Magnetic Fields in Gamma-Ray Bursts and Their Polarised Emission

Abstract

A widely accepted model for producing gamma-ray bursts (GRBs) is based on the dissipation of an ultra-relativistic jet. Synchrotron emission is believed to be the dominant radiation process for the main emission components of GRBs, i.e. prompt gamma-rays, reverse shock emission and afterglow. Although the presence of strong magnetic fields in the emission regions is crucial to produce the bright emission, the origin and its role in the dynamics are still unknown. Understanding the nature of the relativistic jet, especially the energy content, acceleration and collimation, is a major focus of international theoretical and observational effort. Polarisation is one of characteristics of synchrotron emission, and polarimetric observations would allow us to study the magnetic field structure and energy distribution in GRB jets.

13.1 Introduction

Gamma-ray bursts (GRBs) are the brightest photon sources in the universe. GRBs provide direct access to regions of extreme physics (e.g. relativistic shocks, strong gravity) as well as acting as important probes of the high redshift universe (e.g. Piran 1999; Zhang and Mészáros 2004). These events are broadly classified as either long or short depending on burst duration. Long bursts are more common and last for more than 2 s, while short bursts last less than 2 s and tend to have harder spectra. The core-collapses of massive stars and the mergers of binary compact stellar objects such as neutron stars and black holes are possible progenitors for long and short bursts, respectively. Long bursts have been linked to star forming galaxies, and in many cases to core-collapse supernovae. The binary neutron star merger GW 170817 was the first multi-messenger event observed in both gravitational and electromagnetic waves (Abbott et al. 2017a,b,c). The gravitational wave detection in coincidence with the short burst GRB 170817A confirms that at least a subset of short GRBs is produced by binary neutron star mergers, although the peak isotropic luminosity of GRB 170817A is abnormally low.

A widely accepted model for producing GRBs is based on the dissipation of a relativistic outflow, the internal energy produced by shocks is believed to be radiated via the synchrotron process. Relativistic outflow from a GRB central engine is conventionally assumed to be a baryonic jet , producing synchrotron emission with tangled magnetic fields generated locally by instabilities in shocks (e.g. Medvedev and Loeb 1999). An alternative magnetic model is now attracting more attention from researchers (e.g., Drenkhahn and Spruit 2002; Fan et al. 2004; Zhang and Kobayashi 2005; Lyutikov 2006; Giannios 2008; Mimica et al. 2009, 2010; Zhang and Pe’er 2009; Zhang and Yan 2011; Narayan et al. 2011; Granot 2012). The rotation of a black hole and an accretion disk might cause a helical outgoing magnetohydrodynamic (MHD) wave which accelerates material frozen into the field lines (Tchekhovskoy et al. 2008; McKinney and Blandford 2009; Komissarov et al. 2009). In the magnetic models, the outflow is expected to be threaded with globally ordered magnetic fields.

13.2 The Fireball Model

Ultra-relativistic motion (Lorentz factor Γ ≳ 100) is an essential ingredient in the standard GRB model, and it was originally introduced into GRB models to bypass the compactness problem (how very compact and energetic sources can emit non-thermal radiation, e.g. Piran 1999). Later observations show that GRBs are actually associated with relativistic jets. The detection of GeV photons by Fermi requires Γ ≳ 10³ based on the opacity arguments for some short and long GRBs (Abdo et al. 2009a,b). The onset times of many afterglow indicate relativistic motion with Γ ∼ a few hundreds. Superluminal motion was also detected for two near-by events, GRB 030329 and the neutron star merger event GW 170817 (Taylor et al. 2004; Mooley et al. 2018b). The standard scenario (the so-called fireball model, see Fig. 13.1) has been extensively tested against observations, and it has become the leading paradigm to understand GRB events (e.g. Kumar and Zhang 2015). The scenario consists of several stages,

• 1. The energy source (or sometimes called the central engine) is believed to be rapid mass accretion onto a newly formed stellar-mass black hole (or highly magnetised neutron star). The energy release drives an ultra-relativistic jet with Lorentz factor Γ ≳ 100. Unlike other astrophysical jets, GRB jets are transient (i.e. explosive events). Since the matter and energy are highly concentrated in a thin layer at the forefront of the “jets”, GRB jets are sometimes called “flying pancakes” (Piran 1999).

• 2. The prompt gamma-ray emission is assumed to be due to internal shocks that arise in an irregular jet (or due to magnetic reconnections in a magnetised jet) (e.g. Rees and Mészáros 1994; Kobayashi et al. 1997; Daigne and Mochkovitch 1998; Zhang and Yan 2011). This dissipation occurs at a large distance from the central engine around 10¹³–10¹⁵ cm. Electrons are accelerated to high energy within the collisionless shocks, and produce the observed gamma-rays via the synchrotron process.

• 3. At further larger distances ∼ 10¹⁶ cm, the jet impacts on the surrounding medium, and the explosion energy is transferred from the original jet (the ejecta from the central engine) to the ambient matter. A short-lived, optical flash is emitted from the reverse shocked ejecta at this stage (Mészáros and Rees 1997; Sari and Piran 1999b; Kobayashi 2000). After the energy transfer, the forward shock ambient medium (a collimated blast wave) carries most of the explosion energy.

• 4. The collimated blast wave further expands and approaches the self-similar Blandford-McKee solution (Blandford and Mckee 1976, strictly speaking, it is described by a conical section of the spherical solution). The blast wave continuously accelerates electrons to high energies. This causes long lasting synchrotron emission which accounts for afterglow . As more material accumulates, the blast wave slows down. The typical frequency of the afterglow is gradually going down to lower frequencies (X-ray, optical and radio).

• 5. Around 10¹⁸ cm, the blast wave collects sufficient material to slow it down to sub-relativistic velocities, and it approaches the Sedov–Taylor solution. This happens about a few month to a year after the burst. The emission is mostly in the radio band in this final epoch.

Fig. 13.1

Schematic of the Fireball model, illustrating a relativistic jet from a collapsing star. The prompt gamma-rays (GRB) is due to an internal dissipation process in the original jet, the X-ray (X), optical (O) and radio (R) afterglow originate from the collimated blast wave

Early X-ray afterglow presents complex temporal behaviour (Nousek et al. 2006; Zhang et al. 2006). The light curve initially decays very steeply (known as the steep decay phase), and then shows a very shallow declining phase (the plateau phase) until 10⁴ − 10⁵ s after the explosion and before settling at the regular decay expected from the blast wave. About 1/3 of GRBs show X-ray flares in the early afterglow phase which are significant re-brightening (up to a factor of 500 in some cases) with short variability timescales of Δt∕t ∼ 0.2 on average where t is the time of the flare’s onset and Δt is the duration of the flares (Burrows et al. 2005). Although this complex behaviour (especially the plateau phase) is not fully understood yet, the highly variable nature of X-ray flares implies that they are likely to be the results of the same mechanism producing the prompt gamma-ray emission.

