# A Reconnection Switch to Trigger Gamma-Ray Burst Jet Dissipation

Jonathan C. McKinney, Dmitri A. Uzdensky1
Department of Physics and Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305-4060, USA
Chandra Fellow
Center for Integrated Plasma Studies, UCB 390, Department of Physics, University of Colorado, Boulder, CO 80309 E-mail:  (JCM); (DAU);
1footnotemark: 1
Accepted 2011 August 29. Received 2011 July 11; in original form 2010 November 02
###### Abstract

Prompt gamma-ray burst (GRB) emission requires some mechanism to dissipate an ultrarelativistic jet. Internal shocks or some form of electromagnetic dissipation are candidate mechanisms. Any mechanism needs to answer basic questions, such as what is the origin of variability, what radius does dissipation occur at, and how does efficient prompt emission occur. These mechanisms also need to be consistent with how ultrarelativistic jets form and stay baryon pure despite turbulence and electromagnetic reconnection near the compact object and despite stellar entrainment within the collapsar model. We use the latest magnetohydrodynamical models of ultrarelativistic jets to explore some of these questions in the context of electromagnetic dissipation due to the slow collisional and fast collisionless reconnection mechanisms, as often associated with Sweet-Parker and Petschek reconnection, respectively. For a highly magnetized ultrarelativistic jet and typical collapsar parameters, we find that significant electromagnetic dissipation may be avoided until it proceeds catastrophically near the jet photosphere at large radii (), by which the jet obtains a high Lorentz factor (), has a luminosity of , has observer variability timescales of order s (ranging from -s), achieves (for opening half-angle ) and so is able to produce jet breaks, and has comparable energy available for both prompt and afterglow emission. A range of model parameters are investigated and simplified scaling laws are derived. This reconnection switch mechanism allows for highly efficient conversion of electromagnetic energy into prompt emission and associates the observed prompt GRB pulse temporal structure with dissipation timescales of some number of reconnecting current sheets embedded in the jet. We hope this work helps motivate the development of self-consistent radiative compressible relativistic reconnection models.

###### keywords:
accretion discs, black hole physics, galaxies: jets, gamma rays: bursts, MHD, instabilities, relativity, methods: numerical
pagerange: A Reconnection Switch to Trigger Gamma-Ray Burst Jet DissipationA Reconnection Switch to Trigger Gamma-Ray Burst Jet Dissipationpubyear: 2011

## 1 Introduction

Gamma-ray bursts (GRBs) are thought to originate from core-collapse events or compact object mergers leading to magnetars or accreting black holes capable of launching ultrarelativistic jets. The prompt emission from standard cosmological long-duration GRBs has an energy of about over a few seconds that is beamed into a jet with an opening half-angle of a few degrees (Frail et al., 2001; Bloom et al., 2003). The prompt emission is typically presumed to occur in internal shocks (Sari & Piran, 1997). The internal shock model is reasonable because such shocks are expected in an unsteady outflow, which then has an observed variability on timescales related to the central engine. Within the collapsar (Woosley, 1993; Paczynski, 1998; MacFadyen & Woosley, 1999) or other GRB models, the observed variability may arise indirectly from activity near the central compact object, indirectly from entrainment driving propagation instabilities (Aloy et al., 2000; Zhang et al., 2004; Morsony et al., 2007; Wang et al., 2008; Bucciantini et al., 2008a), or directly from relativistic turbulence (Lyutikov & Blandford, 2003; Narayan & Kumar, 2009; Zhang et al., 2009),

However, the internal shock model has some unresolved problems. For example, highly relativistic relative motion between interacting shells is required in order to efficiently generate photons (Kobayashi & Sari, 2001; Maxham & Zhang, 2009) ; only a small fraction of electrons should be accelerated in order to obtain consistency with the observed peak energy (Shen & Zhang, 2009) ; internal shocks produce a steeper spectral slope than observed (Ghisellini et al., 2000; Asano & Terasawa, 2009) ; the afterglow energy would dominate the prompt energy as opposite to observed (Willingale et al., 2007) ; particle-in-cell (PIC) simulations show that shocks are dominated in energy by Maxwellian electrons whose emission may be inconsistent with the observed double power-law Band function (Giannios & Spitkovsky, 2009) ; and if GRB jets contain strong toroidal fields, then the shocks should inefficiently dissipate the kinetic energy (Kennel & Coroniti, 1984; Zhang & Kobayashi, 2005; Narayan et al., 2011) and inefficiently accelerate particles to high energies (Sironi & Spitkovsky, 2009). One alternative to the internal shock model includes dissipation from nuclear and Coulomb collisions (Beloborodov, 2010a).

In light of these issues with shock models and since GRB jets are expected to be highly magnetized, it is interesting to explore alternatives for which electromagnetic dissipation directly leads to acceleration and emission (Romanova & Lovelace, 1992; Thompson, 1994; Usov, 1994; Levinson & van Putten, 1997; Lyutikov & Blackman, 2001; Lyutikov, 2006b; Thompson, 2006). Such investigations are motivated by studies of pulsar winds (Kennel & Coroniti, 1984; Coroniti, 1990; Lyubarsky & Kirk, 2001; Kirk & Skjæraasen, 2003a; Nagata et al., 2008), field reversals in active galactic nuclei (AGN) jets (Lovelace et al., 1994, 1997), and magnetars (Thompson & Duncan, 1995; Lyutikov, 2003, 2006a). Electromagnetic dissipation mechanisms are appealing because they do not require the generation of highly relativistic relative motion between shells of matter or highly relativistic turbulent motion. Dissipation can instead proceed in situ. Indeed, high-resolution 3D magnetohydrodynamical (MHD) simulations of jet propagation show that toroidal magnetic fields shield jets against many of the shear instabilities noticed in purely hydrodynamical simulations (Keppens et al., 2009; Mignone et al., 2010), which suggests that it is difficult for a magnetized jet with a strong toroidal field to produce numerous shells moving at varying relativistic speeds as required by the internal shock model to generate temporal variability and efficient dissipation. Further, modern simulations of magnetized GRB jets suggest that even if relatively relativistic shells were generated, internal shocks would be unable to explain the high efficiency of prompt GRB emission (Narayan et al., 2011). This implies that local electromagnetic dissipation may be the only efficient dissipation mechanism possible in highly-magnetized jets.

Further, several observations point to a requirement of strong electromagnetic fields in GRB jets. The typical absence of a thermal photospheric emission component requires the jet to contain a significant low-entropy (e.g. magnetized) component, and the presence of non-pair-producing GeV photons requires emission to be at larger radii than expected in the internal shock model (Zhang & Pe’er, 2009). The typical non-detection by Fermi of a GeV spectrum excess due to synchrotron self-Comptonization (SSC) in prompt GRBs can be achieved by a highly magnetized jet since fewer electrons are required to support a strong electromagnetic field, as compared to the number of electrons required by the internal shock model that should generate an SSC component (Fan, 2009). Possible measurements of highly polarized gamma-rays from GRBs could require an ordered magnetic field in the emitting region (Lyutikov et al., 2003). The absence of bright optical flashes in the very early afterglow may require the reverse shock region to be somewhat magnetized (Mizuno et al., 2009; Mimica et al., 2009).

