Sub-photospheric shocks in relativistic explosions

Sub-photospheric shocks in relativistic explosions

Andrei M. Beloborodov Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street, New York, NY 10027, USA
Abstract

This paper examines the mechanism of internal shocks in opaque relativistic outflows, in particular in cosmological gamma-ray bursts. The shocks produce neutrino emission and affect the observed photospheric radiation from the explosion. They develop from internal compressive waves and can be of different types depending on the composition of the outflow: (1) Shocks in “photon gas,” with negligible plasma inertia, have a unique structure determined by the force-free condition—zero radiation flux in the plasma rest frame. Radiation dominance over plasma inertia suppresses formation of collisionless shocks mediated by collective electromagnetic fields. (2) If the outflow is sufficiently magnetized, a strong collisionless subshock develops, which is embedded in a thicker radiation-mediated structure. (3) Waves in outflows with a free neutron component lead to dissipation through nuclear collisions. At large optical depths, shocks have a thickness comparable to the neutron free path, with an embedded radiation-mediated and collisionless subshocks. The paper also presents first-principle simulations of magnetized flows filled with photons, demonstrating formation of shocks and their structure. Simple estimates show that magnetized sub-photospheric shocks are efficient producers of photons and have a great impact on the observed photospheric radiation. The shock structure changes as the outflow expands toward its photosphere. The dissipation is accompanied by strong pair creation, and the -dressed shock carries the photosphere with it up to two decades in radius, emitting a strong pulse of nonthermal radiation.

Subject headings:
magnetohydrodynamics (MHD) Ж neutrinos —- radiation mechanisms: non-thermal –Ð radiative transfer –Ð shock waves —- gamma-rays bursts: general

1. Introduction

Astrophysical explosions and jets generate shock waves, which produce radiation. Their radiative properties are determined by the dissipation mechanism that sustains the velocity jump in the shock and by its ability to generate nonthermal particles. This paper examines the mechanism of internal shocks in gamma-ray bursts (GRBs) that occur before the GRB jets become transparent to radiation. The approach and some of the results may also be of interest for other explosions, e.g. in novae or supernovae.

1.1. Internal shocks in GRB jets

The main features of GRB explosions may be summarized as follows: the outflow is relativistic, it carries magnetic fields frozen in fully ionized plasma, and a large fraction of its energy is carried by neutral particles — photons and free neutrons. GRB outflows start very opaque near the central engine of the explosion and become transparent at a large “photospheric” radius . Internal shocks can develop below and above the photosphere.

Early works proposing internal shocks in GRBs focused on shocks above the photosphere (Rees & Mészáros 1994; Kobayashi et al. 1997; Daigne & Mochkovitch 1998). They can only be collisionless, i.e. mediated by collective electromagnetic fields. Their mechanism has been studied in detail using particle-in-cell simulations, and it was found that the presence of transverse magnetic fields renders the shock unable to accelerate particles (Sironi & Spitkovsky 2011): the postshock electron-ion plasma is in a two-temperature state, , with electrons and ions forming nearly Maxwellian distributions. This may, however, change if the electron-ion outflow is loaded with plasma. Ultra-relativistic shocks in -loaded plasma with transverse magnetic field were found capable of accelerating positrons (Hoshino et al. 1992; Amato & Arons 2006; Stockem et al. 2012).

Shocks in opaque plasma below the photosphere are less explored and may be key to understanding GRB emission (Mészáros & Rees 2000b; Pe’er et al. 2006; Beloborodov 2010; Levinson 2012). There is significant evidence that GRB radiation is mainly produced below the photosphere (Ryde et al. 2011; Beloborodov 2013; Yu et al. 2015), and detailed simulations of radiative transfer in opaque heated jets give spectra consistent with GRB observations (Vurm & Beloborodov 2016). Internal shocks provide a plausible mechanism for sub-photospheric heating invoked by these models.

Internal shocks may result from the fast variability of the central engine or the outflow interaction with the progenitor star (e.g. Lazzati et al. 2013; Ito et al. 2015). At later stages of ballistic expansion with a high Lorentz factor , internal shocks at radius can develop from velocity variations on scale (measured in the outflow rest frame). This scale also sets the characteristic optical depth seen by photons in the expanding outflow, , where is Thomson cross section and is the proper density of electrons and positrons.

The present paper is motivated by the following questions:
(1) Can sub-photospheric shocks be collisionless? This is assumed in models of TeV neutrino emission from the jet-progenitor interaction (Razzaque et al. 2003), however the assumption is questionable (Murase & Ioka 2013).
(2) Is the shock capable of producing high-energy particles? The presence of high-energy electrons at large optical depths would have a strong effect on the photospheric radiation (Pe’er & Waxman 2004; Beloborodov 2010; Vurm & Beloborodov 2016).
(3) How does the shock evolve as it emerges from the photosphere and what is its observational appearance?

1.2. Radiation-mediated shocks (RMS)

Since GRB jets carry a large number of photons per electron, sub-photospheric shocks are naturally expected to be mediated by radiation (Levinson 2012). Then dissipation and the profile of the velocity jump are controlled by photon scattering.

