*6.2. Reverse Shock Emission*

As the GRB outflow sweeps up enough external medium, it is decelerated by a reverse shock, while a strong relativistic forward shock propagates into the external medium powering the long-lived afterglow emission. (If the GRB ejecta are still highly magnetized at the deceleration radius *R*dec, *σ*(*R*dec) & 1, this may suppress the reverse shock, making it weak or even completely nonexistent.) Most of the outflow's energy is transferred to the shocked external medium when the reverse shock finishes crossing the ejecta shell at the deceleration radius, *R*dec, corresponding to the deceleration (apparent) time, *t*dec, which therefore signals the peak or onset of the afterglow emission e.g., [254–259]. For the "thick shell" case where the reverse shock is at least mildly relativistic, this time is comparable to the prompt GRB duration, *t*dec ∼ *t*GRB, while, for the "thin shell" case (where the reverse shock gradually transitions from Newtonian to mildly relativistic), *t*dec > *t*GRB. For frequencies that are above the cooling break frequency *ν<sup>c</sup>* of the reverse shock emission at *t*dec, which may include the optical for a sufficiently large *n*ext(*R*dec) (e.g., as expected for the stellar wind of a massive star progenitor in long GRBs), once the reverse shock finishes crossing the ejecta shell the emission from the LOS sharply drops and the flux decays rapidly (∼ *t* −3 ), corresponding to high-latitude emission. Otherwise, for frequencies in the range max(*νa*, *νm*) < *ν* < *νc*, where *ν<sup>a</sup>* is the break frequency corresponding to synchrotron self-absorption, a slightly less steep flux decay of about *t* −2 is expected, as the emission is dominated by the material along the line of sight where the shocked electrons cool adiabatically. Therefore, the optical emission typically peaks on a timescale of tens of seconds and then sharply drops—the **optical flash** e.g., [183,260–264]. The radio, however, is typically below the self-absorption frequency *ν<sup>a</sup>* at *t*dec (while *ν<sup>m</sup>* < *νa*), and its flux

keeps rising until *ν<sup>a</sup>* sweeps past the radio band, roughly after a day or so—the **radio flare** (e.g., [148,150,180,265–267]).

In terms of the polarization properties of the reverse shock emission, it is important to keep in mind the following points:


The **optical flash** emission typically peaks on a timescale of ∼10–100 s, and the ejecta Lorentz factor Γ is only somewhat lower than during the prompt GRB emission with <sup>Γ</sup> <sup>∼</sup> <sup>10</sup>2–102.5. The ejecta are decelerated by the reverse shock, typically reducing <sup>Γ</sup> down to <sup>∼</sup> <sup>1</sup> 2 Γ∞, where Γ<sup>∞</sup> is its value during the coasting phase (it can be lower than this for a highly relativistic reverse shock). However, the prompt GRB emission in photospheric models can arise from Γ < Γ∞, at which point the outflow is still accelerating and has not yet reached Γ∞. The optical flash is therefore expected to probe a comparable (i.e., only somewhat larger) region of angle <sup>∼</sup> 1/<sup>Γ</sup> <sup>∼</sup> <sup>10</sup>−2.5–10−<sup>2</sup> rad around our line of sight. Nonetheless, optical polarization measurements are more reliable than in gamma rays, and the optical flash is almost certainly synchrotron, which enables a cleaner and more robust inference of the ejecta magnetic field structure within this region.