While electromagnetic fields are commonly understood to be dynamically important in the jets produced by GRB engines (Narayan et al., 1992; Thompson, 1994; Usov, 1994; Vlahakis & Königl, 2003; Lyutikov, 2006b; Uzdensky & MacFadyen, 2006, 2007), only recently have self-consistent magnetohydrodynamic (MHD) simulations been able to model the formation and large-scale axisymmetric structure of such jets (Komissarov et al., 2009; Lyubarsky, 2010a), including what happens beyond a stellar envelope in the collapsar model (Tchekhovskoy et al., 2010b). Such large-scale idealized MHD jet simulations are complemented by small-scale general relativistic MHD (GRMHD) axisymmetric and 3D simulations that consider the role of magneto-rotational instability (MRI) driven turbulence (Balbus & Hawley, 1991) and magnetic field geometries close to the black hole (McKinney, 2006b; Beckwith et al., 2008; McKinney & Blandford, 2009) and by realistic GRMHD simulations of the engine in the collapsar model (Mizuno et al., 2004; Barkov & Komissarov, 2008; Nagataki, 2009). Of relevance to GRBs is that a dipolar field appears required to launch a jet from the black hole (Beckwith et al., 2008; McKinney & Blandford, 2009), yet magnetic field advection from a presupernova core collapsing onto a black hole leads to too small of a magnetic flux to generate the power of a cosmological GRB (Komissarov & Barkov, 2009). The MRI must then be invoked (Akiyama et al., 2003), but that only generates small-scale (not dipolar) field. One solution is for a magnetic dynamo to generate quasi-periodic large-scale dipolar fields over tens of dynamical times as occurs for toroidal fields in accretion disks (Davis et al., 2010), but this necessarily implies one must understand electromagnetic dissipation at the interface between each successive flip of the jet’s dipolar field.

However, it remains difficult to understand how the electromagnetic field dissipates in GRB jets. The most advanced simulations of relativistic jets from collapsing stars forming black hole accretion flows use the MHD approximation (Komissarov & Barkov, 2009), so they cannot incorporate all the plasma effects observed in PIC simulations that are required to self-consistently study magnetic reconnection. Even the most advanced PIC simulations are inapplicable to GRB jets that require relativistic reconnection in the presence of pair creation and annihilation, neutrinos, photons, radiative cooling in both optically thick and thin regimes, relativistic compression, etc. There is still no coherent physical theory of relativistic reconnection (see, e.g., Blackman & Field 1994; Lyutikov & Uzdensky 2003; Lyubarsky 2005; Jaroschek & Hoshino 2009; and also section 5.5 of Uzdensky & MacFadyen 2006). Therefore, quasi-analytical approaches must be used to investigate the role of electromagnetic dissipation in GRB jets.

In prior quasi-analytical work, electromagnetic dissipation through magnetic reconnection in GRB jets has been suggested as a possible source of acceleration and emission at large radii from the central compact object (Thompson, 1994; Spruit et al., 2001; Drenkhahn & Spruit, 2002a; Lyutikov, 2006b; Giannios & Spruit, 2006). It was considered problematic for ideal MHD processes to efficiently accelerate and collimate jets, so non-ideal electromagnetic dissipation was favored to accelerate GRB jets (Thompson, 1994; Drenkhahn & Spruit, 2002a) or be a source of energy for emission in GRB jets (Thompson, 1994). Prior works also assume that the flow is already relativistic () and continues to speed-up with radius as based upon one-dimensional jet models. The dissipation rate was assumed to be Alfvénic or some fraction of it, which assumes a fast reconnection mode.

However, it is now known that a relativistic quasi-conical outflow (i.e. split-monopole near compact object and otherwise unconfined) accelerates efficiently near the polar axis in ideal MHD (Tchekhovskoy et al., 2009), and any sufficiently globally collimated outflow is efficiently accelerated (Komissarov et al., 2009; Tchekhovskoy et al., 2010b). This dramatically changes the behavior of the Lorentz factor as a function of radius compared to prior one-dimensional models (for a review, see Spruit 2010.) In addition, jets following ideal MHD acceleration near the compact object start with until reaching the Alfvén surface. So, if dissipation were always nearly Alfvénic, then (unabated by time dilation) fast reconnection could occur near the jet base. Reconnection near the jet base can severely quench the Blandford-Znajek effect (Beckwith et al., 2008; McKinney & Blandford, 2009), which may be required to drive the relativistic electromagnetic jet in the first place. Also, if reconnection were quite close to Alfvénic, then the jet could dissipate deep inside the photosphere leading to a hot fireball instead of an electromagnetic jet. This would lead to a dominant thermal spectrum, which is inconsistent with observations.

Prior quasi-analytical works have also investigated the potential role of the current-driven magnetic kink instability, which is assumed to operate rather efficiently on Alfvénic timescales (Spruit et al., 2001; Giannios & Spruit, 2006). However, modern stability analyses have shown that the kink instability operates somewhat slower than comoving Alfvén timescales for collimating flows (Narayan et al., 2009), and much of a relativistic quasi-conical flow can be out of causal contact with itself to eliminate the instability except within narrow regions right around the polar axis that contain little electromagnetic energy flux (Spruit et al., 2001; McKinney & Blandford, 2009).

Prior work also considered other aspects of electromagnetic dissipation. For example, a violation of ideal MHD occurs when the plasma density drops below the Goldreich-Julian density, and this leads to dissipation (Lyutikov & Blackman, 2001). However, for reasonable lower-limits on the baryon-loading (Levinson & Eichler, 2003), such a violation of ideal MHD only occurs at (Spruit et al., 2001). Reconnecting layers in accelerated jets have been found to be potentially unstable to magnetic interchange instabilities leading to enhanced reconnection (Lyubarsky, 2010b), although they considered a sharp pressure boundary instead of a distributed radiative photosphere as present in GRB jets. Turbulence-driven electromagnetic dissipation from magnetic instabilities induced by internal shocks has also been recently considered (Zhang & Yan, 2010), although the conditions for turbulence in highly magnetized jets remains undetermined.

One aspect of electromagnetic dissipation not investigated so far is the role of critical transitions between collisional and collisionless reconnection and its association with, respectively, rather slow resistive-MHD reconnection (perhaps as slow as Sweet-Parker reconnection; Sweet 1958; Parker 1957) and fast Petschek or Petschek-like reconnection (Petschek, 1964). (For a review, see Priest & Forbes 2000.)

Prior simulations and laboratory experiments show that collisional plasmas avoid the fast Petschek type reconnection in favor of the slow Sweet-Parker type reconnection. For collisional plasmas the Petschek reconnection configuration promptly collapses to a thin Sweet-Parker layer (Uzdensky & Kulsrud, 1998, 2000). No fast (inflow velocity of order the Alfvén velocity) Petschek reconnection is seen to occur in collisional plasmas (Biskamp, 1986a; Ma & Bhattacharjee, 1996; Ji et al., 1998; Uzdensky & Kulsrud, 2000; Breslau & Jardin, 2003; Malyshkin et al., 2005; Kulsrud, 2001).

On the other hand, a collisionless plasma allows for non-ideal MHD effects that can force a plasma into a fast Petschek-like reconnection regime occurring on the ion skin depth scale and operating independently from the classical resistivity. (For a review, see Kirk et al. 1994; Kulsrud 2005; Zweibel & Yamada 2009.) Simulations of pure pair plasmas also show fast reconnection but on the electron skin depth (instead of the ion skin depth) as perhaps due to the electron pressure effect (Bessho & Bhattacharjee, 2007). Petschek reconnection plays an important role in the Sun, where nanoflares remain the most plausible source of coronal heating (Parker, 1988a; Klimchuk et al., 2009). In fact, the solar corona may even exist in a balanced marginally-collisionless state governed by the density-controlled transitions between these two modes of reconnection (Uzdensky, 2007c, a). There is also observational evidence that the transitions between the collisional and collisionless reconnection regimes may also control coronal heating in black-hole accretion disks (Goodman & Uzdensky, 2008). Solar and stellar flares could also be triggered by these critical transitions (Cassak et al., 2006, 2008). Therefore, it is important to determine whether a reconnecting system is collisional or collisionless in order to determine whether current sheets operate in the fast reconnection mode (Malyshkin et al., 2005; Cassak et al., 2005; Yamada et al., 2006; Uzdensky, 2007c, a; Bessho & Bhattacharjee, 2007).

