) and observed (t) TOAs:
Here
is the position of the Earth with respect to the barycentre,
is a unit vector in the direction towards the pulsar at a
distance
d, and
c
is the speed of light. The first term on the right hand side of
this expression is the light travel time from the Earth to the
solar system barycentre. For all but the nearest pulsars, the
incoming pulses can be approximated by plane wavefronts. The
second term, which represents the delay due to spherical
wavefronts and which yields the trigonometric parallax and hence
d, is only presently measurable for four nearby millisecond
pulsars [87
,
41,
136]. The term
represents the Einstein and Shapiro corrections due to general
relativistic effects within the solar system [19]. Since measurements are often carried out at different
observing frequencies, each with its own dispersive delays, the
TOAs are generally shifted back to the equivalent time that would
be observed at infinite frequency. This shift corresponds to the
term
and may be calculated from equation
1
.
Following the accumulation of about ten to twenty barycentric
TOAs from observations spaced over at least several months, a
surprisingly simple model can be applied to the TOAs and
optimised so that it is sufficient to account for the arrival
time of any pulse emitted during the time span of the
observations and predict the arrival times of subsequent pulses.
The model is based on a Taylor expansion of the angular
rotational frequency
about a model value
at some reference epoch
. The model pulse phase
as a function of barycentric time is thus given by:
where
is the pulse phase at
. Based on this simple model, and using initial estimates of the
position, dispersion measure and pulse period, a ``timing
residual'' is calculated for each TOA as the difference between
the observed and predicted pulse phases.
Early sets of residuals will exhibit a number of trends
indicating a systematic error in one or more of the model
parameters, or a parameter not initially incorporated into the
model (e.g.
). From equation
9, an error in the assumed
results in a linear slope with time. A parabolic trend results
from an error
. Additional effects will arise if the assumed position of the
pulsar (the unit vector
in equation
8) is incorrect. A position error of just one arcsecond results in
an annual sinusoid with a peak-to-peak amplitude of about 5 ms
for a pulsar on the ecliptic; this is easily measurable for
typical TOA uncertainties of order one milliperiod or better.
Similarly, the effect of a proper motion produces an annual
sinusoid of linearly increasing magnitude.
After a number of iterations, and with the benefit of a
modicum of experience, it is possible to identify and account for
each of these various effects to produce a ``timing solution''
which is phase coherent over the whole data span. The resulting
model parameters provide spin and astrometric information about
the neutron star to a precision which improves as the length of
the data span increases. The latest observations of the original
millisecond pulsar, B1937+21, spanning almost 9 yr (exactly
165,711,423,279 rotations!) measure a period of
ms [87,
85] defined at midnight UT on December 5 1988! Astrometric
measurements based on these data are no less impressive, with
position errors of
arcsec being presently possible.
|
Binary and Millisecond Pulsars
D. R. Lorimer (dunc@mpifr-bonn.mpg.de) http://www.livingreviews.org/lrr-1998-10 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
