Effect of Fluid Composition on a Jet Breaking out of a Cocoon in Gamma-Ray Bursts: A Relativistic de Laval Nozzle Treatment

Round 1

Reviewer 1 Report

Referee Report "Effect of fluid composition on a jet breaking out of a cocoon in
Gamma Ray bursts : A relativistic de Laval nozzle treatment"

The current paper investigates the jet dynamics of a GRB jet as a function of jet composition and jet mass. The work is based on parametrized jet shape for which the relativistic dynamical equations are written down analytically and the final set of equations is then solved numerically.

This work is in general rather interesting, but at some points clarification is needed:

- What is the origin of Eq 1? References 40&60 don't show this specific formula? 
- The comments on coordinates at line 98 are somewhat misleading. Surely the jet is following the z-axis, but cartesian coordinates are only used in this paragraph
- Between Eq 6&7 it seems that the assumption is made that the jet properties do not change across its diameter? How is this justified? Does it mean that the jet behaves like a step function at its radial borders?
- The system is given in a conservative form before Eq 13&14. Why are Eq 13&14 used for the numerical solution rather than the conservative form?
- How is stability and convergence checked for the coupled system of differential equations?
- The comment on the acceleration of nonthermal particles in the presence of a magnetic field is not very useful. The presence of a magnetic field would not only accelerate particles but also change the jet confinement AND the Rankine-Huginot jump conditions

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The authors have developed a good analytic treatment on jet-cocoon dynamics for GRBs, which is interesting and useful for understanding jet propagation without the time-consuming numerical simulations. However, some assumptions look oversimplified and several issues need to be addressed properly before I can recommend for publication. The details of my major comments and some suggestions are given as follows.

1. Generally, a “cocoon” is formed due to the jet interacting with the dense medium. The jet is decelerated by the envelope and forward-reverse shocks are produced. Shocked envelope and shocked jet material flow sideways and form a cocoon structure that wraps up the jet. In this sense, the definition in Introduction (page 1, paragraph 1, line 5-6) is not very accurate.
2. The term “internal shock” appear several times in the text, and I suppose the authors are referring to recollimation shock. However, in GRB field the “internal shock” is designated to a shock produced by the collision of two shells with different velocities. Therefore, I suggest not to use it in order to avoid misleading.
3. I do not find the descriptions for the constants A_1,A_2, d, and c_j in Eq.(1). Are their values adopted somewhat arbitrarily? Accordingly, the difference in jet geometry in Figure 1 for three m values seems not very prominent, then why can it represent cocoon strength? Physically, the cocoon strength is characterized by its ability to affect the jet properties, for instant, to what extent it can alter the initial jet opening angle. Is is possible to give some physical meaning for m? I guess the change in opening angle is larger with increasing m.
4. While it matches our expectation well that high-E jet is almost unaffected by the cocoon, can the authors briefly explain why low-E jet seems also less affected compared to moderate-E cases? It should be decelerated more easily and might be chocked as well.
5. The authors assume that the jet is initially sub-relativistic, then how about the relativistic jet with initial Lorentz factor ~10 (e.g. Mizuta_2013_ApJ_777_162 )? Does the analytic solutions also apply? If yes, what kinds of correction should be made? If no, the authors can mention in section 6 or7.
6. The shock radius is pretty close to the central engine and much smaller than the typical radius for GRB prompt emission. Also, the dissipated energy at the shock is not enough. Therefore, the emission from non-thermal electrons accelerated by recollimation shocks is unlikely responsible for GRB prompt phase, therefore it does not make sense to compare the index s with observed GRB spectra.
7. At last, I suggest the authors to compare their analytic solutions with some previous numerical simulations or observations in order to make the results more convincing.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors have addressed all the issues properly and I have no further comments.