2.1. Relativistic RMS

In sufficiently fast RMS ($\beta _{u} \gtrsim 0.5$) the temperature is high enough to allow rapid creation of $e^+e^-$ pairs inside the shock (Levinson & Bromberg Reference Levinson and Bromberg2008; Budnik et al. Reference Budnik, Katz, Sagiv and Waxman2010; Katz et al. Reference Katz, Budnik and Waxman2010; Ito, Levinson & Nagataki Reference Ito, Levinson and Nagataki2020a; Ito, Levinson & Nakar Reference Ito, Levinson and Nakar2020b). For $\gamma _{u}>1$ the pair multiplicity approaches $\gamma _{u} m_{p}/m_{e}$, completely dominating the shock opacity. Under such conditions electrostatic forces cannot provide coupling between pairs and ions since the electric field required to decelerate the ions exerts opposite forces on electrons and positrons. To illustrate this, a multifluid model for unmagnetized, relativistic RMS (RRMS) propagating in a pure hydrogen plasma has been developed recently (Levinson Reference Levinson2020). The analysis indicates that once the density of newly created positrons approaches the baryon density, which in RRMS occurs at the onset of the shock transition layer, the charge density, and ultimately the electric field, reverse sign. This leads to decoupling of the different species early on, and to the development of relative drifts between the different beams. The presence of a strong enough background magnetic field perpendicular to the shock velocity can lead to tight coupling, preventing velocity spreads (Mahlmann et al. Reference Mahlmann, Vanthieghem, Philippov, Levinson, Nakar and Fiuza2023).

The velocity separation between the different plasma constituents imposed by the radiation force in unmagnetized RRMS, is expected to induce a rapid growth of plasma instabilities. Linear stability analysis (Vanthieghem et al. Reference Vanthieghem, Mahlmann, Levinson, Philippov, Nakar and Fiuza2022) indicates a growth of various plasma modes, which ultimately become dominated by a current filamentation instability driven by the relative drift between the ions and the pairs. Particle-in-cell simulations (Vanthieghem et al. Reference Vanthieghem, Mahlmann, Levinson, Philippov, Nakar and Fiuza2022) validate these results and further probe the nonlinear regime of the instabilities, elucidating the ion-pair coupling by the microturbulent electromagnetic field. These simulations are local, in the sense that they encompass a kinetic-scale region inside the shock transition layer. The radiation force is modelled as a prescribed force acting solely on the pairs. The relevancy of such simulations to realistic RMS stems from the huge scale separation between radiation and kinetic physics.

An interesting result found in Vanthieghem et al. (Reference Vanthieghem, Mahlmann, Levinson, Philippov, Nakar and Fiuza2022) is a dependence of the coupling length on the pair multiplicity $\mathcal {M}$ (it scales roughly as $\mathcal {M}^{1/2}$). For the large multiplicity anticipated well inside the shock, $\mathcal {M}\approx \gamma _u m_{p}/m_{e}$, it becomes macroscopic, ${\sim }10^6 l_p$. This is still much smaller than the width of an infinite, planar RRMS, but might exceed the shock width during the breakout phase (see § 3.2 for further discussion).

While small-scale simulations allow us to capture more realistic values of the effective radiation force acting on leptons, they do not account for the kinetic and dynamical coupling between microturbulence, pair loading and spatially varying radiation force. Here, we extend the previous hybrid description to investigate this effect, by accounting for the contribution of Compton scattering and pair production from an idealized photon distribution as described in Levinson (Reference Levinson2020). Specifically, the photon density of the beam propagating away from the shock provides the magnitude of the lepton recoil from Compton scattering, while the density of each beam shapes the local injection rate. In this cold beam approximation, we do not resolve the energy-momentum tensor of the photon distribution, and hence, the energy of the pairs at injection is not constrained. As one would expect in a realistic configuration, we assume that the pairs are injected at rest into the local lepton frame with an ad hoc temperature. Below, we discuss a first attempt to perform global hybrid simulations of the precursor of a relativistic radiation-mediated shock with fluid photons and kinetic electrons, positrons and ions.

