Multiwavelength View of the Type IIn Zoo: Optical to X-Ray Emission Model of Interaction-powered Supernovae

We consider a core-collapse SN occurring in a dense CSM. When the SN ejecta collide with the CSM, two shocks are formed: a forward shock (FS) separating the shocked and unshocked CSM and a reverse shock (RS) separating the shocked and unshocked SN ejecta (Chevalier 1982). The interface of the shocked CSM and SN ejecta is called a contact discontinuity (see Figure 1). We consider the case where homologously expanding cold SN ejecta with a power-law density profile (ρ_ej ∝ v⁻ⁿ with n > 5) run through a CSM of a wind density profile

$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{\rm{w}}} & = & \displaystyle \frac{\dot{M}}{4\pi {v}_{{\rm{w}}}{r}^{2}}\sim 5\times {10}^{-15}{\rm{g}}\,{\mathrm{cm}}^{-3}\left[\displaystyle \frac{\dot{M}/{v}_{{\rm{w}}}}{{\left(\dot{M}/{v}_{{\rm{w}}}\right)}_{* }}\right]\\ & & \times {\left(\displaystyle \frac{r}{{10}^{15}\mathrm{cm}}\right)}^{-2},\end{array}\end{eqnarray} \tag{ 1 }$

where constants $\dot{M}$ and v_w are, respectively, the mass-loss rate and wind velocity. We adopt a normalization parameter for the density, ${\left(\dot{M}/{v}_{{\rm{w}}}\right)}_{* }={10}^{-4}{M}_{\odot }{\mathrm{yr}}^{-1}/(\mathrm{km}\,{{\rm{s}}}^{-1})$. The number density and velocity at the immediate downstream can be obtained from the self-similar solution of the shock dynamics (Chevalier 1982). The details of the shock propagation are described in Appendix A. We here overview how the dissipated kinetic energy at these shocks is converted to radiation and how the radiation emanates from the CSM.

The immediate upstream, either the unshocked ejecta or the unshocked CSM, is a cold but ionized gas due to strong UV/X-ray emission from the shocked region. When this crosses the shock, thermal relaxations between particles of the same species first occur; for a gas of density ρ the electron–electron and proton–proton collision timescales (Padmanabhan 2000),

$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{ee}} & \approx & \frac{1}{4\pi }\frac{{m}_{{\rm{e}}}^{1/2}{\left({k}_{{\rm{B}}}{T}_{{\rm{e}}}\right)}^{3/2}}{{e}^{4}(\rho /{m}_{{\rm{p}}})\mathrm{ln}{\rm{\Lambda }}}\\ & \sim & 4\times {10}^{-5}\,{\rm{s}}{\left(\frac{\rho }{{10}^{-13}{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\frac{{T}_{{\rm{e}}}}{{10}^{6}{\rm{K}}}\right)}^{3/2}\end{array}\end{eqnarray} \tag{ 2 }$

$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{pp}} & \sim & {\left(\displaystyle \frac{{m}_{{\rm{p}}}}{{m}_{{\rm{e}}}}\right)}^{1/2}{\left(\displaystyle \frac{{T}_{{\rm{p}}}}{{T}_{{\rm{e}}}}\right)}^{3/2}{t}_{\mathrm{ee}}\\ & \sim & 60\,{\rm{s}}{\left(\displaystyle \frac{\rho }{{10}^{-13}{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{{T}_{{\rm{p}}}}{{10}^{9}{\rm{K}}}\right)}^{3/2},\end{array}\end{eqnarray} \tag{ 3 }$

are generally much shorter than the dynamical timescales,

$\begin{eqnarray}&&{t}_{\mathrm{dyn}}=\frac{r}{{v}_{\mathrm{fs}(\mathrm{rs})}}\sim 10\,\mathrm{days}\left(\frac{r}{{10}^{15}\,\mathrm{cm}}\right){\left(\frac{{v}_{\mathrm{fs}(\mathrm{rs})}}{{10}^{4}\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-1}.\end{eqnarray} \tag{ 4 }$

Here k_B is the Boltzmann constant, m_e (m_p) is the electron (proton) mass, e is the electron charge, and $\mathrm{ln}{\rm{\Lambda }}\approx 30$ is the Coulomb logarithm. Then the temperature T_e (T_p) of electrons (protons) at the immediate downstream of the shock fronts is obtained as a function of the shock velocities v_fs(rs) from the jump condition with an adiabatic index γ = 5/3 as

$\begin{eqnarray}&&{T}_{{\rm{e}},\mathrm{fs}(\mathrm{rs})}=\displaystyle \frac{3}{16}\displaystyle \frac{{m}_{{\rm{e}}}}{{k}_{{\rm{B}}}}{v}_{\mathrm{fs}(\mathrm{rs})}^{2}\sim {10}^{6}\,{\rm{K}}{\left(\displaystyle \frac{{v}_{\mathrm{fs}(\mathrm{rs})}}{{10}^{4}\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2},\end{eqnarray} \tag{ 5 }$

$\begin{eqnarray}&&{T}_{{\rm{p}},\mathrm{fs}(\mathrm{rs})}=\displaystyle \frac{3}{16}\displaystyle \frac{{m}_{{\rm{p}}}}{{k}_{{\rm{B}}}}{v}_{\mathrm{fs}(\mathrm{rs})}^{2}\sim 2\times {10}^{9}\,{\rm{K}}{\left(\displaystyle \frac{{v}_{\mathrm{fs}(\mathrm{rs})}}{{10}^{4}\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}.\end{eqnarray} \tag{ 6 }$

Most of the energy dissipated at the shock goes to protons at the immediate downstream.

