On the Curvature Effect of a Relativistic Spherical Shell

TL;DR

This paper derives an analytical expression for the spectral flux of a relativistic spherical shell and investigates how deviations from constant Lorentz factor affect the high-latitude emission relation between temporal and spectral indices.

Contribution

It provides a simple analytical model for spectral flux and explores how acceleration or deceleration of the shell alters the high-latitude emission relation in gamma-ray burst afterglows.

Findings

The relation $oxed{ ext{α} = 2 + ext{β}}$ holds only for constant Lorentz factor shells.

Acceleration causes initially steeper decay in light curves, gradually returning to the standard relation.

Deceleration results in initially shallower decay, also eventually restoring the relation.

Abstract

We consider a relativistic spherical shell and calculate its spectral flux as received by a distant observer. Using two different methods, we derive a simple analytical expression of the observed spectral flux and show that the well-known relation $\hat \alpha = 2+\hat \beta$ (between temporal index $\hat \alpha$ and spectral index $\hat \beta$) of the high-latitude emission is achieved naturally in our derivation but holds only when the shell moves with a constant Lorentz factor $\Gamma$. Presenting numerical models where the shell is under acceleration or deceleration, we show that the simple $\hat \alpha = 2+\hat \beta$ relation is indeed deviated as long as $\Gamma$ is not constant. For the models under acceleration, we find that the light curves produced purely by the high-latitude emission decay initially much steeper than the constant $\Gamma$ case and gradually resume the $\hat \alpha = 2+\hat \beta$ relation in about one and half orders of magnitude in observer time. For the models under deceleration, the trend is opposite. The light curves made purely by the high-latitude emission decay initially shallower than the constant $\Gamma$ case and gradually resume the relation $\hat \alpha = 2+\hat \beta$ in a similar order of magnitude in observer time. We also show that how fast the Lorentz factor $\Gamma$ of the shell increases or decreases is the main ingredient determining the initial steepness or shallowness of the light curves.