While the usual radio afterglow rise on timescale of weeks or months, some events have a radio flare peaking at about 1 day, rising quickly before the peak and decaying quickly after the peak. These flares have been interpreted as the emission from the ejecta earlier shocked by the reverse shock. The same particles in the ejecta producing the optical flash have been cooled adiabatically, and their emission shifts quickly toward lower frequencies while weakening (Kulkarni et al. 1999b; Sari and Piran 1999a,b; Kobayashi and Sari 2000).

13.3 Afterglow and Shock Instability

The production of synchrotron radiation requires the presence of a magnetic field in the shocked ejecta and shocked ambient material, but the degree of magnetisation and the configuration of the magnetic fields still remain a matter of debate (e.g. Piran 2005; Sironi et al. 2015).

For the afterglow study, it is usually assumed that the energy densities of the relativistic electrons and of the magnetic field can be characterised by equipartition parameters, 𝜖 _e and 𝜖 _B that measure the ratios of these energies to the internal energy produced by shocks. The Blandford-McKee self-similar solution combined with the synchrotron radiation model (Sari et al. 1998) provides an excellent description of the afterglow light curves and spectra. Possibly this is the most successful part of the GRB theory. Afterglow modelling has been carried out mainly for long GRBs, because they produce brighter afterglow due to the larger explosion energy and higher ambient density. Fits to the observed afterglow spectra suggest 𝜖 _e = 10⁻² − 10⁻¹ and 𝜖 _B = 10⁻⁴ − 10⁻² for most events (Panaitescu and Kumar 2001). Considering that the internal energy density in the blast wave (the forward shock ambient matter) is e = 4 Γ² nm _p c ² (Blandford and Mckee 1976; Sari and Piran 1995) where n ∼ 1 proton/cm³ is the ISM density and m _p is the proton mass, the magnetic field strength in the blast wave is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} B=(32\pi m_p\epsilon_Bn)^{1/2}\Gamma c \sim 0.4 ~n \left({\frac{\epsilon_B}{10^{-3}}}\right)^{1/2}\left({\frac{\Gamma}{30}}\right) \mbox{G}. \end{array} \end{aligned} $$

(13.1)

This is much larger than the value obtained by a simple shock compression of the intergalactic magnetic field of ∼ μG.

Magnetic fields in massive stars might be dragged out with the stellar wind before the explosion, and the blast wave could propagate in magnetised wind medium. However, since the strength of the fields quickly decay at large radii (e.g Usov and Melrose 1992), it is not sufficiently strong in the afterglow region R ∼ 10¹⁶–10¹⁷ cm. In other words, the magnetisation of the massive stars would not be high enough to produce a highly magnetised wind medium (see Walder et al. 2012, for magnetic fields in massive stars and their winds). Afterglow modelling has been carried out by assuming the wind density distribution $n = \dot {M}/4\pi m_p v r^2 \propto r^{-2}$ or the ISM distribution n =  constant, where $\dot {M} \sim 10^{-5} M_\odot \mbox{year}^{-1}$ is the mass loss rate of the massive star and v ∼ 10³ km s⁻¹ is the stellar wind velocity (Li and Chevalier 2001). Interestingly, such modelling often favours the ISM ambient medium (Panaitescu and Kumar 2002), although massive stars are supposed to be the progenitors of long GRBs.

The central engine (e.g. a black hole accretion disk system) might produce highly magnetised ejecta at the explosion. However, the afterglow originates from a different emission region (i.e. the forward shock ambient medium rather than the reverse shock ejecta). The strong magnetic field of ∼ 1 G, which the afterglow modelling requires, is believed to be generated locally in the shock. In the context of GRB studies, the most discussed instability is the Weibel instability which develops through a charge separation in the background plasma, triggered by magnetic fluctuations. It has been shown that such an instability could lead to growth of the magnetic field and to particle acceleration (e.g. Medvedev and Loeb 1999; Gruzinov and Waxman 1999; Nishikawa et al. 2003; Spitkovsky 2008; Sironi et al. 2015). However, it is not well-understood yet how the magnetised turbulence evolves in the downstream of the shock, and it has a direct implication to afterglow observations.

It has been suggested (Gruzinov and Waxman 1999; Gruzinov 2001) that the coherence length of the fields generated in Weibel instability should be comparable to the plasma skin-depth δ, and that magnetic power on such small scales is expected to decay rapidly. Magnetic fields would maintain its equipartition magnitude only over a skin depth δ scale in the downstream, while modelling of afterglow observations requires that magnetic fields should persist over scales at least several orders of magnitudes larger than δ. To reconcile the problem, a few scenarios have been discussed (Sironi et al. 2015): the small-scale turbulence might evolve to larger scales through inverse cascade effects (Medvedev et al. 2005), it might be dissipated but at a rate which allows to explain the observations (Lemoine 2013), or large-scale fields might be seeded in the downstream by external instabilities (Sironi and Goodman 2007; Levinson 2009).

13.4 Forward and Reverse Shock Modelling

Determination of the magnetic field properties of the original jets (the original ejecta from the central engine) is a key to understanding the driving mechanism of the explosion. Two methods, (1) the forward and reverse shock modelling of early afterglow and (2) early afterglow and prompt gamma-ray polarimetry have been performed in this context. Since the former is sensitive to the strength of magnetic fields, while the latter depends on the coherence length and geometry of the magnetic fields, the two methods are complementary. We discuss the method (1) in this section, and polarimetric studies will be discussed in the next section.

Because of the relativistic beaming effect, the radiation from a jet can be described by a spherical model with an isotropic explosion energy E. The actual energy in the jet with a solid angle Ω is given by ( Ω∕4π)E. The original jet decelerates when a significant fraction of the energy is transferred to the ambient medium. Equalising the energy E with the blast wave energy $(4/3)\pi m_pc^2 n R_{d}^3\Gamma ^2$, we can estimate the deceleration radius R _d of the ejecta. One factor of Γ in the blast wave energy comes from the bulk expansion of the blast wave, and the other from the shock heating. It takes R _d∕βc for the jet to reach to the deceleration radius where β = (1 − 1∕ Γ²)^1∕2 is the jet velocity in the unit of the speed of light c. Since the jet propagates toward the observer, the apparent deceleration time t _d for the observer is shorter by a factor of 1 − β ∼ 1∕2 Γ². Using E ₅₂ = E∕10⁵² erg, we get the deceleration time as

$$\displaystyle \begin{aligned} t_d \sim \frac{R_d}{2\Gamma^2c}=\left({\frac{3E}{32\pi \Gamma^8n m_pc^5}}\right)^{1/3} \sim 90 ~\left({E_{52}/n}\right)^{1/3} \left({\frac{\Gamma}{100}}\right)^{-8/3}\mbox{s}. {} \end{aligned} $$

