3.2 Correcting the observed pulsar 3 The Galactic Pulsar Population3 The Galactic Pulsar Population

3.1 Selection effects in pulsar searches 

3.1.1 The inverse square law and survey thresholds 

The most prominent selection effect at play in the observed pulsar sample is the inverse square law, i.e. for a given intrinsic luminosity Popup Footnote, the observed flux density falls off as the inverse square of the distance. This results in the observed sample being dominated by nearby and/or bright objects. Beyond distances of a few kpc from the Sun, the apparent flux density falls below the flux thresholds tex2html_wrap_inline9219 of most surveys. Following [71], we write:


where SNR tex2html_wrap_inline9221 is the threshold signal-to-noise ratio, tex2html_wrap_inline9223 and tex2html_wrap_inline9225 are the receiver and sky noise temperatures, G is the gain of the antenna, tex2html_wrap_inline9229 is the observing bandwidth, tex2html_wrap_inline9231 is the integration time, W is the detected pulse width and P is the pulse period.

3.1.2 Pulse dispersion and scattering 

It follows from Equation (3Popup Equation) that the sensitivity decreases as W /(P - W) increases. Also note that if tex2html_wrap_inline9239, the pulsed signal is smeared into the background emission and is no longer detectable, regardless of how luminous the source may be. The detected pulse width W will be broader than the intrinsic value largely as a result of pulse dispersion and scattering by free electrons in the interstellar medium. As discussed above, the dispersive smearing scales as tex2html_wrap_inline9243, where tex2html_wrap_inline9245 is the observing frequency. This can largely be removed by dividing the pass-band into a number of channels and applying successively longer time delays to higher frequency channels before summing over all channels to produce a sharp profile. This process is known as incoherent dedispersion.

The smearing across the individual frequency channels, however, still remains and becomes significant at high dispersions when searching for short-period pulsars. Multi-path scattering results in a one-sided broadening due to the delay in arrival times which scales roughly as tex2html_wrap_inline9247, which can not be removed by instrumental means. A simple scattering model is shown in Fig.  13 in which the scattering electrons are assumed to lie in a thin screen between the pulsar and the observer [210].


Click on thumbnail to view image

Figure 13: Pulse scattering caused by irregularities in the interstellar medium. The difference in path lengths and therefore in arrival times of the scattered rays results in a ``scattering tail'' in the observed pulse profile which lowers its signal-to-noise ratio.

Dispersion and scattering are most severe for distant pulsars in the inner Galaxy where the number of free electrons along the line of sight becomes large. The strong frequency dependence of both effects means that they are considerably less of a problem for surveys at observing frequencies tex2html_wrap_inline9249  [56, 109Jump To The Next Citation Point In The Article] compared to the usual 400 MHz search frequency. An added bonus for such observations is the reduction in tex2html_wrap_inline9225, since the spectral index of the non-thermal Galactic emission is about -2.8 [126]. Pulsars themselves have steep radio spectra. Typical spectral indices are -1.6 [136], so that flux densities are roughly an order of magnitude lower at 1400 MHz compared to 400 MHz. Fortunately, this can usually be compensated for by the use of larger receiver bandwidths at higher radio frequencies. For example, the 1370-MHz system at Parkes has a bandwidth of 288 MHz [147Jump To The Next Citation Point In The Article] compared to the 430-MHz system, where 32 MHz is available [160Jump To The Next Citation Point In The Article].

3.1.3 Orbital acceleration 

Standard pulsar searches use Fourier techniques to search for a-priori unknown periodic signals and usually assume that the apparent pulse period remains constant throughout the observation. For searches with integration times much greater than a few minutes this assumption is only valid for solitary pulsars, or those in binary systems where the orbital periods are longer than about a day. For shorter-period binary systems, as noted by Johnston & Kulkarni [106], the Doppler-shifting of the period results in a spreading of the signal power over a number of frequency bins in the Fourier domain, leading to a reduction in signal-to-noise ratio. An observer will perceive the frequency of a pulsar to shift by an amount aT /(Pc), where a is the (assumed constant) line-of-sight acceleration during the observation of length T, P is the (constant) pulse period in its rest frame and c is the speed of light. Given that the width of a frequency bin is 1/ T, we see that the signal will drift into more than one spectral bin if tex2html_wrap_inline9265 . Survey sensitivities to rapidly-spinning pulsars in tight orbits are therefore significantly compromised when the integration times are large.


Click on thumbnail to view imageClick on thumbnail to view image

Figure 14: Left: A 22.5-min Arecibo observation of the binary pulsar B1913+16. The assumption that the pulsar has a constant period during this time is clearly inappropriate given the drifting in phase of the pulse during the integration (grey scale plot). Right: The same observation after applying an acceleration search. This shows the effective recovery of the pulse shape and a significant improvement in the signal-to-noise ratio.

As an example of this effect, as seen in the time domain, Fig.  14 shows a 22.5-min search mode observation of Hulse & Taylor's famous binary pulsar B1913+16 [102Jump To The Next Citation Point In The Article, 239Jump To The Next Citation Point In The Article, 240Jump To The Next Citation Point In The Article]. Although this observation covers only about 5% of the orbit (7.75 hr), the effects of the Doppler smearing on the pulse signal are very apparent. While the standard search code (seeking constant periodicity) nominally detects the pulsar with a signal-to-noise ratio of 9.5 for this observation, it is clear that the Doppler shifting of the pulse period seen in the individual sub-integrations results in a significant reduction in signal-to-noise.

It is clearly desirable to employ a technique to recover the loss in sensitivity due to Doppler smearing. One such technique, the so-called ``acceleration search'' [168], assumes the pulsar has a constant acceleration during the integration. Each time series can then be re-sampled to refer it to the frame of an inertial observer using the Doppler formula to relate a time interval tex2html_wrap_inline9267 in the pulsar frame to that in the observed frame at time t, as tex2html_wrap_inline9271 . Searching over a range of accelerations is desirable to find the time series for which the trial acceleration most closely matches the true value. In the ideal case, a time series is produced with a signal of constant period for which full sensitivity is recovered (see right panel of Fig.  14). Anderson et al. [5Jump To The Next Citation Point In The Article] used this technique to find PSR B2127+11C, a double neutron star binary in M15 which has parameters similar to B1913+16. Camilo et al. [47Jump To The Next Citation Point In The Article] have recently applied the same technique to 47 Tucanae to discover 9 binary pulsars, including one in a 96-min orbit around a low-mass (tex2html_wrap_inline9273) companion. This is currently the shortest binary period for any known radio pulsar.

For the shortest orbital periods, the assumption of a constant acceleration during the observation clearly breaks down. Ransom et al. [199Jump To The Next Citation Point In The Article] have developed a particularly efficient algorithm for finding binaries whose orbits are so short that many orbits can take place during an integration. This phase modulation technique exploits the fact that the pulses are modulated by the orbit to create a family of periodic sidebands around the nominal spin period of the pulsar. This technique has already been used to discover a 1.7-hr binary pulsar in NGC 6544 [199Jump To The Next Citation Point In The Article]. The existence of these short-period radio pulsar binaries, as well as the 11-min X-ray binary X1820-303 in NGC 6624 [226], implies that there must be many more short-period binaries containing radio or X-ray pulsars in globular clusters that are waiting to be discovered by more sensitive searches.

3.2 Correcting the observed pulsar 3 The Galactic Pulsar Population3 The Galactic Pulsar Population

image Binary and Millisecond Pulsars at the New Millennium
Duncan R. Lorimer
© Max-Planck-Gesellschaft. ISSN 1433-8351
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