The dispersion in pulsar $γ$-ray efficiency

TL;DR

This work tackles the broad dispersion in pulsar γ-ray observational efficiency, $\eta_{obs}$, by factorizing it as $\eta_{obs} = \eta_{rad}\,\epsilon_I\,\epsilon_d\,f_{\Omega}^{-1}$ and using a geometrical/spectral model to map $f_{\Omega}$ and NS-population distributions to a posterior for the intrinsic mechanism efficiency $\eta_{rad}$. Through Monte Carlo sampling and KS testing against the observed $\log_{10}\eta_{obs}$ distribution, they find $\log_{10}\eta_{rad}$ peaks at $\sim 0.7$–$1.2$ (i.e., $\eta_{rad} \approx 5$–$15\%$) with a dispersion of about one dex, indicating $\eta_{rad}$ is the dominant source of variance. The inferred dispersion is only weakly sensitive to distance-error modeling and to the assumed correlation with spin-down power, suggesting intrinsic emission efficiency is the key factor. The approach is readily applicable to alternative beaming geometries, and the results are broadly consistent with magnetospheric simulations that dissipate a substantial fraction of Poynting flux near the current sheet.

Abstract

The observational efficiency of pulsars, defined as the ratio of the observationally derived isotropic-equivalent luminosity, $4πd_{obs}^2 F_{obs}$, where $F_{obs}$ is the average pulsed energy flux of a pulsar and $d_{obs}$ is its estimated distance, to its energy budget, shows a wide range of values. This dispersion is believed to be a combination of beaming effects, different geometries, and case-by-case variability of the emission mechanism efficiency, but it is not clear in what proportion. In this work we focused on the gamma-ray range and analysed the four main ingredients that likely contribute to this dispersion: the geometrical term arising from the anisotropic emission (beaming), viewing and inclination angles, the uncertainty on the pulsar distance, the uncertainty on the moment of inertia, and the intrinsic efficiency of the mechanism producing the gamma-ray emission. Estimating the expected ranges of the moment of inertia and the distance errors, and considering a geometrical and spectral model that we have recently used to fit the light curves and spectra of the entire gamma-ray pulsar population, we estimate the a priori distribution of the first three ingredients in order to obtain the a posteriori distribution of the intrinsic efficiency of the mechanism. We found the latter to peak at $\sim 5-15 \%$ and to have a dispersion of around one order of magnitude. That is, we found the intrinsic efficiency of the mechanism to be the leading factor in the observed dispersion. In addition, we found little sensitivity of these results on different distributions of the estimated pulsar distance errors, and saw that the weak, alleged correlation with the spin-down power can only explain part of the observed dispersion. This methodology can be easily applied to other geometrical models of the emission, to test the sensitivity of these results on the beaming distribution.

Figures (3)

• Figure 1: Probability density of the terms forming $\log_{10}\eta_{obs}$. Top: $\log_{10}f_{\Omega}$ (left) and $\log_{10}\epsilon_I$ (right). Bottom: $\log_{10}\epsilon_d$ from different prescriptions, $P(\Delta)=\mathcal{N}$ (left) and $P(\epsilon_d)=\mathcal{N}$ (right).
• Figure 2: Results of the fitting to find the distribution of $\eta_{rad}$. The top (middle) row shows those obtained using the distance uncertainty distributed as $P(\Delta)$=$\mathcal{N}$, with a Gaussian (skew normal) as the trial distribution. The bottom row were obtained with the distance uncertainty distributed as $P(\epsilon_d)$=$\mathcal{N}$, with a Gaussian as the trial distribution. From left to right: contour plot of $D_{KS}$ in the space of parameters $\mu_{rad}$-$\sigma_{rad}$ (in the skew normal case we collapsed all the contours for the different $\alpha$ considered taking the smallest $D_{KS}$ for each pair $\mu_{rad}$-$\sigma_{rad}$), best-fitting distribution of $\log_{10}\eta_{rad}$, overlapped distribution of $\log_{10}\eta_{obs}$ (red), and joint distribution (blue).
• Figure 3: Results of the fitting to find the distribution of $\eta_{rad}$, analogous to Fig. \ref{['fig:results_fitting_mechanism']}. The top (bottom) row shows those obtained with a $\log_{10}\eta_{obs}$ distribution de-trended from the correlation $\propto \dot{E}^\beta$, with $\beta$ equal to $-0.5$ ($-0.32$). Both cases were obtained using the distance uncertainty distributed as $P(\Delta)$=$\mathcal{N}$ and with a Gaussian as the trial distribution.