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 [130]. 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 [108,
53], 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
with time
t
:
where
and
are a reference phase and time, respectively, and the pulse
frequency
is the time derivative of the pulse phase. Note that the fitted
parameters
and
and the magnetic dipole model can be used to derive an estimate
of the surface magnetic field
:
where
is the inclination angle between the pulsar spin axis and the
magnetic dipole axis,
R
is the radius of the neutron star (about
), and the moment of inertia is
. In turn, integration of the energy loss, along with the
assumption that the pulsar was born with infinite spin frequency,
yields a ``characteristic age''
for the pulsar:
Here
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
, with DM in units of
and the observing frequency
f
in MHz. The Roemer term
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
accounts for the time dilation and gravitational redshift due to
the Sun and other masses in the solar system, while the Shapiro
delay
expresses the excess delay to the pulsar signal as it travels
through the gravitational well of the Sun - a maximum delay of
about
at the limb of the Sun; see [11] for a fuller discussion of these terms. The terms
,
, and
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
[110]. 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 [127]. 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.
where
is the reference value of
at time
. The delay terms then become:
Here
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
and the advance of periastron
, 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
and
, 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-neutron-star tests of GR in Section
4
.
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Testing General Relativity with Pulsar Timing
Ingrid H. Stairs http://www.livingreviews.org/lrr-2003-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |