Radio Emission from Supernovae in the Very Early Phase: Implications for the Dynamical Mass Loss of Massive Stars

Figure 1 shows the initial density structure of our models. We use an RSG progenitor model which is evolved from a zero-age main-sequence star with the initial mass 15M_⊙, by using the stellar evolution code MESA (Paxton et al. 2011, 2013, 2015, 2018). We then attach a "confined" CSM to this progenitor structure by hand, to mimic the observationally inferred distribution (Yaron et al. 2017). We examine three "confined" CSM models; high-, intermediate-, and low-density models corresponding to $\dot{M}={10}^{-2},{10}^{-3},{10}^{-4}\ {M}_{\odot }$ yr⁻¹, assuming the wind velocity of u_w = 100 km s⁻¹. The CSM density outside the "confined" CSM is set identical for the three models, which corresponds to $\dot{M}={10}^{-6}\ {M}_{\odot }$ yr⁻¹ (a typical RSG wind). In addition, a model without the "confined" CSM, which has a smooth RSG wind-type CSM ($\dot{M}={10}^{-6}\ {M}_{\odot }$ yr⁻¹), is also examined. Hereafter, we name these models "Mdot-2confined," "Mdot-3confined," "Mdot-4confined," and "Mdot-6smooth" (Table 1).

We employ the 1D Lagrangian spherical symmetric hydrodynamics simulation open code SNEC (Morozova et al. 2015) and simulate the thermal bomb explosion of the SN for the four models. The explosion energy (E_kin) and the ejecta mass (M_ej) are set as E_kin = 10⁵¹ erg and M_ej = 11M_⊙, respectively.

By following adiabatic hydrodynamics evolution of the system, we trace the position of the shockwave as the shock propagates outward along the Lagrangian coordinate. The simulations provide the information necessary to compute the nonthermal emission properties, e.g., the structure of the shocked CSM. We do not take into account the radiative cooling process such as free–free emission. The radiative cooling timescale is longer than the dynamical timescale in the models "Mdot-3confined," "Mdot-4confined," and "Mdot-6smooth." The radiative cooling timescale is estimated as follows:

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}\sim \displaystyle \frac{5{k}_{{\rm{B}}}T}{n{{\rm{\Lambda }}}_{\mathrm{cool}}(T)}\sim 7\times {10}^{5}{\left(\displaystyle \frac{n}{{10}^{10}{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{T}{{10}^{9}{\rm{K}}}\right)}^{0.5}\ {\rm{s}},\end{eqnarray} \tag{ 1 }$$

where k_B, T and n are Boltzmann constant, the temperature and the number density at the shock front. Λ_cool(T) ∼ 1.0 × 10⁻²² (T/10⁹ K)^0.5 erg s⁻¹ cm³ is the cooling function and for the temperature T ∼ 10⁹ K free–free emission is dominant (Chevalier & Fransson 2017). In the most extreme model "Mdot-2confined," this radiative cooling timescale could be shorter than the dynamical timescale. To further investigate the effect of the radiative cooling on the dynamics we conduct radiation hydrodynamics simulation for the model "Mdot-2confined" by SNEC, noting that this might overestimate the effect because this simulation assumes the full thermalization and the blackbody radiation. As a result, we find that the shock velocity is ∼15% lower than that in the adiabatic hydrodynamics simulation, but the other physical properties, such as the density structure, have little difference between these two simulations. Thus, in this study we conduct the adiabatic hydrodynamics simulation as a first approximation even for the "Mdot-2confined" model.

Figure 2 shows the evolution of the forward shock. In all other models, the forward shock is accelerated at t ∼ 2 days, which announces the shock breakout from the progenitor surface. In addition, the second acceleration of the forward shock is observed at t ∼ 10 days for the models with the "confined" CSM. This is due to the steep decline of the "confined" CSM density at r ∼ 10¹⁵ cm. It is seen that in the phase t ≳ 20 days, the shock velocity in all the models converges to the analytic solution (Chevalier 1982a, 1982b), where the shock velocity is described as follows:

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{sh}} & = & 2.2\times {10}^{4}{\left(\displaystyle \frac{\dot{M}}{{10}^{-6}{M}_{\odot }{{\rm{yr}}}^{-1}}\right)}^{-0.15}{\left(\displaystyle \frac{{u}_{{\rm{w}}}}{100{{\rm{km}}{\rm{s}}}^{-1}}\right)}^{0.15}\\ & & \times \,{\left(\displaystyle \frac{{E}_{\mathrm{kin}}}{{10}^{51}{\rm{erg}}}\right)}^{0.425}{\left(\displaystyle \frac{{M}_{\mathrm{ej}}}{11{M}_{\odot }}\right)}^{-0.275}{\left(\displaystyle \frac{t}{10{\rm{days}}}\right)}^{-0.15}\,{{\rm{km}}{\rm{s}}}^{-1}.\end{array}\end{eqnarray} \tag{ 2 }$$