The FS (RS) downstream is slightly slower (faster) than the contact discontinuity, so for both shocks the incoming gas is gradually advected to the contact discontinuity. The advection timescale is roughly equivalent to the dynamical timescale t_ad ≈ t_dyn. ⁴ During the advection, the electrons in the shocked region exchange their energies with the protons due to Coulomb interaction with an equipartition timescale of

$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{ep}} & \sim & {\left(\displaystyle \frac{{m}_{{\rm{p}}}}{{m}_{{\rm{e}}}}\right)}^{1/2}{t}_{\mathrm{pp}}\\ & \sim & 2\times {10}^{3}\,{\rm{s}}{\left(\displaystyle \frac{\rho }{{10}^{-13}{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{{T}_{{\rm{p}}}}{{10}^{9}{\rm{K}}}\right)}^{3/2}.\end{array}\end{eqnarray} \tag{ 7 }$

The electrons also either gain or lose energy by emitting or absorbing photons via bremsstrahlung processes and by inelastic Compton scattering (Weaver 1976; Katz et al. 2010). Since the timescales in Equations (2) and (3) are shorter than that in Equation (7) and other relevant cooling and heating timescales, the shocked regions can be treated as two temperature plasmas. On the other hand, since the advection of the shocked plasma (Equation (4)) is the slowest process, we can approximate that the radiative shock structure is quasi-steady. For each epoch we obtain the hydrodynamic parameters at the shock interface (shock radius, shock velocity, and upstream density), and use these to solve the electron/proton temperatures inside each region swept by the FS and RS.

Photons are predominantly generated by bremsstrahlung at the immediate shock downstream of the FS and RS. The spectral energy distributions are modified by Compton scattering and free–free and bound–free absorption while propagating through the shocked regions. Radiation escapes from both regions after a diffusion time passes, and half of the radiation that escapes the FS region reaches the observer (see Figure 1). We solve the radiation transfer to calculate the escaping photon spectrum consistently with the two temperature structures in the shocked regions.

In this section, we review the basic equations that govern the structure of the shocked region.

As the scale heights of the downstream parameters are much smaller than r, the geometric factor r² can be treated as a constant in a narrow shocked region. Assuming a steady state in the shock rest frame the hydrodynamic equations are described as (e.g., Takei & Shigeyama 2020)

$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}(\rho v)=0,\end{eqnarray} \tag{ 8 }$

$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}(\rho {v}^{2}+p)=0,\end{eqnarray} \tag{ 9 }$

$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}\left[\rho v\left(\displaystyle \frac{1}{2}{v}^{2}+U+\displaystyle \frac{p}{\rho }\right)\right]=-\displaystyle \frac{d}{{dr}}\int d\nu \,{F}_{\nu },\end{eqnarray} \tag{ 10 }$

where ρ is the mass density of the shock downstream, and the pressure p and specific internal energy U are defined as

$\begin{eqnarray}&&p=\displaystyle \frac{\rho }{{m}_{{\rm{p}}}}{k}_{{\rm{B}}}[{{xT}}_{{\rm{e}}}+{{xT}}_{{\rm{p}}}+(1-x){T}_{{\rm{H}}{\rm\small{I}}}]+\displaystyle \frac{1}{3}\int d\nu \,{e}_{\mathrm{rad},\nu },\end{eqnarray} \tag{ 11 }$

$\begin{eqnarray}&&\rho U=\displaystyle \frac{3}{2}\displaystyle \frac{\rho }{{m}_{{\rm{p}}}}{k}_{{\rm{B}}}[{{xT}}_{{\rm{e}}}+{{xT}}_{{\rm{p}}}+(1-x){T}_{{\rm{H}}{\rm\small{I}}}]+\int d\nu \,{e}_{\mathrm{rad},\nu },\end{eqnarray} \tag{ 12 }$

where x is the ionization fraction. We set x = 1 when protons and electrons have not reached equipartition yet. Once T_e = T_p, assuming local thermodynamic equilibrium the value of x can be obtained from the Saha equation. The gas is assumed to consist purely of hydrogen for simplicity, with the temperatures of neutral and ionized hydrogen assumed to be equal, T_{H i} = T_p, in the entire shocked region. F_ν and e_rad,ν are the energy flux and density of radiation, respectively. Flux-limited diffusion (e.g., Levermore & Pomraning 1981) gives the flux as a function of e_rad,ν by the following equation:

$\begin{eqnarray}&&{F}_{\nu }=-\displaystyle \frac{c{\lambda }_{\nu }}{{\kappa }_{\nu }\rho }\displaystyle \frac{{{de}}_{\mathrm{rad},\nu }}{{dr}}\end{eqnarray} \tag{ 13 }$

where κ_ν is the opacity, c is the speed of light, and λ_ν is a parameter between 0 and 1/3, the limits corresponding to the optically thin and thick cases, respectively.