From the observational perspective, since the optical flash usually has significant temporal overlap with the early optical afterglow emission from the shocked external medium, this requires a detailed modeling of both the total flux and the polarized flux as a function of time from these two distinct emission regions in order to properly disentangle between them and derive stronger and more robust constraints on the underlying properties of the GRB ejecta and its magnetic field structure. Most (but not all, e.g., [268]) of the early optical polarimetric observations relevant for the optical flash were done by the RINGO polarimeters on the Liverpool telescope [269–277]. Combining photometric and polarimetric observations [278], they concluded that their data clearly indicates that all epochs in which significant (linear) polarization was measured were dominated by emission from the reverse shock (while the optical afterglow emission from the forward external shock was sub-dominant). Here are a few examples. In GRBs 101112A and 110205A [275], a polarization of Π = 6 +3 <sup>−</sup><sup>2</sup> % and <sup>13</sup>+<sup>13</sup> <sup>−</sup><sup>9</sup> %, respectively, were measured at the optical peak time of *T*dec ∼ 299 s and ∼ 1027 s, respectively, which appeared to be dominated by the reverse shock because of the sharp rise to the peak (as ∼ *t* 4.2 and <sup>∼</sup> *<sup>t</sup>* 4.6, respectively). In both GRBs, *T*dec *T*GRB, indicating a thin shell (with *T*GRB ≈ *T*<sup>90</sup> ∼ 9.2 s and 249 s, respectively). One of the best examples so far is GRB 120308A [277], in which Π = 28% ± 4% was detected at 240 s < *t* < 323 s, which gradually decreased down to Π = 16+<sup>4</sup> <sup>−</sup>5% at 575 s <sup>&</sup>lt; *<sup>t</sup>* <sup>&</sup>lt; 827 s, as the emission gradually transitioned from reverse-shock- to forward-shock-dominated (see *left* panel of Figure 16).

**Figure 16.** (**Left**): Evolution of optical polarization (degree P (**a**), and position angle *θ* ((**b**); degrees east of north) and brightness ((**c**) in red (555–690 nm) light using RINGO2 and RATCam, in GRB 120308A (from [277]). (**Right**): 3 *σ* upper limits on the linear polarization of the radio flare emission from three different GRBs overlaid on the theoretical polarization light curves for a toroidal magnetic field in the GRB ejecta (from [111]). The top two panels are for a uniform (top-hat) jet where the different lines, from top to bottom, are for *θ*obs/*θ<sup>j</sup>* = 0.9, 0.8, ..., 0.1, while *α* = −*d* log *Fν*/*d* log *ν* is the spectral index (in the observed radio band) and Πmax = (*α* + 1)/(*α* + 5/3). In the top panel, the Lorentz factor of the ejecta is assumed to remain equal to that of the freshly shocked fluid just behind the forward shock ("FS"), while in the middle panel it is assumed to follow the Blandford and McKee [279] self-similar solution. The bottom panel is for a "structured" jet, in which the energy per solid angle drops as *θ* <sup>−</sup><sup>2</sup> outside some small core angle.

The **radio flare** emission, e.g., [148,150,180,265–267], typically peaks on a timescale of a day or so (∼10<sup>5</sup> s). By this time, the shocked GRB ejecta shell settles in the back of the Blandford and McKee [279] self-similar solution, and its Γ (∼5–10) is smaller by a factor of up to ∼1.5–1.8 compared to the material just behind the forward shock that dominates the afterglow emission at the same observed time [111,256]. This corresponds to a visible region of angle ∼0.1–0.2 rad around our line of sight, which is significantly larger than during the optical flash. Moreover, it often includes the entire jet (for a simple top-hat jet model) as suggested by the fact that the radio flare peak time is often comparable to the jet break time in the afterglow lightcurve. Granot and Taylor [111] have used VLA data of radio flares from three GRBs (990123, 991216, and 020405) to constrain its polarization, finding only upper limits for both linear and circular polarization. Their best limits are for GRB 991216, for which they found 3*σ* upper limits on the linear and circular polarization of 7% and 9%, respectively. These limits provide interesting constraints on GRB models and in particular are hard to reconcile with a predominantly ordered toroidal magnetic field in the GRB outflow together with a "structured" jet, where the energy per solid angle drops as the inverse square of the angle from the jet axis (see *right* panel of Figure 16). The polarization of the radio flare may be affected by the location of the observed frequency *ν* relative to the synchrotron self-absorption break frequency *ν<sup>a</sup>* (polarization is suppressed when *ν* < *νa*, during the rising phase of the radio flare) or by Faraday depolarization on the way from the source to us (both are discussed in [111]) and may also be subject to plasma propagation effects within the source (as discussed below, at the end of this section).