(13.2)

The energy transfer from the ejecta to the ambient medium is due to two shocks: a forward shock propagating into the ambient medium and a reverse shock propagating into the original ejecta (Sari and Piran 1995; Kobayashi et al. 1999; Kobayashi and Zhang 2007). At the deceleration time, the two shocked regions separated by the contact discontinuity have the same bulk Lorentz factor and internal energy density e. The typical frequency of synchrotron emission depends on the Lorentz boost Γ and the magnetic field B and the electron’s random Lorentz factor γ _e as $\nu _m\propto \Gamma B \gamma _e^2$ (e.g. Sari et al. 1998). The first two factors are the same for the two shocked regions if 𝜖 _B is the same for the two regions, while the electron’s Lorentz factor is different by a factor of Γ (the two shocked regions contain the same amount of the internal energy, but the ejecta is heavier by a factor of Γ. The internal energy is distributed to a larger number of electrons. Then, the effective temperature is lower by a factor of Γ). Since the typical frequencies of the reverse shock emission is lower by a factor of Γ², if the forward shock emits X-rays, the reverse shock would emit optical emission (i.e. optical flash).

Since the ejecta might be endowed with primordial magnetic fields at the central engine, introducing a new parameter $\mathcal {R}_B=\epsilon _{B,r}/\epsilon _{B,f}$ where the subscripts f and r indicate forward shock and reverse shock, respectively, we have a relation for the typical frequencies of the forward and reverse shock emission at the deceleration time (Zhang et al. 2003; Kobayashi and Zhang 2003),

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\nu_{m,r}}{\nu_{m,f}} = \Gamma^{-2} \mathcal{R}_B^{1/2}. {} \end{array} \end{aligned} $$

(13.3)

We can give similar discussion on the peak flux of the two shock emission (the peaks in the spectral domain at the deceleration time). The peak flux F _ν,max is proportional to the number of accelerated electrons N _e, the magnetic field B and the Lorentz boost Γ (e.g. Sari et al. 1998). From the energy conservation, the mass of the ejecta is larger by a factor of Γ at the deceleration time than that of the ambient medium swept by the forward shock. Since the number of electrons is proportional to the mass, we get another relation at the deceleration time t _d,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{F_{\nu,\mbox{max},r}}{F_{\nu,\mbox{max},f}} = \Gamma \mathcal{R}_B^{1/2}. {} \end{array} \end{aligned} $$

(13.4)

After this onset of the afterglow at t _d, the reverse shock optical emission decays as ∼ t ⁻², while the forward shock optical emission starts to decay as ∼ t ⁻¹ later when the typical frequency of the forward shock emission ν _m,f crosses the optical band (see the top-left panel of Fig. 13.2; Kobayashi 2000, Zhang et al. 2003). If ν _m,f is already below the optical band at t _d (it means ν _m,r is in the microwave or radio band), the forward shock emission also peaks at t _d, and the reverse shock emission is likely to be masked by the forward shock emission (the bottom-left panel Harrison and Kobayashi 2013). If the deceleration time t _d (the onset of afterglow) is detected, the Lorentz factor of the original jet can be estimated by using Eq. (13.2) (e.g. Melandri et al. 2010; Liang et al. 2010; Cui et al. 2018).

Fig. 13.2

Early optical afterglow light curves with different forward reverse shock contributions. Forward shock emission (black dashed lines) and reverse shock emission (red solid lines). The two left panels show non-magnetised cases $\mathcal {R}_B \sim 1$. If ν _m,f is above the optical band at t _d, the forward shock emission peaks when ν _m,f crosses the optical band (top-left panel). If ν _m,f is below the optical band at t _d, the forward shock emission also peaks at t _d (bottom-left panel). The right panel shows a magnetised case $\mathcal {R}_B \gg 1$

The strength of the reverse shock emission depends on the magnetisation of the ejecta. If the ejecta is endowed with primordial magnetic fields, the optical flash could be more prominent as illustrated in the right panel of Fig. 13.2. Using the relative strength of the two emission components and Eqs. (13.3) and (13.4), we can estimate the magnetisation parameter $\mathcal {R}_B$ (see Zhang et al. 2003; Gomboc et al. 2008; Harrison and Kobayashi 2013; Gao et al. 2015, for the details). The magnetisation parameter has been evaluated for the events which show a clear reverse shock signature (e.g. an early steep decay t ⁻² in the optical afterglow light curve) including GRB 990123, GRB 021004, GRB 021211, GRB 041219A, GRB 061126, GRB 090102, GRB 090510, GRB 110205A, GRB 120308A, GRB 130427A, GRB 140512A, GRB 161219B (e.g. Fan et al. 2002, 2005; Kumar and Panaitescu 2003; Gendre et al. 2012; Zheng et al. 2012; Laskar et al. 2013, 2018; Japelj et al. 2014; Zhang et al. 2015; Fraija et al. 2016; Huang et al. 2016). Although $\mathcal {R}_B \lesssim 10$ have been obtained for most of them, very high values $\mathcal {R}_B = 200-8000$ are reported for GRB 990123, GRB 090102 and GRB 140512A (Zhang et al. 2003; Harrison and Kobayashi 2013; Huang et al. 2016). This implies that at least some fraction of GRB ejecta carry significant primordial magnetic fields (Fan et al. 2002; Zhang et al. 2003). Nevertheless, since 𝜖 _B,f ∼ 10⁻⁴ − 10⁻² is usually inferred from afterglow modelling (Panaitescu and Kumar 2001), and a further lower value of 𝜖 _B,f ∼ 10⁻⁶ is obtained for GRB 990123 (Panaitescu and Kumar 2004), it follows that 𝜖 _B,r ≪ 1. Therefore, all above cases correspond to mildly magnetised outflow and not a Poynting flux dominated outflow at the deceleration radii. The ratio σ between the Poynting flux energy and the kinetic energy is estimated as $\sigma \lesssim 10^{-3}$ for these events (Harrison and Kobayashi 2013).

The detection of bright optical flash suggests that a moderately magnetised jet in which the magnetic field is strong enough to enhance the reverse shock emission but not strong enough to suppress the formation of reverse shock or the Fermi particle acceleration (Zhang and Kobayashi 2005; Giannios et al. 2008; Mimica et al. 2009; Sironi and Spitkovsky 2010; Sironi et al. 2015). If future events indicate high values σ ≳ 0.1, an interesting possibility to reconcile the problem is that optical flash (and the prompt gamma-rays) would be produced through a dissipative MHD process rather than shocks (Giannios and Spruit 2006; Lyutikov 2006; Giannios 2008; Zhang and Yan 2011).

