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Temporal Data Analysis A.A. 2011/2012
Figure 6.13: Pulse period of the X–ray binary pulsar GX 301–2 obtained by means of the epoch
folding technique
determined. When multiplied by the instantaneous pulse period, that phase yields a time offset
that can be added to a high-precision reference point on the profile (for example, the edge of
the profile) to create the time-of-arrival or TOA, as shown in Figure 6.14. The general procedure
to derive information on the source from the measured TOAs is depicted in Figure 6.15.
The TOA of the pulse number n is, by definition,
t n = t 0 + n P (6.45)
where t 0 is a reference time (usually the start of the observation). When including intrinsic
period variations we can perform a Taylor expansion in P and write down
t n = t 0 + n P + 1 2 n2 PṖ + 1 6 n3 P 2 ¨P + ··· (6.46)
Equation (6.46) can be inverted and expressed in terms of the pulse phase ϕ at time (t − t 0 )
ϕ = ϕ 0 + f 0 (t − t 0 ) + 1 f ˙(t − t 0 ) 2 + 1 f ¨(t − t 0 ) 3 + ··· (6.47)
2
6
where f 0 = 1/P and f is the frequency. The precision with which a TOA can be determined
is approximately equal to the duration of a sharp pulse feature (e.g., the leading edge) divided
by the signal-to-noise ratio of the average profile. It is usually expressed in terms of the width
of the pulse features W f in units of the period P, the pulse period P, and the signal-to-noise
102 M.Orlandini