In this analytic solution, the density distribution of the outer part of the SN ejecta is assumed as ρ ∝ t⁻³v⁻ⁿ, where v is the ejecta velocity coordinate. The index is set as n = 8.67, as obtained by our simulations.

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The CSM and ejecta are respectively swept up by the forward shock and reverse shock. These two regions are potential sites for the particle acceleration and synchrotron emission. However, the particle acceleration efficiency at the reverse shock front is believed to be low because the relative velocity of the ejecta in the rest frame of the reverse shock is much lower than that in the forward shock region (see e.g., Chevalier & Fransson 2017). In this study, therefore, we neglect the contribution of the reverse shock to the synchrotron emission.

Because we perform adiabatic hydrodynamics simulations, the thermal electron temperature (${T}_{e}$) in the unshocked CSM is uncertain. This quantity is indeed important in computing the free–free absorption (FFA) coefficient. In this study, we simply assume T_e = 10⁵ K, a typical value conventionally used in the previous study (see discussion in Fransson & Björnsson 1998).

In each Lagrangian zone, we compute the "amplified" magnetic field, only once the shock has reached the zone under consideration. This process is modeled as follows:

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{\mathrm{sh}}^{2}}{8\pi }={\epsilon }_{{\rm{B}}}{\rho }_{\mathrm{sh}}{V}_{\mathrm{sh}}^{2},\end{eqnarray} \tag{ 3 }$$

where _B, B_sh and ρ_sh are the amplification efficiency, the amplified magnetic field strength, and the gas density just behind the forward shock. With B_sh given as the initial condition, the subsequent time evolution of B(m_r) in each zone (where m_r is the mass coordinate), at the shock downstream, is then computed, assuming that the magnetic field flux within the zone is conserved:

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\left[2\pi r({m}_{r}){\rm{\Delta }}r({m}_{r})B({m}_{r})\right]=0,\end{eqnarray} \tag{ 4 }$$

where r(m_r) is the radius of the zone and Δr(m_r) is the thickness of the zone.

Here we consider the energy distribution of the nonthermal particle populations in the shocked region, including electrons, protons, and positrons. The electrons and protons are assumed to be accelerated through the DSA mechanism (the primary particles). The relativistic protons can further collide with thermal protons,¹ producing electrons and positrons through pion decay (the secondary particles; Petropoulou et al. 2016). Hereafter, we use the subscript i to refer to electrons, protons, or positrons; i = e⁻, p, e⁺.

We denote the number density of the particles i, whose energy is in the range E_i ∼ E_i + dE_i, as $\tfrac{{{dN}}_{i}}{{{dE}}_{i}}{{dE}}_{i}$. These densities are traced in all the Lagrangian zones within the shocked CSM region, and obtained by using the following equations:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\displaystyle \frac{{{dN}}_{i}}{{{dE}}_{i}}\right)=\displaystyle \frac{\partial }{\partial {E}_{i}}\left(\displaystyle \frac{{E}_{i}}{{t}_{i,\mathrm{loss}}}\displaystyle \frac{{{dN}}_{i}}{{{dE}}_{i}}\right)+{\left(\displaystyle \frac{d\dot{{N}_{i}}}{{{dE}}_{i}}\right)}_{\mathrm{in}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{d\dot{{N}_{i}}}{{{dE}}_{i}}\right)}_{\mathrm{in}}={\left(\displaystyle \frac{d\dot{{N}_{i}}}{{{dE}}_{i}}\right)}_{\mathrm{prim}}+{\left(\displaystyle \frac{d\dot{{N}_{i}}}{{{dE}}_{i}}\right)}_{\sec }.\end{eqnarray} \tag{ 6 }$$

The first term in the right-hand side of Equation (5) accounts for the cooling processes (for the description of the energy loss timescale ${t}_{i,\mathrm{loss}}$, see Appendix A). For protons, the energy losses by inelastic proton collisions, adiabatic expansion, and coulomb interactions are considered. For the electrons and positrons, synchrotron and inverse Compton (IC) emissions are considered together with the adiabatic expansion and the coulomb interaction as the cooling processes. To determine the IC emission timescale, the seed photon energy density must be specified, and we consider two templates for the bolometric light curve as shown in Figure 3; "default" and "high." We apply the "default" light curve to all of the CSM models. The effect of the bolometric light curve of seed photons on the radio properties will be discussed in Section 5, where we apply the "high" light-curve model to "Mdot-2confined" model. We thereby confirm that our conclusions are insensitive to the choice of the bolometric light-curve templates.