Comparing the polarization of the optical flash and radio flare for the same GRB would enable us to study the magnetic field in the GRB ejecta over a wide range of angular scales, probing magnetic structures with a coherence length over this angular range, 10−2.5 . *θ<sup>B</sup>* . 10−<sup>1</sup> . Measuring the reverse-shock emission polarization at intermediate

times and frequencies, such as at sub-mm with ALMA (e.g., [265,267]), would provide a better coverage of this wide range. A particularly interesting example is GRB 190114C, which was also detected at TeV energies [280]. ALMA measured its sub-mm (97.5 GHz) total intensity and linear polarization at 2.2 – 5.2 h after the burst, when the emission was dominated by the reverse shock [267], detecting linear polarization at ≈ 5*σ* confidence, decreasing from Π = 0.87% ± 0.13% to Π = 0.60% ± 0.19%, while the position angle evolved from 10◦ ± 5 ◦ to −44◦ ± 12◦ . This was the first detection and measurement of the temporal evolution of polarized radio/millimeter emission in a GRB. Using the measured linear polarization, Laskar et al. [267] constrained the coherence scale of tangled magnetic fields in the ejecta to an angular size of *<sup>θ</sup><sup>B</sup>* <sup>≈</sup> <sup>10</sup>−<sup>3</sup> rad, while the rotation of the polarization angle rules out the presence of large-scale, ordered axisymmetric magnetic fields and, in particular, a large-scale toroidal field, in the jet.

#### *6.3. Afterglow Emission*

Linear polarization at the level of a few percent has been detected in the optical or NIR afterglow of about a dozen GRBs [22,281–291]. Higher levels of polarization (10% . Π . 30%) have been measured mostly in the very early afterglow, likely being dominated by reverse-shock emission, as discussed above (see, however, [292]). The linear polarization of the afterglow emission was considered as a confirmation that it arises primarily from synchrotron radiation, as was already suggested by its spectral energy distribution.

A variety of **models** have been suggested for GRB afterglow polarization: emission from different patches of uniform but mutually uncorrelated magnetic field, either with microlensing [293] or without it [63], or emission from a random magnetic field within the plane of the afterglow shock together with scintillation in the radio [64] or with a jet viewed not along its symmetry axis [223,294,295], possibly with the addition of an ordered component that pre-exists in the external medium and which is compressed by the afterglow shock and/or a tangled magnetic field that is not purely in the plane of the shock and may even be predominantly in the direction of the shock normal [109,296] or due to clumps in the external medium or a similarly inhomogeneous outflow [109,230].

The most popular models for GRB afterglow polarization feature an axis-symmetric jet viewed not along its symmetry axis along with a tangled shock-produced magnetic field that is symmetric about the local shock normal [109,112,143,222,223,294,295,297–299]. In such models, the only preferred direction on the plane of the sky is that connecting the jet symmetry axis and our LOS, and therefore the net polarization of the unresolved image must lie either along this direction or transverse to it. Indeed, the tell-tale signature of such models for a uniform top-hat jet is a 90◦ change in the polarization PA *θ<sup>p</sup>* as Π vanishes and reappears rotated by 90◦ , around the time of the jet break in the afterglow lightcurve [223,294]. On the other hand, for a structured jet viewed from outside of its narrow core, a constant *θ<sup>p</sup>* is expected. Overall, in such models the linear polarization and its temporal evolution depend on: **(i)** the **jet's angular structure**, **(ii)** the local **structure of the shock-generated magnetic field** about the shock normal, and **(iii)** our viewing angle *θ*obs from the jet symmetry axis. Therefore, afterglow linear polarization observations can teach us both about the jet's angular structure and about the shock-produced magnetic field structure. However, there is a significant **degeneracy** between the two, which usually requires making large assumptions about one of them in order to significantly constrain the other.

