Core-collapse supernovae as cosmic ray sources

Abstract

Core-collapse supernovae produce fast shocks which pervade the dense circumstellar medium (CSM) of the stellar progenitor. Cosmic rays (CRs) if accelerated at these shocks can induce the growth of electromagnetic fluctuations in the foreshock medium. In this study, using a self-similar description of the shock evolution, we calculate the growth time-scales of CR-driven instabilities. We select a sample of nearby core-collapse radio supernova of type II and Ib/Ic. From radio data, we infer the parameters which enter in the calculation of the instability growth times. We find that extended IIb SNe shocks can trigger fast intra-day instabilities, strong magnetic field amplification, and CR acceleration. In particular, the non-resonant streaming instability can contribute to about 50 percent of the magnetic field intensity deduced from radio data. This results in the acceleration of CRs in the range 1–10 PeV within a few days after the shock breakout. In order to produce strong magnetic field amplification and CR acceleration, a fast shock pervading a dense CSM is necessary. In that aspect, IIn supernovæ are also good candidates. But a detailed modelling of the blast wave dynamics coupled with particle acceleration is mandatory for this class of object before providing any firm conclusions. Finally, we find that the trans-relativistic object SN 2009bb even if it produces more modest magnetic field amplification can accelerate CRs up to 2–3 PeV within 20 d after the outburst.

1. INTRODUCTION

High-energy cosmic rays (CRs) are likely accelerated in fast shocks produced in very energetic events (Bell 1978). CRs above an energy of 10¹⁷–10¹⁸ eV are expected to arise from extragalactic sources. Below this energy, the sources are thought to be Galactic. However, few Galactic sources meet the energy confinement limit constraint E_conf = ZeRB (Hillas 1984), where the particle Larmor radius R_L for a particle of charge Ze in a magnetic field of strength B equals the size of the source R. If the magnetic field strength is typical of interstellar medium (ISM) values, i.e. B ~ 3μG then the source size has to be large, at least ~10 pc for protons, in order for them to remain confined at energies up to ~10¹⁷ eV. This could be the case in superbubbles (Bykov 2001; Parizot et al. 2004). High-energy CRs can also be confined in sources with higher magnetic fields. This seems to be the case in young supernova remnants (SNRs). Many historical SNRs show thin X-ray filaments (see Parizot et al. 2006, and references therein). The width of the X-ray filaments allows one to deduce a lower limit on the post-shock (downstream) magnetic field. Some objects (Cassiopeia A, Tycho, and Kepler) have magnetic field strengths larger than 100 μG, much larger than the field that could be generated by a simple compression at the SNR forward shock of the ISM magnetic field.

The process of amplification of the magnetic field is unknown, although many theoretical ideas have been proposed. One argument is that TeV electrons which produce non-thermal X-rays are accelerated at the shock front with a magnetic field amplified by the acceleration of high-energy ions (mainly protons; Park, Caprioli & Spitkovsky 2015). Magnetic field amplification (MFA) then originates from plasma instabilities driven by CR ions (Bell 2004; Pelletier, Lemoine & Marcowith 2006; Marcowith, Lemoine & Pelletier 2006; Zirakashvili, Ptuskin & Völk 2008; Zirakashvili & Ptuskin 2008; Amato & Blasi 2009; Bykov, Osipov & Ellison 2011), a possibility that has been further tested numerically (Reville et al. 2008; Riquelme & Spitkovsky 2010; Reville & Bell 2013; Caprioli & Spitkovsky 2014b; Bai et al. 2015; van Marle, Casse & Marcowith 2018). It appears that while MFA is observed in a wide range of cases, final amplitudes of the magnetic field remain uncertain, mainly because of the limited-time dynamics of simulations that prevent them from reproducing the long time-scales on which particle acceleration evolves. An important argument raised in Bell (2004) is that the fastest instability, induced by CR current streaming ahead the shock front, has a growth rate ${\Gamma }_{\text{g}}\propto n_0^{1/2}{V}_{\text{sh}}^3$, where n₀ and V_sh are the ambient gas density and the shock velocity, respectively. Hence, the largest magnetic field fluctuation growth rates produced by energetic particles at an energy E are obtained in dense environments pervaded by fast shocks. Some authors (Schure & Bell 2013; Marcowith et al. 2014; Cardillo, Amato & Blasi 2015) have therefore pointed to the earliest stages of SN evolution (within months to years of explosion) as possible PeVatron accelerators. At the time of this study, the only available data on related high-energy gamma-ray emission is an upper limit from the very young SNR SN 1987A obtained with the HESS (High Energy Stereoscopic System) Observatory (H.E.S.S. Collaboration et al. 2015).

One possibility would be to search for gamma-ray emission at a very early expansion stage, when the forward shock is interacting with a very dense circumstellar medium (CSM). This is the case for core-collapse SNe (CCSNe), which evolve in the winds of their progenitor stars. For a constant mass-loss rate and wind velocity, the density of the surrounding CSM decreases as R⁻², and is thus highest at radii close to the star. If the wind parameters are not constant, the density decrease could be parametrized as R^−s, and the slope differs from the value of 2. The higher the stellar mass-loss rate, and lower the wind velocity, the higher will this density be at a given radius.

Recent theoretical efforts in this area concentrate on trans-relativistic SNe, superluminous SNe (SLSNe), and especially SNIIn, which are objects where either shock speeds exceed 0.1c or shocks pervade very dense CSM resulting from strong mass-loss rates ~10^−3/−2M_⊙ yr⁻¹ (Chakraborti et al. 2011). After the shock breakout, the forward shock travelling in the CSM becomes collisionless and CR may be accelerated efficiently, producing broadband non-thermal emission and high-energy neutrinos (Murase et al. 2011; Katz, Sapir & Waxman 2012; Ellison, Warren & Bykov 2013; Murase, Thompson & Ofek 2014; Zirakashvili & Ptuskin 2016), see however Giacinti & Bell (2015) for an alternative scenario. Budnik et al. (2008) and Ellison et al. (2013) pointed out that the maximum acceleration efficiency likely occurs in the trans-relativistic regime for in particular Ib/Ic SNe with shock Lorentz factors β_shγ_sh > 1. GeV gamma-rays and neutrinos appear to be the best opportunities to test-particle acceleration and CR production in SNe (Murase et al. 2014). However GeV photons associated with interaction-powered SNe have not been detected in a Fermi–LAT data search of a sample of 147 SNe of type IIn and Ibn, with the closest being at a distance of ~50 Mpc (Ackermann et al. 2015). A search of 45 SLSNe with the Fermi–LAT telescope (Renault-Tinacci et al. 2017) also did not find any excess γ-rays at the SLSN positions. The same conclusion has been found for TeV photons using the HESS Observatory (Simoni et al. 2017).

A proper evaluation of the gamma-ray and related multiwavelength/multimessenger emission during the early phase of blast wave expansion is the main purpose of this series of papers. In this first paper, we investigate particle acceleration efficiency at the SN blast wave shock evolving into a dense CSM. Following the approach adopted in Dwarkadas (2013), we derive a general formalism including SN dynamics and wind properties, which can be applied to any SN type where self-similar solutions (Chevalier 1982) are applicable. In this paper, we assume that the self-similar solutions are applicable even when some of the SN energy is expended in accelerating particles. This is a reasonable assumption provided that the CR pressure does not exceed ~10 percent of the gas pressure (Chevalier 1983; Kang & Ryu 2010). For the young SNe considered in this work, this would most likely be the case, as shown for SN 1993J (Tatischeff 2009).

The main hypothesis driving our study is that the previously invoked CR-driven plasma instabilities are at the origin of the magnetic field strength deduced from radio monitoring of SNe (see Marcowith et al. 2014 for a preliminary discussion and Bykov et al. 2018). Starting from this assumption, we adapt the theory of diffusive shock acceleration (DSA, Drury 1983; Berezhko & Ellison 1999) to the case of fast moving forward shocks expanding into the CSM produced by the wind of massive star SN progenitor. Within the adopted formalism, we discuss the different instabilities that may lead to MFA and test CR acceleration efficiency at the forward shock for different types of CCSNe. We also include an accurate treatment of the evolution of the CR maximum energy with time. However, it should be noted that the origin of the magnetic field deduced from radio observations remains elusive. It can either result from a dynamo process which develops at the contact discontinuity separating the shocked ejecta from the shocked CSM medium. The decelerating contact discontinuity can become Rayleigh–Taylor unstable, producing fingers of ejecta that stretch into the shocked CSM (Jun & Norman 1996; Björnsson & Keshavarzi 2017).

The results obtained herein are quite general, and applicable to any CCSN whose ejecta density profile can be described by a power law, and which propagates in a medium whose density also decreases as a power law, such that the resultant shock wave can be described by a self-similar solution. Specific calculations are made for the case of SN 1993J, mainly because the parameters for this SN are well known from extensive observations, and it has therefore often been used as a testbed for radio and gamma-ray observations (see Section 2).

The layout of this article is as follows. Section 2 describes the properties of SN 1993J, which is the fiducial object in this study. Section 3 details our model of shock dynamics and CSM, including wind density profile and wind magnetic field strengths. In particular Section 3.3 describes shock and magnetic field dynamics of a sample of radio SNe selected on the basis of the quality of their radio data. Section 4 presents the model of particle acceleration and MFA in SN shock waves. The maximum energy reached by CR particles is studied in Section 5. The main results of this work are discussed in Section 6 and a conclusion is finally given in Section 7.

2. SN 1993J

Supernova 1993J was discovered in a spiral arm of the galaxy M81 (Ripero et al. 1993) at a distance of 3.63 Mpc (Freedman et al. 2001). It subsequently became the optically brightest SN in the Northern hemisphere, and one of the brightest radio SNe ever detected. It resulted from the explosion of a massive star in a binary system with a progenitor mass ranging in the interval 13–20 M_⊙ (Maund et al. 2004). The star then evolved into a red supergiant (RSG) phase with a mass-loss rate of ~10⁻⁶ to 10⁻⁵M_⊙ yr⁻¹ and a slow wind V_w ~ 10 km s⁻¹ (see Tatischeff 2009).

SN 1993J is the best monitored SN at radio wavelengths (Bietenholz et al. 2010a; Martí-Vidal et al. 2011a). It is of particular interest to test-particle acceleration and gamma-ray radiation in fast shocks. Kirk, Duffy & Ball (1995) developed a model of DSA for a shock propagating in the dense CSM of SN 1987A and SN 1993J. For SN 1993J, the authors considered the case of a non-stationary wind producing a radial dependence of the CSM density ∝R^−3/2 (van Dyk et al. 1994). In that case, the gamma-ray flux produced by p–p interaction is enhanced and Kirk et al. (1995) predicted a peak gamma-ray flux F_γ (> 1 TeV) ~ 2 × 10⁻¹² photons cm⁻² s⁻¹. This first calculation appeared to be an overestimation because (1) subsequent calculations (Fransson & Björnsson 1998) showed that a constant mass-loss profile resulted in dynamics that were more consistent with radio measurements and (2) the mass-loss rate assumed was higher than in later publications.

Tatischeff (2009, T09 hereafter) used the non-linear DSA model of Berezhko & Ellison (1999) coupled with the self-similar hydrodynamics expansion solutions of Chevalier (1982) to calculate the radio synchrotron emission produced at the blast wave. He found that the magnetic field was strongly amplified in the blast wave region shortly after the explosion. Adopting the non-resonant streaming (NRS) instability (Bell 2004) as the main process for MFA, T09 found an upstream magnetic field strength in the shock precursor of B_u ~ 50(t/1 d)⁻¹ G for a shock propagating in a wind with constant mass-loss properties, and therefore a density profile ρ₀ ∝ R⁻². T09 also found that during the first ~8.5 yr after the explosion, about 19 percent of the total energy processed by the forward shock was used up in accelerating CRs. In this model, maximum CR energies are quickly reached only 2 d after the SN outburst with a peak energy E_max ~ 20 PeV for protons. Finally, accounting for the absorption of gamma-rays by the soft photons from the SN photosphere, using an isotropic gamma–gamma opacity, T09 found a peak gamma-ray flux above 1 TeV of F_γ (> 1 TeV) ~ 4 × 10⁻¹⁵ photons cm⁻² s⁻¹ about 270 d after the explosion, too low to be detected by the current Cherenkov telescope facilities. However, in the GeV domain gamma–gamma absorption is almost negligible, and T09 derived a peak flux F_γ (> 1 GeV) ~ 2 × 10⁻⁹ photons cm⁻² s⁻¹, still more than one order of magnitude below Fermi–LAT sensitivity.