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The second term on the right-hand side in Equation (5) describes the relativistic particle injection. This is divided into two processes (Equation (6)). The "primary injection" is for the electrons and protons through the DSA, with the power-law energy distribution. This mechanism is triggered only just behind the forward shock and once per zone. The energy distribution is given as follows:

$$\begin{eqnarray}{\left(\displaystyle \frac{d{\dot{N}}_{i}}{{{dE}}_{i}}\right)}_{\mathrm{prim}}=\left\{\begin{array}{ll}{C}_{i}{E}_{i}^{-{\alpha }_{i}},{E}_{i,\min }\leqslant {E}_{i}\leqslant {E}_{i,\max } & (i\ ={e}^{-},p)\\ 0 & (i\ ={e}^{+})\end{array}\right.\end{eqnarray} \tag{ 7 }$$

The normalization coefficient C_i is determined by the total energy content of the injected electrons and protons. We define the acceleration efficiency ${\epsilon }_{{e}^{-}}$ and _p as a fraction of the energy dissipated at the shock transferred to the nonthermal particles:

$$\begin{eqnarray}&&{u}_{i}={\epsilon }_{i}{\rho }_{\mathrm{sh}}{V}_{\mathrm{sh}}^{2}\ (i={e}^{-},p),\end{eqnarray} \tag{ 8 }$$

where u_i is the energy density of the primary injected particles. Then, C_i can be estimated by the following integrals:

$$\begin{eqnarray}&&\int \displaystyle \frac{d{\dot{N}}_{i}}{{{dE}}_{i}}{{dE}}_{i}=\displaystyle \frac{4\pi {R}_{\mathrm{sh}}^{2}{n}_{i}{V}_{\mathrm{sh}}}{{ \mathcal V }},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{u}_{i}\int \displaystyle \frac{d{\dot{N}}_{i}}{{{dE}}_{i}}{{dE}}_{i}={n}_{i}\int {E}_{i}\displaystyle \frac{d{\dot{N}}_{i}}{{{dE}}_{i}}{{dE}}_{i}\ (i={e}^{-},p),\end{eqnarray} \tag{ 10 }$$

where ${ \mathcal V }$ is the volume of the zone at the shock front.

The maximum energy ${E}_{i,\max }$, is computed by imposing three physical criteria. The first one is set by the requirement that the acceleration timescale $({t}_{i,\mathrm{acc}})$ cannot be longer than the energy loss timescale $({t}_{i,\mathrm{loss}})$. Thus, the maximum energy is limited by

$$\begin{eqnarray}&&{t}_{i,\mathrm{acc}}={t}_{i,\mathrm{loss}}\ (i={e}^{-},p).\end{eqnarray} \tag{ 11 }$$

The acceleration timescale is given as follows (Drury 1983):

$$\begin{eqnarray}&&{t}_{i,\mathrm{acc}}=\displaystyle \frac{20}{3}\displaystyle \frac{{c}^{2}}{{V}_{\mathrm{sh}}^{2}}\displaystyle \frac{{E}_{i}}{{{qB}}_{\mathrm{sh}}c}\ (i={e}^{-},p).\end{eqnarray} \tag{ 12 }$$

Here, q is the elementary charge. The second limit comes from the current SN age which is the maximum time available for the acceleration of charged particles since the SN shock breakout. Therefore, we impose the constrain on the maximum energy as follows:

$$\begin{eqnarray}&&{t}_{i,\mathrm{acc}}=t-{t}_{\mathrm{sbo}}\ (i={e}^{-},p),\end{eqnarray} \tag{ 13 }$$

where t_sbo is the time at the shock breakout. Finally, escape of the accelerated particles also limits ${E}_{i,\max }$, as the diffusion length is longer for particles with higher energy. We impose this constrain on the maximum energy by equating the diffusion length (L_diff) to the separation between the shock and the upstream free-escape-boundary (L_feb, Caprioli et al. 2009; Lee et al. 2012; Yasuda & Lee 2019) as follows:

$$\begin{eqnarray}&&{L}_{\mathrm{diff}}={L}_{\mathrm{feb}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{L}_{\mathrm{diff}}=\displaystyle \frac{\eta }{3}\displaystyle \frac{c}{{V}_{\mathrm{sh}}}\displaystyle \frac{{E}_{i}}{{qB}}\ (i={e}^{-},p),\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{L}_{\mathrm{feb}}={f}_{\mathrm{esc}}{R}_{\mathrm{sh}},\end{eqnarray} \tag{ 16 }$$

where η is related to nature of the magnetic field turbulence. Here, we assume the Bohm limit η = 1 (see e.g., Drury 1983). L_feb is parametrized by f_esc, and we set f_esc = 0.1, which is a typical value inferred from recent models of young supernova remnants such as Tycho (Slane et al. 2014) and Vela Jr. (Lee et al. 2013). The maximum energy ${E}_{i,\max }$ is set as the smallest value among those obtained by the three criteria. We note that the maximum energy is regulated dominantly by the synchrotron cooling for the electrons, and the escape limit for the protons, in the dense "confined" CSM.

The minimum energy is fixed as ${E}_{i,\min }=2.5{m}_{i}{c}^{2}$ where m_i is the mass of the particle i. Finally, the power-law index α_i is chosen as 3. This is commonly used to explain the radio emission from SNe (Chevalier & Fransson 2006; Chevalier et al. 2006; Maeda 2013).

The second mechanism of the high-energy particle injection is through pion decay, triggered by proton inelastic collisions. We focus on the reaction path where electrons and positrons are generated. In this paper, we call these electrons and positrons "secondary particles." These secondary particles are produced not only at the forward shock, but also in all the zones within the shocked region; the accelerated protons remain energetic not only just behind the shock but also in the downstream, and keep producing the secondary particles through pion decay. The injection rate of the secondary particles is written as follows:

$$\begin{eqnarray}{\left(\displaystyle \frac{d{\dot{N}}_{i}}{{{dE}}_{i}}\right)}_{\sec }=\left\{\begin{array}{ll}{n}_{{\rm{H}}}c\int {{dE}}_{p}\tfrac{{{dN}}_{p}}{{{dE}}_{p}}\tfrac{d{\sigma }_{{\rm{p}},2}({E}_{p},{E}_{i})}{{{dE}}_{i}} & (i\ ={e}^{-},{e}^{+})\\ 0 & (i\ =p),\end{array}\right.\end{eqnarray} \tag{ 17 }$$

where n_H is the number density of target protons. $d{\sigma }_{p,2}({E}_{p},{E}_{i})/{{dE}}_{i}$ is the inclusive cross section of p–p collision that produces the secondary particles having an energy E_i. The fitting formulae for the proton inelastic collision cross section have been developed by some previous works based on experiments or Monte Carlo simulations. The formalization by Kamae et al. (2006) is frequently used in computing the proton collision in SN remnants. In the present simulation, however, the maximum energy of protons can reach to PeV,² which is beyond the applicable energy range of the parametrized formulae in Kamae et al. (2006). The formalism by Kelner et al. (2006) is applicable to protons more energetic than 0.1 TeV. In this study, we adopt these two formalizations in a hybrid way.

Free parameters in our simulations are the following: ${\epsilon }_{{\rm{B}}},{\epsilon }_{e},{\epsilon }_{p},{\alpha }_{{e}^{-}},{\alpha }_{p},$ and E_i,min. These parameters are in principle determined by the shock acceleration physics, but there still remains debate on typical values which represent the real situations (Caprioli et al. 2015). When we compare our calculations with the observational data in future works, these parameters will be constrained and used as feedback to first principle calculations. However, in this study we will focus on the general characteristics of the radio light curves instead of matching any particular observation, and thus simply fix them as follows: ${\epsilon }_{{\rm{B}}}=0.1,{\epsilon }_{e}=0.1,{\epsilon }_{p}=0.1,{\alpha }_{{e}^{-}}=3,{\alpha }_{p}=3$ and ${E}_{i,\min }=2.5{m}_{i}{c}^{2}$.