There do exist purely resistive-MHD alternatives to fast collisionless Petschek reconnection, including due to externally driven MHD turbulence (Lazarian & Vishniac, 1999; Kowal et al., 2009b; Loureiro et al., 2009) and plasmoid-dominated reconnection (Loureiro et al., 2007; Kowal et al., 2009a; Samtaney et al., 2009; Cassak et al., 2009; Huang & Bhattacharjee, 2010; Uzdensky et al., 2010). However, relativistic Poynting-flux dominated jets are strongly electromagnetically-dominated and relativistically expanding flows, and hence they may only be susceptible to turbulence or similar cascades in regions that remain in causal contact across the jet – such as very close to the polar axis where there is little total electromagnetic energy and so little possible emission. These alternatives require further consideration in such relativistic regimes.

The present paper builds upon three novel key elements. The first key element of this work is that we consider the role of transitions between slow collisional and fast collisionless reconnection. The fact that the reconnection rate depends upon the collisionality of the plasma is particularly important for GRB jets that are collisional at the jet base and out to large radii. We compute the properties of the collisional and collisionless reconnecting layers to determine which reconnection mode dominates at each radius and angle within the GRB jet. Once collisionless reconnection is initiated at some “transition radius,” reconnection proceeds at a rate much faster than the collisional rate and can initiate prompt GRB dissipation and emission.

The second key element of this work is that the collisional layer’s properties are determined as due to collisions among species of ions, photons, electrons, positrons, and neutrinos. The current layer structure is treated similarly as done for radiative accretion disks (see, e.g., Kohri et al. 2005). We also make use of relativistic reconnection work by Lyubarsky (2005), which shows that the relativistic Sweet-Parker layer has a reconnection rate as expected from non-relativistic theory. We incorporate radiative effects into the reconnection switch model by following Uzdensky & McKinney (2011), who showed that the current layer compresses in the strong cooling limit of non-relativistic radiative reconnection.

The third key element of this work is that the large-scale jet is modelled using the latest ideal MHD models of ultrarelativistic jets rather than assuming inefficiently accelerated one-dimensional flows. We also extend the study of Spruit et al. (2001) and determine a range of possible ways for field polarity reversals to occur that lead to current sheets in the substructure of the jet.

A basic version of the overall argument of the paper is provided in §2. The GRB jet structure is described in §3, some possible field substructures are described in §4, the reconnection models are described in §5, results for GRBs and other jet systems are described in §6, a discussion is in §7, and conclusions are provided in §8. In Appendix A, the generalized full jet structure solution that is used throughout the paper is obtained. In Appendix B, the equation of state for all species within the radiative current layer is presented. In Appendix C, the collisional and collisionless reconnection models are discussed. We assume a flat space-time in spherical polar coordinates (), an orthonormal basis for all vectors, and Gaussian-cgs-Kelvin-radian units.

## 2 Basic Argument

In this section, a basic argument is presented to demonstrate the existence of a reconnection switch mechanism that leads to dissipation near the GRB jet photosphere.

Figure 1 shows a basic picture of the reconnection switch model for a jet containing multiple current sheets each corresponding to a layer wherein oppositely-directed magnetic field lines are dissipated. Near the central engine, dissipation proceeds via slow collisional reconnection due to the high collisional rate. This allows the electromagnetic field to avoid significant dissipation despite the presence of current sheets, and this allows the generation of a baryon-pure strong electromagnetic field for launching an ultrarelativistic jet. At large radii, pairs annihilate and the densities go down leading to infrequent collisions. Collisionless plasma effects (e.g. due to electron-proton or electron-positron decoupling) can then initiate the much faster collisionless reconnection mode (operating in a Petschek-like geometry) that disrupts the slow collisional mode (operating in a Sweet-Parker-like geometry) (Cassak et al., 2005). This leads to a reconnection switch that triggers jet dissipation, which (as shown below) initiates prompt emission near the GRB jet photosphere. As mentioned in the introduction, the concept of a reconnection switch has been applied to many astrophysical phenomena, and in this paper we simply apply the same concept to GRB jets.

The relevant length scale for fast collisionless reconnection in baryon-dominated plasmas is the proton skin depth

 dp=cωpp, (1)

for proton plasma frequency , proton number density , charge , proton mass , and speed of light . The relevant scale for pair-dominated plasmas is the pair skin depth

 de=cωpe, (2)

with pair plasma frequency , electron+pair number density , and electron mass .

Collisions ensure that resistive MHD applies, which forces the current layer to avoid the fast Petschek-like geometry in favor of a Sweet-Parker-like geometry (see., e.g., Uzdensky & Kulsrud 2000). The Sweet-Parker solution requires pressure equilibrium across the current sheet. Assume, as valid for most of this paper, that the reconnecting field is not significantly weaker than the guide field. Then, the Sweet-Parker solution without a guide field can be used to estimate any quantities to order unity. The electromagnetic pressure (, where is the electromagnetic energy density) and thermal gas pressure (which, as borne out in this paper, is dominated by photon pressure , where is the photon energy density) balance via

 pEM∼pγ. (3)

Next, assume, as also borne out in this paper, that the collisional resistivity against current-carrying electrons and positrons is dominated by Compton scattering, then the corresponding magnetic diffusivity is

 η≈(4/3)d2e(uγσTc)/(mec2), (4)

(Goodman & Uzdensky, 2008), for Thomson scattering cross section . Then, for electromagnetically-dominated jets with relativistic Alfvén speeds, the collisional Sweet-Parker current sheet thickness is

 δSP∼√L0ηc, (5)

where is the comoving length of the current layer as estimated by the scale for variations in the electromagnetic field.

The fast Petschek-like collisionless reconnection mode takes over when for baryonic-dominated plasmas, as discussed above. Using the above equations, the condition for fast collisionless reconnection is then given by

 1≲(dpδSP)2∼(1+npairs/ne)28τγ,sca~μ, (6)

where is the electromagnetic energy per baryon rest-mass energy in the jet, is the baryon rest-mass density for a proton number density assuming protons and neutrons are at equal number densities, is the baryonic-associated electron density from charge neutrality, is the scattering optical depth, and corresponds to the density scale-height for photons, electrons, and positrons for an emitting slab of length (Popham & Narayan, 1995). For a pair-dominated plasma in the limit that baryons play no role, the condition for fast reconnection becomes

 1≲(deδSP)2∼(memp)(1+npairs/ne)8τγ,sca~μ. (7)

Future studies can clarify how fast reconnection operates in a plasma with baryons that typically dominate the rest-mass energy even if pairs have a non-negligible number density.

Assume , then the condition for transition to fast reconnection simply becomes

 τγ,sca<(8~μ)−1. (8)

Because MHD jets for GRBs have at large radii (Tchekhovskoy et al., 2010b), fast reconnection in GRB jets is predicted to occur near the photosphere. Such a dissipation is required by dissipative photosphere models (see, e.g., Rees & Mészáros 2005). Further, once fast dissipation starts, the value of drops, which forces the fast collisionless reconnection condition to be maintained. Even if fast reconnection were relativistic with speed order , dissipation is still suspended until large radii so that a baryon-pure ultrarelativistic jet can form.

The rest of this paper computes the radius where the reconnection switch occurs, the photon optical depth at this radius to determine whether quasi-thermal photospheric emission is possible, and some other observables. Realistic MHD GRB jet models are used since the polar jet’s ideal MHD acceleration is substantially different than considered in prior reconnection jet models. The important pair contribution is non-trivial to compute because near the photosphere the photons and pairs are only marginally optically thick. Also, the plasma temperature near the transition to fast reconnection ends up low enough that pairs are deep in the suppressed regime. A simple exponential suppression factor would lead to more than an order of magnitude error in the transition radius, so pairs must be treated more accurately. Despite pairs being in the suppressed regime, the pair number density can greatly exceed the baryonic-associated electron number density and so crucially affect the transition radius as seen from Equation (6). We must also consider neutrino cooling near the compact object, because strong cooling can lead to an effective drop in (Uzdensky & McKinney, 2011) and so trigger fast reconnection causing the jet to dissipate before it is launched. Overall, this requires us to compute the properties of a jet filled with a complex of multiple reconnecting slabs, each consisting of baryons with electrons, pairs, and neutrinos at arbitrary optical depths. This is achieved by treating the current layer structure as similarly done for radiative accretion disks (see, e.g., Kohri et al. 2005).

