Do Pulsar Radio Fluxes violate the Inverse-Square Law?

Abstract: Singleton et al (2009) have argued that the flux of pulsars measured at 1400 MHz shows an apparent violation of the inverse-square law with distance ($r$), and it is consistent with $1/r$ scaling. They deduced this from the fact that the convergence error obtained in reconstructing the luminosity function of pulsars using an iterative maximum likelihood procedure is about $10^5$ times larger for a distance exponent of two (corresponding to the inverse-square law) compared to an exponent of one. When we applied the same technique to this pulsar dataset with two different values for the trial luminosity function in the zeroth iteration, we find that neither of them can reproduce a value of $10^5$ for the ratio of the convergence error between these distance exponents. We then reconstruct the differential pulsar luminosity function using Lynden-Bell's $C^{-}$ method after positing both an inverse-linear and an inverse-square scalings with distance. We show this method cannot help in discerning between the two exponents. Finally, when we tried to estimate the power law exponent with a Bayesian regression procedure, we do not get a best-fit value of one for the distance exponent. The model residuals obtained from our fitting procedure are larger for the inverse-linear law compared to the inverse-square law. Moreover, the observed pulsar flux cannot be parameterized only by power-law functions of distance, period, and period derivative. Therefore, we conclude from our analysis using multiple methods that there is no evidence that the pulsar radio flux at 1400 MHz violates the inverse-square law.