*2.1. What Are the Outflow Composition and Dynamics?*

The main dissipation and radiation mechanisms that produce the GRB prompt emission are dictated by the composition of the outflow. The two most widely discussed scenarios invoke an outflow that is either kinetic-energy-dominated (KED) [30] or Poyntingflux-dominated (PFD) [31,32]. In the former, most of the energy is initially thermal (fireball) and is eventually transferred to the kinetic energy of the cold baryons, while in the latter the main energy reservoir is the (likely ordered) magnetic field that drives the expansion and acceleration of the flow. If the radiation mechanism is indeed synchrotron (see Section 2.3), then the level of polarization in both types of flows depends on the structure of the magnetic field that is either generated in situ, e.g., in internal shocks in a KED flow, or survives at large distances from the central engine, which could happen in both types of flows. Our theoretical understanding of the B-field structure in the emission region in a given type of flow is still limited and rather speculative. Any measurement of polarization will put strong constraints on the B-field structure. Therefore, in combination with polarization measurements, spectral and temporal (pulse profiles) modeling will allow us to constrain the composition.

The distinction between a KED and PFD flow can be conveniently parameterized using the magnetization parameter,

$$\sigma \equiv \frac{w\_B^{\prime}}{w\_m^{\prime}} = \frac{B^{\prime 2}}{4\pi \left(\rho^{\prime}c^2 + \frac{\hat{\gamma}}{\hat{\gamma}-1}P^{\prime}\right)} \xrightarrow[\text{cold}]{B^{\prime 2}} \frac{B^{\prime 2}}{4\pi \rho^{\prime}c^2} ,\tag{1}$$

which is defined as the ratio of the comoving (all quantities measured in the comoving/fluid frame are primed) magnetic field enthalpy density, *w* 0 *<sup>B</sup>* = *B* <sup>0</sup>2/4*π*, to that of matter, *w* 0 *<sup>m</sup>* = *ρ* 0 *c* <sup>2</sup> + *γ*ˆ *γ*ˆ−1 *P* 0 or *w* 0 *<sup>m</sup>* = *ρ* 0 *c* <sup>2</sup> when it is cold (*P* <sup>0</sup> *ρ* 0 *c* 2 ). Here *B* 0 is the magnetic field strength, and we assumed here for simplicity that the baryons dominate the total rest mass with density *ρ* 0 = *mpn* 0 , where *n* 0 is the particle number density, *m<sup>p</sup>* is the proton mass, and *c* is the speed of light. The baryons were assumed to be cold with an adiabatic index *γ*ˆ = 5/3 (*γ*ˆ = 4/3 for a relativistic fluid) and negligible pressure *P* 0 when compared with the particle inertia. A KED flow will have *σ* < 1; magnetic fields, if present, are weak and randomly oriented with short coherence length scales and are unimportant in governing the dynamics of the outflow. On the other hand, a PFD flow will have *σ* > 1, and the magnetic field is much more ordered where it is responsible for accelerating the flow.

A prime example of a KED flow is the standard "fireball" scenario [33,34], in which total energy *E* is released close to the central engine, launching a radiation-dominated and optically thick outflow, with Thomson optical depth *τ<sup>T</sup>* 1. The temperature at the base of the flow is typically *kBT* & MeV, which leads to copious production of *e* ±-pairs via *γγ*-annihilation that further enhances the optical depth. The enormous radiation pressure causes the flow to expand adiabatically, thereby converting the radiation field energy to the kinetic energy of baryons, which are inefficient radiators due to their large Thomson

cross-sections. The bulk Lorentz factor (LF) of the fireball grows linearly with the radius, Γ(*R*<sup>0</sup> < *R* < *Rs*) ≈ *R*/*R*0, where *R* = *R*<sup>0</sup> is the launching radius, while its comoving temperature declines as *T* 0 (*R*) ∝ *R* −1 . The amount of baryon loading, i.e., the amount of baryons with total mass *M<sup>b</sup>* entrained in the flow of given energy *E*, determines the terminal LF, Γ<sup>∞</sup> = *E*/*M<sup>b</sup> c* 2 , which is attained at the saturation radius *R* = *R<sup>s</sup>* ∼ Γ∞*R*<sup>0</sup> at which point the growth in the bulk Γ saturates and the flow simply coasts at Γ = Γ∞. The kinetic energy of the baryons is tapped at a large distance (*R* > *Rs*) from the central engine via internal shocks (see below).