Basic features of radiation-mediated shocks (RMS) were studied in the 1950s (see Zeldovich & Raizer 1966 and refs. therein). The RMS propagation is sustained by radiation diffusion: radiation generated by the shock diffuses upstream and pre-heats the upstream gas. This creates a pressure gradient, a kind of a “pillow” that allows the gas to smoothly decelerate, avoiding the collapse of the shock thickness to the collisionless scale (the ion Larmor radius).

The first RMS models assumed that radiation is everywhere in local thermodynamic equilibrium. This assumption can be strongly violated in astrophysical explosions, as the timescale to establish thermodynamic equilibrium can be much longer than the time it takes the gas to cross the shock. Models relaxing the equilibrium assumption have been developed and applied to supernova shock breakout (e.g. Weaver 1976; Sapir et al. 2013). The RMS model was also extended to relativistic shocks (Levinson & Bromberg 2008; Budnik et al. 2010; Bromberg et al. 2011; Levinson 2012). The highest temperature achieved in the RMS depends on the photon number carried by the upstream through the shock. Levinson (2012) emphasized the low efficiency of photon production by the RMS in GRB jets and developed a shock model with a conserved photon number.

The RMS thickness is large, comparable to or larger than the photon mean free path. This inhibits diffusive acceleration of charged particles. In particular, electrons radiate energy faster than they can cross the shock. The RMS is only capable of a slow diffusive acceleration of photons up to the MeV band (in the shock frame).

The RMS picture of internal shocks has, however, a few caveats. Previous work did not take into account that the outflow is magnetized, and the magnetic field can change the RMS structure and the dissipation mechanism. In addition, GRB explosions are expected to carry free neutrons; their collisions can play a key role in shaping the shock waves at large optical depths and offer an additional mechanism for producing high-energy particles and neutrinos.

1.3. Outline of the paper

We begin with basics of shock formation. Section 2 examines how a super-sonic compressive wave steepens and launches a pair of shock waves. We first describe shock formation in a polytropic gas using the hydrodynamic approximation (zero mean free path of all particles and photons). Then we relax this approximation and discuss the role of photon diffusion in the formation of RMS and collisionless shocks. We consider a “cold” gas with sound speed and formulate two conditions for the immediate RMS formation (vs. formation of a collisionless shock). We also discuss flows with large , including the extreme regimes where the flow inertia is dominated by radiation () or magnetic fields ().

Section 3 describes the general jump conditions for shocks in media with any thermal pressure and magnetization. A moderate magnetization of the flow changes the jump conditions and we argue that this leads to the formation of a thin collisionless subshock, even at large optical depths. We evaluate the region in the parameter space where a strong collisionless subshock must exist.

Section 4 focuses on shocks in “photon-gas” with sub-dominant magnetic fields and negligible plasma inertia (). This regime may occur in GRB explosions at their early stages, during the jet breakout and its acceleration by radiation pressure. We use a self-consistent simulation of time-dependent radiative transfer and obtain the solution for the shock structure.

Then Section 5 presents the RMS structure at later stages when the plasma inertia becomes important. As the main tool, we use direct Monte-Carlo simulations of time-dependent radiative transfer coupled with the flow dynamics. We first investigate RMS formation in a weakly magnetized flow, and discuss the effect of bulk Comptonization and creation inside the shock front. Then we turn to shocks in a magnetized fluid and demonstrate the formation of a strong collisionless subshock embedded in the RMS, as anticipated in Section 3.

Section 6 investigates how plasma heating in a collisionless (sub)shock results in “breeding” of pairs. Section 7 discusses the emergence of a -dressed shock from the photosphere.

Section 8 describes shocks in outflows with a free neutron component. Due to their large free paths, neutrons introduce a large effective viscosity. Nuclear collisions produce ultra-relativistic pairs and neutrinos. They sustain broad shock fronts until the jet reaches the neutron decoupling radius where most neutrons begin to flow freely without collisions.

The results and their implications for GRB models are discussed in Section 9.

2. Formation of shocks

2.1. Ballistic approximation and caustics

Consider an outflow with internal supersonic motions. Such motions can be approximately described as ballistic: each fluid element is moving with a constant velocity. It is well known that ballistic flows create caustics — surfaces where density diverges (in cosmology such surfaces are called Zeldovich pancakes).

The flow near the caustic is approximately plane-parallel (one-dimensional). It is convenient to view the flow in the rest frame of the caustic and choose the -axis normal to it, so that the flow converges toward along the -axis with velocity . Since may approach unity in relativistic flows, it is useful to introduce dimensionless momentum where . Velocity is related to by

(1)

A simple example of a converging flow is provided by an initial profile,

(2)

The ratio describes the initial steepness of the wave, and describes its amplitude. The wave is non-relativistic if . The characteristic timescale of the profile evolution is . On this timescale the ballistic wave steepens (Figure 1) and the caustic forms at , where is maximum.

Figure 1.— Fluid velocity (in units of ) and density (arbitrary units) evolving in the ballistic compressive wave with the initial profile given by Equation (2), with . The initial velocity profile is shown by the dotted curve, and the next snapshot is taken at the time of caustic formation . The dashed curve shows density at ; the initial density at was uniform.