Dwarkadas (2013) developed a general formalism to derive the gamma-ray flux including self-similar type ejecta and wind profiles. For the case of SN 1993J, he derived an unabsorbed gamma-ray flux one order of magnitude above the flux given by T09 at day 1 after outburst, but decreasing somewhat faster, as t^−1.17 rather than t⁻¹. Dwarkadas (2013) also confirmed that actual GeV/TeV flux levels are not detectable by any active gamma-ray facilities.

3. SHOCK DYNAMICS AND PROGENITOR WIND PROPERTIES

3.1. Shock dynamics

In this study, shock radius and velocity are assumed to evolve as a power law with time. The initial time after the SN outburst is t₀ and the corresponding shock radius is R₀. We have:

$$R_{\text{sh}}(t)=R_0\times {\left(\frac{t}{t_0}\right)}^m$$ (1)

and

$$V_{\text{sh}}(t)=\frac{R_{0}m}{t_0}\times {\left(\frac{t}{t_0}\right)}^{m-1}$$ (2)

We note V₀ = R₀m/t₀.

This formalism can be generalized to the case where the ejecta and surrounding medium are power laws, and a self-similar solution can be used to describe the evolution (Chevalier 1982). If we write the ejecta density of the SN as ρ_ej = At⁻³v^−k, and the surrounding medium density as ρ_cs = C r^−s, then a self-similar solution for the evolution of the forward shock can be written as (Chevalier & Fransson 1994):

$$R_{\text{sh}}(t)=\beta {\left(\frac{\alpha A}{C}\right)}^{1/(k-s)}{t}^{(k-3)/(k-s)}$$ (3)

where α is a constant given in Chevalier (1982), β is the ratio of the forward shock to the contact discontinuity radius R_sh/R_CD, and m, k, and s are linked by m = (k − 3)/(k − s). The value of k is inferred for different progenitors by Matzner & McKee (1999). It is close to 10.2 for RSGs, while it is assumed to be smaller for more compact Wolf–Rayet (WR) stars. The value of s needs to be inferred for each SN from the observations. Often, the value of s can be deduced from the X-ray light curves (Dwarkadas & Gruszko 2012). Type IIn SNe have values of s > 2 at late times after about 3 yr, while IIP SNe may have s ≈ 2.

If we consider the time in days t_d, we can write this as

$$R_{\text{sh}}(t)={(86400)}^{(k-3)/(k-s)}\beta {\left(\frac{\alpha A}{C}\right)}^{1/(k-s)}{t}_{\text{d}}^{(k-3)/(k-s)}.$$ (4)

By comparing equations (1) and (4), we can write:

$$R_0={(86400)}^m\beta {\left(\frac{\alpha A}{C}\right)}^{1/(k-s)}.$$ (5)

For SN 1993J, s = 2 and m = 0.83 (T09), thus k ~ 7.88. For a mass-loss rate of 3.8 × 10⁻⁵M_⊙ yr⁻¹ and a wind velocity of 10 km s⁻¹, we have C = 1.92 × 10¹⁴ g cm⁻¹. According to equation (2.4) in Chevalier & Fransson (1994), with the explosion energy as 10⁵¹ erg, and the ejected mass as 2.2 M_⊙ (T09), A=7.6 × 10⁷⁵. With α = 0.15 and β = 1.265 (Chevalier 1982), we get¹ R₀ ~ 3.43 × 10¹⁴ cm and V₀ ≃ 3.29 × 10⁹ cm s⁻¹.

In the following, we neglect the dynamical impact on the circumstellar wind of the radiation emitted at shock breakout. In reality, the flash of photons from breakout accelerates the layers of the wind close to breakout radius, R_bo (R_bo = R_⋆ for an optically thin wind), to a substantial fraction of the shock velocity at breakout. See, for example, Chevalier & Klein (1979). This reduces the size of the velocity discontinuity at the collisionless shock at early times, and thence the energy processed by this shock. However, photons are diluted as 1/R² at R > R_bo. Therefore, the impact of the radiation becomes negligible once the collisionless shock has reached a distance of only a few R_bo. For optically thin winds, this occurs at t < 1 d at most, which justifies our assumption.

3.2. Properties of the circumstellar medium

3.2.1. Wind density profile

The wind mass density scales as a power law with an index s which depends on the mass-loss history of the progenitor. For a steady wind (constant mass-loss rate and wind velocity) s = 2. The mass density experienced by the forward shock at a time t is, using equation (1),

$${\rho }_{\text{CSM}}(t)={\rho }_0{\left(\frac{R_{\text{sh}}(t)}{R_0}\right)}^{-s}={\rho }_0{\left(\frac{t}{t_0}\right)}^{-ms}$$ (6)

and

$${\rho }_0=\frac{\dot{M}\left(R_0\right)}{4\pi V_{\text{w}}\left(R_0\right)R_0^2}\simeq 1.3m_{\text{p}}{n}_{\text{H},0}$$ (7)

where the factor 1.3 accounts for the presence of a medium containing 90 percent H and 10 percent He, and m_p and n_{H, 0} are the proton mass and hydrogen density at t₀.

Numerically, we have for the CSM mass density at a radius R₀

$${\rho }_0\simeq \left[\frac{5.0\times {10}^{13}}{R_0^2}\text{ g }{\text{cm}}^{-3}\right]{\dot{M}}_{-5}\left(R_0\right)V_{\text{w},10}{\left(R_0\right)}^{-1}$$

where, the shock radius at t₀ is expressed in cm, the progenitor mass-loss rate $\dot{M}$ is expressed in units of 10⁻⁵M_⊙ yr⁻¹, and the wind asymptotic speed V_w is in units of 10 km s⁻¹. The mass-loss rate is derived at a fix radius R_ref = 10¹⁵ cm (see the discussion in Fransson, Lundqvist & Chevalier 1996). The mass-loss rate at R₀ is by definition given by $\dot{M}\left(R_0\right)=\dot{M}\left(R_{\text{ref }}\right){\left(R_{\text{ref }}/R_0\right)}^{2-s}$.

We consider the wind velocity to be constant with the radius. This assumption is justified as soon as min(R₀, R_ref) is larger than the stellar radius R_⋆. The main mechanism which drives RSG winds is not known yet: radiation pressure on dust grains, effect of magneto-acoustic waves, turbulent pressure due to convection and radiative pressure on molecular lines may contribute to mass ejection in these objects (Josselin & Plez 2007; Haubois et al. 2009; Aurière et al. 2010). The wind velocity evolution in RSG is therefore uncertain. Josselin & Plez (2007) use a tomography technique to probe line velocity profiles in the atmosphere of a sample of RSG. They find speeds in the range 10–30 km s⁻¹. We assume that beyond a few stellar radii, the wind is accelerated to its velocity at infinity. Hence, if min(R₀, R_ref) > R_⋆ then V_w(R₀) = V_w(R_ref). Unless otherwise specified, we choose R_⋆ ≃ 10³ R_⊙ ≃ 6.96 × 10¹³cm for RSG stellar radius. In the case of SN 1993J, we adopt ${\dot{M}}_{-5}\left(R_{\text{ref}}\right)\simeq 3.8$ and V_{w, 10}(R_ref) = 1 (see the discussion in T09). Both mass loss and wind velocity are taken identical at R₀ as R₀ ~ 5R_⋆ and R_ref ~ 14R_⋆ and because we set s = 2.

WR winds can be described using a CAK (Castor, Abbott & Klein 1975) profile for a radiation-driven wind, namely V_w(r) = V_w(∞)(1 − R_⋆/R)^b. The index b is not well constrained. Hillier (2003) invokes b values in the range 1–3 but also argues that WR winds have a structure different from O star radiation-driven winds, which implies that a single value of b cannot reproduce the wind velocity profile properly. Nugis & Lamers (2002) propose a model for the optically thick part of the wind where b depends on the WR type (WN or WC) and lies in a range between 2.9 and 6.5. Vink et al. (2011) find lower values for b in the range 1.5–2. Since WR have lost their outer envelopes and are more compact than RSG stars, the stellar radius R_⋆ is much smaller than R_ref and R₀, we identify V_w(R_Ref) and the wind velocity at infinity V_w(∞) and we assume V_w(R₀) = V_w(R_Ref).

The mass-loss properties of SN 1993J are not shared with the entire class of Type IIb SNe. Chevalier & Soderberg (2010) have suggested that there exist two sub-classes of Type IIb SNe, those with extended (eIIb) and those with compact (cIIb) progenitors. SN 1993J falls in the former category. The ones in the latter category may arise from compact progenitors such as WR stars. Their mass-loss rates are lower, and their wind velocities could be significantly higher by up to two orders of magnitude, leading to wind densities that are almost two orders of magnitude lower than that of SN 1993J. An example is SN 2008ax, which had a mass-loss rate a few to 10 times lower (Chornock et al. 2011; Roming et al. 2009), and whose spectrum bears similarity to Ib SNe, which are expected to arise from WR stars. More generally speaking, the wind properties of SN progenitor stars vary considerably with SN types. Mass-loss rates of massive stars are described in Smith (2014). Type IIP SNe are associated with RSG progenitors, and their wind velocities are similar to those of the type IIb SNe, lying between 5 and 20 km s⁻¹. One would expect their mass-loss rates to also span the total range of mass-loss rates of RSGs, from 10⁻⁷ to 10⁻⁴M_⊙ yr⁻¹ (Mauron & Josselin 2011). However, their X-ray emission indicates that their mass-loss rates lie on the lower end of the range, and do not appear to exceed 10⁻⁵M_⊙ yr⁻¹ (Dwarkadas 2014). Type Ib/c SNe are thought to arise from WR stars, whose radiatively driven winds have velocities of 1000–3000 km s⁻¹, and mass-loss rates ranging from 5 × 10⁻⁷ to 5 × 10⁻⁵M_⊙ yr⁻¹. The most diverse class is the type IIn SNe, which have the highest optical and X-ray luminosities, indicative of high mass-loss rates. A mass-loss rate of 10⁻³M_⊙ yr⁻¹ was noted for SN 2005kd (Dwarkadas et al. 2016), and a rate as high as 10⁻¹M_⊙ yr⁻¹ has been found for SN 2010jl (Fransson et al. 2014). Such high rates are not easily explained by stellar winds of RSG or WR stars, but are perhaps characteristic of eruptive outbursts from luminous blue variable stars. Not all type IIn SNe have such high mass-loss rates however. Fitting the radio light curves of SN 1995n suggested a mass-loss rate of 6 × 10⁻⁵M_⊙ yr⁻¹ (Chandra et al. 2009a), whereas hydrodynamical and X-ray modelling of SN 1996cr indicated an even lower mass-loss rate < 10⁻⁶M_⊙ yr⁻¹. The wind velocities of type IIn SNe are not well calibrated either. A velocity of ~100 km s⁻¹ is deduced from the narrow component of spectral lines in SN 2010jl. The expansion velocity of the CSM was found to be ~90 km s⁻¹ for SN 1997ab (Salamanca et al. 1998), ~45 km s⁻¹ for SN 1998S (Fassia et al. 2001), ~160 km s⁻¹ for SN 1997eg (Salamanca, Terlevich & Tenorio-Tagle 2002), and ~100 km s⁻¹ for SN 2002ic (Kotak et al. 2004). This suggests that velocities of 100 ± 50 km s⁻¹ for the winds of Type IIn SNe are common, but it should be emphasized that there could be much wider variation in these velocities. Mass-loss rates and wind velocities can also vary over time, sometime episodically, as often seems to happen near the end of a star’s life (Foley et al. 2007; Margutti et al. 2014). Type IIn SNe progenitors winds hence show complex structures which are difficult to properly account in a self-similar model. This statement should be kept in mind while considering some of these objects in Section 3.3.