13.5 Polarisation Signals

Since synchrotron emission is believed to be the dominant radiation process for the main emission components of GRBs, i.e. prompt gamma-rays, reverse shock emission and afterglow, these components could be polarised. The observed degree of linear polarisation (PD_L) and polarisation position angle (PA) depend on the properties and geometry of the emitting region. Polarimetric observations allow us to study the magnetic field structure and energy distribution in GRB jets and their dynamics (Lazzati 2006; Toma 2013; Covino and Götz 2016; Gill et al. 2018). Since circular polarisation is expected to be zero for optically thin synchrotron emission, we mainly discuss linear polarisation.

13.5.1 Visible Region

A unique feature of GRB jets is the small angular extent of the visible region, which is much smaller than the actual jet opening angle θ _j (see Fig. 13.3). The visible region is an important concept when we discuss polarised emission from a relativistic outflow. As the blast wave is described by a conical section of the spherical Blandford-McKee solution, the jet head has a curvature, and each fluid element at the jet head propagates radially with a high Lorentz factor. Because of the relativistic beaming effect, it radiates practically only in the radial direction (i.e. their propagation direction). Since the radiation is beamed in a forward cone of opening angle of ∼ 1∕ Γ ≪ 1, it is not visible for the observer if the propagation direction of the fluid element makes an angle large than 1∕ Γ with the line of the sight. In Fig. 13.3, the emission from element A is visible, but the one from element B is not visible. The observer cannot see the entire surface of the jet head. The visible region is the small area with an angular size of 1∕ Γ, located around the point at which the line of sight intersects the jet head. In the figure, the edge of the visible region is indicated by the red dotted lines.

Fig. 13.3

The visible region is bound by the two red dotted lines in this figure. For the observer (the telescope in the figure), the visible region is a small circle, and the line of sight goes through the centre. The emission from each fluid element is beamed in a forward cone of opening angle ∼ 1∕ Γ. The relativistic jet is characterised by the opening angle θ _j and Lorentz factor Γ

13.5.2 Afterglow Emission

Synchrotron emission is highly polarised if magnetic field is coherent in the visible region. For a power-law distribution of electron energies with an index p ∼ 2, the degree of linear polarisation is PD_L,0 = (p + 1)∕(p + 7∕3) ∼ 70% (Rybicki and Lightman 1979). However, if magnetic field is fully tangled and isotropic at any point (or in any point-like region), the emission is unpolarised, regardless of the geometry of the visible region. To produce polarised emission, there must be asymmetry in the configuration of magnetic field.

In the afterglow phase, magnetic fields in the emission region (i.e. the blast wave) are produced by shock instabilities, and expected to be tangled. However, the magnetic field generation in collisionless shocks is not fully understood yet, the ultimate fate of the field many skin depths behind the shock front is not clear. If the coherence length of the magnetic field grows at about the speed of light in the local fluid frame, independent, coherent patches of magnetic fields might be created in the visible region (Loeb and Perna 1998; Gruzinov and Waxman 1999). The observed degree of linear polarisation is

$$\displaystyle \begin{aligned} \begin{array}{rcl} PD_{\mathrm{L}}= \frac{PD_{\mathrm{L,0}}}{\sqrt{N}} \sim 10\%, \end{array} \end{aligned} $$

(13.5)

where N ∼ 50 is the number of coherent patches within the visible region of the blast wave (Gruzinov and Waxman 1999). The degree and direction of polarisation should depend on time as the process is stochastic, and the polarisation coherence time is Δt ∼ t (the time since the explosion).

Another asymmetric configuration of magnetic field also has been discussed (Gruzinov 1999; Ghisellini and Lazzati 1999; Sari 1999). Since at any point in the shock front, the shock normal (the radial direction) is a special direction, the magnetic fields parallel and perpendicular to the shock normal could have significantly different averaged strengths (e.g. Medvedev and Loeb 1999). To discuss how anisotropic field leads to polarised emission, we first consider a small region around the shock front (the dashed circle in the right panel of Fig. 13.4). If the region is sufficiently small, the curvature of the shock front would be neglected, the shocked ambient behind the shock front can be described by a slab of plasma (see the left panel of Fig. 13.4). We first discuss the emission in the slab’s rest frame. If the slab is observed face-on (the radial direction θ ^′ = 0), magnetic field would be completely tangled and isotropic. The emission should be unpolarised. However, if the strength of the magnetic field parallel to the shock normal B _∥ is different (let us say it is significantly weaker than that of the perpendicular one B _⊥), the magnetic field has some degree of alignment along the shock front. If the slab is observed edge-on (θ ^′ = π∕2), the emission should be polarised and the polarisation vector (the electric field vector of the radiation) is in the shock normal direction. The linear polarisation degree at a viewing angle θ ^′ is given (Gruzinov 1999; Sari and Piran 1999a,b) by

$$\displaystyle \begin{aligned} \begin{array}{rcl} PD_{\mathrm{L}} (\theta^\prime)=PD_{\mathrm{L,0}} \sin ^2\theta^\prime \frac{2\langle B_\|{}^2 \rangle-\langle B_\bot^2\rangle} {2\sin ^2\theta^\prime \langle B_\|{}^2 \rangle+(1+\cos^2\theta^\prime) \langle B_\bot^2\rangle }. {} \vspace{-2pt} \end{array} \end{aligned} $$

(13.6)

If we choose the z-direction of the local fluid frame coordinate to be in the shock normal direction, we have $\langle B_\bot ^2\rangle = \langle B_x^2\rangle + \langle B_y^2\rangle$ and $\langle B_\|{ }^2 \rangle =\langle B_z^2 \rangle$. We obtain PD_L = 0 for three-dimensional isotropic field. If the field is in the shock plane B _⊥≫ B _∥, the estimate reduces to $PD_{\mathrm {L}}/PD_{\mathrm {L,0}}=-\sin ^2\theta ^\prime /(1+\cos ^2\theta ^\prime )$.