## 3 Large-Scale Jet Structure

In this section, the large-scale axisymmetric structure of relativistic GRB jets from black holes or magnetars is presented. The large-scale jet structure acts as a key constraint on the reconnection physics by fixing the radial dependence of magnetic pressure. During reconnection, the magnetic pressure balances the thermal pressure within current layers, which determines the contributions of baryons, photons, pairs, and neutrinos.

Figure 2 shows the basic elements of the central engine and the production of an axisymmetric jet. The jet could be confined laterally (up to a radius ) by an accretion disk, disk corona, disk wind, stellar envelope, or some ambient medium. A generalized jet solution that applies for both small and large radii is obtained in Appendix A, which collects together results from recent analytical works and simulations (Tchekhovskoy et al., 2008, 2009, 2010b). However, the full solution can be cumbersome for obtaining simple scaling laws for how the results depend upon model parameters.

Therefore, this section outlines a simplified jet structure that approximates the full jet structure model and helps to highlight the basic elements of the full solution. The radius is assumed to be much larger than the size of the compact object (i.e. ), at which a field line attaches at a footpoint radius () and at a footpoint angle () across the surface of the compact object (see Figure 2). The radius is also assumed to be larger than the deconfinement radius (i.e. ). The following expressions apply for with an assumed equatorial symmetry.

The degree of collimation inside is given by a parameter , such that for angles (i.e. cylindrical radius ) one can show that the field lines obey

 rrfp≈θ−2/νf(2sinθfp2)2/ν, (9)

where the field collimates with out to after which it follows the type radial geometry on a particular field line with fixed opening half-angle of giving

 θf≈(rmonorfp)−ν/2(2sinθfp2), (10)

where the total jet opening angle () corresponds to the value of for the largest value of that is allowed by the presence of a confining medium near the compact object.

For , the radial field strength is

 Br≈Br,fp(rmonorfp)ν−2(rrmono)−2. (11)

The -component is small and so can be neglected. The toroidal -component is

 Bϕ≈Br,fp(−2rfpΩFc)(rmonorfp)ν−1(rrmono)−1tan(θf/2), (12)

where for rapidly rotating black holes or neutron stars and is the field rotation frequency one can set at each foot point. For a black hole angular rotation rate of with horizon radius and dimensionless black hole spin , the value of the field line angular rotation rate is . For Lorentz factor of , one can show that , which allows one to obtain the electromagnetic pressure and energy density via , where is the comoving field strength. The baryonic rest-mass density is

 ρb∼ρb,fp(BrBr,fp)(1γ), (13)

where is some estimate of the Lorentz factor for . We define a magnetization parameter as

 ζ≡B2r,fp8πρb,fpc2, (14)

which is used to define from . Near the compact object, the value of is similar to the electromagnetic energy per particle.

Overall, for a given , , , , and an estimate of , one can determine the radial and angular dependence of , , , and , and other quantities defined in Appendix A; such as the electromagnetic energy per unit rest-mass-energy (), the electromagnetic energy flux per unit rest-mass flux (), the electromagnetic energy flux per unit mass-energy flux (), and the jet power (). This MHD jet solution only has to be applicable up to the radius where significant dissipation occurs, because determining the dissipation radius is the primary goal of this work.

For all models, we choose a rapidly rotating black hole foot point radius of ( is black hole mass) with a rapid rotation rate such that . The collapsar case with a black hole of mass is chosen. For the collapsar model is typically chosen (i.e. radius of the progenitor star), while for the short-duration GRB compact object merger model is chosen (i.e. extent of newly formed disk-corona-wind that helps collimate the jet ; Tchekhovskoy et al. 2008, 2010b).

A black hole is chosen instead of a neutron star because it is more likely to be a generator of a powerful baryon-pure jet. First, the black hole cleans magnetic field lines of mass (MacDonald & Thorne, 1982; Levinson & Eichler, 1993), so the magnetization can be quite high in the black hole case. The magnetization could be limited by neutron diffusion (Levinson & Eichler, 2003), which still leads to quite high magnetizations of (McKinney, 2005a) (corresponding to ). Second, a black hole can produce a more powerful jet since its Kerr parameter is . For the magnetar case, the outflow only becomes highly magnetized at late time when the power has significantly diminished, and neutron stars have a Kerr parameter only up to before break-up. These issues make it potentially more difficult for the magnetar to operate as both an efficient and powerful engine of a highly magnetized ultrarelativistic jet (Metzger et al., 2007; Bucciantini et al., 2008b; Metzger et al., 2010).

One must use Appendix A to obtain an accurate dependence for all MHD quantities (i.e. including ) within the jet as a function of radius and angle. Consider typical collapsar model parameters that would lead to an ultrarelativistic jet with . If , G, cm, and , then , , , , and . At there is about times less kinetic energy than electromagnetic energy, which can be tapped for prompt GRB emission if there exists some mechanism to dissipate the energy.

## 4 Jet Field Substructure: Generalized Striped Wind

In this section, we consider the process whereby small-scale field reversals and current sheets become embedded within the large-scale jet structure. The comoving length scale, , of these jet field substructures plays a prominent role in later calculations because the dissipation rate due to collisional reconnection is dominated by the smallest value of .

Figure 3 shows the different types of field geometries that are considered. All of these substructure types probably coexist to some degree. The jet structure solution presented in section 3 was formally constructed with a single polarity for the electromagnetic field where the return field is presumed to connect back to the region slightly beyond the compact object. However, the polarity of the field can have arbitrary reversals without changing the large-scale force balance of the jet as long as the current sheets are inserted so that only the field direction changes. For an assumed field perturbation of approximately with cylindrical radius for the entire jet, Maxwell’s equations give an overall smallest characteristic electromagnetic field length scale of . The comoving length scale () generally depends upon the substructure type.

Substructure type A corresponds to a time-dependent alternating polarity for a dipolar field near the compact object. The injection of field reversals occurs at an angular rate of near the compact object for some spherical harmonic quantum number . Such an injection can occur when the compact object’s rotation axis is misaligned with the magnetic field axis as in the standard striped wind (Coroniti, 1990; Thompson, 1994; Spruit et al., 2001; Palenzuela et al., 2010), a neuron star’s internal large-scale poloidal field dynamo flips the sign of the field, the compact object is fed alternating polarity from the accretion disk (Lovelace et al., 1997), or a large-scale poloidal field accretion disk dynamo determines the field near the black hole. Values of are allowed since a magnetic dynamo can operate on timescales much longer than the dynamical timescale. For example, an accretion disk could have (Davis et al., 2010).

Substructure type B corresponds to a time-independent non-axisymmetric () multipolar field threading the compact object or accretion disk (Akiyama et al., 2003). The length scale of the electromagnetic field is determined by reversals at different foot point angles in the -direction near the compact object, and so type B is also related to the angular rate of .

At large distances, substructure types A and B both have field reversals that are radially stacked such that the lab-frame . In the lab-frame, the pattern of reversals in is advected to large radii at roughly the speed of light, so that the comoving scale is (Spruit et al., 2001). There may be many “stripes” separated radially by .

