An unbiased statistic for pulsar parameter estimation via dual-band light curve fitting

Abstract

The wealth of multiwavelength pulsar data has stimulated the development of emission models that predict light curves (LCs) over multiple wavebands, most notably radio and gamma-ray. Using established statistical methods to fit these model LCs to data can prove ineffectual if the data from one waveband are substantially more precise. This waveband—typically radio—dominates the fit and biases the inferred model parameters. We re-examine the use of Pearson’s chi-squared statistic for joint fits, and introduce a new, derived statistic. Alongside the intuitive reasoning we provide in support of this new statistic, we also construct a formal mathematical framework for pulsar LC fitting, within which our new statistic arises naturally. This framework also provides significant geometric tools by means of which pulsar LC fitting can be better understood intuitively, and the quality of dual-band fits can be evaluated objectively. The core insight that this statistic encodes is that the component single-band chi-squared values (for each waveband) implicitly express goodness of fit in units of the respective LC uncertainties. The resulting implicit weighting that the dual-band chi-squared carries is eliminated by expressing these values in a shared unit before calculating their sum, derived by effectively standardising the scaled pulsar-associated flux across the two wavebands. While our new statistic tends to yield dual-band fits that are not formally good (according to the standard chi-squared statistic), it also tends to yield dual-band fits that better reproduce the broad structure of the observed radio and gamma-ray LCs; this means that the parameter estimates it yield are also better. We use a Monte Carlo method to derive constriants on our parameter estimates, in the form of inclusion contours in the model’s parameter space. Using newly developed quantities to characterise non-colocation of best fits for each band as well as the relative dominance of the respective bands as a function of each band’s precision, we show that chi-squared and our new statistic converge to the same constraints as the precision disparity dissipates. As a first test, we fit two amalgamated dual-band pulsar models to 23 Fermi LAT pulsars and compare the resulting parameter constraints to earlier independent results derived using the same data and similar models. Our fits consistently show no radio dominance, and our constraints more strongly correlate with those derived by eye. We next use PSR J2039-5617 as a case study where we perform joint fits on its radio and gamma-ray LCs, the constraints of which can then be used to infer the pulsar mass. Lastly, we explore two more applications of the new statistic: attempting to recover a preliminary trend between macroscopic conductivity vs. spin-down luminosity for the FIDO pulsar model, and performing a joint fit to three datasets (spectrum and brightness / spectral index profiles) associated with pulsar wind nebulae. For the first application, we conclude that spectral data are needed to solidify this trend, in addition to LC data, while for the second, we accept that sub-optimal fits to the available datasets are not due to the statistical fitting method, but rather point to model revision.