Fig. 13.4

Anisotropic magnetic field and blast wave. Left panel: a small region around the shock front in the local fluid frame. Right panel: the visible region of the shock front in the lab frame (the rest frame of the progenitor which caused an explosion). The polar angles θ ^′ (the local fluid frame) and θ (the lab frame) are measured from the radial direction (the shock-normal direction)

Now we consider the emission from the slab in the lab frame in which the slab is moving in the radial direction with Lorentz factor Γ. Because of the relativistic aberration of photons, the photons emitted at θ ^′ = π∕2 (the edge-on polarised emission) in the slab’s co-moving frame are observed at θ ∼ 1∕ Γ in the lab frame. Considering that the viewing angle of a fluid element is θ = 1∕ Γ if it is located at the edge of the visible region (the red dotted lines in the right panel of Fig. 13.4), the visible region would be highly polarised at the edge. The polarisation vector is also rotated by the Lorentz transformation (e.g. Lyutikov et al. 2003). The polarisation vector is perpendicular (tangential) to the edge if PD_L < 0 (PD_L > 0). The linear polarisation degree inside the visible region can be evaluated by combining Eq. (13.6) and the aberration formula $\cos \theta ^\prime =(\cos \theta -\beta )/(1-\beta \cos \theta )$.

The visible region appears as a ring on the sky especially at frequencies above the typical synchrotron frequency ν _m due to a relativistic limb-brightening effect (Waxman 1997; Panaitescu and Mészáros 1998; Sari 1998; Granot et al. 1999a,b). Although each segment of the ring is polarised, the net polarisation is zero because of the circular geometry (see the left panel of Fig. 13.5). However, if the surface brightness of the jet (and consequently that of the ring) is not uniform or/and the entire ring is not inside the jet (Gruzinov 1999; Waxman 2003), the net polarisation would be expected. As shown in the right panel of Fig. 13.5, if the line of sight runs slightly inside the jet edge (the black solid line), the bottom part of the ring is located outside the jet, and there is actually no emission from this part. Considering that the polarisation vector in the missing part is vertical, the net polarisation is in the horizontal direction. On the other hand, if the line of sight runs slightly outside the jet edge (the black dashed line), the emission comes only from the top part of the ring. The net emission is dimmer, but more polarised. The polarisation direction is vertical this time.

Fig. 13.5

The visible region. Left panel: The blue arrows indicate the polarisation vectors for PD_L < 0. The cross shows the point at which the line-of-sight intersects with the visible region. Right panel: a part or most part of the ring is missing if the line of sight runs almost along the jet edge (the black solid line or the black dashed line)

Afterglow observations suggest that the median opening angle of long GRB jets is about θ _j ∼ 0.2 rad. It is not well constrained for short GRBs, but they also might have similar or slightly wider opening angles (Fong et al. 2015). Since the angular extent of the visible region is very small $1/\Gamma \lesssim 0.01$ in the prompt gamma-ray or early afterglow phase, this off-axis mechanism requires fine tuning of the viewing angle to produce high polarisation in the early phases. Such a viewing angle happens by chance possibly only for narrow jets θ _j ∼ 1∕ Γ (Gruzinov 1999; Waxman 2003; Nakar et al. 2003b). However, the blast wave decelerates as it expands and the angular extent of the visible region grows in the afterglow phase. A favourable viewing angle could be naturally achieved in the afterglow phase (Sari 1999; Ghisellini and Lazzati 1999).

When the Lorentz factor $\Gamma (t) \sim 6(E_{52}/n)^{1/8}t_{\mathrm{day}}^{-3/8}$ of the blast wave is sufficiently large, the afterglow emission is well described by the spherical model where t _day is the observer time in unit of day. However, once the growing visible region runs off the edge of the jet (the left panel of Fig. 13.6), the emission becomes dimmer than what is expected in the spherical model. This induces a steepening (jet break) in the afterglow light curve (Rhoads 1999; Sari et al. 1999; Panaitescu and Mészáros 1999). Since the typical viewing angle is ∼ 2θ _j∕3 for the on-axis observer, equalising the viewing angle with the angular extent of the visible region ∼ 1∕ Γ(t), we obtain the jet break time as

$$\displaystyle \begin{aligned} t_{j} \sim 1.6 (E_{52}/n)^{1/3}(\theta_j/0.2)^{8/3}\mbox{days}. \end{aligned} $$

(13.7)

If the jet break time is detected, we can estimate the jet opening angle $\theta _j \propto (E/n)^{-1/8}t_j^{3/8}.$ If the collimated blast wave expands sideways at the speed of light in its own rest frame, the jet opening angle is given by θ _j ∼ θ _j,0 + ct _p∕R ∼ θ _j,0 + 1∕ Γ where t _p is the proper time in the rest frame and R is the blast wave radius in the lab frame. The sideways expansion could become significant at the jet break time. However, for a modest jet opening angle, the sideways expansion is known to be rather slow (e.g. Zhang and MacFadyen 2009; Meliani and Keppens 2010; van Eerten et al. 2010; Granot and Piran 2012). Here we discuss the non-spreading case.

Fig. 13.6

Jet break and polarisation position angle. The black circles show the jet seen face-on. The yellow rings show the visible regions at an early time (left panel) and a late time (right panel)

Since it is unlikely that the observer is directed exactly at the centre of the jet, the afterglow emission is expected to be polarised around the jet break time. The light curve of the degree of polarisation would have two maxima, with the polarisation position angle changing by 90^∘ between the first and the second maximum (Ghisellini and Lazzati 1999; Sari 1999). This can be easily understood as follows, even if the entire ring is initially inside the jet (zero polarisation), eventually a part of the visible region runs off the edge of the jet (see the left panel of Fig. 13.6), and the net polarisation is in the horizontal direction in the figure. Since the visible region further expands, at a later time only the top part which is polarised in the vertical direction is inside the jet (the right panel). During the ring expands from the left configuration to the right one in Fig. 13.6, there should be a moment of zero polarisation at which a half of the ring is inside the jet. Although the maximal polarisation degree around the jet break depends on the viewing angle measured from the jet axis θ _obs∕θ _j, it would be about PD_L = 10 − 20% if B _⊥≫ B _∥. If the surface brightness of the jet is not uniform but structured (e.g. explosion energy per unit solid angle is a function of the angle from the jet axis. Rossi et al. 2002; Zhang and Mészáros 2002), as the dominant emission always comes from the jet-centre side of the (distorted) ring, the polarisation position angle remains constant throughout the whole evolution. The polarisation temporal behaviour is expected to be sensitive to the luminosity distribution of the jet (Rossi et al. 2004).

Optical linear polarisation measurements have been carried out for many GRB afterglows typically several hours to a few days after the prompt gamma-ray emission (Covino and Götz 2016 and the references therein). This is the period in which a jet break is expected to occur. As we will discuss below, the detection or upper limits of the linear polarisation degree are generally low $\lesssim$ a few percent, although at least two events show a 90^∘ rotation of the polarisation position angle around a possible jet break. The low polarisation results imply that B _⊥ and B _∥ have a similar strength in general. In the discussion below, time t indicates the time since the prompt gamma-ray emission.