The exceptional case of the short GRB 170817A, which was associated with the first gravitational wave detection of the binary neutron star merger, GW 170817, has allowed us to break this degeneracy. This event was observed from a large off-axis viewing angle, and its low-luminosity prompt gamma-ray emission and subsequent long-lived afterglow emission could be observed thanks to its relatively small distance (*D* ≈ 40 Mpc). The combination of an extremely well-monitored afterglow from radio to X-rays e.g., [168–170], and the super-luminal motion of its radio flux centroid (h*β*appi = h*v*appi/*c* = 4.1 ± 0.5

between 75 and 230 days after the burst [300]) has allowed a good determination of our viewing angle and of the jet's angular structure (e.g., [159,160,171–178,301,302]). This has enabled making robust predictions for the linear polarization that depend on the shockproduced magnetic field structure [172]. Shortly thereafter a linear polarization upper limit, |Π| < 12% (99% confidence), was set in the radio (2.8 GHz) at *t* = 244 days [303]. Assuming emission from a two-dimensional surface identified with the afterglow shock front, this has led to a constraint of 0.7 . *b* . 1.5 on the magnetic field anisotropy parameter, *b* ≡ 2h*B* 2 k i/h*B* 2 ⊥ i [172,303], which was introduced by [109], where *<sup>B</sup>*<sup>k</sup> and *B*<sup>⊥</sup> are the magnetic field compenents parallel and perpendicular to the shock normal direction *n*ˆ sh, respectively, and *b* = 1 corresponds to an isotropic field in 3D (for which the local and global polarizations vanish). A more detailed analysis [112] accounted for the emission from the whole 3D volume behind the afterglow shock, with the global angular jet structure implied by the GRB 170817A/GW 170817 observation and a local radial hydrodynamic profile set by the Blandford and McKee [279] self-similar solution. The magnetic field was modeled as an isotropic field in 3D that is stretched along *<sup>n</sup>*<sup>ˆ</sup> sh by a factor *<sup>ξ</sup>* ≡ *<sup>B</sup>*k/*B*⊥, whose initial value *<sup>ξ</sup> <sup>f</sup>* = *<sup>B</sup>*k, *<sup>f</sup>* /*B*⊥, *<sup>f</sup>* describes the field that survives downstream on plasma scales *R*/Γsh, and it is evolved downstream according to the [279] solution assuming flux freezing (i.e., no further magnetic dissipation or amplification far downstream of the shock front). In a local coordinate system where *n*ˆ sh = *z*ˆ, in the above definition of *b* we have h*B* 2 k i = h*B* 2 *z* i and h*B* 2 ⊥ i = h*B* 2 *<sup>x</sup>* + *B* 2 *y* i = 2h*B* 2 *x* i due to the *B*-field's symmetry about *n*ˆ sh, while here in the definition of *ξ*, *B*<sup>⊥</sup> represents either *B<sup>x</sup>* or *B<sup>y</sup>* but not (*B* 2 *<sup>x</sup>* + *B* 2 *y* ) 1/2 (while *<sup>B</sup>*<sup>k</sup> = *<sup>B</sup>z*). Gill and Granot [112] found that the shock-produced magnetic field has a finite, but initially sub-dominant, parallel component: 0.57 . *ξ <sup>f</sup>* . 0.89 (see Figure 17).

**Circular polarization** at the level of Πcirc = 0.61% ± 0.13% has been reported in the optical afterglow of GRB 121024A [304] at *t* = 0.15 days after the burst, when the linear polarization was Πlin ≈ 4%, implying a relatively high circular-to-linear polarization ratio of Πcirc/Πlin ≈ 0.15. Nava et al. [305] performed a detailed analysis of the expected Πcirc and Πlin in GRB afterglows, finding that while ad-hoc configurations may allow large local Πcirc values, after transformations to the observer frame and integration over the whole visible region are performed, Πcirc/Πlin remains vanishingly small in any realistic optically thin synchrotron afterglow emission model and thus concluding that the origin of the observed Πcirc in GRB 121024A cannot be intrinsic.