When electrons and protons have different temperatures, Coulomb interaction works between them to reduce the difference. The energy exchange rate of protons and electrons P_e−p (erg cm⁻³ s⁻¹) is (Katz et al. 2011)

$\begin{eqnarray}&&{P}_{{\rm{e}}-{\rm{p}}}\approx \mathrm{ln}{\rm{\Lambda }}\sqrt{\frac{2}{\pi }}\frac{{m}_{{\rm{e}}}}{{m}_{{\rm{p}}}}{k}_{{\rm{B}}}({T}_{{\rm{p}}}-{T}_{{\rm{e}}}){n}_{i}^{2}{\sigma }_{{\rm{T}}}c{\left(\frac{{k}_{{\rm{B}}}{T}_{{\rm{e}}}}{{m}_{{\rm{e}}}{c}^{2}}\right)}^{-3/2},\end{eqnarray} \tag{ 14 }$

where n_i = xρ/m_p is the number density of protons and electrons, and σ_T is the Thomson cross section. Noting that electrons are dominantly affected by the photon field, the energy equation can be rewritten into the following equations for T_e and T_p:

$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}\left[\displaystyle \frac{5}{2}\displaystyle \frac{\rho {k}_{{\rm{B}}}}{{m}_{{\rm{p}}}}{{vxT}}_{{\rm{e}}}\right]={P}_{{\rm{e}}-{\rm{p}}}-\displaystyle \frac{d}{{dr}}\int d\nu \left[\displaystyle \frac{4}{3}{{ve}}_{\mathrm{rad},\nu }+{F}_{\nu }\right],\end{eqnarray} \tag{ 15 }$

$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}\left[\displaystyle \frac{5}{2}\displaystyle \frac{\rho {k}_{{\rm{B}}}}{{m}_{{\rm{p}}}}{{vT}}_{{\rm{p}}}\right]=-{P}_{{\rm{e}}-{\rm{p}}}-\displaystyle \frac{d}{{dr}}\left(\displaystyle \frac{1}{2}\rho {v}^{3}\right).\end{eqnarray} \tag{ 16 }$

The equation of radiation transfer in spherical symmetry along the advection flow of the downstream plasma is

$\begin{eqnarray}&&\displaystyle \frac{1}{c}\displaystyle \frac{\partial {I}_{\nu }}{\partial t}+\cos \theta \displaystyle \frac{\partial {I}_{\nu }}{\partial r}+\displaystyle \frac{\sin \theta }{r}\displaystyle \frac{\partial {I}_{\nu }}{\partial \theta }=\displaystyle \frac{{j}_{\nu }}{4\pi }-{\alpha }_{\nu }^{{\prime} }{I}_{\nu },\end{eqnarray} \tag{ 17 }$

where I_ν is the specific intensity, θ is the angle between the light ray's propagation direction and the radial direction, and

$\begin{eqnarray}&&{j}_{\nu }\equiv {j}_{\nu ,\mathrm{ff}}\end{eqnarray} \tag{ 18 }$

$\begin{eqnarray}&&{\alpha }_{\nu }^{\mbox{'}}\equiv {\alpha }_{\nu ,\mathrm{ff}}-\frac{{j}_{\nu ,\mathrm{IC}}}{{{ce}}_{\mathrm{rad},\nu }}\end{eqnarray} \tag{ 19 }$

are, respectively, the emissivity and absorption coefficient, which are related to the free–free and Comptonization processes. Taking the zeroth moment of Equation (17) we obtain

$\begin{eqnarray}&&\displaystyle \frac{\partial {e}_{\mathrm{rad},\nu }}{\partial t}+\displaystyle \frac{\partial {F}_{\nu }}{\partial r}={j}_{\nu }-{\alpha }_{\nu }^{{\prime} }{{ce}}_{\mathrm{rad},\nu }.\end{eqnarray} \tag{ 20 }$

Here we neglect the geometric factor 2F_ν /r because the relevant scale lengths are much shorter than the radius r. Transferring to the Euler description and in the shock's rest frame, ∂/∂t → (∂/∂t + v∂/∂r). From the steady-state assumption we obtain

$\begin{eqnarray}&&v\displaystyle \frac{{{de}}_{\mathrm{rad},\nu }}{{dr}}+\displaystyle \frac{{{dF}}_{\nu }}{{dr}}={j}_{\nu }-{\alpha }_{\nu }^{{\prime} }{{ce}}_{\mathrm{rad},\nu }.\end{eqnarray} \tag{ 21 }$

