Pulsars and neutron stars/Pulse profiles

Folded profiles
With a good model for the observed (i.e., measured at a telescope) pulses it is possible to fold a single pulse train to form a folded profile. Off-line software packages (e.g., dspsr, sigproc and presto) exist that enable an existing search-mode time series to be folded. Most observatories that carry out pulsar observations also provide a backend instrument that can fold the data during the observation. In order to account for small errors in the predicted pulse phases it is common to fold only a small time segment (e.g., 1 minute) and then record those data into the file. Each segment is known as a sub-integration. For each sub-integration a number of observing frequency channels are recorded. For each frequency channel the systems record a pulse profile for each polarisation that is available. The pulse profile is simply the folded pulse shape (and so the x-axis can be represented in phase from 0 to 1 - corresponding, in time, to the period of the pulsar). Each profile is divided into a number of phase bins. Modern backend instruments can record many thousands of phase bins.

Example folded profile


In the figure we provide an example folded pulse profile from the PULSE@Parkes outreach project for PSR J1717-4054. The top panel shows the pulse profile. Note that for most of the pulse phase the profile is simply noise. The actual pulse is observed around pulse phase 0.8. In this case the position of the pulse is arbitrary, but often backend instruments use the pulsar timing model in order to place the pulse at phase 0. The central panel shows individual subintegrations (started at time 0 at the bottom of the plot and increasing upwards). In this case the pulse profile amplitude is shown using the colour scale. This pulsar is an intermittent pulsar. Note that the pulsar is not detectable near the end of the observation, but it switches "on" again just before the end. The bottom plot shows the pulse (in colour scale) as a function of phase (x) and observing frequency (f). Note that the pulse is dispersed in frequency.

This figure also demonstrates the following:


 * Wide-band impulsive RFI is seen as a vertical line in the frequency-pulse phase plot around pulse phase 0.3 (weaker RFI is seen at other pulse phases)
 * Sensitivity is lost at the band edges. For observations with the Parkes telescope it is common to remove 5% of the signal at the band edges.
 * The pulse profile flux is in arbitrary units. In order to convert to Jy it is necessary to carry out flux calibration procedures.

Parameterising the folded profile
Dai et al. (2015) published the polarisation profiles for a set of millisecond pulsars. Their analysis was based on measurements of the Stokes parameters (I, Q, U and V). Stokes V was defined as $$I_{\rm LH} - I_{\rm RH}$$ using the IEEE convention.


 * First a baseline region was determined using the Stokes I profile
 * Baselines for the I, Q, U and V profiles were individually set to zero mean
 * The linear polarization $$L$$ was calculated from $$L = \sqrt{Q^2 + U^2}$$
 * The above expression leads to a noise bias. Everett & Weisberg (2001) describe how to remove this bias. A similar bias in |V| is described by Yan et al. (2011).
 * The position angles (PAs) of the linear polarization is referred to the centre of the observing band and calculated using: $$\Psi = \frac{1}{2}\tan^{-1}\left(\frac{U}{Q}\right)$$. In Dai et al. (2015) such PAs were only calculated when the linear polarization exceeded four times the baseline root mean square (rms) noise level.
 * Errors on the PA angles are estimated as described by Everett & Weisberg (2001).

Pulse profile sharpness
Cordes & Shannon (2010) and Shannon et al. (2014) showed how the sharpness of a profile, $$W_s$$ can be measured as (this assumes that the profile has been normalized to have a peak intensity of 1):

$$W_s = \frac{\Delta \phi}{\sum_i \left[I(\phi_{i+1}) - I(\phi_i)\right]^2}$$

where $$\Delta \phi$$ is the phase resolution (in time units) of the profile.