# A Fundamental Plane for Gamma-Ray Pulsars

## TL;DR

This paper identifies a fundamental plane relating gamma-ray pulsar observables, supported by data and theory, revealing insights into their emission mechanisms and predicting how spectral cut-off energy varies with spin-down power.

## Contribution

The study establishes a pulsar fundamental plane linking gamma-ray luminosity, spectral cut-off energy, magnetic field, and spin-down power, supported by observational data and theoretical modeling.

## Key findings

- The fundamental plane relation fits 88 pulsars from Fermi data.
- The observed relation closely matches the theoretical curvature radiation model.
- The scatter around the plane is partly due to observational errors and geometric variations.

## Abstract

We show that the $\gamma$-ray pulsar observables, i.e., their total $\gamma$-ray luminosity, $L_{\gamma}$, spectral cut-off energy, $\epsilon_{\rm cut}$, stellar surface magnetic field, $B_{\star}$, and spin-down power $\dot{\mathcal{E}}$, obey a relation of the form $L_{\gamma}=f(\epsilon_{\rm cut},B_{\star},\dot{\mathcal{E}})$, which represents a 3D plane in their 4D log-space. Fitting the data of 88 pulsars of the second Fermi pulsar catalog, we show this relation to be $L_{\gamma}\propto \epsilon_{\rm cut}^{1.18\pm 0.24}B_{\star}^{0.17\pm 0.05}\dot{\mathcal{E}}^{0.41\pm 0.08}$, a pulsar fundamental plane (FP). We show that the observed FP is remarkably close to the theoretical relation $L_{\gamma}\propto \epsilon_{\rm cut}^{4/3}B_{\star}^{1/6}\dot{\mathcal{E}}^{5/12}$ obtained assuming that the pulsar $\gamma$-ray emission is due to curvature radiation by particles accelerated at the pulsar equatorial current sheet just outside the light cylinder. Interestingly, the FP seems incompatible with emission by synchrotron radiation. The corresponding scatter about the FP is $\sim 0.35$dex and can only partly be explained by the observational errors while the rest is probably due to the variation of the inclination and observer angles. We predict also that $\epsilon_{\rm cut}\propto \dot{\mathcal{E}}^{7/16}$ toward low $\dot{\mathcal{E}}$ for both young and millisecond pulsars implying that the observed death-line of $\gamma$-ray pulsars is due to $\epsilon_{\rm cut}$ dropping below the Fermi-band. Our results provide a comprehensive interpretation of the observations of $\gamma$-ray pulsars, setting requirement for successful theoretical modeling.