The density of the ballistic flow diverges at the caustic. Its evolution is determined by the relation

(3)

where is the initial position of the fluid slab at times , and is the initial velocity profile. The slab is contracting by the factor . Therefore, the evolution of baryon density is described by

(4)

where is the density at . The compression rate is highest at and here density diverges at time

(5)

At this moment, becomes discontinuous at .

2.2. Pressure build-up in the converging flow

True caustics form in flows with zero pressure. A small initial pressure qualitatively changes the picture: it can be strongly amplified in the converging flow near and the generated pressure gradient stops the flow.

The deceleration of the converging flow around accelerates the steepening of the velocity profile on each side of the caustic (Figure 2). As a result, at some time and locations two shocks form and continue to propagate away from . The type of the nascent shock depends on the physical conditions in the region . Below we discuss the pressure build up in the converging flow, then estimate the location of shock formation and the corresponding maximum compression.

Figure 2.— Snapshot of the evolved compressive wave with the same initial conditions as in Figure 1 but with a finite initial pressure. The flow has the initial sound speed and adiabatic index . The snapshot is taken at , shortly before the formation of a pair of shocks.

One source of pressure is the thermal motions of plasma particles. It grows in the converging flow, however its contribution to the total pressure is limited by fast radiative cooling, which converts plasma heat to radiation. In a local thermodynamic equilibrium, radiation strongly dominates the heat capacity of GRB jets, because the photon density greatly exceeds the plasma density. At small radii, where the population is in annihilation equilibrium with Planck radiation, one finds (Svensson 1984); the abundance is decreasing exponentially in the expanding and adiabatically cooling jet. At larger radii, the particle density (ions, , or ) does not exceed and, in a local thermodynamic equilibrium, this implies a plasma pressure . Here we examine compressive waves in a medium that is initially not too far from thermal equilibrium111Shocks create strong deviations from thermodynamic equilibrium, and these deviations become long-lived in the region of moderate optical depth, around and above the photosphere. New shocks in this region will develop in the plasma with hot (thermally decoupled) ions, preheated by previous shocks. and thus has . Then the two main sources of pressure that can be amplified in the wave are radiation and the transverse magnetic field.

Magnetic fields are expected to carry a significant fraction of the jet energy. Comparison of theoretical GRB spectra with observations suggests (Vurm & Beloborodov 2016). The jet plasma is an excellent conductor, so the magnetic field is frozen in it and advected by the flow. In an internal compressive wave, the frozen transverse field is compressed together with the plasma: or , where is the magnetic field measured in the fluid frame, is the proper density of the fluid, and is its Lorentz factor. The magnetic pressure grows in the converging flow as222 and are measured in the same (fluid) frame. Pressure and internal energy density are always measured in the fluid frame and we omit tilde to simplify notation.

(6)

The growth of radiation pressure depends on its ability to diffuse out if the compressed region, which depends on the optical depth. If the flow is sufficiently opaque to photons, the radiation will be trapped and

(7)

In the opposite limit, when the compressed region is transparent to photons, there is no significant amplification of .

Equations (6) and (7) both have a polytropic form , with or . A similar relation could also be used for compressive waves in a medium that is far from thermal equilibrium with radiation and filled with hot, thermally decoupled, ions (protons). The proton pressure in the compressive wave follows the relation with as long as the proton temperature is non-relativistic, .

2.3. Shock formation in non-relativistic polytropic gas

Let us first consider a non-relativistic gas, . Suppose that initially the gas has uniform pressure and density , and is set in motion with velocity that corresponds to momentum profile given e.g. by Equation (2). We assume that the peak of velocity profile is much greater than the sound speed . In the ballistic approximation, the profile would develop a caustic at at time . We wish to know how the finite pressure changes the flow dynamics, in particular what is the maximum pressure achieved in the compressed region before a shock forms, and where the shock formation occurs.

Even if the compressive wave is relativistic, , the condition implies that the shocks form not far from where is non-relativistic. Therefore, the shock formation can be examined using Newtonian hydrodynamics around , so we will use , , and .

The evolution of the gas is convenient to view on the - plane (Figure 3). Each streamline is described by where is the Lagrangian coordinate — the position at . Initially, a small fraction of gas is in the subsonic region near . The streamlines that start outside this region are initially supersonic and eventually become subsonic.

Figure 3.— The streamlines on the spacetime diagram. The gas has the initial velocity profile given by Equation (2) with , the initial sound speed , and the adiabatic index . The boundary of the subsonic region is shown by the red curves. The red curve is dashed where the deceleration to the subsonic speed occurs smoothly and solid where the deceleration occurs through a shock. The shock forms at and .

There is a critical Lagrangian coordinate . Streamlines that start at will become subsonic without a shock: the compressed gas is gradually decelerated as its specific kinetic energy gets transformed into enthalpy , where is the local speed of sound. This “compressive deceleration” to a subsonic speed occurs when the compression factor satisfies

(8)

Approximating the streamline before this moment as ballistic, one can estimate . Therefore, the deceleration time at which the streamline with Lagrangian coordinate becomes subsonic may be estimated from the condition,

(9)

The corresponding location on the streamline is

(10)

The smooth compressive deceleration is only possible for streamlines with sufficiently small . For large one finds , and the compressive deceleration becomes impossible — the ballistic flow does not have a chance to compress enough before it hits the existing subsonic region near . Then the deceleration occurs through a shock.