3.2.2. Wind magnetic field

Magnetic field strength and topology in massive star winds are difficult to measure. Walder, Folini & Meynet (2012) review magnetic fields on the surface of massive stars and in their winds. Fields can be deduced from maser polarization observations using different type of tracers probing different media around the star. At growing distance from the star SiO₂, H₂O, and OH masers are used successively (Vlemmings, Diamond & van Langevelde 2002). There is usually no clear trend on the distance dependence of measured magnetic fields in the CSM of evolved stars; profiles with B ∝ R^−α with α = 1–3 can give reasonable fits to the data. Vlemmings et al. (2017) perform polarization analysis of circumstellar dust and molecular lines in the RSG star VY CMa. The authors found a polarization consistent with a toroïdal magnetic geometry, but higher angular resolution are necessary to confirm this trend. The magnetic field strength has some uncertainty but could be as high as 1–3 G. Aurière et al. (2010) obtain a longitudinal (along the line of sight) magnetic field strength of the order of 1 G in α Ori (Betelgeuse), the most well-studied RSG star. Gauss-level field strength has been confirmed in two other RSGs by Tessore et al. (2017). We can compare the above values to a magnetic field strength at the stellar surface obtained by a balance between magnetic field energy density and wind kinetic energy density (Fransson & Björnsson 1998; ud-Doula & Owocki 2002)²

$$B_{\text{eq},0}\simeq \left[\frac{2.5\times {10}^{13}}{R_0}\text{G}\right]{\dot{M}}_{-5}^{1/2}{V}_{\text{w},10}^{1/2}.$$

With stellar values appropriate for Betelgeuse, R₀ = 10R_⋆ ≃ 8.3 × 10¹⁴ cm, ${\dot{M}}_{-5}\simeq 0.3$, and V_{w, 10} ≃ 1.5 (Smith, Hinkle & Ryde 2009), we get B_eq ≃ 0.03 G. Hence, this value extrapolated at R_⋆ produces a magnetic field ≳0.3 G. But this extrapolation is highly sensitive to R₀ and to the radial dependence of B_eq. An uncertainty of an order of magnitude of the wind magnetic field with respect to B_eq is assumed in this study.

WR stars have fast winds which produce strong line broadening and hence make magnetic field measurements difficult. Hubrig et al. (2016) report on magnetic field measurements in a set of five WR stars, with strengths for the line of sight component in the range 200–300 G. As an example, we consider the particular object WR 6 (class WN4) in this sample, which shows a longitudinal magnetic component of B_z = 258 ± 78 G. As a matter of comparison, we can evaluate the equipartition magnetic field B_{eq, 0} using the stellar parameters derived from Nugis & Lamers (2000): R₀ = 10R_⋆ ≃ 6 × 10¹¹ cm, ${\dot{M}}_{-5}\simeq 0.62$ , V_{w, 10} ≃ 146, and assuming s = 2, for the wind speed profile, we find B_{eq, 0} ≃ 400 G. It is however hazardous to compare directly the two values as the observations provide only a mean longitudinal magnetic field strength.

From the above considerations, we assume in this study a CSM magnetic field strength proportional to B_eq with

$$B_{\text{w}}(t)\simeq \varpi B_{\text{eq},0}{\left(\frac{t}{t_0}\right)}^{\frac{-ms}{2}}$$ (8)

where we assume the ratio ϖ = B_w(t₀)/B_{eq, 0} to be in the range 0.1–10. The time dependence arises from the radial dependence of the wind density as mentioned in Section 3.1.

In equation (8) as soon as R(t) ≫ R_⋆, the wind magnetic field scales as 1/R_sh which is expected in case of a toroidal geometry. As discussed above, there is an important uncertainty on magnetic field topology and radial dependence in the wind of evolved massive stars. RSG winds also show quite inhomogeneous and turbulent structures (Smith et al. 2009). Except for the case ϖ ≫ 1, the magnetic field should also reflect such inhomogeneity and depart from a simple toroidal configuration.³

The ambient Alfvén velocity $V_{\text{A,CSM}}=B_{\text{W}}/\sqrt{4\pi {\rho }_{\text{CSM}}}=\varpi V_{\text{w}}$ and the CSM magnetization M = (V_A,CSM/c)² is

$$M\simeq \left[1.1\times {10}^{-9}\right]{\varpi }^2{V}_{\text{w},10}^2.$$

Considering ϖ to be in the range 0.1–10, we always obtain M ≪ 1 whatever the type of progenitor.

3.2.3. Ionization of the circumstellar medium

Another important parameter entering in the calculation of particle acceleration efficiency is the degree of ionization of the pre-shock medium. Neutrals have a strong impact over magnetic fluctuations which could raise around the shock front (O’C Drury, Duffy & Kirk 1996; Reville et al. 2007).

The ionization of the medium is a complex problem, since it depends not only on the progenitor star but on the presence of any nearby companions, or its location in an association or cluster of stars, which may also serve to ionize the medium.

While the SN explosion itself tends to ionize the medium, the densities close to the stellar surface are so high that recombination occurs quickly. As shown, for example in Dwarkadas (2014), for the mass-loss rates and velocity assumed for the wind medium around SN 1993J, the medium around the star that the shock traverses in the first few months can be considered to have recombined. However, the X-ray emission from the SN itself can ionize the medium. This depends on the quantity χ = L_x/(nR²), where L_x is the X-ray luminosity, n is the density, and R is the radius from the shock (Kallman & McCray 1982). Dwarkadas et al. (2016) write this in terms of the mass-loss rate and velocity. For a steady wind with constant mass-loss parameters, this reads as:

$$\chi =2\times {10}^{-38}{L}_{\text{x}}{\xi }^{-2}{\left[\frac{{\dot{M}}_{-5}}{V_{\text{w},10}}\right]}^{-1},$$ (9)

where ξ = [1 + 2n(He)/n(H)]/[1 + 4n(He)/n(H)] ~ 0.9. If we use the X-ray luminosities given in Chandra et al. (2009b), we find that χ > 100 for at least 11 d but less than 20 d. χ > 100 is required for ionization of intermediate elements like C, N, and O; ionization of heavier elements like Fe requires χ > 1000.

However this is an underestimate, because these luminosities are only in the 0.3–8 keV band. The X-ray emission early on has much higher temperatures, so the luminosity in this range is a small fraction of the total X-ray luminosity. In the case of SN 1993J, Leising et al. (1994) and Fransson et al. (1996) indicate that the total luminosity at 12 d was about 5.5 × 10⁴⁰ erg s⁻¹ in the 50–150 keV range, and even at day 28.5, it was 3.0 × 10⁴⁰ erg s⁻¹. This would suggest that the intermediate elements were ionized at least for the first month, and presumably longer. It is unlikely however that the medium was fully ionized except possibly in the first week – heavy elements such as Fe and Ni would presumably be partially but not fully ionized after a week.

An added complication is the presence of the companion star to SN 1993J. Fox et al. (2014) have shown that this is likely a B2 star, and as such may provide some additional UV ionizing flux. However, without more accurate details regarding its distance and observing surface temperature, it is difficult to estimate its effect.

Equation (9) can be written for a more general CSM density profile as (Dwarkadas 2014):

$$\chi =2L_{{\text{x}}_{38}}{\xi }^{-2}{\left[\frac{{\dot{M}}_{-5}}{V_{\text{w},10}}\right]}^{-1}{V}_4^{s-2}{\left[\frac{t_{\text{d}}}{8.9}\right]}^{s-2}.$$ (10)

where L_x38 is the X-ray luminosity in units of 10³⁸ erg s⁻¹, V₄ is the maximum ejecta velocity scaled to 10⁴ km s⁻¹, and t_d is the time in days. Since the X-ray luminosities of even the most luminous SNe are at most 10⁴² erg s⁻¹, we have $L_{{\text{x}}_{38}}\lesssim {10}^4$. The ionization parameter then depends on the slope of the density profile. For s < 2, the last two terms are raised to a negative power and the ionization parameter decreases with time, whereas for s > 2, it is increasing with time. However, the other quantities are generally decreasing with time. It is clear that the medium will rarely be fully ionized outside of the first few weeks at best, although partial ionization is likely for many SNe for a couple of years.

Neutrals can contribute to partially quench the growth of CR-driven instabilities (Reville et al. 2007). From the above discussion, the ionization fraction X = n_i/n_tot defined as the ratio of ion to total medium densities is likely close to 1 at least during the weeks after the explosion because light elements are fully ionized by the blast wave X-rays. Hence, CR-driven instabilities should be weakly corrected with respect to the fully ionized solution (Reville et al. 2007). However, a description of a long yearly term evolution of the shock environment and CR-driven instabilities would require more accurate modelling, which is beyond the scope of this work, but will be explored in future.

3.3. Properties of a selected sample of SNe

Today about 200 SNe have been detected at radio wavelengths, but only a few are sufficiently close to show well-resolved light curves. A first list of type II SNe detected by very long baseline interferometry (VLBI) includes: SN 1979C (SN IIL), SN 1986J (SN IIn), SN 1987A (SN IIpec), SN 1993J (SN IIb), SN1996cr (SN IIn), SN 2008iz (possibly SNIIb, see Mattila et al. 2013), and SN 2011dh (SN IIb) (Bartel, Karimi & Bietenholz 2017, and references therein).⁴ In this list, we discard SN 1987A and SN 1996cr: the former is interacting with a high-density Hii region and shell close in to the shock (Blondin & Lundqvist 1993; Chevalier & Dwarkadas 1995; Dewey et al. 2012), which cannot be reproduced in a self-similar model; while the latter has too scarce data at early epochs to carry out a proper evaluation of shock dynamics and magnetic fields (Bauer et al. 2008). We add to this list SN 2001gd (SN IIb) (Pérez-Torres et al. 2005; Stockdale et al. 2007) detected by VLBI, Very Large Array (VLA), and GMRT (Giant Metre Radio Telescope) facilities. We select a list of type Ib/c SNe based on the same data quality criterion: SN 1983N (SN Ib), SN 1994I (SN Ic), SN 2003L (SN Ic) (Weiler, Sramek & Panagia 1986; Soderberg et al. 2005). We also include SN 2009bb a relativistic Ibc SN (Soderberg et al. 2010).

The modelling of the radio light curves at different wavebands requires accounting for a number of processes: synchrotron self-absorption (SSA), free–free absorption (FFA) by the ambient thermal plasma (internal FFA) or by CSM matter (external FFA), and plasma processes like the Razin–Tsytovich effect (see Fransson & Björnsson 1998), although the last one has generally not been found to be important in supernovæ. Radio observations are important for many aspects of particle acceleration modelling. First, the spectral turnover produced by SSA leads to an estimate of the magnetic field intensity of the synchrotron emitting zone. Second, the synchrotron spectral index provides a constraint on the electron distribution function and then on the acceleration process. Third, radio images are used to derive the SN shell dynamics and the time evolution of the shock radius and velocity, which are mandatory for any microphysical calculations of particle acceleration efficiency (see T09). The shell radius and speed can also be compared to a self-similar expansion model (see Section 3.1 and Chevalier 1982).

The properties of the radio emission depend on the SN type: Type Ib/c SNe show steep spectral indices (α > 1, with a radio flux scaling as S_ν ∝ ν^−α), and have similar radio luminosity peaking before optical maximum at a wavelength around 6 cm, while Type II SNe show flatter spectra (α < 1) with a wider range of radio luminosities usually peaking at 6 cm significantly after optical maximum (Weiler et al. 2002). We summarize in Tables 1 and 2, the main properties of the above SNe (when available) that can be deduced from VLBI observations. Apart from α, we use the following notations: the magnetic field strength is characterized by its amplitude B₀ at a reference time t₀ to be specified and by its time dependence given by B(t) = B₀(t/t₀)⁻ⁿ. The VLBI shell radius is characterized by its radius at t₀, R₀, and by the index m > 0 (see equation 1). We also show the deduced progenitor mass-loss rate in units of 10⁻⁵ M_⊙ yr⁻¹. To derive this last parameter, it is usually necessary to make an assumption on the progenitor wind speed. This is also specified in both tables.