Based on the energy distribution of electrons and positrons, the synchrotron emission from the shocked CSM region is calculated. The synchrotron intensity I_ν is computed by integrating the following one-dimensional radiative transfer equation from the CD to infinity,

$$\begin{eqnarray}&&\displaystyle \frac{{{dI}}_{\nu }}{{dr}}=-{\alpha }_{\nu }{I}_{\nu }+{j}_{\nu }.\end{eqnarray} \tag{ 18 }$$

The synchrotron emissivity (j_ν) and synchrotron self-absorption (SSA) coefficient (${\alpha }_{\nu ,\mathrm{SSA}}$) are defined as follows:

$$\begin{eqnarray}&&{j}_{\nu }=\displaystyle \frac{1}{4\pi }\displaystyle \sum _{i={e}^{-},{e}^{+}}\int {P}_{\nu }({E}_{i})\displaystyle \frac{{{dN}}_{i}}{{{dE}}_{i}}{{dE}}_{i},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{\alpha }_{\nu ,\mathrm{SSA}}=\displaystyle \frac{{c}^{2}}{8\pi {\nu }^{2}}\displaystyle \sum _{i={e}^{-},{e}^{+}}\int \displaystyle \frac{\partial }{\partial {E}_{i}}\left[{E}_{i}^{2}{P}_{\nu }({E}_{i})\right]\displaystyle \frac{1}{{E}_{i}^{2}}\displaystyle \frac{{{dN}}_{i}}{{{dE}}_{i}}{{dE}}_{i},\end{eqnarray} \tag{ 20 }$$

in all of the zones in the shocked CSM region (Rybicki & Lightman 1979). We note that we include the term for positrons. P_ν(E) is the synchrotron radiation power per unit frequency (see Appendix B). In the preshock CSM region, free–free absorption (FFA) is important. FFA coefficient is given as follows:

$$\begin{eqnarray}&&{\alpha }_{\nu ,\mathrm{ff}}=0.018{T}_{e}^{-3/2}{Z}^{2}{n}_{e}{n}_{i}{\nu }^{-2}{\tilde{g}}_{\mathrm{ff}}\,{{\rm{cm}}}^{-1},\end{eqnarray} \tag{ 21 }$$

where T_e, Z, and ${\tilde{g}}_{\mathrm{ff}}$ are thermal electron temperature, the charge of thermal ions, and the free–free Gaunt factor, and the variables in this equation are given in cgs unit. We assume T_e = 10⁵ K (see Section 2.2). For the Gaunt factor we employ the value given by Rybicki & Lightman (1979).

Figure 4 shows the centimeter light curves for four models, at the frequency 22 GHz, frequently employed in radio observations of SNe. Our simulations show that the centimeter emission in the first 10 days is weaker for the "confined" CSM with higher density. The synchrotron centimeter emission is damped by SSA or FFA. It will lead to an observational challenge; without robust detection, a large fraction of the information about the nature of the CSM is lost. One will therefore need to go for the higher frequencies, e.g., millimeter, to provide a useful tracer of the "confined" CSM.

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Figure 5 shows the millimeter light curves for four models at 100 and 250 GHz. The peak luminosity in the millimeter emission is enhanced with the "confined" CSM. The shape of the light curve shows different behavior depending on the CSM density. It is the "Mdot-4confined" model that gives a robustly detectable sign of the millimeter emission in the first 10 days since the shock breakout. The "Mdot-2confined" model shows the millimeter light curve damped in the first 10 days, similar to the case for the centimeter emission. An important difference for the centimeter is the fast variation around t ∼ 10 days. This phenomenon is caused by FFA in the preshocked CSM region. If the CSM density is low or the synchrotron frequency is high, FFA becomes negligible before the shockwave fully sweeps up the "confined" CSM. In the opposite situation, the synchrotron emission is masked by FFA. When the shock reaches to the outer edge of the "confined" CSM, the synchrotron emission suddenly becomes optically thin, thus we can observe the millimeter as a fast transient whose variation timescale is a few days. We can analytically investigate this behavior as follows, assuming that FFA is effective only in the "confined" CSM,

$$\begin{eqnarray}&&{\int }_{{R}_{\mathrm{sh}}}^{{R}_{\mathrm{CSM}}}{\alpha }_{\nu ,\mathrm{ff}}{dr}=1\iff {R}_{\mathrm{sh}}({\tau }_{\mathrm{ff}}=1)={R}_{\mathrm{CSM}}{\left\{1+{ \mathcal S }\right\}}^{-1/3}.\end{eqnarray} \tag{ 22 }$$

The characteristic quantity, ${ \mathcal S }$, depends on the nature of the CSM and the synchrotron frequency as follows:

$$\begin{eqnarray}&&{ \mathcal S }\sim {10}^{-3}{\left(\displaystyle \frac{\dot{M}}{{10}^{-2}{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{-2}{\left(\displaystyle \frac{\nu }{100{\rm{GHz}}}\right)}^{2}{\left(\displaystyle \frac{{R}_{\mathrm{CSM}}}{7\times {10}^{14}{\rm{cm}}}\right)}^{3}.\end{eqnarray} \tag{ 23 }$$