Substructure type C corresponds to a time-independent axisymmetric (e.g. spherical harmonic corresponding to dipolar, quadrupolar, etc.) multipolar field threading the compact object or disk. Notice that cannot be zero because jet formation requires a non-zero axial field at the jet base. The scale of the electromagnetic field is determined by reversals at different foot point angles in the -direction near the compact object. The field reversal scale is . The scale of changes in the field is perpendicular to the flow direction so that . If , then the jet becomes causally disconnected across in due to finite wave speeds for any finite magnetization. The causally connected region has a size , so physical models effectively have when using the original formula. The model described in figure 2 of Romanova & Lovelace (1992) corresponds to this substructure type C with and with a prescribed structure in . Another example of type C is a magnetic tower model, where there is a single large current sheet that separates two regions (Lynden-Bell, 1996; Uzdensky & MacFadyen, 2006, 2007). This substructure is qualitatively different than substructure types A and B because while the other substructures do not depend upon once is roughly constant.

Substructure type D corresponds to an ordered dipolar field at the jet base, but the jet becomes unstable at large radii. Such instabilities could be due to intrinsic jet properties leading to the, e.g., current-driven magnetic kink instability or due to turbulence at boundary layers such as the jet-wind or jet-envelope boundary. For example, in the relativistically magnetized limit of MHD jets, if the comoving toroidal field exceeds the comoving poloidal field by factors of a few to tens, then the growth rate for the kink instability is generating structure from to (Narayan et al., 2009). Such kink modes would be stabilized by jet expansion or finite mass loading leading to causal disconnection across much of the jet except within (Spruit et al., 2001). In general, the scale of the perturbations could be randomly oriented across and generated on a time-scale of in the lab-frame. Hence, this substructure corresponds to some mixture of substructures A-C for and within .

Figure 4 shows the jet substructures at large radii. Types A and B are similar, but in detail they differ. For types A-C, at all radii (including near the central compact object), the comoving poloidal field is at most order unity comparable in magnitude to the comoving toroidal field. For types A-C, the toroidal field dominates in the lab and comoving frames for . Generally, substructure D might correspond to having a strong poloidal field. However, magnetic kink driven modes are expected to typically leave the jet’s comoving toroidal and poloidal fields as comparable (i.e. when the so-called “safety factor” is of order unity in comoving frame), because that is when the instability is initiated. Overall, one concludes that any guide field (which does not reconnect) in the current sheet is not much stronger than the reconnecting field. This means the analysis in section 2 and in later sections that assume a “weak” guide field are accurate to order unity or better, because the Sweet-Parker analysis is only significantly modified if the guide field is significantly stronger than the reconnecting field.

In summary, the jet can develop various substructures that have comoving scale satisfying

 1Δ0(l,m)∼lπRj+mΩFγ2πc, (15)

for some real numbers and . Only dynamo and jet-medium interaction computations can determine (the power spectrum of) , but many dynamos and instabilities saturate at low . The length scale () over which the comoving electromagnetic field changes is a natural scale over which any dissipation term enters into the comoving induction equation, and so plays a central role in determining the comoving physics of reconnection discussed in the next section. At large radii, these modes correspond to the jet having current sheets with a guide field that is not significantly stronger than the reconnecting field, which allows one to estimate (to order unity) the properties of the collisional layer using the Sweet-Parker analysis without a guide field.

## 5 Reconnection Physics

In this section, the conditions for a plasma to undergo collisional or collisionless reconnection in the comoving frame of the jet are computed. As described in section 4, the characteristic length scale of the electromagnetic field (before reconnection occurs) is given by . As discussed below, such current distributions can be unstable to collapse to a much thinner size . The jet is assumed to be filled with a complex of multiple current layers each of thickness . Instead of seeking a solution that is in global force balance within the jet, all fluid quantities ( in the background jet solution) are assumed to be roughly homogeneous within the current layer at any given radius.

In section 5.1, the formation and length of current sheets is discussed, because the length enters into models of slow and fast reconnection. In section 5.2, the equation of state and emission rates for the current layer are summarized (see Appendix B for details). In section 5.3, the transition from slow to fast reconnection is discussed (see Appendix C for more details). In section 5.4, how the electromagnetic field is modified by fast dissipation is computed. Finally, in section 5.5, the timescales for observed dissipation/emission variability are estimated.

### 5.1 Layer Formation and Length

First, we must establish that current sheet formation is possible on timescales shorter than the jet propagation timescale to distances relevant for prompt GRB emission. Current layer formation starts with a distributed current of size that spans a null point in the magnetic field strength. Such current distributions can collapse to current sheets on fast Alfvén crossing time scales through a variety of mechanisms: 1) exponential collapse of the distributed current until the thermalized gas pressure within a thickness can support the external electromagnetic pressure (Kulsrud, 2005) ; 2) collapse as part of the evolution of the global magnetic field as occurs for the equatorial current sheet in a pulsar wind ; or 3) intrinsically unstable collapse at an X-point (Dungey, 1953, 1958; Imshennik & Syrovatskiǐ, 1967; Waelbroeck, 1993; Loureiro et al., 2005). For example, X-type collapse forms current sheets on a timescale

 tr∼Δ02vAlnS, (16)

(Priest & Forbes, 1986), where is the Alfvén speed outside the newly forming current layer, , is the upstream comoving electromagnetic field strength, is the upstream gas internal energy, and is the upstream gas pressure. The relevant controlling parameter is the global Lundquist number, , describing the ratio of the resistive time ( for resistivity ) to the typical electromagnetic advection time (). Typically and so ideal MHD is valid on scales of order . The above timescale shows that current sheet formation only depends logarithmically on the resistivity. For rapidly rotating black holes generating jets with , half-opening angle , substructure types A or B with , and , current sheet formation occurs by cm as comparable with the expected radius of prompt GRB emission. Thus, current sheet formation is a relatively fast process that can occur during jet propagation.

Second, the length of the current layer is needed because this is what enters the slow and fast reconnection models. The length of the current layer in the comoving frame () can be estimated from the characteristic reversal length scale () of the electromagnetic field, because this is the natural length scale that enters the MHD equations. The X-type collapse described above can feed off of perturbations introduced initially in the jet, producing structures with (Priest & Forbes, 1986). In addition, the presence of a complex of multiple current layers can itself induce structures with (Otto & Birk, 1992; Yan et al., 1994). Ideal MHD type processes such as magnetic Rayleigh-Taylor instability can also produce structure along the layer with (Lyubarsky, 2010a). Even in the absence of any current layers (i.e. and ), the jet still has a fundamental characteristic scale of .

These arguments suggest that the length of the current layer should be within the range . Because is represented by and for most of the paper never appears explicitly independent from , for simplicity, the allowed range for is subsumed as an allowed variation in and . With the length of the layer determined, one determine the thickness of the slow collisional reconnection Sweet-Parker layer in section 5.3.

### 5.2 Current Layer Pressure, Temperature, Density, and Emission Rates

The reconnection physics is determined not only by the externally imposed magnetic field, rest-mass density, internal energy density, field reversal scale, and length of the current layer as described in the previous sections. In addition, the reconnection physics is determined by the form of energy that the magnetic energy is dissipated into.

The solution for the multi-species equation of state (EOS) and emission rates are described in Appendix B, which follows similar physics used for radiative accretion disks. In summary, the gas EOS consisting of baryons, photons, non-pair-produced electrons, pair-produced electrons-positrons, and neutrinos is given by the total internal energy density

 ug=ub+uγ+ue+upairs+uν, (17)

and total thermal gas pressure

 pg=pb+pγ+pe+ppairs+pν, (18)

where each species is listed respectively and is a function of temperature () and baryon density ().

Given the proton number density (), the baryon number density (), charge neutrality such that the electron number density is , then the electron fraction is . A value of is assumed (see Appendix B). Also, , where and are the total electron and total positron number densities, respectively. Let the total electron+positron number density be , then the pair-only number density is .