Figure 1 shows the results of a hybrid simulation performed using the OSIRIS code (Fonseca et al. Reference Fonseca, Silva, Tsung, Decyk, Lu, Ren, Mori, Deng, Lee and Katsouleas2002, Reference Fonseca, Vieira, Fiuza, Davidson, Tsung, Mori and Silva2013). In this example, plasma is injected into the right-hand side boundary with a Lorentz factor of $\gamma =10$, weak background magnetic field of $eB_z=0.1m_{e} \omega _{pe} c$ and reduced mass ratio of $m_{i}=25m_{e}$. We resolve the plasma skin depth $\Delta x = 0.7c/\omega _{pe}$ where $\omega _{pe}^2=4 {\rm \pi}\gamma _\infty n_\infty e^2/m_{e}$, $c\Delta t=0.5\Delta x$ corresponding to the Courant–Friedrichs–Lewy condition of the electromagnetic field solver introduced in Blinne et al. (Reference Blinne, Schinkel, Kuschel, Elkina, Rykovanov and Zepf2018) and extensively tested in Grošelj, Sironi & Beloborodov (Reference Grošelj, Sironi and Beloborodov2022). The lepton recoil is artificially enhanced to observe significant deceleration over the box size. The combined lepton deceleration and pair loading lead to progressive current filamentation of the background, followed by anomalous coupling of the different species.

Figure 1.

Structure of the RMS precursor for fiducial parameters described in the text. We determine the position $x=0$ as the tip of the photon beam. (a) Transverse magnetic field profile. (b) Density of the ions. (c) Density profile of the injected positrons in log-scale. As seen, the combined lepton deceleration and pair loading lead to progressive current filamentation of the background, followed by anomalous coupling of the different species.

2.2. Shocks propagating in multi-ion plasma

In reality, the medium into which the shock propagates in most of the aforementioned systems contains multiple ion species with different charge-to-mass ratio. In such situations, the RMS physics may be considerably altered. The main point to note is that the deceleration rate of ions inside the shock depends on the charge-to-mass ratio, and since the charge conservation condition in a multi-ion plasma is degenerate, a large velocity separation between the different ions in the postdeceleration zone is, in principle, allowed, even in subrelativistic RMS that are devoid of positrons, provided Coulomb collisions are not effective. As in RRMS, sufficiently strong magnetic fields can couple the ions; however, since the gyroradius of an ion of mass $Am_{p}$ is larger by a factor $Am_{p}/Zm_{e}$ than that of an electron (or positron), much stronger magnetic fields may be needed to couple all ions. Such strong fields may not be present in most relevant systems.

The structure of a Newtonian, multi-ion RMS has been explored recently using a semianalytic, multifluid shock model that can incorporate any number of ion species (Granot, Levinson & Nakar Reference Granot, Levinson and Nakar2023). The model invokes the diffusion approximation for the transfer of radiation through the shock, and computes the electrostatic coupling between the ions and electrons in a self-consistent manner, by solving the energy and momentum equations of the radiation and the multifluid plasma, together with Maxwell's equations, taking into account the electrostatic force acting on the charged fluids. An example is shown in figure 2 for two ion species. As seen, a substantial velocity separation is developed inside the shock, and is maintained in the post deceleration zone, where all forces (radiation and electrostatic) vanish. In practice, the relative drift between the different ion beams (and the electrons) is expected to lead to generation of plasma turbulence that will ultimately couple the ions, transferring their energy to radiation in the downstream. However, the dominant wave modes should differ from those found in a single-ion RRMS (Vanthieghem et al. Reference Vanthieghem, Mahlmann, Levinson, Philippov, Nakar and Fiuza2022).

Figure 2.

Velocity profiles of electrons (dashed line) and two ion species, $\alpha 1$ (solid red) and $\alpha 2$ (solid blue) inside a shock moving at a velocity of $\beta _u=0.3$. The charge-to-mass ratio in this example is $Z_{\alpha _1}/A_{\alpha _1}=0.47$ and $Z_{\alpha _2}/A_{\alpha _2}=0.4$, and the density ratio is $n_{\alpha _1}/n_{\alpha _2} = Z_{\alpha _2}/Z_{\alpha _1}$, where $Z_{\alpha _i}$ and $A_{\alpha _i}$ denote the atomic and mass numbers of species $\alpha _i$, respectively. The solid black line gives the velocity separation, $\beta _{\alpha _2} - \beta _{\alpha _1}$.