Table 1.

Magnetic field and shock radius evolution for a set of type II SNe. The following references have been used for the different sources in the list: SN 1979C (Weiler et al. 1991; Marcaide et al. 2009; Martí-Vidal, Pérez-Torres & Brunthaler 2011b; Lundqvist & Fransson 1988), SN 1986J (Weiler, Panagia & Sramek 1990; Bietenholz, Bartel & Rupen 2010b; Martí-Vidal et al. 2011b), SN 1993J upper row (Fransson & Björnsson 1998) and lower row (T09), SN2001gd (Pérez-Torres et al. 2005), SN 2008iz (Kimani et al. 2016; Marchili et al. 2010), and SN 2011dh upper row (Horesh et al. 2013) and lower row (Krauss et al. 2012; Yadav, Ray & Chakraborti 2016). The spectral index α corresponds to the optically thin synchrotron spectrum. SN 1979C and SN 1986J: the magnetic field strength derived by Martí-Vidal et al. (2011b) is obtained at t₀ = 5 d from a synchrotron model. Upper and lower values for B₀ are associated with the uncertainties of α and m. The value of n is obtained from the solution of propagation in a wind with constant mass-loss rate with s = 2. The radius R₀ at day 5 is extrapolated from angular size measurements obtained by VLBI observations. SN 1993J: Fransson & Björnsson (1998) use an expansion index m = 0.74 at t > 100 d, Martí-Vidal et al. (2011a) present a long-term radio survey where the expansion law index at frequencies above 1.7 GHz varies from m_< = 0.925 ± 0.016 before t_br = 360 ± 50 d to m_> = 0.87 ± 0.02 after. As the index is not specified at time-scales close to the breakout, we use m = 1. SN2001gd: the magnetic field strength derived by Pérez-Torres et al. (2005) in the same manner as in Martí-Vidal et al. (2011a). Mass-moss rates values are consistent with the results presented in Stockdale et al. (2007) where a value for s = 1.61, i.e. different from a steady mass loss, has been used. SN2008iz: Kimani et al. (2016) derive the equipartition magnetic field strength, the upper and lower limits depend on whether protons are accounted for in the estimation or not. SN 2011dh: we keep the two expansion parameter solutions derived by Horesh et al. (2013) and Krauss et al. (2012), respectively. Note that de Witt et al. (2016) reported on a value of the expansion parameter m = 0.91 ± 0.01 compatible with the Horesh et al solution.

SN name	α	t0(d)	(R0(cm), m)	$\dot{M}\left({10}^{-5}{\text{M}}_{\odot }{\text{yr}}^{-1}\right)$	Vw(10 km s−1)	(B0(G), n)
SN 1979C	${0.74}_{-0.08}^{+0.05}$	5	(8.7(e14), 0.91 ± 0.09)	12	1	([20 – 30], −1.00)
SN 1986J	${0.67}_{-0.08}^{+0.04}$	5	(3.2(e15), 0.69 ± 0.03)	4–10	1	([30 – 50], −1.00)
SN 1993J	1.00	10	(1.9(e15), 1.00)	5	1	(25.5, −0.93 ± 0.08)
	0.90	100	(1.6(e16), 0.829 ± 0.005)	3.8	1	(2.4 ± 1.0, −1.16 ± 0.20)
SN	1.0 ± 0.1	500	(3.6(e16), 0.845)	2–12	1	(0.05–0.35,−)
2001gd
SN 2008iz	1.00	36	(8.8(e15), 0.86 ± 0.02)	3.7	1	([0.4 – 3.2], −1.00)
SN	1.15	4	(5.0(e14), 1.14 ± 0.24)	0.01	1	(5.9, −1.00 ± 0.12)
2011dh
	0.95	15	(3.1(e15), 0.87 ± 0.07)	0.07	2	(1.1, −1.00)

Table 2.

Magnetic field and shock radius evolution for a set of type Ib/Ic SNe, adapted from the following references: SN 1983N (Sramek, Panagia & Weiler 1984; Weiler et al. 1986; Slysh 1992), SN 1994I (Weiler et al. 2011; Alexander, Soderberg & Chomiuk 2015), SN 2003L (Soderberg et al. 2005), and SN 2009bb (Soderberg et al. 2010; Chakraborti et al. 2011). SN 1983N: the magnetic field strength dependence with time is not available, and is therefore assumed to be ∝t⁻¹. Slysh (1992) only derives estimates of an upstream magnetic field strength of 0.9 G which is multiplied by 4 to account for shock compression. Sramek et al. (1984) derive a mass-loss rate of ~5 × 10⁻⁶M_⊙ yr⁻¹ with V_w = 10 km s⁻¹ but decide to fix V_w = 1000 km s⁻¹, a value which seems more reasonable for this type of SN. SN 1994I: the fitted parameters of the model derived by Alexander et al. (2015) do not show a simple power-law time dependence for both B and R. Alexander et al. (2015) invoke two extreme mass-loss rates depending on the progenitor terminal wind speed. Weiler et al. (2011) deduce from radio observations a mass-loss rate of $\dot{M}~2\times {10}^{-7}{\text{M}}_{\odot }{\text{yr}}^{-1}$ for V_w = 10 km s⁻¹. We again select the solution corresponding to a wind velocity V_w = 1000 km s⁻¹. Alexander et al. (2015) use an expansion index m = 0.88. However, Björnsson & Keshavarzi (2017) criticized this assumption deduced from a self-similar solution. We assume by default this parameter to be 1.0 (corresponding to the free expansion solution). SN 2003L: the solution corresponds to model 1 of Soderberg et al. (2005). The authors provide two other models with faster shock deceleration. Their model 2 also includes a shallower CSM density profile with ρ ∝ r^−1.6. SN 2009bb: Soderberg et al. (2010) do not specify the terminal wind speed assumed in order to calculate the mass-loss rate. We take V_w = 1000 km s⁻¹ by default.

SN name	α	t0(d)	(R0(cm), m)	$\dot{M}\left({10}^{-5}{\text{M}}_{\odot }{\text{yr}}^{-1}\right)$	Vw(1000 kms−1)	(B0(G), n)
SN 1983N	1.03 ± 0.06	13	(2.3(e15), 0.81)	50	1	(3.6, −1.00?)
SN 1994I	1.22	10.125	(2.4(e15), 1.00)	3	1	(2.28, −)
SN 2003L	1.1	10	(4.3(e15), 0.96)	0.75	1	(4.5, −1.00)
SN 2009bb	1.0	20	(4.4(e16), 1.00)	0.2 ± 0.02	1 ?	(0.6, −1.00)

4. DIFFUSIVE SHOCK ACCELERATION AND MAGNETIC FIELD AMPLIFICATION

This section describes the model of particle acceleration and MFA after the SN outburst.

In Sections 4.1 and 4.2, we apply the model to SN 1993J, using the R and B parameters derived by T09, but extrapolated at t₀ = 1 d. It should be kept in mind that these extrapolated values may not be precise, as T09 and Fransson & Björnsson (1998) only derived them at t₀ = 100 and 10 d, respectively, from the modelling of radio light curves (see also the footnote of Section 3.1).

4.1. Acceleration models

We adopt a model for particle acceleration at collisionless shocks based on the theory of DSA (Drury 1983). The highest CRs have an upstream diffusion coefficient κ_u which fixes the length scale of the CR precursor ℓ_u = κ_u/V_sh. The time-scale to advect the frozen CR-magnetized fluid to the shock front is

$$T_{\text{adv},\text{u}}=\frac{{\kappa }_{\text{u}}}{V_{\text{sh}}^2}.$$ (11)

CRs at energies close to E_max stream ahead of the shock and simultaneously generate electromagnetic fluctuations. The upstream diffusion coefficient at these energies can be expressed with respect to the diffusion coefficients parallel and perpendicular to the background wind magnetic field. This coefficient depends on two parameters (Jokipii 1987): η, the ratio of the parallel CR mean-free path to CR Larmor radius R_L, and θ_B the magnetic field obliquity. We write the parallel diffusion coefficient as κ_∥ = ηR_Lv/3, where v is the particle speed. The minimum value η = 1 corresponds to the Bohm diffusion limit. In parallel shocks (θ_B = 0) κ_u = κ_u, while in perpendicular shocks (θ_B = π/2), it matches the perpendicular diffusion coefficient, i.e. κ_u = κ_⊥. Without considering magnetic field line wandering in the wind turbulent medium, we have κ_⊥ = κ_∥/(1 + η²). Hence, if η ≫ 1 diffusion is suppressed in the perpendicular shock case. If the magnetic field in the wind is purely toroidal and weakly turbulent the advection time-scale T_{adv, u} drops. On the other hand, if the wind medium has some level of turbulence (see the discussion in Section 3.2), then we can expect to have a diffusion coefficient close to Bohm (η ~ 1), and to have a non-negligible portion of the shock in the parallel configuration.

We define as models P and T, the two extreme configurations described above. In model P, the wind magnetic field is assumed to be parallel. The advection time is in this case

$$T_{\text{adv},\text{u},\text{P}}\simeq \frac{{\eta }_{\text{P}}{R}_{\text{L}}v}{3V_{\text{sh}}^2}.$$ (12)

We account for some turbulence in the wind medium and include a contribution due to perturbations in the wind magnetic field, δB_u, which is assumed to be in equipartition with the mean field strength $B_{\text{w},0}:B_{\text{w}}^2=\delta B_{\text{u}}^2+B_{\text{w},0}^2$. This turbulence is assumed to be injected at large wind scales, typically the wind termination shock radius, and δB_u ≃ B_{w, 0} at the highest CR energies.

Using equation (8) for the wind mean magnetic field and expressing the proton Larmor radius R_L ≃ E/eB_w for a particle energy of 1 PeV as

$$R_{\text{L}}\simeq 3.3\times {10}^{12}{E}_{\text{PeV}}{B}_{\text{w},\text{G}}^{-1}\text{cm}$$ (13)

we find an advection time (expressed in seconds)

$$\begin{gathered} T_{\text{adv},\text{u},\text{P}}\simeq \left[\frac{1.3\times {10}^9{\eta }_{\text{P}}{R}_{0,\text{ cm}}}{V_{0,\text{ cm}/\text{s}}^2\varpi }\text{s}\right] \\ \times \frac{E_{\text{PeV}}}{{\dot{M}}_{-5}^{1/2}{V}_{\text{w},10}^{1/2}}{\left(\frac{t}{t_0}\right)}^{2(1-m)+m\frac{s}{2}}. \end{gathered}$$ (14)

In model T, the wind mean magnetic field is assumed to be toroidal and weakly perturbed with fluctuations of strength δB_{u, w} < B_{w, 0} ≃ B_w. The advection time is in this case

$$T_{\text{adv},\text{u},\text{T}}\simeq \frac{R_{\text{L}}v}{3{\eta }_{\text{T}}{V}_{\text{sh}}^2}.$$ (15)

For the parameters adopted for SN 1993J, we have $T_{\text{adv,u,T}}\simeq (0.24\text{ d})\times \left(1/{\eta }_{\text{T}}\varpi \right){\text{E}}_{\text{PeV}}{\text{t}}_{\text{d}}^{1.17}$ and $T_{\text{adv,u,P}}\simeq (0.24\text{ d})\times \left({\eta }_{\text{P}}/\varpi \right){\text{E}}_{\text{PeV}}{\text{t}}_{\text{d}}^{1.17}$, hereafter t_d is the time in units of days after the SN explosion.