The reconnecting layer is treated as a radiating slab, with a total cooling rate () for photons, pairs, and neutrinos of

 Qg=Qγ+Qpairs+Qν, (19)

for each species respectively. The respective optical depths in the jet are given by , , and , which are separated into scattering and absorption optical depths (e.g., and , respectively). A non-negligible occurs when as allowed when pairs leave the current layer by travelling along field lines.

### 5.3 Transition from Slow Collisional to Fast Collisionless Reconnection

In this section, we discuss how the current layers within the GRB jet make a transition from collisional to collisionless reconnection, which triggers the Petschek-like reconnection geometry leading to fast reconnection. Then, dissipation and emission from fast reconnection can contribute significantly to prompt GRB emission.

When the collisionless reconnection layer thickness is larger than the Sweet-Parker thickness (Sweet, 1958; Parker, 1957), the layer enters a Petschek-like reconnection geometry (Petschek, 1964) allowing for fast reconnection (see., e.g., Ma & Bhattacharjee 1996; Uzdensky & Kulsrud 2000; Kulsrud 2001; Cassak et al. 2005; Yamada et al. 2006). For current layers where plasma , the condition that the plasma no longer obeys the resistive MHD equations on the scale of the current layer can be shown to be equivalent to

 δSP′

where is the ion plasma frequency (Kulsrud, 2005). Here, , where is the nominal Sweet-Parker thickness that is determined using a balance of gas pressure () and electromagnetic pressure () across the layer by following the Sweet-Parker analysis. The compression ratio (where is the dissipation rate in the Sweet-Parker layer) accounts for radiative cooling in the layer (Uzdensky & McKinney, 2011). Typically, is found in this paper. The value of the Lundquist number, , is obtained from Spitzer and Compton drag resistivities. For more details see Appendix C.

When the above condition is satisfied, the reconnection operates in a fast collisionless mode. The transition radius to fast collisionless reconnection is then given by

 rtrans=r[δSP′=di]. (21)

This transition radius is used to identify where fast magnetic reconnection starts and so leads to dissipation and emission.

One interesting caveat is that once collisionless reconnection is initiated, the higher dissipation rate may lead to different plasma parameters such as the temperature. The loss of magnetic energy may raise the local temperature in the optically thick limit, but it can lower the temperature at slightly larger radii if the flow is already optically thin because a loss of magnetic energy leads to lower gas pressures. If resistivity increases due to dissipation, then the Sweet-Parker thickness increases and may lead to recovery of the collisional mode of reconnection. In such a situation, the plasma may enter into a “marginally collisionless” reconnection mode in which is maintained as the plasma dissipates and continues to enter and exit the fast and slow reconnection modes. It is difficult to estimate the average rate of reconnection in such a scenario (Uzdensky, 2007b; Goodman & Uzdensky, 2008). However, in some studies, the transition back to slow reconnection was shown to be avoided by a hysteresis effect (Cassak et al., 2005).

Slow collisional reconnection can be as slow as Sweet-Parker, giving an inflow velocity of , or it may be as fast as (Uzdensky et al., 2010). Various studies have found that the fast collisionless reconnection rate is of order or even (for details see Appendix C.4). For simplicity and definiteness, the reconnection rate is chosen to be in all cases. The exact number used for only moves the expected dissipation radius and does not significantly affect the transition radius or any other calculations. Once significant dissipation has occurred, the value of drops below , but in all cases we will find it only drops to by the time the slow collisional mode would recover. Also, since dropping below indicates most of the electromagnetic energy flux has been dissipated (i.e. the goal of this work), we simply avoid detailed discussions about the solution at large radii where this occurs.

### 5.4 Effect of Magnetic Reconnection on the Jet Structure

In this section, we determine how the jet electromagnetic energy density, , vs. is modified by fast dissipation once the transition to fast reconnection has been triggered. The resulting function for vs. radius modifies the original large-scale jet structure dependence given in section 3 and Appendix A. The radial extent over which significant dissipation occurs also determines the photon/pair/neutrino opacity integrals used to determine the equation of state and emission rates for the radiative current layer in Appendix B.3.

The electromagnetic energy dissipated in the jet is given by the electromagnetic enthalpy flux into both sides of a current layer where each side harbors a region of size from which magnetic flux can be accumulated for each sheet. The comoving electromagnetic dissipation energy density rate is then

 −QEM≡ddtco(b28π)=−2b24πvrΔ0, (22)

where is the comoving rate of reconnection, corresponding to the inflow of magnetic flux into the current layer. Notice that the relevant length scale is , the distance between layers instead of the length of the layer, because magnetic flux must be brought in from the space between layers.

For the jet structure, the relativistic MHD energy-momentum equations of motion are dominated by the radial electromagnetic field advection term111 This calculation can be compared to that in Lyubarsky & Kirk (2001); Kirk & Skjæraasen (2003b), and also compared with Drenkhahn (2002) who made some simplifications to obtain their equation 31 giving their equation 37. This simplification was also used to obtain equation 1 in Drenkhahn & Spruit (2002b). Drenkhahn et al. concluded that no more than of the electromagnetic energy flux can be converted to optically thin radiation and that the rest goes into kinetic energy flux due to jet acceleration. However, no works have yet fully included radiative cooling that (in the optically thin regime) can remove thermal energy on a timescale comparable to the timescale for heating and jet acceleration. For simplicity, we assume dissipation only leads to radiation (as might occur in a fully radiative calculation) but does not lead to jet acceleration. This issue does not affect any of our interpretations of results. for either an electromagnetically-dominated jet with or for and roughly constant as occurs for (as applicable to all astrophysically relevant models studied in this paper). Then, Equation (22) is a sink term to the quasi-steady relativistic MHD energy equation given by

 0=−(1r2sinθ)∂r(r2sinθb2γ2c)−4πQEMγ. (23)

For (i.e. ) corresponding to the ideal MHD jet solution, the comoving electromagnetic energy density is a power-law dependence with . In general, Equation (23) can be solved for to give

 b2(r)b2(r0)=exp(−∫rr0dr[−2r−2vrcγΔ0]), (24)

which assumes as relevant for electromagnetically-dominated jets. Once , significant dissipation (i.e. the goal of this work) has already occurred, and then we simply avoid detailed discussions about the overall solution for such radii rather than seeking the correct vs. in the weakly magnetized regime.

For modes, the value of is fixed (i.e. does not scale with for ), and then the jet solution has an exponential term such that,

 b2b2[r=rtrans]=(rrtrans)−2exp(−(r−rtrans)(2vrcγΔ0)), (25)

for a radius of transition to fast reconnection and .

On the other hand, for modes , and then the dissipation does not lead to exponential decay and instead leads to a modified power-law given by

 b2b2[r=rtrans]=(rrtrans)−2(1+lvrπcγsinθf), (26)

for . In general, one obtains for a fraction of magnetic energy dissipated (e.g. for , , , and ). A simple conclusion would be that negligible change in occurs for large enough as due to a given -mode becoming causally disconnected across the jet. However, the modes generate plasmoids with radial size that may remain fixed in size once significant dissipation occurs. This alternative assumes, perhaps optimistically, that for modes the magnetic flux can be accumulated and dissipated from all directions instead of only in the direction. So instead of assuming some modes fail to dissipate, an alternative is also considered that the modes produce plasmoids similar to substructures A and B except with an effective whose value is fixed with radius as evaluated at the transition radius.

Once fast reconnection starts at , then Equation (25) shows that reconnection completes a single e-folding of dissipation/emission by

 rdiss≈rtrans+γcΔ02vr, (27)

assuming modes or the optimistic case for modes. For example, consider a model with , , a rapidly rotating BH such that , , and . This model gives , which results in a reasonable radius for prompt GRB emission for small enough . For , most of the dissipation has completed and leaves relatively few photons or pairs. So, this radial extent can be used to determine the opacity integral for photons, pairs, and neutrinos within a relativistic jet as done in Appendix B.3.