• GRB 990123 was a remarkable burst with very high prompt gamma-ray fluence and with very bright optical flash (Akerlof et al. 1999; Sari and Piran 1999a). The reverse shock also produced the radio flare at t ∼ 1 day (Kulkarni et al. 1999b; Sari and Piran 1999a; Kobayashi and Sari 2000). A jet break was detected at t _j ∼ 2 days (Kulkarni et al. 1999a). Polarisation of the optical afterglow was measured at t ∼ 18 h, and an upper limit of 2.3% on the linear polarisation was obtained by using the Nordic Optical Telescope (NOT) (Hjorth et al. 1999).

• GRB 990510: The well-sampled optical light curves and simultaneous early time radio observations can be well fit by a simple model for the jet evolution with a jet break time t _j ∼ 1 day Harrison (1999). The linear polarisation of the optical afterglow was measured by two independent teams at a level of 1.7% ± 0.2% at t =  18.5 h (Covino et al. 1999), and 1.6% ± 0.2% at t =  20.6 h (Wijers et al. 1999) with the Very Large Telescope (VLT, Antu). The polarisation degree and angle show no evidence of change.

• GRB 990712: Rol et al. (1999) report the detection of polarisation in the optical afterglow with VLT on three instances t =  0.44, 0.70 and 1.45 days. The linear polarisation degrees are 2.9% ± 0.4%, 1.2% ± 0.4% and 2.2% ± 0.7%, respectively, and the polarisation position angle does not vary significantly during these observations. Although optical follow-up observations were performed for more than 30 days, there is no evidence of the jet break in the light curve.

• GRB 991216: A gradual optical steepening around t _j ∼ 2 days has been discussed in the context of the jet break. The steep X-ray afterglow at earlier times might imply that the higher energy emission is concentrated in a narrower jet core that began to spread earlier (Halpern et al. 2000). VLT observations provide upper limits of linear polarisation degree < 2.7% at t =  35 h and < 5% at t =  60 h (Covino et al. 2004).

• GRB 010222: The light curve of the optical afterglow shows the decay of the brightness with a broken power law, of which the indices are −0.7 before and −1.3 after the break. The break point of the light curve is t ∼ 0.5 day (Masetti et al. 2001; Watanabe et al. 2001). Polarimetric observations were obtained at NOT at t ∼ 22.65 h, the linear polarisation degree is 1.36% ± 0.64% (Björnsson et al. 2002).

• GRB 011211: The optical afterglow decays as a power law with a slope of −0.83 for the first ∼ 2 days after the burst, at which time there is evidence of a break (Holland et al. 2002). Polarimetric observations performed with the VLT on the afterglow at t ∼ 35 h, the observations yielded an upper limit of 2.7% (Covino et al. 2002).

• GRB 020405: The optical light curves of the afterglow are consistent with a single power-law decay of temporal index −1.54 between 1 and 10 days after the prompt event (Bersier et al. (2003a) reports a much steeper decay −1.72 between t = 1.24 days and 4.3 days). The red bump detected at t ∼ 20 days has been modelled with an emerging SN. The light curve at t ≳ 20 days is consistent with a power-law decay with index −1.85 (Masetti et al. 2003). Linear polarisation measurements were carried out with VLT at t = 29.5 h (1.5% ± 0.4%) by Masetti et al. (2003) and at t = 52.0 h (1.96% ± 0.33%) and t = 76.2 h (1.47% ± 0.43%) by Covino et al. (2003). Interestingly, Bersier et al. (2003a) measured a high polarisation 9.9% ± 1.3% at t = 31.7 h by using the Multiple Mirror Telescope (MMT) . The patchy emission region model (e.g. Gruzinov and Waxman 1999) is not consistent with the rapid change in the polarisation degree (1.5–9.9% in a short period of Δt∕t ∼ 0.1) nor the almost constant polarisation position angle. Although early radio data indicate that the ejecta underwent a jet break at t ∼ 1 day (Berger et al. 2003), there is no signature of a jet break in the optical light curve. The jet-edge models also do not work.

• GRB 020813: The optical light curves of the afterglow indicate a jet break 0.33 < t _j < 0.88 days (Gorosabel et al. 2004 and the references therein). The linear polarisation degree is measured at a level of 1.8–2.4% during t = 4.7 − 7.9 h by Barth et al. (2003) with the Keck telescope. VLT polarimetric monitoring campaign of the optical afterglow carried out in three nights, from t =  0.88–4.20 days (Gorosabel et al. 2004, see also Lazzati et al. 2004). The linear polarisation is detected at ∼ 1% level with an almost constant polarisation position angle, ruling out the 90^∘ rotation expected in a (homogeneous) jet-break model.

• GRB 021004: The optical light curves of the afterglow can be interpreted as a power-law decay with 10–20% fluctuations on several timescales, ranging from a few hours down to 20–30 min (Bersier et al. 2003b). The fluctuations can be explained in several models including fluctuations in the ambient matter density, energy injections (refreshed shocks) and angular fluctuations in the jet energy (e.g. Lazzati et al. 2003; Nakar et al. 2003a; Heyl and Perna 2003). A late time steepening can be modelled as a jet break with t _j = 9 days (Mirabal 2003). Rol et al. (2003) performed polarimetric observations of the optical afterglow, obtained with NOT and VLT between t = 8 and 17 h. The linear polarisation degree is almost constant at a level of 1–2%, but the polarisation position angle rotated by about 45^∘ between the NOT and VLT observations. Late VLT observations even show a 90^∘ change between t = 9 and 89 h (Lazzati et al. 2003). The main features in both the light curve and the polarisation fluctuations could be reproduced by the patchy-shell model (Nakar and Oren 2004).

• GRB 030226: The optical light curve shows a jet break around t = 0.8 days. VLT polarimetry provides an upper limit of linear polarisation degree < 1.1% at t ∼ 1 day (Klose et al. 2004).

• GRB 030328: The light curve of the optical afterglow shows the decay of the brightness with a broken power law, of which the indices are −0.76 before and −1.50 after the break time 0.48 days. VLT polarimetry shows a liner polarisation degree of 2.4% ± 0.6% at t ∼ 18.5 h (Maiorano et al. 2006).

• GRB 030329: This is a nearby event (z ∼ 0.17). The very bright optical afterglow makes it the most well-studied afterglow to date. The optical emission steepens from a decay index of −0.873 to −1.97 at t _j ∼ 0.48 days. The light curve also shows a very large variability a few days after the burst possibly due to refreshed shocks (Granot et al. 2003). VLT polarimetry shows linear polarisation degree at a level of 0.3–2.5% throughout a 38-day period, with significant variability in polarisation degree and angle on timescales down to hours Greiner et al. (2003).

