Intrinsic variability and field statistics for the Vela pulsar -- III. Two-component fits and detailed assessment of stochastic growth theory

Abstract

The variability of the Vela pulsar (PSR B0833–45) corresponds to well‐defined field statistics that vary with pulsar phase, ranging from Gaussian intensity statistics off‐pulse to approximately power‐law statistics in a transition region and then lognormal statistics on‐pulse, excluding giant micropulses. These data are analysed here in terms of two superposed wave populations, using a new calculation for the amplitude statistics of two vectorially combined transverse fields. Detailed analyses show that the approximately power‐law and lognormal distributions observed are well fitted at essentially all on‐pulse phases by Gaussian–lognormal and double‐lognormal combinations, respectively. The good fits found, plus the smooth but significant variations in fitting parameters across the source, provide strong evidence that the approximately power‐law statistics observed in the transition region are not intrinsic. Instead, the data are consistent with the normal emission of the pulsar having lognormal statistics whenever the pulsar is detectable. This is consistent with generation in an inhomogeneous source obeying stochastic growth theory (SGT) and with the emission mechanism being purely linear (either direct or indirect), with no evidence for non‐linear processes. A non‐linear mechanism is viable only if it produces lognormal statistics when suitably ensemble‐averaged. Variations in the SGT fit parameters with phase are consistent with the radiation being relatively more variable near the pulse edges than near the centre, consistent with earlier work. In contrast, the giant micropulses of Vela come from a very restricted phase range and have power‐law statistics with indices (6.7 ± 0.6) not inconsistent with non‐linear wave collapse. These results are consistent with normal pulses coming from a source and generation mechanism different from giant micropulses, as suggested previously on other grounds. Analysis of field statistics thus emphasizes the richness of pulsar physics, the apparently widespread applicability of SGT, and connections between variability and the generation mechanism.

The High Time Resolution Universe Pulsar Survey - V. Single-pulse energetics and modulation properties of 315 pulsars

“…There exist a great number of viable plasma‐state models, a few of which predict pulse energy distributions; the most commonly proposed predictions are of Gaussian, log‐normal and power‐law distributions. Cairns, Johnston & Das (2003a) and Cairns et al (2003b), and references therein, provide discussion on these models. Energy distributions have been examined in detail for only a few pulsars (Cognard et al 1996; Cairns, Johnston & Das 2001, 2004), resulting in the conclusion that those pulsars obey log‐normal statistics.…”

Strong Langmuir turbulence in a pulsar emission region: statistical analysis

“…Even so, we are able to derive numerically from it the behaviour of the probability distribution of the ensemble of Langmuir solitons considered in our statistical treatment of the Langmuir turbulence. The comparison of the latter probability distribution to the lognormal distribution proposed and obtained by Cairns et al (2001, 2003a,b), fitting the bundle of intensities observed in the central region of an individual pulse emission window, shows that such a comparison is only possible in a domain of variation of the intensities from which the region of extremely high intensities is excluded. However, the model we did use does not privilege any region within an individual pulse emission window.…”

“…To compare with the results obtained by Cairns et al (2001, 2003a,b), who successfully fitted the observed intensity in the region of maximum intensities using a lognormal distribution , it is thus necessary to integrate their specific density function, namely the lognormal density (1/ E ) exp − (log E ) 2 /2. We assume that the variable E , representing the energy, has a lognormal density, with parameters m = 0 and σ 2 = 1.…”

“…This means that the contribution to the distribution function of the intensities larger than I eff max is less than ε, that is to say, negligible. Then, we are able to perform the scaling of the intensities, changing the intensity variable I to I eff1 max / I eff2 max , where I eff1 max and I eff2 max correspond to the cases treated either by us, or by Cairns et al (2001, 2003a,b). The result of this change of scale is shown in Fig.…”

“…The shape of the curve giving the variation with time of the probability distribution of the mean intensity could be confronted with pulsar radio observations of the intensity in a relatively large part of the pulse window, if one excepts the region of extremely high intensities. Recently, Cairns, Johnston & Das (2001, 2003a) and Cairns et al (2003b) did correctly picture the distribution of high intensities observed in the central region of a pulse emission window by means of a lognormal distribution. In fact, comparison on a large interval of intensities, of the behaviour of our distribution function of the intensity and of the one chosen by Cairns et al, is limited because we have introduced for technical reasons a finite maximum value for the intensity.…”