On the Curvature Effect of a Relativistic Spherical Shell

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 $\widehat{\alpha }=2+\widehat{\beta }$ (between temporal index $\hat \alpha$andspectralindex$\hat \beta$) of the high-latitude emission is achieved naturally in our derivation but holds only when the shell moves with a constant Lorentz factor $\mathrm{\Gamma }$. Presenting numerical models where the shell is under acceleration or deceleration, we show that the simple $\hat \alpha = 2+\hat \beta$relationisindeeddeviatedaslongas$\Gamma$ 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 $\mathrm{\Gamma }$ case and gradually resume the $\widehat{\alpha }=2+\widehat{\beta }$ 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 $\mathrm{\Gamma }$ case and gradually resume the relation $\widehat{\alpha }=2+\widehat{\beta }$ in a similar order of magnitude in observer time. We also show that how fast the Lorentz factor $\mathrm{\Gamma }$ of the shell increases or decreases is the main ingredient determining the initial steepness or shallowness of the light curves.