The critical Lagrangian coordinate at which the shock forms is given by (see Appendix A),

(11)

The compression of gas with the Lagrangian coordinate is determined by Equation (8) with evaluated at . In the limit of , the compression along this streamline is given by

(12)

It determines the maximum pressure developed in the flow before the shock is launched,

(13)

The maximum pressure is comparable to the peak kinetic energy density of the wave, even though only develops in a small region near the caustic where the flow momentum is much smaller than . This is because is controlled by the curvature of the velocity profile (described by , see Appendix A), which depends on .

At and the sound speed of the ballistic flow is not much below its bulk speed , so the nascent shock is not strong. Then the shock propagates through the ballistic gas with increasing Lagrangian coordinate where the upstream velocity is higher, and the shock compression ratio quickly approaches the strong-shock limit .

2.4. Shock formation in relativistic polytropic gas

Shock formation in relativistic gas may be examined in a similar way. This regime occurs in relativistic explosions at small radii where radiation dominates the gas inertia. Then (Landau & Lifshitz 1959), and one must consider relativistic compressive waves with .

The flow is initially subsonic in the zone where . Outside this zone the flow is approximately ballistic and its density is growing with time as , where . The compressive deceleration of the relativistic gas is quite efficient: a large fraction of the bulk kinetic energy is converted to enthalpy when the gas is compressed by only a factor of .

However, even such a moderate compression is difficult to achieve in the relativistic ballistic flow, because the gas with () has a small and hence it is compressed slowly. The maximum time allowed for ballistic compression is and the corresponding maximum compression factor is

(14)

Gas with a relativistic ballistically hits the subsonic region before it has a chance for compressive deceleration. Thus, the shock must form at Lagrangian coordinate such that , not far from the boundary of the initial subsonic zone . The time and location of shock formation are and . The shock forms with a mildly relativistic amplitude; it becomes ultra-relativistic when it propagates into the gas converging with .

One can also consider shock formation in a magnetically dominated gas and . Then and it is convenient to define . Shocks form in compressive waves with , which corresponds to Lorentz factor .

In the limit of strong magnetization, the sound speed becomes equal to and shocks do not form. In this case, the dynamic equations read with the stress-energy tensor components

(15)

(the magnetic field is assumed to lie in the - plane perpendicular to the fluid velocity). The neglect of the plasma contribution to defines so-called force-free electrodynamics, where plasma only serves to conduct electric currents demanded by and supplies no inertia. The plasma velocity is related to the electric field by and . Adding and subtracting the energy and momentum conservation laws,

(16)

one obtains

(17)

The initial profiles of determine . This gives explicit solutions for and , demonstrating their smooth behaviour, with no caustics or shocks.

2.5. Radiation diffusion and formation of RMS

Radiation diffusion is an essential ingredient of an RMS, since it is the mechanism of shock propagation. However, too fast diffusion would let radiation escape, inhibiting the RMS formation. A shock wave is usually assumed to be radiation-mediated if two conditions are satisfied:

(A) The jump conditions give in the downstream , so that a large fraction of energy generated by the shock is carried by radiation (Zeldovich & Raizer 1966).

(B) The medium has optical depth , so that the shock generates radiation faster than it could diffuse out of the system of size . For instance, in a supenova explosion one could take as the radius of the expanding ejecta (e.g. Tolstov et al. 2013).

In fact, these conditions do not guarantee that the shock is mediated by radiation. The velocity profile connecting the upstream and downstream may contain a “subshock” — a sharp jump mediated by the plasma on a scale much smaller than the photon free path to scattering. In non-relativistic shocks () satisfying conditions (A) and (B) the velocity profile is smooth, with no subshock (Zeldovich & Raizer 1966). However, in the relativistic case, , a weak subshock was reported (Budnik et al. 2010). In addition, in the above condition (B) one should be careful with what is meant by the “size of the system.”

Consider a compressive wave of a mildly relativistic amplitude and length . A characteristic optical depth may be defined as

(18)

where is the opacity of the gas. Suppose the unperturbed gas has a non-relativistic sound speed . Section 2.3 described how at time two shocks form near the caustic, at the Lagrangian coordinate . Thus, the region of shock formation has the optical depth

(19)

For a radiation-dominated flow, and . In this case, however, Equation (19) can only be used if radiation is trapped, i.e. unable to diffuse out of the region of shock formation on the timescale . This requires

(20)

where is the flow velocity upstream of the nascent shock. The trapping condition is satisfied if

(21)

For the flow with the upstream pressure dominated by radiation, one can use the relation

(22)

Then condition (21) may also be written as . If this condition is satisfied, a propagating jump in radiation pressure will develop at , and the nascent shock will be mediated by photons, i.e. an RMS will be launched.