Between the transition and dissipation radii most of the jet’s electromagnetic power, , is lost as radiation from plasma blobs moving at Lorentz factor being observed at some angle away from the jet axis. For a lab-frame radial dissipation range of , lab-frame cross-sectional area , and lab-frame dissipation rate , the total isotropic jet luminosity is

 Lj∼(γQEM)(γcΔ0/vr)(πR2j)∼γ2b2cR2j∼P(EM)j≈Pj, (28)

which shows that the lab-frame loss rate of electromagnetic energy equals the lab-frame electromagnetic jet power. This also shows that the reconnection rate, , does not determine the luminosity from the region . Instead, a larger number of layers within the range dissipate simultaneously for smaller , so a fixed jet power leads to a fixed total isotropic luminosity.

### 5.5 Variability from Fast Reconnecting Layers

In this section, we compute various timescales from reconnecting layers that may be imprinted in observations of the prompt GRB emission. The jet substructure type determines the temporal structure of the prompt GRB emission via the number of dissipating layers. Also, emission timescales can be linked to the total event timescale, the event timescale per unit number of reconnecting current layers embedded in the jet, the dissipation timescale of a single current layer, and the thermal photon radiation emission timescale.

Consider a reconnecting layer (plasma blob) that begins to dissipate at a transition radius with a comoving dissipation timescale of . This is related to the lab-frame timescale via . Photons emitted at the start of fast reconnection travel outward as the plasma blob also travels outward. The observed photon time difference from beginning to end of current sheet dissipation is , such that the observed photon timescale accounting for light-travel effects is .

For the modes and for the optimistic case for modes, the comoving dissipation timescale is , corresponding to an observed timescale of

 dtobs,diss=Δ08γvr. (29)

Each layer present in the jet over the duration of the event gives one expected emission timescale. For substructure types A and B the lab-frame cycle time to produce a single slab is . Therefore, one expects a variability timescale of

 dtobs,m∼2πmΩF, (30)

associated with each shell that emerges from . For substructure type C, there are simultaneous emitting layers that in principle could dissipate stochastically in time rather than going off simultaneously. In that case, for an event duration , one expects a variability timescale of

 dtobs,l∼Tl. (31)

Reconnection is expected to proceed stochastically, but it will also have a short timescale associated with the transit time of fluid through a single reconnection layer of exhaust length . The comoving transit time is order , such that each reconnection event has an observed time scale of

 dtobs,tra=L02γvA. (32)

The thermal photon radiation emission timescale is

 dtobs,γ∼uγ2γQγ, (33)

corresponding to the timescale over which the thermal radiation contributes fully to its component of force balance against the magnetic pressure in the current layer complex.

For scale-height and mean free path of , the photon diffusion timescale is

 dtobs,d=12γmax[3H2λc,Hc], (34)

where for one has , the optical depth. The observer absorption diffusion timescale is denoted when setting , while the observer scattering diffusion timescale is denoted when setting .

## 6 Results: Jet Dissipation

This section computes several results obtained with the procedure:

• Given the independent variables , , , , and and the jet structure from section 3 (specifically, using Appendix A), one obtains , , , , and .

• Given , and and the jet substructure from section 4, one obtains the single current layer spacing of and layer length of at any point in the jet.

• The multiple current layer complex equations determined in section 5, with the non-ideal MHD jet solution for (equations 25 and 26) used after , are iteratively solved as a set of simultaneous equations for the temperature , pair number density , and the compression ratio . A single (unique) solution is always found.

• Given , , and , remaining quantities are iteratively computed.

The solution at each radius is obtained independently222Each solution takes hours to compute on a modern Xeon core, where over all models about CPU-hours are required.. The transit time through the current layer is roughly the equilibrium timescale, which is sufficiently small compared to the jet flow time for this to be a valid approximation except for radii much larger than the transition or dissipation radii.

In what follows, a fiducial model is discussed in detail in section 6.1, a model parameter exploration is presented in section 6.2, the simplified jet structure given in section 3 is used to write down scaling laws for results as functions of independent variables, a comparison is made to prior work in section 6.4, and finally in section 6.5, a discussion is provided about how the reconnection switch model applies to other (non-GRB) jet systems in order to identify whether GRB jets are special.

### 6.1 Results: Fiducial Model with Full Jet Structure

Consider a fiducial model of a rapidly rotating black hole with a magnetization that leads to ; a radial magnetic field at the compact object of G that leads to ; a collimating field geometry with as roughly consistent with a shocked stellar envelope (see discussion in Tchekhovskoy et al. 2008, 2010b) ; a deconfinement radius of cm as roughly the radius of a Wolf-Rayet star in the presupernova phase ; and a foot point opening half-angle of as applicable to a neutrino-dominated accretion disk that takes on a quite thin geometry near a rotating black hole where the jet is launched (see, e.g., Kohri et al. 2005; Chen & Beloborodov 2007 and for simulations of such magnetized thin disks see, e.g., Shafee et al. 2008; Penna et al. 2010). The chosen corresponds to an electromagnetic energy flux per unit rest-mass flux of and an electromagnetic energy per baryon rest energy at the field line foot point of . For simplicity is assumed, as applicable for a rapidly rotating neutron star or black hole.

Given the ambiguity in whether small (with ) modes actually efficiently dissipate (see end of section 5.4), let the fiducial model correspond to (dipolar jet with no separated current sheets in the -direction), and let (low order non-axisymmetric mode). The choice of is motivated by recent studies of accretion disks showing non-axisymmetric modes generated on tens of dynamical times (Davis et al., 2010).

This fixes the values of all the free model parameters, defining our fiducial model explored in this section. The figures show positions in radius along a field line within the jet. The effects of varying the model parameters on the solution are explored in the next section.

Figure 5 shows the structure of the jet along a single field line from the compact object to large radii. At a radius of , the opening angle rapidly becomes constant and the Lorentz factor jumps up due to a rarefaction wave that forces the jet towards the monopolar solution. The jet is cold in the sense that and up to the dissipation radius, so the cold ideal MHD jet structure solution is roughly valid at large radii even when current sheets are dissipating.

For MHD jets, the Lorentz factor is limited such that strictly at all radii. The Lorentz factor is dominated by radial motion for our jet solutions. As discussed later, the loss of electromagnetic energy occurs near the jet photosphere, so no dominate thermal fireball is generated. This loss of electromagnetic energy causes the total energy flux per unit rest-mass flux () to drop down to the bulk Lorentz factor.

Ideal MHD jets are characterized by starting with a poloidal field stronger than the toroidal field at the foot point on the compact object, while at larger radii the toroidal field dominates. In the present model, the field near the compact object is in the super-critical QED regime. The comoving field at large radii is , before significant dissipation occurs because in the monopole regime beyond , and remains roughly constant. In the -direction the electromagnetic pressure is constant as required by force balance.

Figure 5 also shows how the ideal MHD jet solution is modified by the current sheet dissipation. Beyond the transition radius at cm, dissipation proceeds due to fast collisionless reconnection. This causes (and so ) to drop exponentially for several e-foldings. Fast dissipation finally ceases once the transition back to slow collisional reconnection occurs due to low temperatures leading to a dominant Spitzer resistivity. Despite the significant drop in the comoving toroidal field, only by cm does it become smaller than the comoving poloidal field. Beyond this radius, any current sheet analysis requires consideration of a guide field, but this is beyond the dissipation radius so that all of our work remains valid.