• GRB 071010A: The later evolution of the afterglow requires a break, consistent with being fully achromatic in the optical/NIR and X-ray bands, at t _j ∼ 1 day, with a post-break decay index of −2.07. VLT polarimetry provides an upper limit of linear polarisation degree < 1.3% at t ∼ 0.9 days (Covino et al. 2008).

• GRB 091018: The X-ray light curve of the afterglow shows at least two breaks, as well as flaring activity. Although the densely sampled optical light curve also shows several flares, an achromatic break at t ∼ 9 h was identified (Wiersema et al. 2012). VLT polarimetric monitoring took place over three observing nights. Initially the linear polarisation degree gradually increase up to ∼ 2%. After the break, the polarisation degree and polarisation position angle are more variable possibly due to the additional flaring components in the afterglow. Although the scatter is large after the break, the polarisation position angle seems to show a 90^∘ rotation at the break (Wiersema et al. 2012, 2014).

• GRB 121024A: Both the X-ray and optical/NIR light curves of the afterglow are best fitted by a broken power law with similar break times t ∼ 10 h (Wiersema et al. 2014; Varela et al. 2016). The occurrence of a light curve break simultaneously in X-rays and optical wavelengths is suggestive of a jet break origin. However, the post-break decay indices are rather shallow −1.67 (X-ray) and −1.25 (optical) even for non-spreading jets. VLT polarimetry shows linear polarisation degrees at a level of 3–5% and a constant polarisation position angle during t ∼ 3 − 6 h. Further observations carried out the night after (and after the jet break) show a lower polarisation level with a clear 90^∘ rotation of the polarisation position angle (Wiersema et al. 2014). This would be another identification of the polarisation position angle rotation expected in the jet break model. Surprisingly, circular polarisation PD _C = 0.61% ± 0.13% is detected at t ∼ 4 h (Wiersema et al. 2014). The linear polarisation degree during the circular polarimetry interval is about 4%, the afterglow shows a ratio PD _C∕PD _L ∼ 0.15. If the electron pitch angle distribution is isotropic, the circular polarisation contributions of electrons with pitch angles θ + 𝜖 and θ − 𝜖 nearly cancel out, and the remaining PD_C scales with the angular size of the beaming cone of the synchrotron emission 1∕γ _e where γ _e is the random Lorentz factor of electrons. Then, PD _C∕PD _L ∼ 1∕γ _e ∼ 10⁻⁴ is expected. The origin of the circular polarisation in the optically thin synchrotron emission is puzzling and unexpected. Even if the electron pitch angle distribution is extremely anisotropic, cancellation would arise from the integration over the emitting region (Nava et al. 2016). Plasma propagation effects (e.g. Faraday conversion) within the source could be strong at low frequencies (close to the synchrotron self-absorption frequency), but negligible at optical frequencies (Matsumiya and Ioka 2003; Sagiv et al. 2004; Toma et al. 2008).

The binary neutron star merger GW 170817 was the first multi-messenger event observed in both gravitational and electromagnetic waves (Abbott et al. 2017a,b). The electromagnetic signal began with a weak, short burst GRB 170817A (Abbott et al. 2017c), which was followed over the next hours and days by the ultraviolet, optical and near-infrared emission from a radio actively powered kilonova (e.g. Arcavi et al. 2017; Coulter et al. 2017; Evans et al. 2017; Pian et al. 2017; Smartt et al. 2017; Tanvir et al. 2017). The late onset of the X-ray, optical and radio afterglow emission suggests emission from a narrow relativistic jet viewed off-axis, initially dominated by material along the line of sight and gradually overtaken by the more energetic core parts of the jet (e.g. Alexander et al. 2017; Granot et al. 2017; Haggard et al. 2017; Hallinan et al. 2017; Kasliwal et al. 2017; Kim et al. 2017; Margutti et al. 2017; Troja et al. 2017a,b; D’Avanzo et al. 2018; Lazzati et al. 2018; Lamb and Kobayashi 2018; Lynman et al. 2018; Mooley et al. 2018a; Nakar et al. 2018; Ruan et al. 2018).

VLT polarimetry shows a linear polarisation degree of 0.50% ± 0.07% at 1.46 days after the detection of the gravitational waves. The polarisation measurement was made when the kilonova was still in its blue phase. The low degree of polarisation is consistent with intrinsically unpolarised emission scattered by Galactic dust, suggesting a symmetric geometry of the emitting region and low inclination of the merger system (Covino et al. 2017). VLA radio polarimetry provides an upper limit of 12% at 244 day since the gravitational waves (Corsi et al. 2018). Within the structured jet scenario, the upper-limit on the radio linear polarisation degree constrains the magnetic field configuration in the blast wave (Corsi et al. 2018; Gill and Granot 2018), and it excludes B _∥ = 0 (magnetic field fully contained in the plane of the shock), and requires $2\langle B_\|{ }^2 \rangle / \langle B_\bot ^2\rangle > 0.5$ (semi-isotropic magnetic field). The magnetic configuration seems to be similar to that in the regular GRB blast wave.

13.5.3 Early Afterglow: Reverse Shock Emission

Polarimetric observations of early afterglow have been carried out in recent years by using fully automated “robotic” telescopes. Once the reverse shock crosses the thin ejecta, no new electrons are accelerated anymore, the reverse shock emission rapidly decays in time. Optical flash lasts only for 10–20 mins after the prompt gamma-ray emission. To detect optical flash from GRB events which happen in random directions and at random times, we need robotic telescopes that promptly respond to satellite triggers (X-ray/gamma-ray satellites keep watching the sky with a large field of view) and make observations without the intervention of a human.

The significance of reverse shock emission observations in the GRB study can be understood if we consider that the late (>20 mins) afterglow originates from the blast wave (the forward shocked ambient medium). Since the dynamics of the blast wave is determined only by the ambient density and the explosion energy (or energy per solid angle for structured jets), the late afterglow is insensitive to the properties of the original ejecta from the central engine. The initial Lorentz factor and/or magnetisation of the ejecta can be examined only through the observations of the prompt gamma-rays, reverse shock emission (i.e. optical flashes and radio flares) or possibly X-ray flares. These emission components are believed to originate from the original ejecta from the central engine.

Figure 13.7 shows the optical polarisation measurements (detections or upper limits) of nine events as a function of the observing time since the GRB trigger. We note that all polarisation detection cases in the figure (GRB101112A, GRB110205A and GRB120308A) were achieved at relatively early times t < 10³ s. This reinforces the point that prompt measurements are essential to characterise the polarimetric properties of GRB afterglow; the polarisation degree decays very rapidly as the tight upper limits at late times show (Steele et al. 2017). We list notable polarisation measurements of early afterglow below.