The dense CSM or the low frequency leads to substantial FFA and delays the emergence of the synchrotron signal. If the "confined" CSM is opaque to the synchrotron emission, a fast transient-like variant is expected due to the rapid decrease of the optical depth as the shock approaches to the outer edge of the "confined" CSM. In fact in "Mdot-2confined" model ${ \mathcal S }$ is smaller than unity and R_sh(τ_ff = 1) ∼ R_CSM is realized. On the other hand, if the density of the "confined" CSM is not to much high (e.g., "Mdot-4confined" model at 250 GHz) then R_sh(τ_ff = 1) < R_CSM; in this case, the system becomes transparent for FFA when the shock is still propagating in the "confined" CSM. These analyses are consistent with our numerical results.

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Figure 6 shows the time evolution of SSA and FFA optical depths since the shock breakout. We can derive the dependences of these absorption coefficients on the density and frequency as follows:

$$\begin{eqnarray}&&{\tau }_{\mathrm{ff}}\propto {\rho }^{2}{\nu }^{-2},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\tau }_{\mathrm{SSA}}\propto {\epsilon }_{e}\rho {B}_{\mathrm{sh}}^{({\alpha }_{{e}^{-}}+2)/2}{\nu }^{-({\alpha }_{{e}^{-}}+4)/2}\\ \quad \propto {\epsilon }_{e}{\epsilon }_{{\rm{B}}}^{5/4}{\rho }^{9/4}{\nu }^{-7/2}\ ({\alpha }_{{e}^{-}}=3).\end{array}\end{eqnarray} \tag{ 25 }$$

We used the relation ${B}_{\mathrm{sh}}\propto {({\epsilon }_{{\rm{B}}}{\rho }_{\mathrm{sh}})}^{1/2}$. The two absorption processes indeed have similar dependence on the density. Therefore, the difference in the mass-loss rate has little effect on the ratio of the optical depths of the two processes. In our models, τ_SSA/τ_ff ∼ 100 is realized. However, we note that τ_SSA is sensitive to the shock acceleration parameters (_e and _B). The smaller value of _e or _B results in smaller τ_SSA. If ${\epsilon }_{e}{\epsilon }_{{\rm{B}}}^{5/4}\sim {10}^{-4}$, then τ_SSA ∼ τ_ff is realized, and then the light curve may be exclusively shaped by FFA.

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In Figure 6, the gray lines show the peak date, and imply that the maximum luminosity is determined by SSA (but see below for the important role of FFA). This can be explained in the same way as the previous SN radio emission studies (e.g., see Chevalier 1998; Chevalier & Fransson 2017). The peak luminosity is given by

$$\begin{eqnarray}&&{L}_{\nu ,\mathrm{peak}}=4\pi {R}_{\mathrm{sh}}^{2}\pi {S}_{\nu },\end{eqnarray} \tag{ 26 }$$

where S_ν is the source function of synchrotron defined as

$$\begin{eqnarray}\begin{array}{rcl}{S}_{\nu ,\mathrm{syn}} & = & \displaystyle \frac{8\pi {m}_{e}}{{\alpha }_{{e}^{-}}+1}{\left(\displaystyle \frac{2\pi {m}_{e}c}{3{qB}\sin \theta }\right)}^{1/2}\\ & & \times \,{\nu }^{5/2}\displaystyle \frac{{\rm{\Gamma }}({\alpha }_{{e}^{-}}/4+19/12){\rm{\Gamma }}({\alpha }_{{e}^{-}}/4-1/12)}{{\rm{\Gamma }}({\alpha }_{{e}^{-}}/4+1/6){\rm{\Gamma }}({\alpha }_{{e}^{-}}/4+11/6)}\end{array}\end{eqnarray} \tag{ 27 }$$

(Rybicki & Lightman 1979). At the peak date the optical depth must be unity,

$$\begin{eqnarray}&&{\alpha }_{\mathrm{SSA}}{\rm{\Delta }}{R}_{\mathrm{sh}}=1,\end{eqnarray} \tag{ 28 }$$

where ΔR_sh is the thickness of the shocked region. Under the standard assumption that the peak luminosity is mostly dominated by the primary electrons (with ${\alpha }_{{e}^{-}}=3$) as confirmed by the simulations, the constraint on the magnetic field is derived as follows:

$$\begin{eqnarray}&&B\propto {\nu }^{7/9}{R}_{\mathrm{sh}}^{-2/9}.\end{eqnarray} \tag{ 29 }$$

