2 General Relativity Measurements from Pulsars
73
span necessary to get a precise determination of the pulsar position by exploiting
the motion of the Earth around the Sun (see Sect. 2.4.2)—to tens of years, if,
for instance, pulsars are used as detectors for the nano-Hertz gravitational waves
emitted by coalescing supermassive black holes (see Sect. 2.5.3). The spacing
between observations also varies: initially, having a very rough determination of
the pulsar ephemeris, a denser coverage (up to few observations a day) is needed;
as the errors on timing parameters gets smaller (see Sect. 2.4.2), the cadence can be
relaxed and one monthly observation is usually enough to derive a coherent timing
solution over the entire dataset (see Sect. 2.4.2).
Single pulses from pulsars, with the exception of a few very bright sources, are
usually below the level of the noise in the data. To get a precise measurement of
a ToA, it is hence necessary to average several pulses. This, besides increasing
the signal-to-noise ratio of the pulse, allows us to get a stable pulse shape, the
peak of which always falls at the same rotational phase, which allows for a correct
determination of the ToA. While single pulses do vary in shape, flux density and
exact location of the peak within each given rotation, the pulse shape obtained by
summing few hundreds of rotations is extremely stable, 3 so that the determination
of the ToA of its peak can be used to precisely keep track of the rotation of the star.
A timing observation is hence obtained by summing a high number of pulse
rotations modulo the expected apparent spin period, as predicted from the start
time of the observation, the location of the telescope and the best pulsar ephemeris
available. A correction for the dispersive delay within the instrument’s bandwidth is
also necessary and is performed on the basis of the pulsar’s DM and the observing
frequency. Each pulse profile is time-tagged using an accurate clock (e.g. an
hydrogen-maser clock) present at the radio telescope, which is regularly corrected
by comparing it with the time distributed by the Global Positioning System (GPS).
In order to remove radio frequency interference (RFI) from the data, the timing
observation is usually divided into a number of time sub-integrations of short
duration (a few seconds to a few minutes) and into frequency sub-bands (usually
up to 1024): in this way, short-duration or narrow-band RFI can easily be mitigated
by deleting only the frequency channel or the small section of time affected. Subintegratios and sub-bands also allow us to verify if the folding period and DM are
correct by checking that the pulse falls at the same rotational phase at all times and in
all frequency channels. Once all sub-bands and sub-integrations contain only clean
pulse profiles all in phase with each other, the data are further summed using the
most recent ephemeris available and one (or more, depending on the length of the
timing observation and on the brightness of the pulsar) high signal-to-noise profile
is obtained.
To obtain the ToAs, the main ingredients of the timing process, these pulse
profiles are finally compared—usually through a convolution method—with a
3 There are rare examples in which even the shape of the integrated profile is variable in time, as for
instance in the case of PSRJ0737−3039B in the double pulsar system [41, 42]; these cases require
the application of more sophisticated timing methodologies
73
span necessary to get a precise determination of the pulsar position by exploiting
the motion of the Earth around the Sun (see Sect. 2.4.2)—to tens of years, if,
for instance, pulsars are used as detectors for the nano-Hertz gravitational waves
emitted by coalescing supermassive black holes (see Sect. 2.5.3). The spacing
between observations also varies: initially, having a very rough determination of
the pulsar ephemeris, a denser coverage (up to few observations a day) is needed;
as the errors on timing parameters gets smaller (see Sect. 2.4.2), the cadence can be
relaxed and one monthly observation is usually enough to derive a coherent timing
solution over the entire dataset (see Sect. 2.4.2).
Single pulses from pulsars, with the exception of a few very bright sources, are
usually below the level of the noise in the data. To get a precise measurement of
a ToA, it is hence necessary to average several pulses. This, besides increasing
the signal-to-noise ratio of the pulse, allows us to get a stable pulse shape, the
peak of which always falls at the same rotational phase, which allows for a correct
determination of the ToA. While single pulses do vary in shape, flux density and
exact location of the peak within each given rotation, the pulse shape obtained by
summing few hundreds of rotations is extremely stable, 3 so that the determination
of the ToA of its peak can be used to precisely keep track of the rotation of the star.
A timing observation is hence obtained by summing a high number of pulse
rotations modulo the expected apparent spin period, as predicted from the start
time of the observation, the location of the telescope and the best pulsar ephemeris
available. A correction for the dispersive delay within the instrument’s bandwidth is
also necessary and is performed on the basis of the pulsar’s DM and the observing
frequency. Each pulse profile is time-tagged using an accurate clock (e.g. an
hydrogen-maser clock) present at the radio telescope, which is regularly corrected
by comparing it with the time distributed by the Global Positioning System (GPS).
In order to remove radio frequency interference (RFI) from the data, the timing
observation is usually divided into a number of time sub-integrations of short
duration (a few seconds to a few minutes) and into frequency sub-bands (usually
up to 1024): in this way, short-duration or narrow-band RFI can easily be mitigated
by deleting only the frequency channel or the small section of time affected. Subintegratios and sub-bands also allow us to verify if the folding period and DM are
correct by checking that the pulse falls at the same rotational phase at all times and in
all frequency channels. Once all sub-bands and sub-integrations contain only clean
pulse profiles all in phase with each other, the data are further summed using the
most recent ephemeris available and one (or more, depending on the length of the
timing observation and on the brightness of the pulsar) high signal-to-noise profile
is obtained.
To obtain the ToAs, the main ingredients of the timing process, these pulse
profiles are finally compared—usually through a convolution method—with a
3 There are rare examples in which even the shape of the integrated profile is variable in time, as for
instance in the case of PSRJ0737−3039B in the double pulsar system [41, 42]; these cases require
the application of more sophisticated timing methodologies