The RMS velocity profile is shaped by the competition between advection of radiation through the shock and its diffusion in the opposite, upstream direction. Therefore, the optical depth of the velocity jump is regulated to

(23)

The RMS propagation involves continual amplification of radiation advected through the shock — the result of photon scattering in the region of a steep velocity gradient. As the hot downstream photons diffuse back into the upstream, they experience “bulk Comptonization” — they are boosted in energy by the factor of . As a result, the energy of radiation advected through the shock is amplified, as required by the jump conditions for a propagating shock.

Launching an RMS at requires an initial build-up of radiation density near the shock front, which is only possible if the trapping condition (21) is satisfied. Otherwise, radiation leaks out of the compressed region to large distances . This may be viewed as a violation of RMS condition (B), as the effective “size of the system” during the shock formation is comparable to . Then the radiation pressure gradient is too weak to control the velocity profile of the flow. Radiation is unable to resist the steepening of the velocity profile, and the width of the velocity jump is quickly reduced to the ion Larmor radius, forming a collisionless shock mediated by the collective electromagnetic field. It may later evolve into an RMS, when the postshock region has accumulated a sufficient optical depth, if there is enough time for that in the expanding outflow, i.e. if the shock forms sufficiently deep below the photosphere.

2.6. Critical magnetization

When both magnetic field and radiation contribute to pressure, there are two contributions to the sound speed, . It is convenient to define the dimensionless enthalpy of the flow,

(24)

A similar quantity for the magnetic field in the fluid frame is

(25)

Suppose the optical depth is large so that the radiation trapping condition is satisfied. The type of the nascent shock is determined by whether the magnetic field or radiation dominates the pressure in the compressed region near the caustic, . An RMS forms if is dominated by radiation; otherwise, a collisionless shock is launched. Since must be the same in either case (see Equation 13), it is sufficient to compare the compressions needed to reach with only magnetic or only radiation pressure: and . This comparison gives an approximate condition for launching a collisionless shock in a cold () and opaque medium,

(26)

Note that this condition only applies to the nascent shock, at the point of maximum ballistic compression in the converging wave near the caustic. As the shock becomes stronger and continues to propagate into the ballistic flow where the upstream is less compressed (and hence less magnetized), its type may change.

An established steady shock structure is determined by the parameters of its upstream. The first step in the analysis of a steady propagating shock is the solution for its jump conditions.

3. Shock jump conditions

Jump conditions for relativistic magnetized shocks were studied by de Hoffmann & Teller (1950). Pulsar wind nebulas and GRBs revived interest to relativistic shocks. The upstream medium is usually assumed to be cold in the sense that its enthalpy is much smaller than the rest-mass energy of the plasma. This condition is not, however, satisfied in the inner regions of GRB jets. Below we write down the general jump conditions for shocks propagating in a hot magnetized plasma filled with radiation, and show their solutions.

Consider a shock wave propagating in a sufficiently extended, optically thick medium. Far upstream and far downstream of the shock the plasma and radiation can be described as an ideal gas with isotropic pressure. The thermal energy density and pressure are related by

(27)

where , as long as is dominated by radiation. When formulating the jump conditions we will keep general, and specialize to in numerical solutions.

In the rest frame of the upstream (pre-shock) fluid, an observer will see the downstream (post-shock) fluid approaching with velocity . The shock front is perpendicular to and approaching with a higher velocity parallel to . The plasma carries a frozen magnetic field (measured in the fluid rest frame). We only consider magnetic fields perpendicular to the fluid velocity; a parallel magnetic field is anyway unchanged by the shock and hence does not affect the jump conditions.

3.1. Stress-energy tensor and sound speed

The stress-energy tensor of a hot magnetized flow with four-velocity is given by

(28)

where is the proper rest-mass density of the baryons, is the metric tensor of Minkowski spacetime, and is the electromagnetic tensor. Its electric and magnetic components in the lab frame, and , are related by , as the plasma is a nearly ideal conductor. Using and , the stress-energy tensor may be reduced to the ideal fluid form,

(29)

with the effective relativistic enthalpy and pressure

(30)

Before considering shocks, it is useful to examine sound waves in a uniform background that has , , and . Let us consider longitudinal (compressive) waves propagating along the -axis. In the linear order, perturbations are described by the four-velocity , four-acceleration , and compression . The linearized equations of motions give (for and )

(31)
(32)

These two equations can be reduced to the wave equation for ,

(33)

where the wave speed is defined by

(34)

As the two main parameters of the fluid, it is convenient to use the dimensionless contributions of enthalpy and magnetic fields to the fluid inertia,

(35)

Then the wave speed defined in Equation (34) may be expressed as

(36)

This general expression reduces to familiar cases in four limits:
(1) : (non-relativistic sound waves),
(2) : (non-relativistic fast MHD modes in a cold plasma),
(3) : (sound waves in a relativistic gas), and
(4) : (force-free limit of the MHD modes).

Internal supersonic motions generate shocks, as discussed in detail in Section 2. In addition, shocks can form through nonlinear steepening of sound waves excited by a subsonic perturbation, (Zeldovich & Raizer 1966). The steepening occurs because is slightly increased in the region compressed by the wave, so the crest of the wave (maximum and maximum ) travels faster than the trough (minimum and minimum ). Using Equation (36) one can verify that , i.e. compression indeed increases the local sound speed. The shock formed through sound-wave steepening propagates super-sonically but has a subsonic velocity jump, i.e. it separates regions with a relative velocity . Such weak shocks are found among the solutions shown below, along with strong shocks formed by supersonic motions .

