Candidate folding

Pulsar searching 49 2.2.4.9 Candidate sifting

After periodicity searching, the candidates are sifted to remove duplicates and to attempt to ensure that all detected signals are of the fundamental frequency. Furthermore, when periodicity searches have been performed for all trial DMs, the candidates can be sifted to remove signals which exhibit RFI-like behaviour. This can be signals which do not appear at a minimum number of adjacent DM trials, signals which appear strongest below a threshold DM¹²or signals which are at known RFI periods. A true pulsar signal should appear at multiple DM trials, with the S/N in each DM trial being dependent on the true DM of the pulsar, its pulse width and its rotational period (see figures 2.5 and 2.6). At this stage the candidates may be examined graphically or ranked on significance, or S/N, to be passed to a phase-folding algorithm.

Pulsar searching 50

0 32 64 96 128 160 192 224 256

T im e (s )

Accelerated

0.0 0.2 0.4 0.6 0.8 1.0

Pulse phase

0 5 10 15 20

S / N

Acceleration corrected

0.0 0.2 0.4 0.6 0.8 1.0

Pulse phase

Figure 2.10: A simulated observation of a 5-ms pulsar experiencing a constant accel-eration of 100 m s⁻². Left-hand panels: Subintegration plot and integrated pulse profile without taking into account the pulsar’s acceleration. Right-hand panels: Subband plot

and integrated pulse profile after correcting for the accelerated motion.

2.2.5.1 Fold optimisation

Having phase-folded a candidate, it is possible and desirable to further improve its period and DM. Optimisation of these parameters is important for correctly identifying pulsars in the data and for producing more accurate initial ephemerides for the purpose of pulsar timing.

A candidate which has been found with a periodP_disc, will exhibit a drift in phase across the course of an observation of (following e.g. Lorimer and Kramer, 2005)

∆φ_P = t_obs Pdisc

P_opt Pdisc −1

, (2.15)

whereP_opt is the true period of the pulsar to within

∆P_opt = P_opt²

n_binst_obs, (2.16)

wheren_bins is the number of phase bins across the a folded profile.

Pulsar searching 51

Figure 2.11: The discovery observation of PSR J2206+6152, found through the High Time Resolution Universe North pulsar survey (see Chapter 4). Top-left: The pulse profile, shown twice. Bottom-left: A subintegration plot showing the behaviour of the pulse profile with time. Middle: A subband plot showing the behaviour of the profile with frequency. Middle-bottom: A plot of reducedχ² as a function of DM. Top-right:

A plot of reduced χ² as a function of period. Middle-right: A plot of reduced χ² as a function of period-derivative. Bottom-right: A period, period-derivative space mapping.

This figure was created using theprestosoftware package (see Section 2.2.7.1)

Similarly, a pulsar which has been found with a DM of DM_disc, will exhibit a drift in time across its bandwidth of (following e.g. Lorimer and Kramer, 2005)

∆t_DM = 4.15×10³ 1

f_bottom² − 1 f_top²

!

(DM_opt−DM_disc) sec, (2.17) where DMopt is the true DM of the pulsar to within

∆DM_opt= 1.2×10⁻³ Pf³

n_bins∆f pc cm⁻³. (2.18) By applying phase rotations to individual subbands or subintegrations, we may examine the signal strength at a range of trial phase offsets in both DM and period to obtain the characteristic period and DM curve for the signal. Both period and DM curve are shown in Figure 2.11.

Pulsar searching 52 2.2.5.2 Folded profile significance

An important measure of a folded profile is its S/N or significance. As the sum of multiple Gaussian distributions is itself a Gaussian distribution, the probability of a signal having a S/N greater than S/N_min is given by,

p(S/N>S/N_min) = n_bins 2 erfc

S/N_min

√2

, (2.19)

where n_bins is the number of bins in the folded profile and erfc is the complimentary error function,

erfc(x) = Z _∞

x

e^−t2dt. (2.20)

Although S/N is not a defining feature of the signal from a pulsar, low signal-to-noise ratios tend to indicate that a signal is the product of random noise processes. For example, in a 1024-bin profile dominated by white noise, the probability of any bin having a S/N > 2 is ∼ 2300%. By comparison, for a good pulsar detection we expect

S/N&8. The probability of such a detection being from Gaussian noise processes is of

the order 10⁻¹¹%.

For a folded profile p_i, the S/N can be defined as (following e.g. Lorimer and Kramer, 2005)

S/N = pW_eq

σ_off ( ¯S_on−S¯_off), (2.21) where ¯S_on and ¯S_off are the mean on- and off-pulse flux densities, σ_off is the off-pulse standard deviation andW_eq is the equivalent width of a ‘top-hat’ function of the same area and peak height as the pulse. In the case of a known pulsar, the S/N is relatively simple to calculate, as the pulse width is already known, making it trivial to determine the on- and off-pulse regions of the profile. However, when the pulse width is unknown, we must apply methods which can accurately determine the on- and off-pulse phases.

Such methods often have varying effectiveness for different pulse shapes, making the process of finding the true on-pulse region difficult. A good estimate for the off-pulse standard deviation can be found by considering the noise statistics of the data prior to folding, as the raw data should be dominated by Gaussian noise of standard deviation σ_T. For a time series which has been foldedn_folds=t_obs/P times, the standard deviation of the resulting off-pulse portion of the profile will be √n_foldsσ_T. Further normalising the profile by dividing each phase-bin by the number of points folded into it, we find σ_off =σ_T/√n_folds =σ_T/p

(P/t_obs).

It is also possible to use the probability-value of theχ² statistic of the profile to obtain a significance in terms ofσ (i.e.a measure of the profiles deviation from pure Gaussian noise, see e.g. Leahy et al., 1983). This method is well suited to use on binned data as

Pulsar searching 53 well as sampled data. However, It should be noted that for highly significant signals, the probability rapidly approaches zero. This imposes a limit on numerically calculated values of σ for the most significant signals (i.e. for signals of > 40-σ significance, the true significance cannot be easily calculated numerically).

After optimisation and determination of the S/N orσvalue of all candidates, the folded data are stored for later analysis by eye or by automatic pulsar selection algorithms (see Section 4.6.3.5).