For simplicity we assume the free–free process is the dominant emission/absorption process and neglect free–bound/bound–free and bound–bound contributions. The emissivity (erg s⁻¹ cm⁻³ Hz⁻¹) and absorption coefficient (cm⁻¹) are given in cgs units by Rybicki & Lightman (1979):

$\begin{eqnarray}&&{j}_{\nu ,\mathrm{ff}}\approx 6.8\times {10}^{-38}{n}_{i}^{2}{T}_{{\rm{e}}}^{-1/2}{g}_{\mathrm{ff}}\exp [-h\nu /{k}_{{\rm{B}}}{T}_{{\rm{e}}}],\end{eqnarray} \tag{ 22 }$

$\begin{eqnarray}&&{\alpha }_{\nu ,\mathrm{ff}}\approx 3.7\times {10}^{8}{T}_{{\rm{e}}}^{-1/2}{n}_{i}^{2}{\nu }^{-3}{g}_{\mathrm{ff}}(1-\exp [-h\nu /{k}_{{\rm{B}}}{T}_{{\rm{e}}}]),\end{eqnarray} \tag{ 23 }$

where h is Planck's constant and g_ff is the Gaunt factor

$\begin{eqnarray}&&{g}_{\mathrm{ff}}(\nu )\sim \max \left(1,\displaystyle \frac{\sqrt{3}}{\pi }\mathrm{ln}\left[\displaystyle \frac{2.25{k}_{{\rm{B}}}{T}_{{\rm{e}}}}{h\nu }\right]\right).\end{eqnarray} \tag{ 24 }$

For the photon frequency we sample a wide range of 0.1 eV < hν < 100 MeV, with 80 bins evenly spaced on a logarithmic scale.

For inverse Compton (IC) scattering, we obtain the rate of change in photon energy density j_ν,IC (erg s⁻¹ cm⁻³ Hz⁻¹) as

$\begin{eqnarray}\begin{array}{rcl}{j}_{\nu ,\mathrm{IC}} & = & (\mathrm{gain}\,\mathrm{term})\,-\,(\mathrm{loss}\,\mathrm{term})\\ & \approx & \displaystyle \frac{\nu {e}_{\mathrm{rad},\nu ^{\prime} }}{\nu ^{\prime} }[{n}_{i}{\sigma }_{\mathrm{KN}}(\nu ^{\prime} )c]\displaystyle \frac{d\nu ^{\prime} }{d\nu }-{e}_{\mathrm{rad},\nu }{n}_{i}{\sigma }_{\mathrm{KN}}(\nu )c,\end{array}\end{eqnarray} \tag{ 25 }$

where σ_KN(ν) is the Klein–Nishina cross section given by Rybicki & Lightman (1979):

$\begin{eqnarray*}\begin{array}{rcl}\frac{{\sigma }_{\mathrm{KN}}(\nu )}{{\sigma }_{{\rm{T}}}} & = & \frac{3}{4}\,\left[\frac{1+\beta }{{\beta }^{3}}\left\{\frac{2\beta (1+\beta )}{1+2\beta }-\mathrm{ln}(1+2\beta )\right\}\right.\\ & & +\left.\frac{\mathrm{ln}(1+2\beta )}{2\beta }-\frac{1+3\beta }{{\left(1+2\beta \right)}^{2}}\right],\end{array}\end{eqnarray*}$

with β = h ν/m_e c², and $\nu ^{\prime} (\nu ,{T}_{{\rm{e}}})$ is the typical frequency of photons whose frequency becomes ν after scattering.

For a nonrelativistic electron (k_B T_e ≪ m_e c²) and a photon energy $h\nu ^{\prime} \ll {m}_{{\rm{e}}}{c}^{2}$, the typical frequency ν after scattering of a photon of energy $\nu ^{\prime}$ is (Rybicki & Lightman 1979)

$\begin{eqnarray}&&\nu =\nu ^{\prime} +\displaystyle \frac{\nu ^{\prime} }{{m}_{{\rm{e}}}{c}^{2}}(4{k}_{{\rm{B}}}{T}_{{\rm{e}}}-h\nu ^{\prime} ).\end{eqnarray} \tag{ 26 }$

Since we consider a photon spectrum at high energies exceeding m_e c² as well, we refine the second term as follows. For a photon scattering with an electron at rest, from the kinematics of the scattering

$\begin{eqnarray*}&&\nu =\displaystyle \frac{\nu ^{\prime} }{1+(h\nu ^{\prime} /{m}_{{\rm{e}}}{c}^{2})(1-\cos \theta )},\end{eqnarray*}$

where θ is the angle between the momenta of scattered and incident photons. Averaging over θ

$\begin{eqnarray*}\begin{array}{rcl}\langle \nu \rangle & = & \displaystyle \frac{1}{2}{\int }_{0}^{\pi }\displaystyle \frac{\nu ^{\prime} \sin \theta d\theta }{1+(h\nu ^{\prime} /{m}_{{\rm{e}}}{c}^{2})(1-\cos \theta )}\\ & = & \displaystyle \frac{{m}_{{\rm{e}}}{c}^{2}}{2h}\mathrm{ln}\left(1+\displaystyle \frac{2h\nu ^{\prime} }{{m}_{{\rm{e}}}{c}^{2}}\right).\end{array}\end{eqnarray*}$

