is inertial relative to the pulsar. TOAs derived in the rest frame of a telescope on the Earth must therefore be translated to such a reference frame before Equation (1 ) can be applied. The best approximation available for an inertial reference frame is that of the Solar System Barycentre (SSB). Even this is not perfect; many of the tests of GR described below require correcting for the small relative accelerations of the SSB and the centre-of-mass frames of binary pulsar systems. But certainly for the majority of pulsars it is adequate. The required transformation between a TOA at the telescope $τ$ and the emission time $t$ from the pulsar is

Here $2 D ∕f$ accounts for the dispersive delay in seconds of the observed pulse relative to infinite frequency; the parameter $D$ is derived from the pulsar’s dispersion measure by $− 4 D = DM ∕2.41 × 10 Hz$ , with DM in units of pc cm^–3 and the observing frequency $f$ in MHz. The Roemer term $ΔR ⊙$ takes out the travel time across the solar system based on the relative positions of the pulsar and the telescope, including, if needed, the proper motion and parallax of the pulsar. The Einstein delay $ΔE ⊙$ accounts for the time dilation and gravitational redshift due to the Sun and other masses in the solar system, while the Shapiro delay $ΔS ⊙$ expresses the excess delay to the pulsar signal as it travels through the gravitational well of the Sun – a maximum delay of about $120 μs$ at the limb of the Sun; see [12 ] for a fuller discussion of these terms. The terms $Δ R$ , $Δ E$ , and $Δ S$ in Equation (4

) account for similar “Roemer”, “Einstein”, and “Shapiro” delays within the pulsar binary system, if needed, and will be discussed in Section 2.3.2 below. Most observers accomplish the model fitting, accounting for these delay terms, using the program tempo [134 ]. The correction of TOAs to the reference frame of the SSB requires an accurate ephemeris for the solar system. The most commonly used ephemeris is the “DE200” standard from the Jet Propulsion Laboratory [126 ]. It is also clear that accurate time-keeping is of primary importance in pulsar modeling. General practice is to derive the time-stamp on each observation from the Observatory’s local time standard – typically a Hydrogen maser – and to apply, retroactively, corrections to well-maintained time standards such as UTC(BIPM), Universal Coordinated Time as maintained by the Bureau International des Poids et Mesures in Paris.

2.3.2 Binary pulsars

The terms $ΔR$ , $ΔE$ , and $ΔS$ in Equation (4 ), describe the “Roemer”, “Einstein”, and “Shapiro” delays within a pulsar binary system. The majority of binary pulsar orbits are adequately described by five Keplerian parameters: the orbital period $Pb$ , the projected semi-major axis $x$ , the eccentricity $e$ , and the longitude $ω$ and epoch $T0$ of periastron. The angle $ω$ is measured from the line of nodes $Ω$ where the pulsar orbit intersects the plane of the sky. In many cases, one or more relativistic corrections to the Keplerian parameters must also be fit. Early relativistic timing models, developed in the first years after the discovery of PSR B1913+16, either did not provide a full description of the orbit (see, e.g., [23 ]), or else did not define the timing parameters, in a way that allowed deviations from GR to be easily identified (see, e.g., [51 , 61 ]). The best modern timing model [35 , 132 , 45 ] incorporates a number of “post-Keplerian” timing parameters which are included in the description of the three delay terms, and which can be fit in a completely phenomenological manner. The delays are defined primarily in terms of the phase of the orbit, defined by the eccentric anomaly $u$ and true anomaly $Ae (u)$ , as well as $ω$ , $Pb$ , and their possible time derivatives. These are related by

[( ) ( )2] T-−-T0- P˙b- T-−-T0- u − esinu = 2 π Pb − 2 Pb , (5 ) [( )1 ∕2 ] A (u ) = 2 arctan 1-+-e tan u- , (6 ) e 1 − e 2 (P ˙ω ) ω = ω0 + --b-- Ae (u), (7 ) 2π

where $ω0$ is the reference value of $ω$ at time $T0$ . The delay terms then become:

Δ = xsinω (cosu − e(1 + δ )) + x (1 − e2(1 + δ )2)1∕2cos ω sinu, (8 ) R r 𝜃 ΔE = γ sinu, (9 ) { [ 2 1∕2 ]} ΔS = − 2r ln 1 − e cosu − s sin ω(cosu − e ) + (1 − e ) cosω sinu . (10 )

Here $γ$ represents the combined time dilation and gravitational redshift due to the pulsar’s orbit, and $r$ and $s$ are, respectively, the range and shape of the Shapiro delay. Together with the orbital period derivative $˙ Pb$ and the advance of periastron $˙ω$ , they make up the post-Keplerian timing parameters that can be fit for various pulsar binaries. A fuller description of the timing model also includes the aberration parameters $δr$ and $δ𝜃$ , but these parameters are not in general separately measurable. The interpretation of the measured post-Keplerian timing parameters will be discussed in the context of double-eutron-star tests of GR in Section 4.