SKA1 Pulsar Timing Resolutions


This memo aims to address and update the following level 1 system requirements for pulsar timing with SKA1_Low and SKA1_Mid:

  • SKA1-SYS_REQ-2961: SKA1_Mid Pulsar Timing resolution. The SKA1_Mid Pulsar timing mode shall have a timing resolution of better than 100 ns.

  • SKA1-SYS_REQ-2962: SKA1_Low Pulsar Timing resolution. The SKA1_Low Pulsar timing mode shall have a timing resolution of better than 100 ns.

We will approach this in two ways: Firstly we will look at the maximum timing resolution that can be envisaged as being required for pulsar timing with SKA1 and secondly we will consider how multi-path scattering imposes further limits on timing resolution as a function of dispersion measure.

Maximum Temporal Resolutions for SKA1

Figure \ref{fig:pulse_spectra} displays fluctuation power spectra derived from the long-term average profiles of two Parkes Pulsar Timing Array (PPTA) pulsars observed at 20 and 50 cm (Dai 2015). In these plots, the fluctuation power of each pulsar drops exponentially as a function of spin harmonic, eventually hitting a white noise floor. PSR J0437\(-\)4715 and PSR J2241\(-\)5236 are the worst case examples in which fluctuation power approaches the Nyquist limit where spin harmonics will be aliased. These plots indicate that 1024 phase bins is sufficient to resolve all of the structure in the mean pulse profiles of the currently known MSPs used for high-precision timing.

Greater telescope sensitivity increases the signal-to-noise ratio, causing more fluctuation power to rise out of the white noise floor; however, these plots indicate that the number of spin harmonics required to resolve all significant power is proportional only to the logarithm of the S/N; i.e. to first order, the maximum harmonic,

\[H_\mathrm{max} \sim (1 - \log S/N) / k,\]

where \(k\) is the slope of the line fit to the spectrum of the pulsar, \(S/N\) is the signal-to-noise ratio at the y-intercept (approximately the \(S/N\) in the first harmonic) and it is assumed that harmonics past \(S/N\sim1\) are not important. By eye, increasing the S/N by two orders of magnitude will add around 100 to 300 harmonics for 0437 and 2241, so 2048 phase bins should suffice for observing these pulsars with SKA1.

\label{fig:pulse_spectra} Fluctuation power spectra for PSR J0437\(-\)4715 at 20 cm and PSR J2241\(-\)5236 at 40 cm exhibit significant power up to the 400\(^{th}\) harmonic of the pulsar’s spin period. The black line is the fluctuation power in the total flux and the red line is the power in the polarized flux. Note that these average profiles have been integrated from 500 hours and 70 hours of observations, respectively.

It is reasonable to assume that more extreme pulsars with faster spin periods than are currently known will be found given the sensitivity of SKA1. If we assume a J0437\(-\)4715-like pulsar with a 500-\(\mu\)s spin period, it would require \(\sim\)200 ns-time resolution observations to be able to fully resolve all of the structure in its pulse profile. Naïvely speaking 200 ns time resolution would imply 5-MHz frequency channels via a reciprocal bandwidth argument. However due to the shape of the impulse response of polyphase filterbanks, it is not possible to achieve 200-ns time resolution with 5-MHz frequency channels while meeting the stringent spectral leakage requirements of SKA1. Through simulation of various polyphase filters, we find that using a 22-tap filter allows us to meet the spectral purity requirements of SKA1 and achieve an effective time resolution of two time samples (i.e. 200 ns for a 10-MHz channel).

The 200-ns resolution described above is required only for high-precision pulsar timing, where we need to resolve the highest spin harmonics in the pulsars we observe. Due to the deleterious effects of the interstellar medium (ISM) we consider 200-ns time resolution to be unnecessarily high for SKA1_Low. The pulsar-related science goals of SKA1_Low (probing the ISM, emission mechanism studies, etc.) do not require such high time resolution; therefore, a larger sampling interval would be acceptable. Currently the LFAA design implies that channel widths of approximately 800 kHz will be delivered to the CSP for processing. By the same arguments as above, this would provide \(\sim\)2.5 \(\mu\)s effective time resolution. This is \(\sim2\) times better than the resolution achievable with LOFAR. We consider this resolution to be acceptable for achieving pulsar science goals with SKA1_Low.

The above frequency and time resolutions allow us to suggest rewordings for requirements 2691 and 2692 such that they are framed in terms of fractional resolutions of spin period with an absolute resolution limit:

  • SKA1-SYS_REQ-2961: SKA1_Mid Pulsar Timing resolution. The SKA1_Mid when in pulsar timing mode shall resolve a pulsar’s pulse profile with up to 2048 equal-width, contiguous phase bins of maximum effective time resolution, 200 ns.

  • SKA1-SYS_REQ-2962: SKA1_Low Pulsar Timing resolution. The SKA1_Low when in pulsar timing mode shall resolve a pulsar’s pulse profile with up to 2048 equal-width, contiguous phase bins of maximum effective time resolution, 2.5 \(\mu\)s.

These two requirements should constrain both the acceptable channel widths provided to the SKA1 pulsar timing instruments and the polyphase filterbank design used to channelise the digitized baseband voltages. At level 2, these requirements will predominantly affect CSP_Mid, CSP_Low, and LFAA. There may also be minor implications for TM, DSH and SaDT. To provide level 3 design constraints for the CSP and LFAA we below define what we mean by “effective time resolution”:

Effective Time Resolution (non image processing):
def. The full-width at 10% maximum of the SKA1_Mid or SKA1_Low impulse response.

While the above requirements would appear to satisfy the needs of the pulsar community, the practicality of these requirements is limited by the computational burden of performing coherent dedispersion at low frequencies on wide frequency channels. This burden only becomes more difficult to manage as we move to larger dispersion measures (DMs). As such, the requirements suggested above need a modifier that allows them to change with DM. This modifier is provided by the physical limitation that interstellar scattering places on the minimum resolvable feature width in a pulse profile. This is discussed in detail below.