Repeating the same argument that led to Equation (26), we obtain

$\begin{eqnarray}&&\nu =\displaystyle \frac{\nu ^{\prime} }{{m}_{{\rm{e}}}{c}^{2}}\cdot 4{k}_{{\rm{B}}}{T}_{{\rm{e}}}+\displaystyle \frac{{m}_{{\rm{e}}}{c}^{2}}{2h}\mathrm{ln}\left(1+\displaystyle \frac{2h\nu ^{\prime} }{{m}_{{\rm{e}}}{c}^{2}}\right).\end{eqnarray} \tag{ 27 }$

The factor 4 is valid only when electrons are nonrelativistic, which is generally satisfied for our model parameters. For each ν we solve Equation (27) implicitly and obtain $\nu ^{\prime}$.

By solving Equations (8), (9), (13), (15), (16), and (21) for both the FS and RS regions, one can in principle obtain the shock structure, i.e., T_e(r), T_p(r), ρ(r), v(r), F_ν (r), and e_rad,ν (r). These equations take into account the radiation feedback onto the dynamics of the advected gas, but this will be self-consistent only if the propagation of the shock is simultaneously modified (Takei & Shigeyama 2020). Our formulation relies on the self-similar solution for the shock propagation and parameters at the immediate downstream, and taking into account the deviation from this self-similar evolution drastically complicates the problem.

In order to simplify the problem, in this work we adopt an assumption that the pressure gradient is zero in each shocked region, i.e., the dynamics is assumed to be one-zone. The plasma that crosses the shock is thus advected by a constant advection velocity, given by the downstream velocity in the shock's rest frame. Takei & Shigeyama (2020), who solved the above equations with the assumption of local thermodynamic equilibrium in the shocked region and by adopting the diffusion approximation, showed that a nearly constant pressure is indeed seen (see their Figure 1).

On the other hand we follow the evolution of T_e, T_p, and e_rad,ν in the region, which are important for the emission. With our assumption the energy equation (10) is simplified to

$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}(\rho {vU})=-\displaystyle \frac{d}{{dr}}\int d\nu {F}_{\nu }.\end{eqnarray} \tag{ 28 }$

Under the one-zone assumption, we average the flux that appears in the energy equation, over the shocked region's volume. The volume average then becomes a surface integral, and the energy equation is modified as

$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}\left(\rho {vU}\right)=\int d\nu ({\dot{e}}_{\mathrm{esc},\nu }-{j}_{\nu ,\mathrm{in}}),\end{eqnarray} \tag{ 29 }$

where e_esc,ν (erg cm^–3 Hz^–1) is the energy density of radiation taken away from the shocked region, and j_ν,in is the source term, which consists of two components explained later in this section. From our constant-pressure assumption v is independent of r, and we can set vd/dr = d/dt, where t is the time that has passed since the downstream plasma crossed the shock. Then with T_p > T_e, the equations for T_e and T_p are

$\begin{eqnarray}&&\displaystyle \frac{3}{2}\displaystyle \frac{\rho {k}_{{\rm{B}}}}{{m}_{{\rm{p}}}}\displaystyle \frac{d}{{dt}}({{xT}}_{{\rm{e}}})={P}_{{\rm{e}}-{\rm{p}}}-\int d\nu \left({\dot{e}}_{\mathrm{rad},\nu }+{\dot{e}}_{\mathrm{esc},\nu }-{j}_{\nu ,\mathrm{in}}\right),\end{eqnarray} \tag{ 30 }$

$\begin{eqnarray}&&\displaystyle \frac{3}{2}\displaystyle \frac{\rho {k}_{{\rm{B}}}}{{m}_{{\rm{p}}}}\displaystyle \frac{{T}_{{\rm{p}}}}{{dt}}=-{P}_{{\rm{e}}-{\rm{p}}}.\end{eqnarray} \tag{ 31 }$

We make T_e and T_p equal by hand (with the total energy conserved) if these two approach each other within 10%. This is merely for numerical purposes, as a finite difference tends to make the Coulomb term P_e−p erroneously large after electrons and protons have cooled down. Once T_e = T_p, thermal relaxation is fast enough that this equality is kept. Then the equations for T_e and T_p are further simplified to

$\begin{eqnarray}&&\displaystyle \frac{\rho {k}_{{\rm{B}}}}{{m}_{{\rm{p}}}}\displaystyle \frac{d[(1+x){T}_{{\rm{e}}}]}{{dt}}=-\displaystyle \frac{2}{3}\int d\nu \left({\dot{e}}_{\mathrm{rad},\nu }+{\dot{e}}_{\mathrm{esc},\nu }-{j}_{\nu ,\mathrm{in}}\right).\end{eqnarray} \tag{ 32 }$

We first obtain [1 + x(T_e)]T_e with this equation, then use the Saha equation and implicitly obtain T_e (= T_p).