We can deduce the acceleration time-scale from the above estimates

$$T_{\text{acc},\text{P}}=g(r)T_{\text{adv},\text{u},\text{P}}=g(r)\frac{{\kappa }_{\text{u}}}{V_{\text{sh}}^2}$$ (16)

where g(r) = 3r/(r − 1) × (1 + κ_dr/κ_u) depends on the shock compression ratio r and on the ratio of the downstream to upstream diffusion coefficients. The ratio κ_d/κ_u depends on the magnetic field obliquity and on the shock compression ratio r. We have ${\kappa }_{\text{d}}/{\kappa }_{\text{u}}=r_{\text{B}}^{-1}$ with r_B = B_d/B_u is the ratio of magnetic fields in the post-shock region and in the wind and g(r) = 3r/(r − 1) × (1 + r/r_B). In the model P, we have r_B ≃ 1 and g(r) = 3r(r + 1)/(r − 1). In the model T, the magnetic field is weakly perturbed and perpendicular to the shock normal and κ_d/κ_u = r⁻¹ and g(r) = 6r/(r − 1).

The calculations in this section have not assumed any MFA. MFA at the shock precursor by the streaming of high-energy CRs produces a reduction of the precursor length. In the case of Bohm diffusion, the reduction factor corresponds to the ratio of the amplified to ambient magnetic field strengths. If the perturbations in the precursor are isotropic then $r_{\text{B}}\simeq \sqrt{\left(1+2r^2\right)/3}$ (Parizot et al. 2006) and g(r) is modified accordingly.

4.2. Magnetic field amplification

In this section, we discuss different CR-driven instabilities that may operate at the SN forward shock and generate magnetic field fluctuations necessary for the DSA process to operate at a high efficiency. All calculations are performed in the framework of model P, but we discuss the case of particle acceleration in model T in Section 4.2.5.

In this work, we do not consider fluid instabilities triggered by the CR pressure gradient in the precursor (Drury & Downes 2012). Fluid perturbations generate magnetic field fluctuations through a small-scale dynamo process (Beresnyak, Jones & Lazarian 2009). The magnetic field growth can be fast if the velocity stretching in the CR precursor is strong enough. This happens when the shock is modified by the CR pressure. We did not consider this possibility in this work and assumed that the shock modification is weak (see T09). We postpone to a future study the case of strongly CR-modified shocks and their impact over the different instabilities which may grow in the CR precursor.

4.2.1. Bell non-resonant streaming instability

Bell (2004) discussed a process of MFA in SNRs driven by currents produced by the streaming of CRs ahead of the shock front. The CR streaming induces a return current in the background plasma, which triggers magnetic fluctuations at scales ℓ ≪ R_L, where R_L is the Larmor radius of the CRs producing the current. This instability is non-resonant and can be treated using a modified magnetohydrodynamic (MHD) model (Bell 2004, 2005; Pelletier et al. 2006).

Growth time-scale.

Bell (2004) gives the minimum growth time-scale (corresponding to the maximal growth rate) for the NRS instability

$$T_{\text{min},\text{NRS}}=\frac{2\varphi }{{\xi }_{\text{CR}}}\frac{R_{\text{L}}c}{V_{\text{sh}}^3}{V}_{\text{A},\text{CSM}}$$ (17)

where ϕ ≃ ln (E_max/m_pc²) is fixed by the maximum CR energy. This expression implicitly assumes that the CR distribution at the shock scales with the particle momentum as p⁻⁴. In other words, the CR distribution is close to the test-particle solution and a softer distribution scaling as p^−k with k > 4 would lead to ϕ ≃ 1/(k−4). T09 finds that this is a reasonable assumption for the case of SN 1993J and we assume that this condition holds true in this study. A more detailed modelling including the effects of CR backreaction is beyond the scope of this paper, but will be addressed in future work. In equation(17), we take ξ_CR to be the fraction of the shock ram pressure imparted to CRs. T09 finds ξ_CR = ξ_CR0 × (t/t₀)^{1 −m} ∝ 1/V_sh. This scaling is in a strict sense only valid as long as CR backreaction over shock dynamics is weak. Then, we can write ${\xi }_{\text{CR}}\propto p_{\text{inj}}/V_{\text{sh}}^2$, where p_inj is the CR injection momentum. It varies as p_th, the momentum of thermal shocked plasma, itself being proportional to V_sh (Blasi, Gabici & Vannoni 2005). Once P_CR is beyond a certain fraction of shock ram pressure, the acceleration process becomes non-linear and CR escaping upstream carry an increasing energy flux which backreacts over the injection process and hence, this scaling should be revised and the injection process becomes less efficient. The inclusion of this non-linear physics is beyond the scope of this simple study and will be addressed in a forthcoming work.

Using equation (8), the NRS growth rate reads (using V_{A, CSM} = ϖV_w)

$$\begin{gathered} T_{\text{min},\text{NRS}}\simeq \left[\frac{2.2\times {10}^{18}{R}_0}{V_0^3}\text{ s}\right]\frac{{\varphi }_{14}}{{\xi }_{\text{CR},0.05}}{E}_{\text{PeV}} \\ \times {\dot{M}}_{-5}^{-1/2}{V}_{\text{w},10}^{1/2}{\left(\frac{t}{t_0}\right)}^{2(1-m)+\frac{ms}{2}}. \end{gathered}$$ (18)

We assume E_max ≃ 10¹⁵ eV and we note ϕ₁₄ = ϕ/14.

Advection constraint.

A necessary condition for the NRS instability to grow is R_NRS = T_min,NRS/T_adv,u < 1, hence using equations (18), this condition reads

$$R_{\text{NRS}}\simeq \left[\frac{{1.710}^9\varpi }{\eta V_0}\right]\frac{{\varphi }_{14}}{{\xi }_{\text{CR}0,0.05}}{V}_{\text{w},10}<1$$ (19)

which is independent of time because ξ_CR scales as 1/V_sh. In the model P, we have η ~ 1 (we drop the subscript p hereafter). In the case of SN 1993J, this gives R_NRS ≃ 0.5ϖϕ₁₄/ξ_CR0,0.05. The equipartition parameter ϖ is important, we see that if ϖ > 2, the NRS instability can not grow. The most favourable condition for the instability to grow is the case of sub-equipartition wind magnetic field.

A useful quantity is the wavenumber corresponding to the maximum NRS instability growth rate. It is given by

$$k_{\text{max}}^{-1}=T_{\text{min},\text{NRS}}{V}_{\text{A},\text{CSM}}.$$ (20)

Using equation (18), we express it in cm units

$$\begin{gathered} k_{\text{max}}^{-1}\simeq \left[\frac{2.2\times {10}^{24}{R}_0}{V_0^3}\text{ cm}\right]\frac{{\varphi }_{14}\varpi }{{\xi }_{\text{CR}0,0.05}}{E}_{\text{PeV}} \\ \times {\dot{M}}_{-5}^{-1/2}\times V_{\text{w},10}^{1/2}{\left(\frac{t}{t_0}\right)}^{2(1-m)+m\frac{s}{2}}, \end{gathered}$$ (21)

which for SN 1993J is

$$k_{\text{max}}^{-1}\simeq 2.1\times {10}^{10}\text{ cm}\frac{{\varphi }_{14}\varpi }{{\xi }_{\text{CR}0,0.05}}{\text{E}}_{\text{PeV}}{\text{t}}_{\text{d}}^{1.17},$$

where again t_d is the time in day units.

In Pelletier et al. (2006), the saturation magnetic field is (see their equation 28)

$$B_{\text{sat},\text{NRS}}^2=12\pi \frac{{\xi }_{\text{CR}}}{\varphi }{\rho }_{\text{CSM}}\frac{V_{\text{sh}}^3}{c}.$$ (22)

Hence, we have

$$\begin{gathered} B_{\text{sat,NRS}}\simeq {\left[224\frac{V_0^3}{R_0^2}\text{G}\right]}^{1/2}\sqrt{\frac{{\xi }_{\text{CR}0,0.05}}{{\varphi }_{14}}} \\ \times {\dot{M}}_{-5}^{1/2}{V}_{\text{w},10}^{-1/2}{\left(\frac{t}{t_0}\right)}^{m-1-\frac{ms}{2}}. \end{gathered}$$ (23)

For s = 2, B_{sat, NRS} varies as t⁻¹. A typical value of the NRS saturation field in the case of SN 1993J is $B_{\text{sat,NRS}}\simeq 16\text{G}\sqrt{{\xi }_{\text{CR}0,0.05}/{\varphi }_{14}}\times {\text{t}}_{\text{d}}^{-1}$, a result already obtained in T09.

Using equations (8) and (23), we define the amplification factor, the ratio of the saturation and wind magnetic fields:

$$\begin{gathered} A=\frac{B_{\text{sat,NRS}}}{B_{\text{w}}}\simeq \left[\frac{6.0\times {10}^{-13}{V}_0^{3/2}}{\varpi }\right]\frac{1}{V_{w,10}} \\ \times \sqrt{\frac{{\xi }_{\text{CR}0,0.05}}{{\varphi }_{14}}}{\left(\frac{t}{t_0}\right)}^{m-1}. \end{gathered}$$ (24)

For SN 1993J, we obtain an amplification factor $A\simeq 113/\varpi \times \sqrt{{\xi }_{\text{CR}0,0.05}/{\varphi }_{14}}\times t_{\text{d}}^{-0.17}$, slowly decreasing with time.

In Sections 4.2.2–4.2.4, we calculate the growth rate of instabilities which produce long-wavelength perturbations. These perturbations are necessary to confine high-energy CRs around the shock front.

4.2.2. Resonant streaming instability

The streaming of CRs faster than the local Alfvén speed is known to produce long-wavelength modes at scales ℓ ~ R_L. To derive the instability growth rate, we follow the treatment presented in Amato & Blasi (2009). We introduce the following parameter $\sigma =3{\xi }_{\text{CR}}{V}_{\text{sh}}^3/\varphi c$. We have

$$\sigma \simeq \left[3.5\times {10}^{-13}{V}_0^3\right]\frac{{\xi }_{\text{CR}0,0.05}}{{\varphi }_{14}}{\left(\frac{t}{t_0}\right)}^{2(m-1)}{\text{cm}}^2{\text{s}}^{-2}$$

and

$$\frac{\sigma }{V_{\text{A,CSM}}^2}\simeq \left[3.5\times {10}^{-25}{V}_0^3\right]\frac{{\xi }_{\text{CR}0,0.05}}{{\varpi }^2{V}_{\text{w},10}^2{\varphi }_{14}}{\left(\frac{t}{t_0}\right)}^{2(m-1)}.$$

We note in passing that the NRS mode exists only in a particular wavenumber interval k ∈ [k₁, k₂] with $k_1{R}_{\text{L}}=\sqrt{\pi }/2\sqrt{\sigma /V_{\text{A,CSM}}^2}$ and $k_2{R}_{\text{L}}=\sigma /V_{\text{A,CSM}}^2$. It disappears if $\sigma /V_{\text{A,CSM}}^2=\pi /4$ (Amato & Blasi 2009).

For long wavelengths (with wavenumbers such that kR_L < 1), the minimum resonant streaming (RS) growth rate in the non-linear regime is given by (see Amato & Blasi (2009), their equation 40)

$$T_{\text{min,RS}}\simeq R_{\text{L}}\sqrt{\frac{8}{\pi \sigma }}.$$ (25)

We find using equation (8)

$$\begin{gathered} T_{\text{min,RS}}\simeq \left[\frac{3.6\times {10}^5{R}_0}{\varpi V_0^{3/2}}\text{ s}\right] \\ \times \frac{E_{\text{PeV}}}{{\dot{M}}_{-5}^{1/2}{V}_{\text{w},10}^{1/2}}{\left(\frac{{\varphi }_{14}}{{\xi }_{\text{CR}0,0.05}}\right)}^{1/2}{\left(\frac{t}{t_0}\right)}^{1-m+m\frac{s}{2}}. \end{gathered}$$ (26)

Similar to the NRS instability case, we define the ratio of the RS growth time to the advection time equation (14)

$$R_{\text{RS}}\simeq \left[\frac{2.7\times {10}^{-4}\sqrt{V_0}}{\eta }\right]{\left(\frac{{\varphi }_{14}}{{\xi }_{\text{CR}0,0.05}}\right)}^{1/2}{\left(\frac{t}{t_0}\right)}^{(m-1)}.$$ (27)

For SN 1993J, this ratio is $~15/\eta \sqrt{{\varphi }_{14}/{\xi }_{\text{CR}0,0.05}}{t}_{\text{d}}^{-0.17}$ but we note that it decreases with time, and it may rapidly be lower than 1 if η is larger than a few.