Formation of shocks through steepening of sound waves is inefficient in the relativistic regimes (3) and (4), as in this case ( or is constant in both cases). In the force-free limit (), shock formation does not occur at all (Section 2.4). In a radiation-dominated medium (), shocks can be launched by a supersonic motion .

3.2. Jump conditions

Jump conditions express the continuity of fluxes of energy, momentum, and baryon number in the rest frame of the shock front. The fluxes of energy and momentum are given by the stress-energy tensor in Equation (29). The baryon flux is described by the four-vector

(37)

The fluxes along the shock normal (the -axis) are given by

(38)
(39)
(40)

where , and can be expressed in terms of : . Equating the fluxes upstream (index “u”) and downstream (index “d”) one obtains the relations,

(41)
(42)

Given the upstream parameters , , , and taking into account that (implied by the flux freezing condition ), one can solve Equations (41) and (42) for the two unknowns and .

Typically, the upstream velocity relative to the downstream, , is a given in the shock problem. Therefore, we chose as an independent parameter instead of . The upstream momentum in the shock frame, , is related to the upstream momentum measured in the downstream frame, , by the Lorentz transformation between the two frames,

(43)

For a given , a trial determines , and the solution of Equations (41) and (42) (which is obtained numerically) yields and together with .

The solutions are shown for and 10 in Figures 4 and 5, assuming (pressure is dominated by radiation). Figure 4 shows the compression ratio (which also determines ) and Figure 5 shows the ratio . The latter determines the dissipation efficiency of the shock: if then a large fraction of the upstream energy goes to heat rather than ends up stored in the compressed magnetic field. The ratio is also interesting for another reason: it is related to the dissipation mechanism in the shock front, as discussed below.

Figure 4.— Compression ratio in a relativistic shock with (upper panel) and (lower level). The compression depends on the upstream enthalpy and magnetization . Solid curves show contours of the function . The region where the shock is weak, , is dotted in blue and bounded by the dashed curve.
Figure 5.— Ratio of the enthalpy and magnetization in the downstream, , for shocks with (upper panel) and (lower panel). This ratio depends on the upstream enthalpy and magnetization ; the dependence is shown using contours on the - plane. The blue region is the same as in Figure 4. Red dots highlight the region where . In this region, the shock is expected to have a strong collisionless jump where the velocity changes on a scale comparable to the gyroradius of the plasma particles.

3.3. Collisionless shocks

The jump conditions do not describe the structure or dissipation mechanism of the shock front. However, they allow one to evaluate the region in the parameter space where dissipation must be mainly collisionless.

We expect the shock to be mainly mediated by collective electromagnetic fields when the downstream enthalpy is dominated by the compressed magnetic field . Then radiation cannot control the shock structure, as its pressure is below the ram pressure of the shock. In particular, in the limit of the downstream can be approximated as a cold magnetized medium with negligible heat, so radiation has no effect on the upstream deceleration and the shock velocity profile; the profile inevitably steepens so that the entire velocity jump occurs in a thin layer on the collisionless plasma scale (gyroradius).

In contrast, in shocks with significant ratio , the diffusion of the postshock radiation into the upstream region creates a precursor that changes the upstream velocity and reduces the amplitude of the collisionless jump. The resulting structure may be described as a collisionless shock with a radiation precursor or, equivalently, an RMS with a collisionless subshock. In the limit of , the subshock becomes weak or non-existent. Section 4 below demonstrates this fact for shocks in relativistic gas, ), and Section 5 will show the shock structure for a moderate with and without a significant magnetic field.

The transition between the two dissipation regimes — mainly mediated by collective electromagnetic fields and mainly mediated by radiation — occurs at (the shock structure in this transition region will be calculated in Section 5). The region where a strong collisionless jump is expected () is highlighted in red in Figure 5. We also show (in blue) the region where the shock is weak () and could only form through steepening of sound waves. As discussed in Section 3.1, such shocks do not easily form in a relativistic fluid ( or ) since steepening takes a long time, typically longer that the expansion time of the outflow. The region between the two curves and is where strong collisionless shocks occur. When only a small fraction of the upstream kinetic energy is dissipated in the shock, and most of it ends up stored in the compressed magnetic field. Therefore, the strongest collisionless dissipation is expected if .

For a medium with a given enthalpy one can define a characteristic magnetization such that a shock propagating in the medium will have (the boundary of the red-dotted region in Figure 5). The magnetization depends on the shock strength (upstream momentum measured in the downstream frame). This dependence is shown in Figure 6. In the ultra-relativistic limit , we find that does not depend on (both and scale as , so their ratio does not depend on ). For non-relativistic shocks with a cold upstream (), scales as . This is because weakly magnetized shocks have downstream enthalpy while .