• GRB 090102: The steep-shallow decay of optical emission is characteristic of an afterglow whose early time light is dominated by fading radiation generated in the reverse shock. Linear polarisation degree at a 10% level was measured with the Liverpool Telescope (LT) at t ∼ 3 mins (Steele et al. 2009). The detection of high polarisation provides the first direct evidence that large-scale magnetic fields are present when significant reverse shock emission is produced.

• GRB 091208B: Optical linear polarisation in the afterglow is measured with the 1.5 m Kanata telescope at t = 149 − 706 s and the polarisation degree is 10.4% ± 2.5% (Uehara et al. 2012). The early time light curve with a decay index −0.75 is consistent with the forward shock emission. Uehara et al. (2012) suggest that the magnetic fields are amplified by the MHD instabilities. Another possibility could be that the forward shock propagated in the magnetised wind from the progenitor star (Granot and Königl 2003), although the afterglow light curve implies that the density profile of the external medium is constant rather than wind type.

• GRB 120308A: Early afterglow polarimetry was performed by using LT (see Fig. 13.8; Mundell et al. 2013). Linear polarisation degree 28% was detected around the peak at t ∼ 300 s, and it declined to 16% by t ∼ 800 s. The polarisation position angle was almost constant over this period. The high polarisation degree and the constancy of the polarisation position angle rule out local shock instability models. Instead, the polarisation properties imply a magnetised baryonic jet with large-scale uniform fields that can survive long after the initial explosion. The amplification of magnetic fields by the rotation of a black hole accretion disk system is often discussed in magnetic jet models. For an axially symmetric field configuration, the poloidal component of the magnetic field decays faster with radius. Therefore, the large-scale magnetic field is likely to be dominated by a toroidal component. Although internal dissipation processes preceding the onset of afterglow can distort the toroidal field (Zhang and Yan 2011; Troja et al. 2017a,b), the visible region with angular scale ∼ 1∕ Γ might have a rather uniform magnetic field. In such a case, the polarisation vector points toward the jet axis, and the polarisation position angle is expected to be constant.

Fig. 13.7

Linear polarisation degree as a function of time after the burst for nine GRBs from the Liverpool Telescope RINGO2 sample. The temporal error bars show the duration of the exposure. From Steele et al. (2017)

Fig. 13.8

Evolution of optical polarisation and brightness in GRB120308A. Evolution of (a) polarisation degree P and (b) polarisation position angle. (c) Optical light curve. Model fits using one peak (blue solid line) or two peaks (broken grey line for each component). Figure from Mundell et al. (2013)

Granot and Taylor (2005) used VLA data to constrain the polarisation of radio flares (the late time reverse shock emission from the original ejecta). They find only upper limits for both linear and circular polarisation. Their best limits are for GRB 991216, for which upper limits on the linear and circular polarisation are 7% and 9%, respectively. The upper limits challenge a structure jet model with a large-scale ordered toroidal fields.

13.5.4 Prompt Gamma-Rays

The claim by Coburn and Boggs (2003) that the prompt gamma-ray emission of GRB 021206 was polarised at 80% level has stimulated the discussion on polarisation models, even though the claim was later contradicted by Rutledge and Fox (2004) and Wigger et al. (2004). A large number of mechanisms potentially can produce highly polarised prompt emission (e.g. Lazzati 2009; Toma et al. 2009; Gill et al. 2018 and the references therein), which include a large-scale magnetic fields in the original ejecta, coherent magnetic patches within the visible region, specific viewing angle with anisotropic magnetic fields, Compton drag and photospheric emission.

• GRB 041219A: Using multiple-detector coincidence events in the 100–350 keV energy band, Kalemci et al. (2007) searched for polarisation in the prompt gamma-ray emission with the SPI instrument on INTEGRAL. Although they reported a linear polarisation degree 98% ± 33%, they could not constrain the systematics of their measurements. The multiple event data was analysed and compared with the predicted instrument response obtained from Monte-Carlo simulations using the GEANT 4 INTEGRAL mass model, McGlynn et al. (2007) measured the linear polarisation in the brightest pulse of the GRB to be about 60%, but at a low level of significance ∼ 2σ. Götz et al. (2009) used the imager on board the INTEGRAL (IBIS) to measure the polarisation of the prompt gamma-ray emission. However, no polarisation signal was found integrating over the same pulse, and the upper limit is 4%.

• GRB 100826A: The average linear polarisation degree of 27% ± 11% was detected with the GAP on board the IKAROS (Yonetoku et al. 2011). By dividing the GRB into two 50 s time intervals, the change of the polarisation position angle was detected.

• GRB 110301A, GRB 110721A: With GAP on board the IKAROS, linear polarisation degrees of 70% ± 22% and $84^{+16}_{-28}\%$ were detected for GRB 110301A and GRB 110721A, respectively. They did not detect any significant change of polarisation position angle for these events, which have shorter durations and dimmer brightness compared with GRB 100826A.

• 151006A, 160106A, 160131A, 160325A, 160509A, 160607A, 160623A, 160703A, 160802A, 160821A, 160910A: Chattopadhyay et al. (2017) presented the polarisation data for the brightest 11 GRBs detected by CZTI on board the AstroSat satellite during the 1st year of operation. Most of the GRBs show clear polarisation signatures with ≥ 3σ detection significance for 4 GRBs and ∼ 2.5σ significance for another 3 GRBs.

13.6 Conclusions

Polarimetric study has been carried out at various stages of the GRB phenomena (almost at the moment of the explosion to a few hundred days after the explosion) and at various frequencies (radio, optical, X-ray/gamma-ray). Although polarimetry at a few percent level can be still demanding for rapidly fading sources, there are already a large number of optical polarisation measurements about several hours to a few days after the prompt gamma-ray emission. The predicted rotation of polarisation position angle has been detected in events which show a possible jet break. The typically low linear polarisation degree of the afterglow emission provides constraints on the plasma/shock physics. The detection of circular polarisation in the optical afterglow of GRB 121024A is totally unexpected, further circular polarisation observations would be needed to understand the origin. High magnetisation parameters obtained by the early afterglow modelling and high polarisation degree of the early optical afterglow imply that ordered magnetic field is advected by the ejecta from the central engine. Although gamma-ray polarimetry is still technically challenging, considering that the high polarisation of the ejecta emission (reverse shock emission), the prompt emission of GRBs should be good targets for gamma-ray polarimeters. High energy polarimetry will enable us to test various GRB models. In coming years, a large number of neutron star–neutron star merges (and possibly neutron star–black hole mergers) will be detected by gravitational wave observatories such as LIGO , Virgo and KAGRA . Optical and radio polarimetry of the electromagnetic counterparts is likely to constrain the geometry of the outflow from the mergers.