2.3 Follow-up timing
2.3.4 Fitting pulsar models
With the observation frames corrected to the SSB, TOA phases should be predominantly determined by a combination of the motion of the pulsar and uncertainty in the pulsar’s position. This being the case, it is now possible to attempt to fit a selection of the pulsar’s parameters.
2.3.4.1 Isolated pulsars
For isolated pulsar systems, we can express the pulsar spin frequency as a Taylor expan-sion
ν(t) =ν0+ν.0(t−t0) +. . . , (2.26) whereν0 and ν.0 are the spin frequency and its derivative at reference epoch t0. As the spin frequency can be simply thought of as the rate of change of pulse phase, we may integrate Equation 2.26 to obtain
φ(t) =φ0+ν0(t−t0) + 1 2
ν.0(t−t0)2+. . . , (2.27) where φ0 is the pulse phase at t0. Through the use of least squares fitting, we may accurately determine bothν0 and ν.0. In least squares fitting the TOAs, we attempt to minimise the expression
χ2 =
NTOA
X
i=0
φ(ti)− bφ(ti)e σi
2
, (2.28)
whereNTOAs is the number of TOAs,φ(ti) is the phase of the ith TOA, bφ(ti)e isφ(ti) rounded to the nearest integer andσi is the TOA uncertainty in units of phase.
In practice, minimisation of Equation 2.28 is done in stages. Initially we take a set of observations which are close enough in time that the spin frequency error calculated
15http://ssd.jpl.nasa.gov
Pulsar searching 59 through the fold optimisation of a single observation is less than a single pulse period over the time between observations. By taking φ0 as the pulse phase from the one or other of the observations and settingν. = 0, we may fit forν across our observation pair.
Repeating this process, adding in consecutive observations with each new fit, will begin to give a very accurate measure of the pulsar’s spin frequency. The quality of the fit will not improve linearly, as with longer timing baselines the effects of spin-down will come into play. Refitting the data for both ν and ν. will allow continued timing of the pulsar by taking into account its spin-down. Similarly for longer timing baselines, the effects of positional uncertainty will become apparent through incorrect calculation of ∆tR
. Such an effect is easily seen as a year long sinusoid in the offsets of all TOAs from the best-fit model for the pulsar’s rotation, otherwise known as thetiming residuals. Figure 2.13 shows a graphical representation of the solving of an isolated pulsar. For initial timing of isolated pulsars, it is enough to fit for only the spin frequency, spin-down and position, assuming that the pulsar shows no significant timing irregularities. For binary systems the process is markedly more complicated, as the orbital motion of the pulsar introduces many more free parameters to the pulsar model.
2.3.4.2 Binary pulsars
Assuming a simple binary system that shows no strong relativistic effects, the orbit may be described by five parameters (see e.g. Lorimer and Kramer, 2005). These are Pb, the orbital period of the pulsar; T0, the epoch of periastron passage; ω, the longitude of periastron passage; e, the eccentricity of the orbit; and asini, the projected semi-major axis, where i is the inclination of the orbital plane w.r.t. the line of sight. All these parameters are contained within the classical R¨omer delay, ∆tRbinary, caused by the motion of the pulsar in its orbit. Blandford and Teukolsky (1976) show this delay to be
∆tRbinary =x(cosE−e) sinω+xsinEp
1−e2cosω, (2.29) wherex=asini/cand E is the eccentric anomaly as given by
E−esinE= 2π
Pb(t−T0), (2.30)
to first order. E may be solved-for numerically using a root-finding algorithm such as the Newton-Raphson method. By fitting for our five orbital parameters, the intrinsic spin frequency and spin-down, we may determine a phase-coherent timing solution for the pulsar.
Due to the high number of degrees of freedom, we require accurate initial guesses at the binary parameters prior to fitting. To estimate the binary parameters, we observe
Pulsar searching 60
0 0 1 1 2 2 3
−0.4
−0.3
−0.2
−0.10
P ul se ph as e
a)
0 10 20 30 40 50 60 70 80
0 0.1 0.2 0.3 0.4
P ul se ph as e
b)
0 20 40 60 80 100 120
0 0.1 0.2 0.3 0.4
P ul se ph as e
c)
0 50 100 150 200
0 0.1 0.2 0.3
P ul se ph as e
d)
0 50 100 150 200
Time elapsed (days)
−0.0015
−0.0005 0.0005 0.0015 0.0025
P ul se ph as e
e)
Figure 2.13: A step-by-step solving of the 922-ms, isolated pulsar J0425+4936. Panel a) shows the first three TOAs acquired. By fitting these TOAs for spin frequency, we can roughly predict the behaviour of the pulsar for ∼80 days, before we lose phase connection. This is shown in panel b). By further fitting the data in panel b) for a constant period derivative (panel c)), right ascension offset (panel d)) and declination offset (panel e)) we obtain a phase-coherent timing solution for the pulsar, which covers
almost 250 days of timing.
the period evolution of the pulsar. As seen in Section 2.2.4.7, the Doppler correction to the spin frequency of a pulsar in a binary system is given by ν(t) = ν0(1−vt/c).
Expressing the line-of-sight velocity, vl.o.s., in terms of the position of the pulsar in its orbit we obtain (see e.g. Lorimer and Kramer, 2005)
ν(t) =ν0(1−vl.o.s.(AT)/c) (2.31)
Pulsar searching 61 where AT is the true anomaly, which describes the position of a body in a Keplerian orbit. The true anomaly is given by
AT(E) = 2 arctan
"r 1 +e 1−e tanE
2
#
. (2.32)
If the line-of-sight velocity is given by (see e.g. Lorimer and Kramer, 2005) vl.o.s.(AT) = 2π
Pb
asini
√1−e2[cos (ω+AT) +ecosω], (2.33) then by least-squares fitting the changes in observed spin frequency, we may generate estimates for all our binary parameters.
In practice we start by assuming all binary parameters to be zero and fit for the intrinsic spin frequency of the pulsar. With a good estimate for this value we may then attempt to fit forPb and asini, while keepinge= 0 , i.e. assuming a perfectly circular orbit. If the χ2 from the fit starts to become large when adding in more period measurements, then the data may be further fit withe,ω and T0 to determine if the orbit is eccentric.
With suitably accurate estimates for the binary parameters, the process of solving the system follows closely the methodology used for isolated pulsars, with Equation 2.27 changed to
φ(t) =φ0+ν0(t−t0) +1 2
ν.0(t−t0)2+ ∆tRbinary, (2.34) where ∆tRbinary is a function of binary phase and the orbit is Keplerian.
It should be noted that in the case of highly circular orbits (i.e. e'0), Equation 2.29 reduces to
∆tRbinary =xsin(E+ω). (2.35)
Timing models based on this equation are unable to decouple the covariance of ω and T0, leading to large uncertainties in these quantities. By redefining the orbit of the pulsar in terms of three new parameters; Tasc, the time of ascending node; η, the first Laplace-Lagrange parameter; andκ, the second Laplace-Lagrange parameter, where
η=e sinω, (2.36)
κ=e cosω, (2.37)
Tasc=T0−ωPb
2π , (2.38)
Pulsar searching 62 Lange et al. (2001) developed a timing model by which ω and T0 may be decoupled.
Here, the R¨omer delay is redefined as
∆tRbinary '
sinφ+κ
2sin 2φ−η
2cos 2φ
, (2.39)
where
φ= 2π Pb
(t−Tasc). (2.40)
This timing model, known as theELL1 model, is important for precision timing of low-eccentricity binaries, such as the majority of MSPs. The application of this model may be seen in Chapter 3, in the timing of PSR J1745+1017.