*2.4. Models of Jet Propagation through the Progenitor Vestige*

The jet launched by the central engine must initially propagate through the dense surrounding region constituted by the progenitor vestige, that is, the stellar envelope in the case of a collapsar or the ejecta cloud in the case of a compact binary merger. The details of the propagation and its final outcome depend on the properties of the jet at its base, its duration, and the properties of the vestige. The main features of the jet evolution during this phase can be understood based on a relatively simple hydrodynamical model ([30,76,114–117], see, e.g., [117–119] for the extension to the highly magnetized jet case, which presents some quantitative differences, despite the general picture remaining similar), whose main features are the following (see Figure 3 for a sketch): the jet is represented by an outflow with luminosity *L*<sup>j</sup> , expanding radially at a speed *β*j,0 within an aperture angle *θ*j,0 at its base. The outflow interacts with the vestige at a height *z* above the central engine, where a forward-reverse shock structure forms, called the jet head. The jet head advancement speed *β*<sup>h</sup> through the vestige, whose density at a height *z* is *ρ*<sup>v</sup> (and which can be expanding outwards at a speed *β*v), is set by the balance between the jet momentum flux that crosses the reverse shock and the ram pressure of the vestige material as seen from the forward shock downstream frame, namely (e.g., [74,75,120,121], neglecting the vestige pressure).

$$
\Gamma\_{\rm j}^{2} \Gamma\_{\rm h}^{2} (\beta\_{\rm j} - \beta\_{\rm h})^{2} \rho\_{\rm j}^{\prime} h\_{\rm j} c^{2} = \Gamma\_{\rm h}^{2} \Gamma\_{\rm v}^{2} (\beta\_{\rm h} - \beta\_{\rm v})^{2} \rho\_{\rm v} c^{2} \,. \tag{4}
$$

where Γ<sup>x</sup> = (1 − *β* 2 x ) −1/2 , *β*<sup>j</sup> is the dimensionless jet speed just before crossing the reverse shock, *πθ*<sup>2</sup> j *z* 2 is the reverse shock working surface, and we used *ρ* 0 j *h*<sup>j</sup> = *L*j/*πθ*<sup>2</sup> j *z* <sup>2</sup>*β*jΓ 2 j *c* 3 . This can be solved for *β*h, which gives

$$
\beta\_{\rm h} = \frac{\beta\_{\rm \tilde{\rm l}} + \tilde{L}^{-1/2} \beta\_{\rm v}}{1 + \tilde{L}^{-1/2}},
\tag{5}
$$

where the dimensionless quantity *L*˜ = Γ 2 j *ρ* 0 j *hj*/Γ 2 <sup>v</sup>*ρ*<sup>v</sup> = *L*j/*πθ*<sup>2</sup> j *z* <sup>2</sup>*β*jΓ 2 <sup>v</sup>*ρ*v*c* 3 is what sets the overall properties of the jet advancement [117]. When the head advancement is subrelativistic, then *<sup>β</sup>*<sup>h</sup> <sup>∼</sup> *<sup>β</sup>*<sup>v</sup> <sup>+</sup> *<sup>L</sup>*˜ 1/2*β*<sup>j</sup> ; when it is relativistic, then <sup>Γ</sup><sup>h</sup> <sup>∼</sup> *<sup>L</sup>*˜ 1/4/ √ 2.

**Figure 3.** Sketch of the main elements in a basic hydrodynamical model of the jet propagation through the progenitor vestige. Adapted from [114].

If the head advancement speed *β*<sup>h</sup> is sufficiently low, Γh*β*<sup>h</sup> . 1/ √ 3*θ*<sup>j</sup> [114], the head is causally connected by sound waves in the transverse direction: the lack of lateral confinement in the head then causes the shocked material (both that of the jet and that of the vestige) to flow laterally and form an over-pressured cocoon that shrouds the jet. The cocoon slowly expands laterally within the vestige at a speed proportional to the square root of the ratio of its pressure to the average vestige density [75,120], but this is typically not sufficient to prevent a pressure build-up as it is filled with an increasing amount of shocked material flowing from the head. When the cocoon pressure becomes comparable to the transverse momentum flux of the jet at its base, a "re-confinement shock" forms [76] where such transverse momentum is dissipated, collimating the jet into a cylindrical flow. The condition for such a self-collimation can be written approximately as *L*˜ . *θ* −4/3 j,0 [75]. The self-collimation reduces the reverse shock working surface, therefore favoring the jet head propagation. The increase in the head speed, on the other hand, reduces the energy flow to the cocoon, therefore affecting its ability to effectively collimate the jet. In self-collimated jets, the final jet opening angle at breakout is thus set by these competing effects.

In presence of a homologous expansion of the vestige (as expected in the case of compact binary merger ejecta), another self-regulation effect arises [122]: if the jet head stalls (i.e., *β*<sup>h</sup> ∼ *β*v) and the jet is self-collimated (so that the head working surface is constant—but this is ensured by the fact that *<sup>β</sup>*<sup>h</sup> <sup>∼</sup> *<sup>β</sup>*<sup>v</sup> implies *<sup>L</sup>*˜ <sup>1</sup> and hence the selfcollimation condition *L*˜ . *θ* −4/3 j,0 is certainly satisfied), the expansion has the effect of easing the head propagation because it reduces *L*˜. If the jet is launched shortly after the onset of the vestige homologous expansion (so that the jet duration is much longer than such delay), the result is that the jet's ability to break out depends solely on the ratio of the jet energy to that of the expanding vestige [122], regardless of the jet duration.

The importance of the jet propagation phase, from the observational point of view, stems from the fact that the jet structure after breakout carries the imprint of the jet-vestige interaction. If the rearrangement of the jet after the breakout (and before the prompt emission is produced) does not erase such memory, it is possible in principle to extract information about the progenitor (and possibly also about the central engine) from the jet structure at a stage when observable emission can be produced (e.g., [74]). The extent to which the resulting structure is determined by the central engine, the jet-vestige interaction, or both, is still a matter of debate. For what concerns binary neutron star mergers, the recent three-dimensional, special-relativistic hydrodynamical numerical simulations by [123] seem to suggest that turbulence at the jet-cocoon interface (see also [124]) tend to erase the

details of the injected jet structure (i.e., the angular structure as initially set by the central engine), which could lend support to the hypothesis that "adult" jets from binary neutron star mergers share a quasi-universal structure (e.g., [24,125]). Yet, the development of such turbulence seems strongly suppressed in magnetohydrodynamic simulations with a significantly magnetized injected jet [126] and, moreover, in simulations that use a different jet injection technique (e.g., [127]) come to the opposite conclusion.