We next present our method of obtaining e_rad,ν and e_esc,ν . The equation for radiation transfer becomes

$\begin{eqnarray}\begin{array}{rcl}{\dot{e}}_{\mathrm{rad},\nu }+{\dot{e}}_{\mathrm{esc},\nu } & = & ({j}_{\nu ,\mathrm{ff}}+{j}_{\nu ,\mathrm{in}})-{\alpha }_{\nu }^{{\prime} }{{ce}}_{\mathrm{rad},\nu },\\ & \equiv & {j}_{\nu }-{\alpha }_{\nu }^{{\prime} }{{ce}}_{\mathrm{rad},\nu },\end{array}\end{eqnarray} \tag{ 33 }$

where we redefine j_ν ≡ j_ν,ff + j_ν,in. This equation reduces to the standard equation of radiation transfer if ${\dot{e}}_{\mathrm{esc},\nu }=0$ is assumed. In this case the formal solution can be obtained by expressing e_rad,ν as ${e}_{\mathrm{rad},\nu }(t)=F(t)\exp [-{\int }_{0}^{t}{ds}\,{\alpha }_{\nu }^{{\prime} }(s)c]$, and solving for F(t) by substituting this into Equation (33).

To determine ${\dot{e}}_{\mathrm{rad},\nu }$ and ${\dot{e}}_{\mathrm{esc},\nu }$ separately we need another constraint, which should be related to the physics of diffusion. As an approximation we treat the diffusion as a hard cutoff over a typical diffusion timescale t_diff. In other words, photons created at time t = s are in the shocked region and contribute to e_rad,ν at time s < t < s + t_diff, then completely escape and contribute to e_esc,ν at t > s + t_diff. Then by analogy of the standard case described above, we find the solution of the form

$\begin{eqnarray}&&{e}_{\mathrm{rad},\nu }(t)={\int }_{t-{t}_{\mathrm{diff}}}^{t}{j}_{\nu }(s){ds}\exp \left[-{\int }_{s}^{t}{\alpha }_{\nu }^{{\prime} }(\tilde{s}){cd}\tilde{s}\right],\end{eqnarray} \tag{ 34 }$

$\begin{eqnarray}&&{e}_{\mathrm{esc},\nu }(t)={\int }_{0}^{t-{t}_{\mathrm{diff}}}{j}_{\nu }(s){ds}\exp \left[-{\int }_{s}^{s+{t}_{\mathrm{diff}}}{\alpha }_{\nu }^{{\prime} }(\tilde{s}){cd}\tilde{s}\right],\end{eqnarray} \tag{ 35 }$

for t > t_diff, and

$\begin{eqnarray}&&{e}_{\mathrm{rad},\nu }(t)={\int }_{0}^{t}{j}_{\nu }(s){ds}\exp \left[-{\int }_{s}^{t}{\alpha }_{\nu }^{{\prime} }(\tilde{s}){cd}\tilde{s}\right],\end{eqnarray} \tag{ 36 }$

$\begin{eqnarray}&&{e}_{\mathrm{esc},\nu }(t)=0,\end{eqnarray} \tag{ 37 }$

for 0 < t ≤ t_diff. To calculate the integrals, we keep records of j_ν and ${\alpha }_{\nu }^{{\prime} }$ throughout the dynamical time with time step t_dyn/24,000. This resolution is limited by our computational resource. In the RS region, we observe cases where the cooling timescale (Equation (46)) becomes shorter than this time resolution, thereby preventing us from obtaining these integrals accurately. In these cases we switch to an analytical treatment for calculating e_rad,ν and e_esc,ν , explained in Appendix B.

We calculate T_e, T_p, e_rad,ν , and e_esc,ν from t = 0 to t_end, the timescale for the gas advection from the shock to the contact discontinuity, in our case equivalent to the dynamical time (see Section 2.1). Thus t_end is roughly equivalent to the epoch ${ \mathcal T }$, or the time from explosion. ⁵ We do this integration for a series of epochs $\{{{ \mathcal T }}_{0},{{ \mathcal T }}_{1},\cdots \}$, with ${t}_{\mathrm{end},N}={{ \mathcal T }}_{N}\,(N=0,1,\cdots )$. For each epoch the conditions at t = 0 (i.e., at the shock) are obtained by the equations in Appendix A. The series should be set so that the dynamics and radiation flow do not significantly change between neighboring epochs. We set the epochs by the following equations:

$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal T }}_{0} & = & \max (1\,\mathrm{day},1.1{{ \mathcal T }}_{\mathrm{CSM}})\\ {{ \mathcal T }}_{N+1}-{{ \mathcal T }}_{N} & = & \left\{\begin{array}{ll}0.5{t}_{\mathrm{diff},N}^{\mathrm{FS}} & ({{ \mathcal T }}_{N}\lt 10{t}_{\mathrm{diff},N}^{\mathrm{FS}})\\ 0.1{{ \mathcal T }}_{N} & ({{ \mathcal T }}_{N}\geqslant 10{t}_{\mathrm{diff},N}^{\mathrm{FS}}).\end{array}\right.\end{array}\end{eqnarray} \tag{ 38 }$