In a PFD flow, large-scale magnetic fields propagate outwards from the central engine with an angular coherence scale *θ<sup>B</sup>* > 1/Γ, where 1/Γ represents the characteristic angular scale over which the flow is causally connected and, as discussed later, also the angular scale into which the emitted radiation is beamed towards the observer from a relativistic flow. While the fireball scenario is well agreed upon and has enjoyed many successes since it is fairly robust, no such *standard model* exists for a magnetized outflow to explain GRB properties. In several works (e.g., [35–39]), ideal-MHD models for a steady-state, axisymmetric, and non-dissipative outflow have been developed in which the flow expands adiabatically due to magnetic stresses. The flow is launched highly magnetized near the light cylinder radius, *RL*, with *σ*(*RL*) = *σ*<sup>0</sup> 1 and bulk LF Γ(*RL*) = Γ<sup>0</sup> ∼ 1. As the flow expands, its magnetization declines with radius, and in the case of a radial wind (i.e., unconfined, with a negligible external pressure) the flow is limited to a terminal LF of Γ<sup>∞</sup> ∼ *σ* 1/3 <sup>0</sup> where the corresponding magnetization of the flow is *σ* ∼ *σ* 2/3 0 [40]. For weak external confinement (an external pressure profile *p*ext ∝ *z* <sup>−</sup>*<sup>κ</sup>* with *κ* > 2, where *z* ≈ *R* = (*z* <sup>2</sup> + *r* 2 cyl) 1/2 is the distance from the central source along the jet's symmetry axis and *r*cyl is the cylindrical radius), the acceleration saturates at a terminal LF of Γ<sup>∞</sup> ∼ *σ* 1/3 0 *θ* −2/3 *j* and magnetization *σ*<sup>∞</sup> ∼ (*σ*0*θj*) 2/3 <sup>∼</sup> (Γ∞*θj*) <sup>2</sup> <sup>1</sup> where *<sup>θ</sup><sup>j</sup>* is the jet's asymptotic half-opening angle [41]. For strong external confinement (*p*ext ∝ *z* <sup>−</sup>*<sup>κ</sup>* with *κ* < 2), the jet maintains lateral causal contact and equilibrium, leading Γ ∼ *r*cyl/*R<sup>L</sup>* ∼ (*z*/*RL*) *κ*/4 , which saturates at Γ<sup>∞</sup> ∼ *σ*0, *σ*<sup>∞</sup> ∼ 1, and Γ∞*θ<sup>j</sup>* ∼ 1. Since prompt GRB observations demand the dissipation region to be expanding ultrarelativistically with Γ<sup>∞</sup> & 100, to avoid the *compactness problem* [25,42], and afterglow observations suggest that typically *θ<sup>j</sup>* & 0.05 − 0.1, which implies Γ∞*θ<sup>j</sup>* & 10 in GRBs. This suggests that the weakly confined regime is most relevant for GRBs; however, it implies *σ*<sup>∞</sup> ∼ (Γ∞*θj*) <sup>2</sup> 1, which suppresses internal shocks. It has been pointed out [43,44] that the sharp drop in the surrounding (lateral) pressure as the jet exits the progenitor star in long GRBs can lead to Γ∞*θ<sup>j</sup>* 1 along with a more modest asymptotic magnetization *σ*<sup>∞</sup> & 1, but even then internal shocks remain inefficient.

