74
M. Burgay et al.
very high S/N standard profile, typically obtained from summing in phase many
observations of the given pulsar. This comparison produces a set of so-called
topocentric ToAs, which are calculated by adding at the reference time of any pulse
profile the fraction of spin period by which the pulse profile is shifted with respect
to the standard profile (e.g. when applying a convolution method, this is the fraction
of rotational period at which the χ 2 of the convolution is minimised).
As an approximate rule-of-thumb, the characteristic rms uncertainty in the
determination of a topocentric ToA scales as the ratio between the width of the
pulse and the S/N of the pulse profile.
2.4.2 Timing Procedure: Modelling the ToAs
In order to fully exploit the precisely repetitive nature of the radio signals from a
pulsar, it is necessary to be able to account for all the arrived pulses (i.e. all of the
NS rotation) from a reference time t ref to a generic time t and, consequently, to
predict the times of arrival of all of the following pulses.
Assuming t ref as the time of arrival of one pulse (this can always be done, since
the exact reference phase in the pulse itself is not important), we can model the
rotational evolution of a pulsar with a power series:
N(t) = ν ref × (t − t ref ) +
1
2
˙
ν ref × (t − t ref )
2
+
1
6
¨
ν ref × (t − t ref )
3
+ . . . .
(2.5)
Here N(t) is the number of rotations occurred from t ref to t. and ν ref , ˙
ν ref , ¨
ν ref , . . .
are the star’s spin frequency and its derivatives at the reference epoch t ref .
The objective of pulsar timing is to derive ν ep , ˙
ν ref , ¨
ν ref , . . . with high enough
precision such that N(t next ) will be very close to an integer (i.e. a complete
rotation) for any future time t next of appearance of a radio pulse. Through the
timing procedure, in summary, we are able to predict ToAs, through the rotational
model of Eq. (2.5), that will match the future observations within the observational
uncertainties. To quantify the accuracy of timing, we can make use of the so-called
timing residuals R(t i ) = N(t i ) − n(t i ), where n(t i ) is the nearest integer to the N(t i )
derived by the model. If R(t i ) 1 for all the observed ToAs in a given time span
Δt span , we have reached a satisfactory coherent timing solution over Δt span .
Note that R(t i ) = R(t i ; α 1 , α 2 , . . . , α m ), where α 1 , α 2 , . . . , α m are the m
parameters of the timing model (ν ref , ˙
ν ep , ¨
ν ref in a simple 3-parameter case).
Operationally, hence, the timing solution is determined and improved through a
multi-parametric least-square fit, by minimising the residual’s χ 2 :
χ
2
= Σ i
R(t i ; α 1 , α 2 , . . . , α m )
i
2
(2.6)
were i is the uncertainty on the i-th ToA.
M. Burgay et al.
very high S/N standard profile, typically obtained from summing in phase many
observations of the given pulsar. This comparison produces a set of so-called
topocentric ToAs, which are calculated by adding at the reference time of any pulse
profile the fraction of spin period by which the pulse profile is shifted with respect
to the standard profile (e.g. when applying a convolution method, this is the fraction
of rotational period at which the χ 2 of the convolution is minimised).
As an approximate rule-of-thumb, the characteristic rms uncertainty in the
determination of a topocentric ToA scales as the ratio between the width of the
pulse and the S/N of the pulse profile.
2.4.2 Timing Procedure: Modelling the ToAs
In order to fully exploit the precisely repetitive nature of the radio signals from a
pulsar, it is necessary to be able to account for all the arrived pulses (i.e. all of the
NS rotation) from a reference time t ref to a generic time t and, consequently, to
predict the times of arrival of all of the following pulses.
Assuming t ref as the time of arrival of one pulse (this can always be done, since
the exact reference phase in the pulse itself is not important), we can model the
rotational evolution of a pulsar with a power series:
N(t) = ν ref × (t − t ref ) +
1
2
˙
ν ref × (t − t ref )
2
+
1
6
¨
ν ref × (t − t ref )
3
+ . . . .
(2.5)
Here N(t) is the number of rotations occurred from t ref to t. and ν ref , ˙
ν ref , ¨
ν ref , . . .
are the star’s spin frequency and its derivatives at the reference epoch t ref .
The objective of pulsar timing is to derive ν ep , ˙
ν ref , ¨
ν ref , . . . with high enough
precision such that N(t next ) will be very close to an integer (i.e. a complete
rotation) for any future time t next of appearance of a radio pulse. Through the
timing procedure, in summary, we are able to predict ToAs, through the rotational
model of Eq. (2.5), that will match the future observations within the observational
uncertainties. To quantify the accuracy of timing, we can make use of the so-called
timing residuals R(t i ) = N(t i ) − n(t i ), where n(t i ) is the nearest integer to the N(t i )
derived by the model. If R(t i ) 1 for all the observed ToAs in a given time span
Δt span , we have reached a satisfactory coherent timing solution over Δt span .
Note that R(t i ) = R(t i ; α 1 , α 2 , . . . , α m ), where α 1 , α 2 , . . . , α m are the m
parameters of the timing model (ν ref , ˙
ν ep , ¨
ν ref in a simple 3-parameter case).
Operationally, hence, the timing solution is determined and improved through a
multi-parametric least-square fit, by minimising the residual’s χ 2 :
χ
2
= Σ i
R(t i ; α 1 , α 2 , . . . , α m )
i
2
(2.6)
were i is the uncertainty on the i-th ToA.