Bell & Lucek (2001) [their equation (14) in regime A⁵] give the level of saturation of the magnetic field if only the resonant mode is destabilized:

$$B_{\text{sat,RS}}=\frac{B_{\text{W}}}{2}{\xi }_{\text{CR}}\frac{V_{\text{sh}}}{V_{\text{A,CSM}}}.$$ (28)

or using equations (6) and (7)

$$B_{\text{sat,RS}}\simeq \left[\frac{6.3\times {10}^5{V}_0}{R_0}\text{G}\right]\frac{{\xi }_{\text{CR}0,0.05}{\dot{M}}_{-5}^{1/2}}{V_{\text{w},10}^{1/2}}{\left(\frac{t}{t_0}\right)}^{-m\frac{s}{2}}.$$ (29)

This estimate accounts for the non-linear growth of the resonant waves. But if the non-resonant mode is also destabilized the total magnetic saturation magnetic field is (Pelletier et al. 2006) (their equation 37)

$$B_{\text{sat,RS}}=B_{\text{sat,NRS}}{\left(\frac{{\xi }_{\text{CR}}c}{V_{\text{sh}}}\right)}^{\frac{1}{4}}$$ (30)

where the multiplicative term is

$${\left(\frac{{\xi }_{\text{CR}}c}{V_{\text{sh}}}\right)}^{1/4}\simeq 197{\left(\frac{{\xi }_{\text{CR}0,0.05}}{V_0}\right)}^{1/4}{\left(\frac{t}{t_0}\right)}^{\frac{(1-m)}{2}}.$$ (31)

Using equation (23), we find in that case that $B_{\text{sat,RS}}\propto {\left(\frac{t}{t_0}\right)}^{\frac{(m-1)}{2}-m\frac{s}{2}}$.

4.2.3. Filamentation instability

Reville & Bell (2012) demonstrate that CRs form filamentary structures in the precursors of SNR shocks due to their self-generated magnetic fields. They show that the filamentation resulted in the growth of a long-wavelength instability, with a minimum growth time (given by their equation 13)

$$T_{\text{min,Fil}}=\sqrt{\frac{\varphi }{{\xi }_{\text{CR}}}}\frac{{\overline{R}}_{\text{L}}c}{V_{\text{sh}}^2}$$ (32)

where ${\overline{R}}_{\text{L}}$ is the CR Larmor radius taken in the amplified magnetic field produced by the NRS streaming instability. We note E_th the threshold energy to trigger the filamentation instability, i.e.

$${\overline{R}}_{\text{L,th}}\ge \frac{V_{\text{sh}}}{k_{\text{max}}c}.$$ (33)

For the parameters adopted for SN 1993J, this threshold is independent of time. Using equations (2), (21), and (23), we find

$$E_{\text{PeV,th}}\simeq \frac{331}{\sqrt{V_0}}\sqrt{\frac{{\varphi }_{14}}{{\xi }_{\text{CR}0,0.05}}}\varpi E_{\text{PeV}}.$$ (34)

At energies E > E_th, the filamentation instability is destabilized and grows with the rate given by equation (32)

$$\begin{gathered} T_{\text{min,Fil}}\simeq \left[\frac{{1.110}^{23}{R}_0}{V_0^{7/2}}\text{ s}\right]\left(\frac{{\varphi }_{14}}{{\xi }_{\text{CR}0,0.05}}\right) \\ \times {\dot{M}}_{-5}^{-1/2}{V}_{\text{w},10}^{1/2}{E}_{\text{PeV}}{\left(\frac{t}{t_0}\right)}^{\frac{5}{2}(1-m)+m\frac{s}{2}}. \end{gathered}$$ (35)

Once the NRS instability is onset and the energy threshold to trigger the filamentation instability is reached, then long-wavelength modes can be very rapidly generated by the production of filaments. For the case of SN 1993J, we find $T_{\text{min,Fil}}~(0.1\text{ d})\times \left({\varphi }_{14}/{\xi }_{\text{CR}0,0.05}\right){\text{E}}_{\text{PeV}}\times {\text{t}}_{\text{d}}^{1.255}$.

4.2.4. Instability generating long oblique modes

Bykov et al. (2011) show that the presence of turbulence with scales shorter than the CR gyroradius enhances the growth of modes with scales longer than the gyroradius for particular polarizations. The mode growth time is (given by their equation 37)

$$T_{\text{ob}}=\sqrt{\frac{4}{\pi A}}\sqrt{\frac{1}{k{k}_{\text{max}}{V}_{\text{A,CSM}}^2}}.$$ (36)

The minimum growth time is obtained for k = 1/(ηR_L)

$$T_{\text{Min,ob}}=\sqrt{\frac{4}{\pi A}}\sqrt{\frac{\eta R_{\text{L}}}{k_{\text{max}}{V}_{\text{A,CSM}}^2}}.$$ (37)

Again we have $k_{\text{max}}^{-1}=T_{\text{min,NRS}}{V}_{\text{A,CSM}}$. Using equations (8), (18), and (24), we find

$$\begin{gathered} T_{\text{min,ob}}\simeq \left[\frac{7.9\times {10}^{11}{R}_0\sqrt{\eta }}{\sqrt{\varpi }{V}_0^{9/4}}\text{ s}\right]{\left(\frac{{\varphi }_{14}}{{\xi }_{\text{CR}0,0.05}}\right)}^{3/4} \\ \times {\left(\frac{1}{V_{\text{w},10}{\dot{M}}_{-5}}\right)}^{1/2}{E}_{\text{PeV}}{\left(\frac{t}{t_0}\right)}^{\frac{3}{2}(1-m)+m\frac{s}{2}}. \end{gathered}$$ (38)

Using the parameters selected for SN 1993J, we find $T_{\text{min,ob}}\simeq (0.6\text{ d})\times \frac{\sqrt{\eta }}{\sqrt{\varpi }}{\left({\varphi }_{14}/{\xi }_{\text{CR}0,0.05}\right)}^{3/4}{\text{E}}_{\text{PeV}}\times {\text{t}}_{\text{d}}^{1.085}$. Following the onset of the NRS instability, the oblique modes grow fast, but still with a slower rate compared to the filamentation modes.

4.2.5. Obliquity effects

In the previous calculations, we have assumed that model P applies, but the orientation of the mean magnetic field with respect to the shock normal can have an impact on the growth of CR-driven instabilities. We consider the case of a CR current j_CR perpendicular to the background magnetic field B_w. The NRS instability still grows unless the modes have wavenumbers k ⊥ B_w. The growth time for modes propagating parallel to the background magnetic field involves both background sound and Alfvén speeds (Bell 2005) $T_{\text{NRS,T}}\simeq T_{\text{NRS,P}}\times \sqrt{c_{\text{s}}{V}_{\text{A}}/2\left(c_{\text{s}}^2+V_{\text{A}}^2\right)}$. In the meantime, the advection time given by equation (15) is shortened because the parameter η > 1. If the factor 1/(ηϖ) ≪ 1, then the NRS instability can be quenched in this field configuration. However, if the wind magnetic field is in sub-equipartition and if 1/(ηϖ) ~ 1, the ratio T_NRS/T_adv < 1 and the NRS instability can grow.

Several recent studies have investigated the efficiency of CR injection and acceleration in high obliquity shocks with some different conclusions. Bell, Schure & Reville (2011) using a Vlasov model including a collision term induced by electromagnetic process find particle acceleration at perpendicular fast SNR shocks. The obliquity effects tend to sharpen the CR distribution with respect to the parallel shock case. Caprioli & Spitkovsky (2014a) using a hybrid code including a kinetic treatment of non-thermal particles find an acceleration efficiency dropping beyond obliquity angles ~45° for shock Mach numbers up to 50. Protons crossing high obliquity shocks are only accelerated by the shock drift process and cannot repeatedly cross the shock front because downstream advection is faster. The typical sound speed in massive star winds is c_s ~ 10 km s⁻¹(T/10⁴ K)^1/2. Early in the SN evolution, shock velocities are of order 10⁴ km s⁻¹, therefore we have typical shock Mach numbers 10³(T/10⁴ K)^−1/2. It would be interesting to test this limit angle of acceleration efficiency for such high Mach number shocks. van Marle et al. (2018) using a PIC (Particle-In-Cell) module for CRs in a MHD shock solutions find that high obliquity shocks produce CR acceleration especially because the particles accelerated by the shock drift mechanism are able to induce enough turbulence downstream do corrugate the shock front. The shock corrugation produces patterns where the local magnetic field is parallel and where particle that cross the shock from downstream can trigger NRS/filamentation instability upstream. It requires long multiscale simulations possibly in 3D to characterize particle acceleration at highly oblique shocks and we consider the issue still opened (see also Caprioli, Zhang & Spitkovsky 2018).

Finally, another point raised by Zirakashvili & Ptuskin (2018) is that the wind magnetic field is radial at close distances to the progenitor star. Then, fast shocks can inject CR efficiently during the early shock propagation phase. Once CRs are injected the onset of CR-driven magnetic fluctuations are able to maintain a parallel magnetic field over some fraction of the shock front, and thereby maintain particle injection.

4.3. CR-driven wave growth in supernovae

We consider first the example of SN 1993. In Fig. 1, we plot the advection time and the different growth time-scales for model P for particles with energies of 1 PeV. We find at every times R_NR <1, hence NRS modes can grow. Large-scale modes, for the adopted set of parameters, can be produced by the filamentation instability. The oblique mode instability and the RS instability have growth time-scales larger by a factor ~2.5 and a factor ~15 than the advection time and, for this set of parameters can not grow. However, these time-scales drop more rapidly with time and at some stage can become shorter than the advection time. They can hence compete with the filamentation instability to produce long-wavelength perturbations.

Figure 1.

Main instability growth time-scales as a function of the time in days for the fiducial case SN 1993. We have assumed η = ϖ = 1, E = 1 PeV, ϕ = 14, and ξ_CR = 0.05.

Table 3 shows values at t = t₀ of all relevant time-scales for the other SNe in our sample. In all cases, the index s = 2 is selected. We can see that in almost all cases, the parameter R_NRS is in the range 0.3–3. In these cases, MFA by the NRS instability seems possible given the uncertainties on the other parameters. It also appears that the saturated magnetic field strength is a fraction, typically about 10–20 percent of B₀ displayed in Tables 1 and 2. As $B_{\text{sat,NR}}\propto \sqrt{{\xi }_{\text{CR}}}$ doubling this percentage would require to convert 50 percent of the kinetic power of the shock into energetic particles which is unlikely. Investigating this regime would require non-linear DSA calculations, which is beyond the scope of this simple study. However, the final saturation value can (and should) be higher if long-wavelength modes are destabilized (note that we have checked that in every cases where R_NR < 1, we have E_th ≪1 PeV, so the filamentation instability can be destabilized at this energy). Finally, the transversal magnetic field component is compressed at the shock front. This means that CR-driven instabilities, if they can be triggered, can contribute to at least 50 percent of the magnetic field strength inferred from modelling the radio observations. Among all sources, SN 1986J, a Type IIn SN, appears to produce the fastest instability growth and produce the highest amplification factor. This effect is due to the high mass-loss rate and high shock velocity in SN 1986J, which has an R₀ a factor of 3.7 larger than SN 1979C at the same time t₀ = 5 d. However, the results for SN 1986J have to be considered to be optimistic. Fast mode growths are mostly due to the small size shock value at t₀ because the expansion index is close to the Sedov value (0.69 versus 2/3). The combination of the radio expansion and the measured VLBI radius lead to a very high value the shock speed V₀. The self-similar solution is questioned by the fit of the X-ray emission (Dwarkadas & Gruszko 2012) as it is the case for most of type IIn SNe. This class requires a dedicated modelling of the shock dynamics and the results obtained here are subject to an important uncertainty. SN 2001gd, SN 2011dh, SN 1983N (low wind speed solution), and SN 1994I (low wind speed solution) show instability growth time-scales larger than t₀ by factors <10. In these cases, the growth of the instability may be limited by the age rather than a condition R_NR < 1.

