Note: This is a part of the gravitational wave page because one of the expected electromagnetic signatures on neutron star mergers is breakout emission from the cocoon surrounding the relativistic GRB jet. Here we explore an order of magnitude estimate of the breakout time and luminosity of this cocoon. The same physics can apply to other breakouts, such as in supernovae.

We are interested in estimating the peak luminosity of a gamma-ray burst or some similar phenomenon where a jet or structure is moving through a medium and at some point "breaks out" of its surroundings. As a setup, picture a spherical medium (radius $ {R} $) the center of which will begin creating two narrow and collimated jets pointing in opposite directions. Around this jet, material is forming in a cocoon that is azimuthally symmetric about the center of each jet. This cocoon will eventually escape with angle $ {\theta{}} $ at the edge of the medium, i.e. when it reaches distance $ {R} $.

For this we need the energy of the burst and the time scale over which the energy is released, $ {L=\frac{E}{t}} $. The timescale will be how long it takes photons to diffuse through the medium ($ {t=t_{diff}} $), and that is dependent on how much of the medium there is and how optically thick it is. This diffusion time is the same as the time for the photon to travel one mean free path ($ {t_{MFP}} $), scaled by the optical depth ($ {\tau{}_{diff}} $), so $ {t_{diff}=\tau{}_{diff}t_{MFP}} $. At the beginning, we expect the surrounding medium to be dense and optically thick ($ {\tau{}>1} $), so $ {\tau{}_{diff}=}max[{\tau{}, \tau{}^{2}]={\tau{}^{2}}} $ (Rybicki & Lightman p.36). This $ {\tau{}} $ is also related to mean free path length scale by $ {\tau{}=\frac{R}{\ell{}_{MFP}}} $, where R is the total path that the photon must travel (in this case, from the center of our medium to the edge) and $ {\ell{}_{MFP}} $ is the mean free path length of the photons in this material. Also, $ {t_{MFP}=\frac{\ell{}_{MFP}}{c}} $ when photons are traveling at c. Now $ {\tau{}_{diff}=\tau{}^{2}=\left(\frac{R}{\ell{}_{MFP}}\right)^{2}} $. Plugging these into $ {t_{diff}} $ yields, $ {t_{diff}=\left(\frac{R}{\ell{}_{MFP}}\right)^{2}\left(\frac{\ell{}_{MFP}}{c}\right)=\left(\frac{R}{\ell{}_{MFP}}\right)\left(\frac{R}{c}\right)\Rightarrow{}\tau{}\frac{R}{c}} $

We can relate $ {\tau{}} $ to the mass absorption coefficient (or opacity coefficient) $ {\kappa{}} $ by $ {\tau{}=\frac{\kappa{}M}{4\pi{}r(\theta{})^{2}}} $, where $ {M} $ is the mass of medium traveled through (assumptions about medium makeup and density enter here) and the denominator is the region into which the cocoon is expanding. It is at first taken to be the full sphere, but this will be reduced, since we are expanding into a ring-like surface around the jet. The surface area of this ring is $ {\pi{}r(\theta{})^{2}=\pi{}R^{2}\theta{}^{2}} $ where we have neglected the tiny area taken by the jet and assumed that $ {\theta{}} $ is small enough that $ {r(\theta{})\approx{}R\theta{}\rightarrow{}\tau{}=\frac{\kappa{}M}{\pi{}R^{2}\theta{}^{2}}} $ and so $ {t_{diff}=\left(\frac{\kappa{}M}{\pi{}R^{2}\theta{}^{2}}\right)\left(\frac{R}{c}\right)=\frac{\kappa{}M}{\pi{}R\theta{}^{2}c}} $. Distance $ {R} $ is traveled in time $ {t_{diff}} $ with speed $ {v} $, $ {R=vt_{diff}} $. This leads to $ {t_{diff}^{2}=\frac{\kappa{}M}{\pi{}v\theta{}^{2}c}} $ or

$ {t_{diff}=\sqrt{\frac{\kappa{}M}{\pi{}v\theta{}^{2}c}}} $

This is the diffusion time for most of the emission since we used mean free path arguments. The bolometric energy at peak ($ {E_{tot}} $) can be measured by a consortium of instruments. $ {L=\frac{E_{tot}}{t}} $ since we are talking about the full bolometric energy.

$ {L=\frac{E_{tot}\sqrt{{\pi{}v\theta{}^{2}c}}}{\sqrt{\kappa{}M}}} $

Now, assuming some values, E=$ {10^{50}} $, $ {\kappa{}=0.4}{cm^{2}g^{-1}} $, M=0.01 $ {M_{\odot{}}} $, v=0.866c, $ {\theta{}=0.1} $. We find $ L=1.7{\times{}10^{44}}ergs/s $