Figure 6 shows the temperature, collisional and collisionless current layer thicknesses, number densities, and optical depths for neutrinos, photons, and pairs as functions of . The sequence of events from small to large radii is as follows: 1) The optically thick current layer’s temperature decreases until pairs drop out cm ; 2) Photons dominate the gas pressure. But after the photon absorption photosphere at cm, the photon densities decrease faster ; 3) The loss of photons causes the temperature to begin to increase at cm, which eventually leads to a reemergence of pairs at cm (with at cm) and then to a slight quenching of the rise in temperature because the pairs suspend the decrease in the photon scattering opacity ; 4) The transition to fast collisionless reconnection occurs at cm, where ; 5) The transition radius is nearly coincident with where photons begin to free-stream leading to at ; 6) Eventually the pairs drop out again at cm due to the pair absorption and scattering optical depths approaching unity, after which the temperature rises rapidly again ; and 7) An e-folding of electromagnetic dissipation occurs by cm, leading to a decrease in temperature due to a loss of electromagnetic and hence thermal pressures.

The key point from Figure 6 is that at cm, marking the transition radius to fast collisionless reconnection. Beyond , the collisional resistivity is no longer effective and dissipation proceeds due to collisionless processes that operate on the larger scale of the ion skin depth. This disrupts the current layer into a Petschek-like geometry, which allows a relatively fast reconnection rate of rather than a slower Sweet-Parker-like rate. This is why we identify this as a catastrophic reconnection switch.

Note that the radial range where electrons are degenerate is far inside the transition radius, so degeneracy does not affect the resistivity nor the value of . Also, note that even with the loss of pairs for cm–cm, the proton-electron plasma remains highly collisional, which forces the transition to fast collisionless reconnection to be at large radii.

The transition to fast collisionless reconnection might lead to a marginally collisionless state that forces . Using the full non-ideal MHD jet solution shows that the collisional layer thickness rises once dissipation occurs, and fast dissipation ceases when the two thicknesses become roughly equal because the collisional mode recovers. However, by this radius, significant electromagnetic dissipation has already occurred – and in fact this is what allows the collisional mode to recover. In addition, the possibility for a return to collisional reconnection was tested by using different up to , which leads to a different radial dependence of . We find that marginal collisionality was unable to be established for any until there has already been significant electromagnetic dissipation. Also, notice that the dissipation included in the model does not itself control the transition to fast reconnection. This was tested by using the ideal MHD jet solution for vs. , which shows that the thickness for the collisional layer remains much smaller than for the collisionless layer once fast collisionless reconnection would have started.

The condition for fast reconnection in electron-positron plasmas is the pair-dominated switch condition . This condition occurs at cm in the ideal MHD jet solution that assumes no dissipation up to this radius. If the baryon-dominated switch condition applies and dissipation did occur, then the pair-dominated condition is reached at cm after significant dissipation has already occurred. However, the pair-dominated switch condition is not expected to be relevant for the fiducial model or for a broad range of variations in model parameters. Pairs only reemerge after the photon scattering opacity becomes of order unity, and the transition to fast collisionless reconnection has occurred to order unity. Also, the total mass-energy of the plasma is found to be carried by protons by the radius where or . So, the condition for fast reconnection may still be dominated by the protons, although future studies should consider how fast reconnection proceeds when pairs dominate in number density but ions dominate in energy density.

Figure 6 also shows the scattering optical depth for neutrinos, photons, and pairs. The neutrinos are never optically thick in this model. The quantity corresponds to the opacity towards the observer accounting for locally generated pairs and downstream baryonic electrons as computed via Equation (84). The photons become optically thin to scattering at , which is consistent with our basic argument presented in section 2. This means that the reconnection switch mechanism occurs once non-thermal and quasi-thermal photons can be produced. Beyond , the non-ideal MHD jet solution has a photon scattering opacity that deviates only slightly from that of the ideal MHD jet solution. So dissipation is not crucial for lowering the photon scattering optical depth at larger radii. The pair scattering opacity is dominated by Coulomb interactions when the temperature is low, but then the pair density is negligible (e.g. beyond leading to a jump in the pair opacity). For the ideal MHD jet solution, the pair optical depth roughly follows that of the photons and continues to drop at all larger radii.

Perhaps the most distinct non-intuitive feature of the solution is the two-step rise and final drop in temperature. Let us consider a simplified calculation to explain why the temperature is forced to rise once the jet becomes optically thin to scattering. Cyclo-synchrotron radiation dominates over free-free emission at all radii, so the optically thin energy density loss rate is

 Q≈ne,tot43σTc(γ2e−1)uEM, (35)

where (recall ) is the electron thermal Lorentz factor. Then, is the absorption optical depth from Kirchhoff’s law, and the scattering optical depth is since pairs (when present) typically dominate the photon scattering opacity. The energy density of photons is obtained from the two-stream approximation using Equation (76), which in the limit and gives

 uγ≈3.2τγ,absu0,γ, (36)

where is the optically thick radiation energy density, and is the radiation pressure that dominates over other pressures. Then, the pressure equilibrium condition across the current layer of determines the temperature implicitly via

 τ−1γ,sca≈2.9(ΘeΓ−1)+1.4(ΘeΓ−1)2 (37)

when (otherwise the temperature is set by ). So a drop in the scattering optical depth leads to a rise in the temperature in this limit. One also obtains

 τγ,abs≈uEMu0,γ, (38)

which indicates that the absorption opacity drops significantly when the electromagnetic energy density is not high enough to produce optically thick photons. Equation (37) shows that once the flow becomes optically thin, the temperature is forced to rise up to K regardless of the temperature in the optically thick regime. The full solution has a slightly lower temperature of K because the cyclo-synchrotron emission is slightly dominated by higher energy electrons than those with the mean temperature. This gives a solution closer to using than in the above simplified equations. Overall, the behavior of the temperature vs. radius can be explained as follows: 1) The temperature decreases with radius due to adiabatic expansion in the optically thick regime ; 2) The temperature rises when the photons become optically thin due to the need to maintain pressure equilibrium with electromagnetic field ; 3) The slight flattening in the temperature during its rise (two-step rise) is because pairs reemerge and increase the scattering opacity, which suppresses the rapid temperature rise ; 4) Eventually the pair opacity also decreases and the pairs drop out, which leads to a continued rise in the temperature as the scattering opacity decreases ; and 5) Finally, the temperature decreases independently of the photon scattering opacity once a significant loss of electromagnetic energy has occurred.

The scattering and absorption photospheres lead to thermal emission and might correspond to the observed peak energy. In addition, low energy electrons’ cyclo-synchrotron emission is self-absorbed and then Comptonization can dominate their emission leading to a Comptonized thermal component as the peak energy corresponding to MeV (Rees & Mészáros, 2005; Thompson et al., 2007). Electromagnetic dissipation at the transition radius (coincident with the photon scattering photosphere) leads to a thermal peak of for the fiducial model value of . The thermal peak obtained at the photon absorption photosphere is smaller at K, giving at this relatively high . This peak energy is roughly consistent with the observed peak energy for cosmological long-duration GRBs.

Figure 7 shows the Lundquist number, observer timescales, energy density loss rates, and pressures within the collisional current layer at each radius.

For the ideal MHD jet solution, one would estimate that the resistivity is dominated by Compton drag at all radii, with the Spitzer (due to proton-electron Coulomb collisions) becoming nearly as important at cm and at cm. For the full dissipative MHD jet solution, Spitzer resistivity begins to play a role only after significant electromagnetic dissipation occurs once there is a significant decrease in the temperature within the layer at cm. The associated Lundquist number is order and eventually drops at large radii due to MHD dissipation.

Figure 7 also shows various observed timescales for photon emission. Once fast collisionless reconnection kicks in at , these timescales give an estimate for observed variability. The rise in the timescales at large radii is due to the jet being optically thin and so the diffusion timescale reaches the light crossing time that increases with radius. The transit time for fluid to pass through the current layer would lead to variable emission directly due to reconnection, while some of the other timescales give variability due to photon diffusion and emission rates. The mode produces many pulses due to new reconnection layers passing beyond . Emission would decay over the timescale for completing dissipation. The observer time for photon absorption diffusion from the current layer photosphere () and the observer time for photon scattering diffusion from the current layer photosphere () roughly follow . Between and , these timescales are s–s, s,
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