Table 3.

Main instability time-scales of our SNe sample in the case of low progenitor wind speed (V_w < 100 km s⁻¹). All values are derived with: η = 1, ϖ = 1, ϕ₁₄ = 1, ξ_{CR, 0.05} = 1, and E_PeV = 1. For SN 1986J and SN 2001gd, we use the mean value of the mass-loss rate given in Table 1. In the case of SN 1993J, we adopt two different set of parameters derived at two different times by Fransson & Björnsson (1998) (t₀ = 10 d) and T09 (t₀ = 100 d).

SN name	Tmin, NRS(d)	RNR	Bsat, NRS (Gauss)	A	Tmin, RS(d)	Tmin, Fil(d)	Tmin, ob(d)
SN 1979C	1.0	0.9	4.7	47.1	13.3	1.2	3.3
SN 1986J	0.2	0.3	4.5	219.2	13.8	0.2	1.6
SN 1993J	2.0	0.8	1.8	61.9	34.3	2.2	7.4
SN 2001gd	992.3	2.4	0.02	11.2	3.0(e3)	1.9(e3)	1.5(e3)
SN 2008iz	8.1	0.7	0.4	72.0	158.2	8.2	31.8
SN 2011dh	28.4	1.0	0.2	40.2	311.0	34.9	83.4
	33.1	0.8	0.1	57.0	514.3	36.3	115.8
SN 2009bb	1.5	6.8	0.06	24.4	10.1	0.5	0.3

If the progenitor wind velocity is high, as it is the case in Type Ib/Ic SNe, the ambient gas density drops and the NRS growth rate also. For instance, one can see in Table 4 that SN 2003L has R_NR ~ 40 (and T_{min, NRS} ~ 12 d). Here, MFA by the NRS instability is not likely because the advection towards the shock is too fast to allow for NR modes to grow unless ϖ ≪ 1. Another point to be mentioned in fast progenitor wind SNe is that the parameter R_R < R_NR. In the case of SN 2003L, we find R_R ~ 5. To illustrate this trend in Table 4, we show values of T_{min, R}, R_R, and R_NR for type Ib/Ic SNe. In the case of SN2009bb, the NRS seems to be able to amplify the magnetic field at the shock precursor (see Table 3). In this object, with respect to type II SNe, the smallness of the $\dot{M}/V_{\text{w}}$ ratio is compensated by a shock reaching a mildly relativistic regime.

Table 4.

Resonant instability time-scales in high progenitor wind SNe with V_w > 100 km s⁻¹, to the exception of SN 2009bb where the shortest growth time-scales are due to the non-resonant instability.

SN name	Tmin, RS(d)	RR	Bsat, RS (Gauss)	RNR
SN 1983N	2.0	11.0	0.3	102.4
SN 1994I	4.0	14.1	0.1	62.0
SN 2003L	6.3	12.0	0.06	35.6

Acknowledging for these results including their limitations, we confirm with our calculations that in order to amplify the magnetic field by CR streaming a fast shock pervading a dense medium is necessary.

5. MAXIMUM COSMIC RAY ENERGIES

The maximum CR (hadrons) energy is fixed by five different processes: the shock age limitation, the finite spatial extend of the shock, the generated current limitation, the nuclear interaction losses, and the adiabatic losses.

To the exception of E_{max, cur} due to the current limitation, all maximum energies expressions can take two different values depending if the background magnetic field in which high-energy CR gyrate is assumed to be the wind magnetic field or the field amplified by the NRS instability. In the latter scenario, calculated maximum energies have to be seen as lower values, because large-scale magnetic perturbations can be destabilized either by filamentation or oblique mode dynamo generation.

5.1. Age-limited maximum energy

We can write the acceleration time as T_acc = (1/EdE/dt)⁻¹ and the maximum energy

$$E_{\text{max}}(t)-E_{\text{max}}\left(t_0\right)={\int }_{t_0}^t\text{ d}t\times \text{d}E/\text{d}t={\int }_{t_0}^t\text{ d}t\times E/T_{\text{acc}}(E).$$ (39)

The acceleration time given by equation (16) is T_acc = T_{adv, u}g(r). In the case of model P, using equation (14), we have

$$\begin{gathered} E_{\text{max,age}}(t)\simeq \left[\frac{7.7\times {10}^{-10}{V}_0^2{t}_0\varpi }{\eta R_0\left(1-2m+m\frac{s}{2}\right)g(r)}\text{PeV}\right] \\ \times {\dot{M}}_{-5}^{1/2}{V}_{\text{w},10}^{1/2}\left(1-{\left(\frac{t}{t_0}\right)}^{2m-1-m\frac{s}{2}}\right) \end{gathered}$$ (40)

if $s>\frac{2}{m}(2m-1)$. Otherwise $E_{\text{max,age }}\propto \left({\left(\frac{t}{t_0}\right)}^{2m-1-m\frac{s}{2}}-1\right)$, and it grows as ln(t/t₀) in the case $s=\frac{2}{m}(2m-1)$.

In the model T, the maximum energy is E_{max, age, T} = E_{max, age, P} × (g_P/g_T) × η_Pη_T. If we consider the background magnetic field as being B_{sat, NRS}, then the previous maximum energy must be multiplied by the amplification factor A and $E_{\text{max,age}}(t)\propto \left(1-{\left(\frac{t}{t_0}\right)}^{3m-2-ms/2}\right)$ if $s>\frac{2}{m}(3m-2)$ and $E_{\text{max,age}}(t)\propto \left({\left(\frac{t}{t_0}\right)}^{3m-2-ms/2}-1\right)$ or E_{max, age}(t) ∝ ln(t/t0) otherwise.

5.2. Geometrical losses maximum energy

Geometrical losses are given by the condition: κ_u = η_escV_shR_sh, where η_esc is a parameter in the range 0.1–0.3 used to mimic the effect of particle loss in spherical geometry (Berezhko 1996). Hence, for model P, we have

$$\begin{gathered} E_{\text{max,esc}}(t)=\left[2.7\times {10}^{-10}{V}_0\text{PeV}\right]\frac{\varpi }{\eta }\frac{{\eta }_{\text{esc}}}{0.3} \\ \times {\dot{M}}_{-5}^{1/2}{V}_{\text{w},10}^{1/2}{\left(\frac{t}{t_0}\right)}^{2m-1-m\frac{s}{2}}. \end{gathered}$$ (41)

We note that if we consider the magnetic field to be given by the saturation value obtained for the NRS instability (see equation 23) the maximum geometrical loss-limited energy is multiplied by the amplification factor A and $E_{\text{max,esc}}(t)\propto {\left(\frac{t}{t_0}\right)}^{3m-2-ms/2}$.

5.3. Current-driven maximum energy

If the NRS instability operates, the maximum energy is fixed by the number N of e-folding times growth of the NRS instability, i.e. t/T_min,NRS = N. We have N ∈ [1, ln A], where the lower limit N = 1 corresponds to the minimum time for the instability to grow in the linear phase, while N = ln A corresponds to the amplification by a factor A of the magnetic field. Following Schure & Bell (2013, their equation 4), the maximum CR energy is given by the relation

$$E_{\text{max,cur }}\text{ln}\left(\frac{E_{\text{max,cur }}}{m_{\text{p}}{c}^2}\right)=\frac{\sqrt{\pi }{\xi }_{\text{CR}}}{N}\times q\sqrt{\rho }{R}_{\text{sh}}\frac{V_{\text{sh}}^2}{c}$$ (42)

where E_max is in erg.

Finally,

$$\begin{gathered} E_{\text{max,cur }}{\varphi }_{14}\simeq \left[\frac{4.5\times {10}^{-19}{V}_0^2}{N}\text{PeV}\right]{\xi }_{\text{CR}0,0.05} \\ \times {\dot{M}}_{-5}^{1/2}{V}_{\text{w},10}^{-1/2}\times {\left(\frac{t}{t_0}\right)}^{2m-ms/2-1} \end{gathered}$$ (43)

5.4. Maximum energy from the nuclear interaction losses

High-energy CRs interact with ambient matter through p–p interaction with a cross-section given by (Kafexhiu et al. 2014)

$$\begin{gathered} {\sigma }_{\text{pp}}\simeq 30\text{mb}\left(1.89+0.18\text{ ln}\left({\text{E}}_{\text{PeV}}\right)+6\times {10}^{-3}{\text{ln}}^2\left({\text{E}}_{\text{PeV}}\right)\right) \\ \times {\left(1-\frac{4\times {10}^{-13}}{E_{\text{PeV}}^{1.9}}\right)}^3=30\text{mb}{\overline{\sigma }}_{\text{pp}}\left({\text{E}}_{\text{PeV}}\right) \end{gathered}$$ (44)

and a loss time-scale T_pp ≃ (K_ppσ_pp〈n_H(t)〉c)⁻¹ with K_pp ~ 0.2. The hydrogen density n_H in the wind is obtained from equation (7), and we account for the residence times of CRs upstream and downstream of the shock T_u/d = 4κ_u/d/(V_u/dc), where V_u/d are the upstream and downstream fluid speeds in the shock rest frame. Then, the mean density experienced by a CR during a Fermi cycle is

$$\langle n_{\text{H}}\rangle =\frac{n_{\text{H}}{T}_{\text{u}}+n_{\text{d}}{T}_{\text{d}}}{T_{\text{u}}+T_{\text{d}}}$$

We use n_d = 4n_H in the case of a weakly modified shock (see discussion in Section 4.2). We then have 〈n_H〉 = 4Fn_H with

$$F=\frac{1+\frac{r_{\text{B}}}{4r}}{1+r_{\text{B}}/r}$$

and n_H is given by equation (7). Finally,

$$T_{\text{pp}}\simeq \left[\frac{6.0\times {10}^{-23}{R}_0^2}{{\overline{\sigma }}_{\text{pp}}\left(E_{\text{PeV}}\right)}\text{s}\right]\times \left(\frac{V_{\text{w},10}}{F{\dot{M}}_{-5}}\right)\times {\left(\frac{t}{t_0}\right)}^{ms}.$$ (45)

Comparing this time with T_acc given by equation(16), we obtain

$$\begin{gathered} E_{\text{max,nuc}}\simeq \left[\frac{4.8\times {10}^{-32}{R}_0{V}_0^2\varpi }{g(r)F\eta {\overline{\sigma }}_{\text{pp}}\left(E_{\text{PeV}}\right)}\text{PeV}\right] \\ \times {\dot{M}}_{-5}^{-1/2}{V}_{\text{w},10}^{3/2}\times {\left(\frac{t}{t_0}\right)}^{2(m-1)+m\frac{s}{2}}. \end{gathered}$$ (46)

5.5. Maximum energy from the adiabatic losses

Due to the rapid flow expansion CRs also suffer from adiabatic losses. To account for the residence of CRs upstream and downstream of the shock, we use equation (1) of Voelk & Biermann (1988) and find

$$T_{\text{Ad}}\simeq \left[\frac{6R_0}{V_0}\frac{r}{4(r-1)}\text{s}\right]\left(1+\frac{r_{\text{B}}}{r}\right)\left(\frac{t}{t_0}\right)$$ (47)

In the case of SN 1993J, we have T_Ad,s ~ 4.4 d(t/t₀). The maximum energy fixed by balancing the acceleration and adiabatic loss time-scales is

$$E_{\text{max,adi}}\simeq \left[3.8\times {10}^{-10}{V}_0\text{PeV}\right]{\dot{\text{M}}}_{-5}^{1/2}{\text{ V}}_{\text{w},10}^{1/2}\frac{\varpi }{\eta }\times {\left(\frac{\text{t}}{{\text{t}}_0}\right)}^{2\text{ m}-1-\text{m}\frac{\text{s}}{2}}.$$ (48)

Again if the background magnetic field is B_{sat, NRS}, then the maximum energies limited by losses have to be multiplied by A, then E_{max, nuc} ∝ (t/t₀)^{3(m− 1) + ms/2}, and E_{max, Adi} ∝ (t/t₀)^{3m− 2 −ms/2}.