Figure 6.— The characteristic upstream magnetization that gives in the downstream. A strong collisionless subshock should form in the RMS if . Each curve shows the dependence of on the shock strength for a fixed upstream enthalpy (indicated next to the curve). The curves are dashed where the shock is weak, .

At large and , the effective sound speed approaches the speed of light (see Equation 36). Shocks easily form if the internal compressive motions are supersonic, i.e. their Lorentz factors exceed . The dissipation in magnetically dominated shocks occurs in a microscopically thin, collisionless shock front. Dissipation is reduced in this regime, as a large fraction of shock energy goes into the compressed magnetic field, however dissipation can still be significant. For example, a shock with Lorentz factor propagating in a medium with upstream enthalpy and has the downstream enthalpy and . In this example, the shock compression ratio is , and adiabatic compression would imply the amplification of by only a factor of , well below 6.8. We note also that the downstream enthalpy is mainly determined by the upstream enthalpy and the amplitude of the shock; it weakly depends on .

4. Shocks in photon gas

In GRB explosion models, sub-photospheric shocks begin to form at early stages, when the jet rest mass is still dominated by radiation, before the jet accelerates to its asymptotic Lorentz factor. This section examines the structure of shock waves in this regime, neglecting magnetic fields.

In essence, we deal here with shocks in the gas of photons, as the plasma inertia is negligible. The plasma role is to provide opacity and thus to couple the photons into a single fluid, with a small mean free path. The plasma particles may be viewed as passive “markers” following the motion of the photon gas.

4.1. Jump conditions

Far upstream and far downstream of the shock, the radiation can be described as ideal fluid with isotropic pressure and the stress-energy tensor

(44)

where we have used the equation of state . Jump conditions express the continuity of (flux of energy) and (flux of momentum) in the rest frame of the shock,

(45)
(46)

where subscript “u” stands for upstream and “d” for downstream; pressure is measured in the fluid frame, and velocity is measured in the shock frame. Dividing Equations (45) and (46), one finds that and satisfy the condition

(47)

Rewriting the definition of as

(48)

one can view and as the two roots of the quadratic equation, and hence they are related by

(49)

Since , one concludes that . This condition merely states that the shock moves supersonically relative to the upstream (recall that the sound speed is ). Using the relation (49) and Equation (45) or (46), one finds the pressure jump across the shock

(50)

The shock compresses the volume measured in the fluid frame by the factor

(51)

which also gives the relation

(52)

For ultra-relativistic shocks, , the jump conditions simplify to and .

4.2. Evolution equation for the photon gas

Formation of shocks in the gas of photons can be simulated numerically. It is convenient to think of this problem as a radiative transfer problem for the bolometric intensity of radiation . Since the stress-energy tensor is dominated by radiation, the plasma is effectively massless and its velocity is controlled by the “force-free” condition: equals the equilibrium value such that the radiation flux in the fluid frame vanishes (zero flux implies zero force applied by radiation to the plasma). This condition leads to a well-defined radiative transfer problem (Beloborodov 1999). It has a simple solution for steady spherically symmetric relativistic outflows (Beloborodov 2011). Here we are interested in shock formation in variable outflows, so the problem is time-dependent.

The shock is thin and locally flat (in the - plane), and we can study its formation in the plane-parallel geometry. Then the bolometric intensity is described by the function where and is the photon angle with respect to the -axis.

The stress-energy tensor of radiation is determined by the moments of the intensity,

(53)

In particular, , , and . The force-free condition reads in the fluid frame, and the transformation of from the lab frame to the fluid frame gives the quadratic equation for velocity,

(54)
(55)

The evolution of intensity is described by the transfer equation,

(56)

where is the number density of electrons/positrons measured in the lab frame, and is the source function. In the simplest case of isotropic scattering, is given by (Beloborodov 1999)

(57)

4.3. Numerical solution

Equation (56) supplemented with the equilibrium velocity condition (Equation 55) can be solved numerically. A sample solution is shown in Figure 7. In this example, the initial state is given by Equation (2) with and . The initial plasma density is uniform in the lab frame, , and the unit length in corresponds to a slab of unit Thomson optical depth. The initial radiation density measured in the fluid frame is uniform in the lab frame, .

Figure 7.— Evolution of the fluid velocity, compression, and radiation energy density (measured in the fluid frame). Unit length in corresponds to a slab of unit Thomson optical depth in the initial state. The indicated times are measured in units where .
Figure 8.— Shock wave in “photon gas” (). The shock structure is shown as a function of Thomson optical depth in the frame where the downstream is at rest. The reference point is chosen at half-maximum of the momentum profile . Energy density is measured in the fluid frame (normalized to its pre-shock value). The solid curves show the shock wave with amplitude . The dashed curve shows for the shock with .

One can see the compression of the converging supersonic flow and the formation of a pair of shocks symmetric about , as described in Section 2. As the two shocks continue to propagate, the downstream fluid comes nearly to rest in the lab frame. The upstream velocity relative to the downstream, , is related to the upstream and downstream velocities measured in the shock frame by

(58)

Using (Section 4.1), one finds

(59)

Once the shock wave is established, its structure becomes independent of the details of the initial conditions. The shock has only one parameter: or . Figure 8 shows the obtained structure of a shock wave with . It is shown as a function of the optical depth measured in the direction. Then the result is independent of the plasma density, so the obtained solution is unique. Using and that correspond to , one finds from Equations (51) and (52) the ratio of downstream and upstream pressures . This asymptotic value of is observed in Figure 8.