The relation between the luminosity and the shock radius is thus expressed as follows:

$$\begin{eqnarray}&&{L}_{\nu ,\mathrm{peak}}\propto {\left({R}_{\mathrm{sh}}\nu \right)}^{19/9}.\end{eqnarray} \tag{ 30 }$$

The shock radius thus determines the peak luminosity. Both "Mdot-2confined" and "Mdot-3confined" models have similar radius at the peak date, and thus show the similar peak luminosities.

Finally, we will comment on why FFA is important in shaping the light curves even if τ_SSA ≫ τ_ff. This is explained by the dependence of the intensity on these optical depths; ${I}_{\nu }\simeq {S}_{\nu }(1-{e}^{-{\tau }_{\mathrm{SSA}}}){e}^{-{\tau }_{\mathrm{ff}}}$. The absorber of SSA, a high-energy electron, is also the emitter. Thus, the large optical depth of SSA leads to the intensity approaching to the source function. The absorber of FFA, on the other hand, is the thermal particles in the preshocked CSM, working as an external absorber. Hence, if τ_ff ≫ 1 is satisfied, the radio emission is completely damped, unlike SSA. This feature is apparent especially in the first 10 days since the shock breakout, t ≲ 10 days.

What kind of particles dominates the synchrotron emission is also an interesting question. Figure 7 shows the time evolution of the luminosity in the four models, divided into contributions from different sources. The emission at peak luminosity is dominated by the primary electrons. For the models with the "confined" CSM, the secondary particles make a large contribution to the synchrotron emission in the phase after the maximum. Especially in "Mdot-2confined" and "Mdot-3confined" models, they overshadow the emission from the primary particles. This is a phenomenon attributed to the existence of the "confined" CSM (see Murase et al. 2019 for a general model with a dense CSM, developed for SNe IIn). The protons in the "confined" CSM obtain relativistic energies by the shock acceleration, but their cooling timescale is much longer than those of electrons and positrons. Thus, for a while after the propagation of the shockwave in the "confined" CSM, the protons remain energetic and later generate the secondary electrons and positrons.

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We note that the time dependence of the luminosity emitted by the secondary particles is steeper than that of the primary particles. This is because the particle injection rate, corresponding to the second term of the right-hand side in Equation (6), has different dependence on the CSM density compared with the primary particles. In case of the primary electrons, it is ∝ρ (see e.g., Chevalier et al. 2006), while it is ∝ρ² for the secondary particles. Therefore, the secondary injection is more sensitive to the CSM density and hence decays faster with time than the primary injection.

As mentioned in Section 3.2, the IC cooling is computed by using the bolometric luminosity L_bol. Type II-P SNe have a variety in the bolometric light curves, but they generally show a plateau lasting for ∼100 days since the explosion (Anderson et al. 2014). However, there is a possibility that the extremely dense and "confined" CSM may alter the bolometric light curve in the earliest phase through the interaction between the ejecta and CSM (see e.g., Morozova et al. 2017).

For "Mdot-2confined" model, we have tested this possibility by replacing the bolometric light curve (the "high" model in Figure 3). This is motivated by the following estimate of the optical depth τ in the initial profile of the CSM to the SN thermal photons,

$$\begin{eqnarray}\begin{array}{rcl}\tau & = & {\int }_{\mathrm{CSM}}\kappa \rho {dr}\simeq 10\left(\displaystyle \frac{\kappa }{0.2\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}}\right)\\ & & \times \,\left(\displaystyle \frac{\dot{M}}{{10}^{-2}{M}_{\odot }\,{{\rm{yr}}}^{-1}}\right){\left(\displaystyle \frac{{u}_{{\rm{w}}}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 31 }$$

where the opacity κ is assumed to be coming from electron scatterings. We see that only "Mdot-2confined" model may change the bolometric light curve, due to the large optical depth. Figure 8 shows that the radio luminosity after the peak date is decreased by a factor of a few. In the earliest phase, 10 days since the shock breakout, it has little effect. This is due to the intense magnetic field in the shocked CSM; the particle cooling time is determined by synchrotron, not by IC cooling. Therefore, if we focus on the observation around the peak luminosity time, the details of the bolometric luminosity evolution do not affect our conclusions.

