A MAD Explanation for the Correlation between Bulk Lorentz Factor and Minimum Variability Timescale

TL;DR

This paper explains the anti-correlation between the minimum variability timescale and the bulk Lorentz factor in gamma-ray bursts using a magnetically arrested disk model, linking theoretical limits to observed data.

Contribution

It introduces a MAD-based explanation for the $MTS$-$\Gamma$ anti-correlation and extends the model to account for $MTS$-$L$ relations in GRBs and AGNs.

Findings

Derived a relationship $t_{MAD} \\propto \\Gamma^{-6}$ consistent with data.

Natural explanation for the $MTS$-$L$ anti-correlation in GRBs.

Observed similar correlations in AGN data.

Abstract

We offer an explanation for the anti-correlation between the minimum variability timescale ($MTS$) in the prompt emission light curve of gamma-ray bursts (GRBs) and the estimated bulk Lorentz factor of these GRBs, in the context of a magnetically arrested disk (MAD) model. In particular, we show that previously derived limits on the maximum available energy per baryon in a Blandford-Znajek jet leads to a relationship between the characteristic MAD timescale, $t_{MAD}$, in GRBs and the maximum bulk Lorentz factor: $t_{MAD} \propto \Gamma^{-6}$, somewhat steeper than (although within the error bars of) the fitted relationship found in the GRB data. Similarly, the MAD model also naturally accounts for the observed anti-correlation between $MTS$ and gamma-ray luminosity $L$ in the GRB data, and we estimate the accretion rates of the GRB disks (given these luminosities) in the context of this model. Both of these correlations ($MTS-\Gamma$ and $MTS-L$) are also observed in the AGN data, and we discuss the implications of our results in the context of both GRB and blazar systems.