**Plasma propagation effects** due to the presence of cooler thermal electrons, which are not shock accelerated and represent a fraction 1 − *ξ<sup>e</sup>* of the total number, may be important if a significant ordered magnetic field component is present in the emitting region [306–308]. Such effects are most prominent in the early afterglow and around the self-absorption frequency and may therefore potentially affect the reverse shock emission (the "optical flash" or "radio flare"), as well as the forward shock emission in the radio up to a day or so [306–308]. These effects may include Faraday conversion of the linear polarization of the emitted radiation to circular polarization or Faraday depolarization of the emitted linear polarization. For typical GRB afterglow microphysical parameters, the latter effect may strongly suppress the linear polarization in the radio but preserve that in the optical. Therefore, simultaneous observations yielding statistically significant measurements of polarization in both optical and radio can be extremely useful to confirm the population of thermal electrons as well as the existence of an ordered B-field. In some GRBs, this effect may manifest in the sub-mm band where comparison between ALMA and VLA measurements can constrain the value of *ξ<sup>e</sup>* [308]. In fact, Urata et al. [309] argued that the unusually low afterglow polarization (Π = 0.27% ± 0.04%) of GRB 171205A in the sub-mm band, as compared to the typical late-time optical polarization, may have been the result of Faraday depolarization. Since the true afterglow shock kinetic energy is given by *E* 0 = *E*/*ξ<sup>e</sup>* [310], where *E* would be the true energy for *ξ<sup>e</sup>* = 1, a constraint on *ξ<sup>e</sup>* would lead to better constraints on the burst energetics.

**Figure 17.** Constraining the magnetic field structure in collisionless relativistic shocks from a radio afterglow linear polarization upper limit in GRB 170817/GW 170817 [112]. (**Left**): Schematic of post-shock magnetic field geometry for different values of the local anisotropy parameter *<sup>ξ</sup>* <sup>≡</sup> *<sup>B</sup>*k/*B*<sup>⊥</sup> <sup>=</sup> *<sup>ξ</sup> <sup>f</sup> <sup>χ</sup>* (7−2*k*)/(8−2*k*) , whose initial value just behind the shock is *ξ <sup>f</sup>* , for an external density profile *ρ*ext ∝ *R* −*k* , where *χ* = 1 + 2(4 − *k*)Γ 2 sh(1 − *r*/*R*) is the Blandford and McKee [279] self-similar variable, *r* is the radial coordinate, and *R* and Γsh are the local radius and Lorentz factor of the afterglow shock front, respectively. (**Top Right**): The corresponding evolution of the magnetic field equipartition parameter, *eB*, with the distance behind the shock (as parameterized through *χ*) for *ξ <sup>f</sup>* = 0, 0.2, 0.4, 0.57, 0.7, 0.89, 1.2, 2, ∞ (from bottom to top). The two extreme values of *ξ <sup>f</sup>* = 0, ∞ are shown as dotted (straight) lines. The light-grey shaded region corresponds to the allowed range found in [112], 0.57 . *ξ <sup>f</sup>* . 0.89. (**Bottom Right**): The linear polarization evolution, Π(*t*), obtained from a volume integration of the flow, shown for different values of *ξ <sup>f</sup>* . The two arrows mark the polarization upper limit, |Π| < 12%. Comparison was made between two jet structures–a Gaussian jet (GJ) and a power-law jet (PLJ). The result from [172], which assumed an infinitely thin shell geometry as well as locally isotropic synchrotron spectral emissivity, is also shown (labeled GG18) for the magnetic field anisotropy parameter *b* = 0.
