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  • SKA1 Pulsar Timing Resolutions

    Aims

    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 co