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According to our results, the millimeter signals from an SN at a distance of several ten Mpc should be detectable with the observed flux density of ${ \mathcal O }(1)$ mJy. We propose that this is an interesting target for ALMA, to robustly probe the existence of the "confined" CSM. Figure 9 shows the time sequence of the simulated synchrotron spectral energy distribution (SED), assuming that the SN explodes at a distance of 30 Mpc. There is a distinct gap between the very early (t ≲ 10 days) and the later phase (t ≳ 10 days), and the degree of spectral difference depends on the density of the "confined" CSM. We can see that the "confined" CSM produces strong signals in the millimeter range in the first 10 days, while the signal is totally damped in the centimeter range (for an example of the practical observation, see Ho et al. 2019).

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In practice, when fitting the observational data, more detailed simulations are required to test several uncertainties in the present models. Especially, accurately deriving the CSM density scale can be difficult, as this is coupled with the unknown parameters, e.g., _e and _B, while this degeneracy can be partly reduced by considering the effects of cooling processes (Maeda 2012). We, however, note that deriving the spatial distribution of the CSM density (whether it is smooth or truncated) is decoupled to these uncertainties, and can be robustly determined.

The multidimensional geometry of the CSM might also have an effect on the observed properties of the nonthermal emission, depending on the viewing angle. Observations indicating a global asymmetry in the CSM have indeed been reported for some SNe (Leonard et al. 2000; Hoffman et al. 2008; Patat et al. 2011; Katsuda et al. 2016). Nonthermal emission modeling based on multidimensional hydrodynamic simulations will thus be of interest. We postpone such investigations to future works.

Finally, it is worth mentioning that while the electron spectral index has been constrained by observations so far (${\alpha }_{{e}^{-}}=3$), the proton spectral index still involves uncertainties. The value we adopted (α_p = 3 in our models) is different from the canonical value of 2 predicted by the test particle theory of DSA for strong nonrelativistic shock (Bell 1978; Park et al. 2015). It is important because the proton spectral distribution is critical for secondary particle production. However, our models predict that if a "confined" CSM is present, the secondary particles dominate the synchrotron emission in the later phase (t ≳ 10 days). Therefore, future radio follow-up observations by ALMA will constrain the proton index which hopefully leads to the better understanding of particle acceleration at the SN shock.

Thanks to recent high-cadence transient surveys and rapid follow-up observations, it has been revealed that some massive stars may release a large amount of their own mass within decades before the SN and form a "confined" CSM in the vicinity of the progenitors. However, previous investigations based on only optical data involve uncertainties in their interpretations. We suggest that radio synchrotron emission can alternatively be a more robust tracer of the "confined" CSM.

Our calculations basically follow general formalisms developed by previous works on the nonthermal emission from SNe: capturing the propagation of the shockwave, solving the particle energy distribution in the post-shocked CSM region, and estimating the synchrotron intensity. We perform spherically symmetric hydrodynamics simulations. Within the first 10 days, the forward shock propagates within the "confined" CSM. In the shocked CSM region, relativistic particles are injected via the particle acceleration at the shock through mechanisms such as DSA or the inelastic collisions of protons. Depending on the degree of the absorption (which depends on the CSM density), the emerging millimeter emission can robustly probe the existence of a "confined" CSM and its density (and spatial extent).

We have shown that strong millimeter emission is expected within the first 10 days since the shock breakout, and that the density of the "confined" CSM alters the behavior of the radio emission. The centimeter emission will be simply damped due to SSA and FFA. On the other hand, the millimeter emission is still detectable thanks to its transparency, and thus it is a robust tracer of the confined CSM. In addition, we have shown that the peak luminosity is dominated by the primary electrons, accelerated by the shockwave, while in the later phase (t ≳ 10 days) the secondary electrons and positrons make a large contribution to the synchrotron radiation if the density of the "confined" CSM is larger than $\rho \gtrsim {10}^{-15}\left(\tfrac{\dot{M}}{{10}^{-3}\ {M}_{\odot }{\mathrm{yr}}^{-1}}\right){\left(\tfrac{\dot{{u}_{{\rm{w}}}}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-1}{\left(\tfrac{\dot{{R}_{\mathrm{CSM}}}}{7\times {10}^{14}{\mathrm{cm}}^{-2}}\right)}^{-2}$ (corresponding to "Mdot-2confined" and "Mdot-3confined"). In summary, we propose that a target of opportunity observation by ALMA can provide strong diagnostics on the existence of the "confined" CSM and its nature.