In general, the anomalous coupling length is expected to depend on the threshold drift velocity for the onset of the instability, on the saturation level of the turbulence, and conceivably other factors. In BNS mergers, where the plasma density ${\gtrsim }10^{25}\,{\rm cm}^{-3}$, the anomalous coupling length may be large enough to allow full velocity separation, leading to nuclear transmutations that might have important implications to kilonova emission (see § 3.3 for further discussion). In other systems, where the density is lower by many orders of magnitude, anomalous coupling likely occurs on scales much smaller that the radiation length. In that case, dissipation via anomalous friction should first lead to formation of collisionless subshocks before the energy will be converted to radiation. If the growth of the instability occurs well after decoupling, the ion temperature should be high enough to allow inelastic ion–ion collisions by random motions. A full kinetic study is needed to compute the detailed shock structure and assess whether nuclear reactions are expected in such systems.

3.1. Generation of non-thermal particle distributions and hard emission

The generation of magnetic turbulence in RRMS can potentially convert a fraction of the dissipated energy to non-thermal particles with power-law distributions. Both the increased non-adiabatic heating found in Vanthieghem et al. (Reference Vanthieghem, Mahlmann, Levinson, Philippov, Nakar and Fiuza2022) and the possible formation of power-law energy spectra can impact the shock breakout emission. In particular, a hard spectral component, extending well beyond current predictions of single-fluid models, may be present in the breakout signal.

3.2. Long range plasma scales at high pair multiplicity

Single-fluid models of finite RRMS (Granot et al. Reference Granot, Nakar and Levinson2018), indicate that during the shock breakout phase the width of shock transition layer decreases dramatically, owing to radiative losses. Since the scale over which pairs and ions couple becomes macroscopic when the pair multiplicity becomes large (see § 2.1), it could be that the coupling length will approach or even exceed the radiation scale during the shock breakout episode. If this indeed happens, it means that the RMS structure will be vastly different than that predicted by current RRMS models, which might have profound implications for the observed shock breakout signal (e.g. in the case of a shock breakout from a stellar surface a wider shock implies a softer and more energetic breakout radiation). Under such circumstances, global shock models may be necessary to compute the breakout dynamics and emission.

3.3. Abundance evolution and kilonova emission in BNS mergers

The expulsion of a relativistic jet following neutron star coalescence drives a fast RMS into the merger ejecta. The breakout of the shock from the high-velocity tail of the merger ejecta can produce a gamma-ray flash that, even though much fainter than a typical short GRB, can overwhelm the jet emission at large enough viewing angles (with respect to the jet axis). According to one scenario (Kasliwal et al. Reference Kasliwal, Nakar, Singer, Kaplan, Cook, Van Sistine, Lau, Fremling, Gottlieb and Jencson2017; Gottlieb et al. Reference Gottlieb, Nakar, Piran and Hotokezaka2018; Pozanenko et al. Reference Pozanenko, Barkov, Minaev, Volnova, Mazaeva, Moskvitin, Krugov, Samodurov, Loznikov and Lyutikov2018; Beloborodov, Lundman & Levin Reference Beloborodov, Lundman and Levin2020), the gamma-ray flash GRB 170817A that accompanied the gravitational wave signal GW 170818 was produced by such a process.

As explained in § 2.2, the propagation of the RMS through the merger ejecta can lead to a significant velocity separation of different rapid neutron-capture process (r-process) isotopes just downstream of the shock transition layer, provided that anomalous friction and Coulomb collisions  are not too effective. An estimate of the scale separation in RMS suggests that, under conditions anticipated in BNS mergers, the anomalous coupling length is likely to exceed the shock width (Granot, Levinson & Nakar Reference Granot, Levinson and Nakar2023). In that case, collisions of the different ion beams will induce nuclear transmutations in regions where the shock velocity exceeds the corresponding activation barriers. Recent analysis (Granot, Levinson & Nakar Reference Granot, Levinson and Nakar2023) indicates that the ion–ion collision length is smaller than the shock width, and that in regions where $\beta _u \gtrsim 0.2$, the collision energy may be large enough to induce fission and fusion of many elements. This can significantly alter the composition profile of r-process material behind the shock and, potentially, the kilonova emission if the change in composition affects the opacity and/or the radioactive energy deposition in the ejecta. In addition to inelastic ion collisions, neutron-rich isotopes downstream of the shock can undergo fission through the capture of free neutrons that cross the shock. The activation energy for neutron-induced fission ranges from practically zero for ${}^{235}U$ to a peak of approximately 40 MeV for elements of mass number $A\approx 100$, so a shock velocity of $\beta _u \gtrsim 0.1$ should give rise to fission of many elements. Whether free neutrons are sufficiently abundant at early times (during shock propagation) to significantly alter the ejecta composition is yet an open issue.