The parameter ${{ \mathcal T }}_{\mathrm{CSM}}$ is the diffusion timescale in the CSM, which controls the light-curve evolution:

$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal T }}_{\mathrm{CSM}} & \approx & \frac{{\kappa }_{\mathrm{scat}}\dot{M}}{{v}_{{\rm{w}}}c}\\ & \sim & 0.7\,\mathrm{days}\left(\frac{{\kappa }_{\mathrm{scat}}}{0.34\,{\mathrm{cm}}^{2}{{\rm{g}}}^{-1}}\right)\left[\frac{\dot{M}/{v}_{{\rm{w}}}}{{\left(\dot{M}/{v}_{{\rm{w}}}\right)}_{* }}\right]\end{array}\end{eqnarray} \tag{ 39 }$

where κ_scat ≈ 0.34 cm² g⁻¹ is the Thomson scattering opacity. The series continues up to 250 days or when the RS sweeps the entire outer ejecta, whichever is earlier. The variables ${t}_{\mathrm{diff}}^{\mathrm{RS}},{t}_{\mathrm{diff}}^{\mathrm{FS}}$ are, respectively, the diffusion timescales in the RS and FS regions. The diffusion time in the FS region ${t}_{\mathrm{diff}}^{\mathrm{FS}}$ is given by its optical depth multiplied by the light-crossing timescale:

$\begin{eqnarray}&&{t}_{\mathrm{diff}}^{\mathrm{FS}}\sim \min [{\kappa }_{\mathrm{scat}}{\rho }_{\mathrm{fs}}{\rm{\Delta }}{r}_{\mathrm{FS}},1]({\rm{\Delta }}{r}_{\mathrm{FS}}/c),\end{eqnarray} \tag{ 40 }$

where the width of the FS region Δr_FS relates to the radius of the contact discontinuity r_sh derived in Equation (A4) as Δr_FS ≈ [(γ − 1)/(γ + 1)]r_sh. The diffusion time in the RS region ${t}_{\mathrm{diff}}^{\mathrm{RS}}$ depends on the parameters for the self-similar solution, but is generally much shorter than ${t}_{\mathrm{diff}}^{\mathrm{FS}}$ due to the much smaller width. For example if we assume n = 10 and γ = 4/3, we find ${t}_{\mathrm{diff}}^{\mathrm{RS}}\approx 0.134{t}_{\mathrm{diff}}^{\mathrm{FS}}$.

In our model j_ν,in has two components. One is radiation entering from the other shocked region, shown as black arrows in Figure 1. For radiation escaping from the FS region, assuming the escaping radiation is isotropic ${r}_{\mathrm{fs}}^{2}/({r}_{\mathrm{sh}}^{2}+{r}_{\mathrm{fs}}^{2})\approx 1/2$ escapes outward (which is observed), while the other half goes inward to the RS region, and is possibly reprocessed there. Under the assumption of cold SN ejecta in which only the vicinity of the shocked region is ionized, we assume that the bulk of the ejecta is transparent and all the escaping radiation from the RS region goes to the FS region as a source term. We calculate the source term from the e_esc,ν in the previous epoch, and inject this at a constant rate for t_dyn.

The other component, which is rather artificial, is e_rad,ν , which should be present in the shocked region at t = 0. Our formulation of e_rad,ν (Equation (36)) implicitly assumes e_rad,ν = 0 at t = 0, which neglects this contribution. We instead include this as a source term estimated from the previous epoch, i.e., leftover e_rad,ν in the end of the previous epoch, e_rad,ν (t = t_end,N−1), is carried over in the present epoch ${{ \mathcal T }}_{N}$ as a source term. We inject this at a constant rate for the first diffusion time at epoch ${{ \mathcal T }}_{N}$.

When setting the injection term from the previous epoch, we make sure that the total injected energy in the shocked region, e_esc,ν (or e_rad,ν ) times the volume of the shocked region, is conserved. Mathematically, j_ν,in is given for the FS and RS regions as

$\begin{eqnarray}\begin{array}{rcl}{j}_{\nu ,\mathrm{in}}^{\mathrm{FS}} & = & \frac{{V}_{N-1}^{\mathrm{FS}}}{{V}_{N}^{\mathrm{FS}}}\left[\left(\frac{{V}_{N-1}^{\mathrm{RS}}}{{V}_{N-1}^{\mathrm{FS}}}\right)\frac{{e}_{\mathrm{esc},\nu }^{\mathrm{RS}}(t={t}_{\mathrm{end},N-1})}{{{ \mathcal T }}_{N}}\right.\\ & & +\left.\frac{{e}_{\mathrm{rad},\nu }^{\mathrm{FS}}(t={t}_{\mathrm{end},N-1})}{{t}_{\mathrm{diff},N}^{\mathrm{FS}}}{\rm{\Theta }}({t}_{\mathrm{diff},N}^{\mathrm{FS}}-t)\right],\end{array}\end{eqnarray} \tag{ 41 }$

