A Novel Relationship Between Gamma Ray Burst Duration And Photospheric Radius

Long Gamma Ray Bursts (lGRBs) are associated with jets in Type Ic broadline supernovae. The Collapsar model provides a theoretical framework for the jet formation from the core collapse of a massive star in such supernovae. The GRB can only be produced after a successful jet break out from the star. Under this formalism the GRB duration ($t_{\rm{90}}$) has been hypothesized to be the difference between the central engine activity duration ($t_{\rm{eng}}$) and the jet breakout time ($t_{\rm{bo}}$), that is $t_{\rm{90}} = t_{\rm{eng}} - t_{\rm{bo}}$. This disallows $t_{\rm{90}} > t_{\rm{eng}}$ and puts a lower bound on successful lGRB jet central engine duration ($t_{\rm{eng}} > t_{\rm{bo}}$), various numerical simulations have shown otherwise. This study considers a photospheric GRB emission from a relativistic jet punching out of a Wolf-Rayet-like star. We use the bolometric lightcurve generated to calculate the lGRB duration ($t_{\rm{90}}$) for varying engine duration. We find for longer engine duration the lGRB lightcurve reflects the jet profile and $t_{\rm{90}} \approx t_{\rm{eng}}$. While for shorter engine duration, the $t_{\rm{90}}$ has photospheric radius ($R_{\rm{ph}}$) dependence. This can be modeled by a relation, $t_{\rm{90}} = t^{\rm{90}}_{\rm{eng}} + 0.03\left(\frac{R_{\rm{ph}}}{c}\right)$, where c is the speed of light, with a lower bound on $t_{\rm{90}}$ for a successful lGRB. This relation should be most relevant for possible low-luminous lGRBs originating from a collapsar with central engine duration comparable to the jet breakout time.