For comparison, Figure 8 also shows the momentum profile for a shock with , obtained from a similar time-dependent simulation. When re-scaled by the factor of , the momentum profile is the same as for .

5. Monte-Carlo simulations of shocks

Time-dependent simulations may also be employed to study shocks in plasma with significant rest mass and magnetic fields. In contrast to the photon gas studied in Section 4, now the radiative transfer equation cannot be closed by the force-free condition . Instead, the fluid acceleration must be calculated together with the radiative transfer. Another complication is the need to follow the evolution of the radiation spectrum, which develops a hard tail extending above inside the shock front; then the scattering cross section is changed by the electron recoil.

Below we solve this problem using a direct numerical experiment that follows individual photons and their interactions with the plasma, so that the transfer of momentum and energy is described on a microscopic level. In the initial state, the flow has a smooth velocity profile described by Equation (2) and carries thermal radiation, which is isotropic in the fluid frame. The flow is opaque and supersonic, which leads to the formation of a pair of shocks propagating in the directions, as described in Section 2.

The flow has two interacting components:
(1) Magnetized plasma. The plasma is assumed to carry a transverse magnetic field, which provides strong coupling between all charged particles, so their dynamics along the axis is well described as a single-fluid motion.333For any realistic magnetic field the Larmor radii of ions and electrons are microscopic, many orders of magnitude smaller than the photon free path. Therefore, we assume strong magnetic coupling of the plasma particles even in the weakly magnetized model that is formally labeled as . In the numerical simulations we use a Lagrangian grid moving together with the fluid of charged particles: the fluid is discretized into shells of equal rest mass and a small scattering optical depth. Besides mass, each shell is characterized by the magnetic flux frozen in it, internal thermal energy, and total pressure. The magnetic flux remains constant while thermodynamic quantities may change as the shell contracts (or expands) and interacts with radiation. Thermal conductivity of the plasma in the -direction is suppressed by the transverse magnetic field and neglected.
(2) Radiation. Radiation is represented by photons which are followed individually. The photons migrate through the plasma shells and occasionally scatter off a thermal electron. The scattering is followed using Monte-Carlo technique, with the exact Klein-Nishina differential cross section and assuming that the thermal electrons are isotropic in the fluid frame. The electrons are assumed to have a Maxwellian distribution with a self-consistently calculated temperature.

A detailed description of the numerical method will be given in an upcoming paper. A possible alternative to the Monte-Carlo method is the solution of the transfer equation for the radiation intensity, as in Section 4 but now including fluid inertia. A similar approach was taken in the recent work by Ohsuga & Takahashi (2016) who simulated the evolution of bolometric intensity assuming Thomson scattering. Thomson approximation may be sufficient only for the RMS with negligible fluid inertia . If , strong bulk Comptonization develops in the RMS and, if treated in the Thomson approximation, leads to runaway in photon energy (Blandford & Payne 1981). Scattering with substantial electron recoil, in the Klein-Nishina regime, becomes inevitable and limits the growth of photon energy. The electron recoil is also essential in maintaining heat exchange between radiation and plasma, even when the photon energies (and the plasma temperature) are well below .

5.1. Sample models

In our two sample simulations the initial flow has dimensionless enthalpy and magnetization (Model A) or (Model B). The initial average photon energy in the fluid frame is everywhere . In both simulations we observed how the compressive wave with amplitude steepened and formed a pair of shocks at time . We chose a sufficiently large optical depth of the steepening region and observed how the magnetic field and the trapped radiation were advected toward , building up a strong pressure maximum. This launched the RMS, as described in Section 2, and the two symmetric shocks continued to propagate away from . The amplitude of the shocks slowly grows as they propagate toward the asymptotic momentum of the converging flow .

Figure 9.— Snapshot of a shock propagating in the flow with upstream radiation enthalpy and upstream magnetization (upper panel) or 0.1 (lower panel). The shock is propagating to the left and the Thomson optical depth is measured from the site of shock formation (caustic of the initial supersonic wave). The solid curves show the profiles of momentum and proper density (normalized to the upstream proper density ). A strong subshock has formed in the magnetized case; it is highlighted by the grey strip. The subshock is resolved (not a discontinuous jump) due to a finite viscosity employed in the simulation of plasma dynamics; the small subshock thickness is controlled by viscosity. Radiation is everywhere simulated directly as a large collection of individual photons whose propagation and scattering is followed using the Monte-Carlo technique. The photon spectra measured at three locations (indicated as sp1, sp2, sp3) are shown in Figure 10.

Figure 9 shows one of the two symmetric shocks at in Model A (upper panel) and Model B (lower panel). By this time the shock has crossed a Thomson optical depth from its formation site, and the upstream momentum has reached . The shock structure is steady and propagating relative to the downstream with speed . The shock exhibits the jump conditions calculated in Section 3. In particular, the shock compression ratio in Model B is . The jump conditions also give a moderate ratio