5.6. Time-dependent maximum CR energy

In Fig. 2, the maximum CR energy limits E_{max, age}, E_{max, esc}, E_{max, cur}, E_{max, nuc}, and E_{max, adi} are shown for model P as a function of time after the shock outburst, for the case when the NRS instability has time to amplify the background magnetic field to B_{NRS, sat}. Note that T09 considered this case only and omitted the effect of pp and adiabatic losses. Fig. 3 shows the maximum CR energy limits E_{max, age}, E_{max, esc}, E_{max, nuc}, and E_{max, adi} in the case when the background magnetic field is assumed to be the wind field.

Figure 2.

Maximum CR energy limits in PeV units for the model P as a function of time after shock breakout for the fiducial case of SN 1993J if the background field has been amplified up to B_{sat, NRS}. The dotted line plots E_{max, nuc}(t), the large dotted–dashed line plots E_{max, adi}(t), the intermediate dotted–dashed line plots E_{max, cur}(t), the small dotted–dashed line plot E_{max, esc}(t), and the solid line plots E_{max, age}(t). The following parameters have been used: ϖ = 1, η = 1, N = 5, ϕ = 14, and ${\overline{\sigma }}_{\text{pp}}=1.87$.

Figure 3.

Maximum CR energy limits in PeV units for the model P as a function of time after shock breakout for the fiducial case of SN 1993J if the background field is B_w. The dotted line plots E_{max, nuc}(t), the large dotted–dashed line plots E_{max, adi}(t), the small dotted–dashed line plots E_{max, esc}(t), and the solid line plots E_{max, age}(t). The following parameters have been used: ϖ = 1, η = 1, and ${\overline{\sigma }}_{\text{pp}}=1.87$.

At a given time t, the maximum CR energy is E_max(t) = Min(E_{max, age}, E_{max, esc}, E_{max, cur}, E_{max, nuc}, E_{max, adi}). If the NRS instability is active in the CR precursor, as soon as E_{max, cur} < E_{max, age-a}, then E_max = E_{max, cur}. This occurs at t > 1.1t₀ in Fig. 2, and the peak value for E_max ~ 2 PeV which then drops as t^−0.17. At t = 10 d (t = 100 d), we have E_max ~ 1.4 PeV (E_max ~ 0.9 PeV). We find E_max ~ 680 TeV after one year, which is comparable, within a factor of 2, with values derived by Schure & Bell (2013) for SN shocks propagating in RSG winds. Note that adiabatic losses dominate over the losses due to pp collisions. If the CR current is not strong enough to generate a strong MFA then E_max = E_{max, esc-w}. This occurs at t > 12t₀ in Fig. 3 where have E_max ~ 300 TeV. Then, E_max drops as t^−0.17 to ~110 TeV after one year. For SN 1986J, at t = 5 d, we find E_max ~ 6 PeV. Again for the reasons addressed in Section 4.3, this number has to be considered as an upper limit. For SN 2009bb, at t = 20 d, we find E_max ~ 2.5 PeV.

6. DISCUSSION

In this study, we show that SNe can produce particles up to multi-PeV energies via the combination of fast shocks (velocity of order 0.1c), a high-density CSM produced by stellar winds, and low wind magnetizations. A high degree of CSM ionization can ease the particle acceleration process, but cannot be assumed and in general is unlikely. Assuming that the background magnetic field has a turbulent component, different instabilities driven by the acceleration process can grow over intra-day time-scales. This model is applied to a set of powerful SNe (both type II and type Ib/Ic) detected at radio wavebands by the VLA and by VLBI.

This first study should be seen as a proof of concept. A full derivation of the time-dependent CR distribution and gamma-ray emission is required to obtain a testable model. Parameters affecting the early gamma-ray emission from SNe include the ratio of the mass-loss rate to the wind velocity $\left(\dot{M}/V_{\text{w}}\right)$, which determines the CSM medium density and thereby affects the CR-driven instability growth rate. The shock velocity controls the growth rate of the instabilities and the acceleration time-scale. The degree of ionization is important for the particle acceleration efficiency, and may also produce element-dependent CR spectra in the case of partial ionization. The background magnetic field is partly responsible for the local magnetization and the shock obliquity. The SN luminosity affects the gamma–gamma absorption process. In case, the CR pressure becomes larger than 10 percent of the shock ram pressure non-linear calculations are mandatory to find the final particle and photon distributions. Finally, the gamma-ray detectability should be restricted to nearby events. The preliminary estimations made in Marcowith et al. (2014) show an horizon of detectability at 1 TeV for the Cherenkov Telescope Array of ~10 Mpc.

Only about 5–6 percent of the local CCSNe have been classified as Type IIb such as SN 1993J (Smartt 2009). There is a further subdivision into two classes of IIb SNe, with compact and extended progenitors (Chevalier & Soderberg 2010). The ones with extended radii, such as SN 1993J, seem to have higher mass-loss rates and lower wind velocities, and are the more promising candidates for high-energy CR acceleration and the early detection of γ-ray emission due to their higher wind density. The compact ones have wind velocities similar to those of WR stars, and thus correspondingly lower wind densities. Therefore, they may not be likely candidates for detecting early γ-ray emission, thus further reducing the observable sample.

Type IIn SNe are probably the most promising targets for gamma-ray telescopes in terms of high ambient density without significantly reduced velocities. In the case of the Type IIn SN 1996cr, it has been deduced, using numerical simulations that managed to reproduce the X-ray spectra over more than a decade of evolution that the shock was interacting with a shell of density ~10⁵ cm⁻³ (Dwarkadas, Dewey & Bauer 2010) a few years after explosion. This provides a high-density target for producing γ-rays via pion production, and is the basis for taking IIns as promising candidates for early γ-ray emission. These densities however are still lower than those suggested by some authors (Murase et al. 2011, 2014) in their calculations. The latter calculations suggest very high γ-ray fluxes. Type IIn SNe would, for the same reasons, also be promising targets for detecting neutrino emission from secondaries. Unfortunately, the number of Type IIn SNe is pretty small, comprising less than 4 percent of the total core-collapse population. It is likely that numerical simulations would need to be done to effectively deal with the SN hydrodynamics in the ambient medium. We will consider Type IIn SNe in a later paper.

SNe IIP comprise the largest class of CCSNe, making up around half the total. Their progenitors are RSG stars, which have wind mass-loss rates ranging from 10⁻⁷ to 10⁻⁴ M_⊙ yr⁻¹ (Mauron & Josselin 2011). However, Smartt (2009) demonstrated that observed progenitors of Type IIP SNe all appear to have masses below about 16.5 M_⊙. Similarly Dwarkadas (2014) showed that IIPs have the lowest X-ray luminosities among all core-collapse SNe, and thus put an upper limit of 19 M_⊙ on the initial mass of their progenitors, with correspondingly lower mass-loss rates (Mauron & Josselin 2011). The low mass-loss rates will result in lower maximum energies than calculated for SN 1993J (see Cardillo et al. 2015).

The rare Ib/Ic SNe harbour the fastest shock waves. They are assumed to arise from WR progenitors, which have wind velocities two orders of magnitude greater than RSGs, and therefore should have corresponding wind densities two orders of magnitude lower. The fast shock velocities are consistent with the lower densities. These shocks tend to accelerate particles more efficiently to higher energies, and their X-ray flux is presumed to be due to Inverse Compton or synchrotron emission (Chevalier & Fransson 2006), suggesting accelerated electrons. It is possible therefore that the shocks are capable of accelerating protons to high energies although our analysis suggests that due to lower wind densities CR instability growth rates can be reduced in such type of objects. Some W-R stars are surrounded by low-density wind-blown bubbles bordered by a high-density shell. If the shell is formed soon before the explosion, as is the case for the SN 2006jc (Foley et al. 2007), then it provides a good target for accelerated protons to collide with. Such W-R stars may be good candidates for the early detection of gamma-ray emission.

There may exist SNe similar to SN 1987A, whose progenitor, a blue supergiant, had a very low wind mass-loss rate wind on order 10⁻⁸ M_⊙ yr⁻¹(Chevalier & Dwarkadas 1995), but which shows evidence for a dense Hii region with density of order 200 particles cm⁻³ (Dewey et al. 2012), surrounded by a dense circumstellar ring with density ~10⁴ particles cm⁻³ at a distance of ~0.2 pc from the SN. Finally, other promising targets would include the class of SLSNe, especially those that are H-rich (Nicholl et al. 2015), as these may be interacting with extremely dense environments. High densities close in to the star could favour fast instability growths and could again provide target material for proton–proton collisions and detectable γ-ray emission at an early age.

Finally, we mention here the extremely rare Type Ia-CSM class, which appear to have the highest ambient densities as a class. It is not clear what the progenitors of these SNe are, and in fact whether they are bona-fide Ia’s, but densities inferred for the surrounding medium are as high as 10⁸ cm⁻³ (Deng et al. 2004; Aldering et al. 2006; Bochenek et al. 2018), at least in the first couple of years. If a close SN of this type were detected, they would be perhaps the most likely to show detectable gamma-ray emission. Unfortunately, these are the rarest class, and much like the IIns, the density structure is not well known but expected to be quite complex, with perhaps a two-component surrounding medium, not amenable to analytic calculations, and thus require detailed modelling.

7. CONCLUSIONS

The main conclusions of the study are as follows:

1. Magnetic field strengths inferred from radio observations of a sample of powerful and nearby type II and type Ib/Ic SNe can be, at least partly, explained by the process of MFA driven by CR driven instabilities.

2. We find that in fast shocks moving in dense CSM as it is the case of many type II SNe, the NRS instability can develop in the shock precursor in parallel shock configuration. Saturated magnetic field strengths can reach up to ~ 50 percent of the magnetic field deduced from the modelling of radio light curves. This number accounts for both MFA in the CR precursor and the transversal magnetic component compression at the shock front. Perpendicular shocks may also trigger CR-driven instabilities but only in the configuration of a sub-equipartition wind magnetic field energy density with respect to the wind kinetic energy density.

3. In our sample, we find that SN 1986J and SN 1993J were the most efficient at generating turbulent magnetic fields and accelerating CRs. We find that for these cases maximum CR energies reach ~1–10 PeV within a few days after the explosion and ~0.1–1 PeV after one year. The upper limits can shift to higher energies if long-wavelength magnetic perturbations can be generated. So Type IIn or compact Type IIb SNe may be good candidates for the Pevatron CR class. However, an accurate shock dynamics modelling of type IIn SN requires to go beyond the self-similar solution adopted in this study. The results obtained for SN 1986J have to be taken as upper limits. Finally, we find that the trans-relativistic SN 2009bb can accelerate CR at energies in the range 2–3 PeV within a few days after the outburst.

In a subsequent paper, we will include a detailed calculation of gamma–gamma opacity. This will allow us to derive a time-dependent gamma-ray flux in the TeV domain, and to make accurate predictions for gamma-ray detectability of SNe with HESS and the future Cherenkov Telescope Array. This modelling will also include the multiwavelength emission produced by secondary particles as a result of charged pion decay. We will calculate the expected high-energy neutrino flux from these SNe and compare it to the flux sensitivity of current and future neutrino facilities. A final study will treat SN IIn separately, as the wind density profile for these objects is more complicated and require more refined treatment than the self-similar calculations used in this study. These objects require numerical modelling in order to derive shock dynamics and evaluate particle acceleration and multimessenger emission efficiencies.