When the steady-state assumption is relaxed, alternative models that consider an impulsive and highly variable flow yield a much larger terminal LF with Γ<sup>∞</sup> ∼ *σ*<sup>0</sup> and may achieve *σ*<sup>∞</sup> < 1 or even *σ*<sup>∞</sup> 1 under certain conditions [45,46]. In this scenario, a thin shell of initial width `<sup>0</sup> is accelerated due to magnetic pressure gradients that causes its bulk LF to grow as Γ ∼ (*σ*0*R*/*R*0) 1/3, where *<sup>R</sup>*<sup>0</sup> <sup>≈</sup> `0, while its magnetization drops as *σ* ∼ *σ* 2/3 0 (*R*/*R*0) <sup>−</sup>1/3. The bulk LF of the shell saturates at *<sup>R</sup><sup>s</sup>* <sup>∼</sup> *<sup>σ</sup>* 2 <sup>0</sup>*R*<sup>0</sup> at which point its Γ ∼ Γ<sup>∞</sup> ∼ *σ*<sup>0</sup> and *σ* ∼ 1. For *R* > *R<sup>s</sup>* , the magnetization continues to drop further as *σ* ∼ (*R*/*Rs*) <sup>−</sup><sup>1</sup> as the shell starts to spread radially. For a large number of shells initially separated by `gap, the radial expansion is limited as neighboring shells collide and one expects an asymptotic mean magnetization of *σ*<sup>∞</sup> ∼ `0/`gap. This scenario offers the dual possibility of magnetic energy dissipation via MHD instabilities when *σ* > 1 at *R* < *R<sup>s</sup>* as well as kinetic energy dissipation via internal shocks when *σ* < 1 at *R* > *R<sup>s</sup>* .

Similar outflow dynamics were obtained in a popular model that makes the magnetohydrodynamic (MHD) approximation and features a striped-wind magnetic field structure [47–51], in which magnetic field lines reverse polarity over a characteristic length scale *<sup>λ</sup>* <sup>∼</sup> *<sup>π</sup>R<sup>L</sup>* <sup>=</sup> *<sup>π</sup>c*/<sup>Ω</sup> <sup>=</sup> *cP*/2 <sup>=</sup> 1.5 <sup>×</sup> <sup>10</sup>7*P*−<sup>3</sup> cm. Here, <sup>Ω</sup> <sup>=</sup> <sup>2</sup>*π*/*<sup>P</sup>* is the central engine's angular frequency with *P* being its rotational period. Close to the central engine the flow may be accelerated by magneto-centrifugal, and to some extent, thermal

acceleration. At distances larger than the Alfvén radius, where *R<sup>A</sup>* & *RL*, these effects are negligible, and when collimation-induced acceleration is ineffective then the properties of the flow can be described using radial dynamics. If a reasonable fraction (the usual assumption is approximately half) of the dissipated energy in the flow goes towards its acceleration, conservation of the total specific energy, while ignoring any radiative losses, yields the relation Γ(*R*)[1 + *σ*(*R*)] = Γ0[1 + *σ*0] for a cold flow, which simplifies to Γ(*R*)*σ*(*R*) ≈ Γ0*σ*<sup>0</sup> for *σ*(*R*) 1. At the Alfvén radius, the four velocity of the flow is *u<sup>A</sup>* = Γ*Aβ<sup>A</sup>* = *σ* 1/2 *<sup>A</sup>* ≈ Γ*<sup>A</sup>* ≈ Γ0*σ*0/*σ<sup>A</sup>* ≈ *σ*0/*σA*, which implies that *σ<sup>A</sup>* ≈ *σ* 2/3 0 and Γ*<sup>A</sup>* ≈ *σ* 1/3 0 . The terminal LF is achieved at the saturation radius *R<sup>s</sup>* when *σ*(*Rs*) ∼ 1, at which point Γ<sup>∞</sup> ≈ Γ0*σ*<sup>0</sup> ≈ *σ*<sup>0</sup> = *σ* 3/2 *A* . In this scenario, the saturation radius is given by *R<sup>s</sup>* = Γ 2 <sup>∞</sup>*λ*/6*<sup>e</sup>* <sup>=</sup> 1.7 <sup>×</sup> <sup>10</sup>13<sup>Γ</sup> 2 <sup>∞</sup>,3(*λ*/*e*)<sup>8</sup> cm, where *e* = *v*in/*v<sup>A</sup>* ∼ 0.1 is a measure of the reconnection rate where it quantifies the plasma inflow velocity *v*in into the reconnection layer in terms of the Alfvén speed. For magnetized flows, *v<sup>A</sup>* = *c* p *σ*/(1 + *σ*), which approaches the speed of light for *σ* 1. Beyond the Alfvén radius, the bulk LF grows as a power law in radius, with Γ(*R*) = Γ∞(*R*/*Rs*) 1/3, while the magnetization declines as *σ*(*R*) = (*R*/*Rs*) −1/3 .

In the regime of high magnetization (*σ* 1), an alternative model that does not make the MHD approximation was considered by Lyutikov and Blandford [32] and Lyutikov [52].