$\begin{eqnarray}\begin{array}{rcl}{j}_{\nu ,\mathrm{in}}^{\mathrm{RS}} & = & \displaystyle \frac{{V}_{N-1}^{\mathrm{RS}}}{{V}_{N}^{\mathrm{RS}}}\left[\left(\displaystyle \frac{{V}_{N-1}^{\mathrm{FS}}}{{V}_{N-1}^{\mathrm{RS}}}\right)\displaystyle \frac{{e}_{\mathrm{esc},\nu }^{\mathrm{FS}}(t={t}_{\mathrm{end},N-1})}{2{{ \mathcal T }}_{N}}\right.\\ & & +\left.\displaystyle \frac{{e}_{\mathrm{rad},\nu }^{\mathrm{RS}}(t={t}_{\mathrm{end},N-1})}{{t}_{\mathrm{diff},N}^{\mathrm{RS}}}{\rm{\Theta }}({t}_{\mathrm{diff},N}^{\mathrm{RS}}-t)\right],\end{array}\end{eqnarray} \tag{ 42 }$

where Θ(t) is the Heaviside step function, and ${V}_{N}^{\mathrm{FS}(\mathrm{RS})}$ is the volume of the FS (RS) region at epoch ${{ \mathcal T }}_{N}$. Variables with superscript FS (RS) denote values in the FS (RS) region. In the self-similar solution the ratio between the volumes of the FS and RS regions is a constant of time. We neglect additional energy sources that can appear as a source term, such as radioactive decay of nickel inside the ejecta.

For the first epoch ${{ \mathcal T }}_{0}$, the calculation is iterated until e_rad,ν becomes stable. To obtain j_ν,in for the next iteration, we use the values of e_rad,ν and e_esc,ν at t = t_end,0 because there is no previous epoch. The energy density at t = t_end,0 becomes stable within 1% after about 10 iterations, from which we move on to epoch ${{ \mathcal T }}_{1}$.

To obtain the observed spectrum, we need to solve the radiation transfer through the unshocked CSM. At each epoch, the luminosity of radiation that escapes from the FS region toward the observer is

$\begin{eqnarray}\begin{array}{rcl}{L}_{\nu }({{ \mathcal T }}_{N}) & \approx & \displaystyle \frac{1}{2}{V}_{N}^{\mathrm{FS}}\displaystyle \frac{{e}_{\mathrm{esc},\nu }^{\mathrm{FS}}(t={t}_{\mathrm{end},N})}{{{ \mathcal T }}_{N}}\\ & \approx & 2\pi {r}_{\mathrm{sh}}^{2}{\rm{\Delta }}{r}_{\mathrm{FS}}\displaystyle \frac{{e}_{\mathrm{esc},\nu }^{\mathrm{FS}}(t={t}_{\mathrm{end},N})}{{{ \mathcal T }}_{N}},\end{array}\end{eqnarray} \tag{ 43 }$

where the factor 1/2 is due to the aforementioned geometrical effect. Once we obtain this luminosity L_ν , we use version 17.02 of CLOUDY (Ferland et al. 2017) to obtain the spectrum after it is processed through the unshocked CSM L_ν,CSM. We assume the CSM extends with a spherically symmetric wind profile from r = r_sh out to 10¹⁷ cm. The metallicity of the CSM is assumed to be solar.

This treatment also assumes a steady state, i.e., the photoionization at a given epoch is determined by the luminosity at that time. For Type IIn SNe this is justified, as the recombination timescale is generally much shorter than the dynamical timescale (Chevalier & Irwin 2012). For example, for a gas of T_e ∼ 10⁴ K the recombination coefficient is α_rec ∼ O (10⁻¹³) cm³ s⁻¹, and the recombination timescale is of the order of 10³ (10⁵) s at day 10 (100) for our fiducial model parameters.

We also include the effect of diffusion inside the CSM delaying the light curve, by the following formula (Chatzopoulos et al. 2012; Equation (4)):

$\begin{eqnarray}&&{L}_{\nu ,\mathrm{obs}}({ \mathcal T })=\frac{{e}^{-{ \mathcal T }/{{ \mathcal T }}_{\mathrm{CSM}}}}{{{ \mathcal T }}_{\mathrm{CSM}}}{\int }_{0}^{{ \mathcal T }}d{ \mathcal T }^{\prime} {e}^{{ \mathcal T }^{\prime} /{{ \mathcal T }}_{\mathrm{CSM}}}{L}_{\nu ,\mathrm{CSM}}({ \mathcal T }^{\prime} ).\end{eqnarray} \tag{ 44 }$

For the contribution of L_ν,CSM at $t\lt {{ \mathcal T }}_{0}$, which is absent in our modeling, we adopt the value at $t={{ \mathcal T }}_{0}$ for simplicity. The actual contribution is uncertain, as this also depends on when the CSM interaction starts. However, the exponential dependence makes this uncertainty unimportant after about a few times ${{ \mathcal T }}_{\mathrm{CSM}}$.

We show the change of electron temperature T_e as a function of time in Figure 2. The value of T_e initially rises due to Coulomb interaction with the protons (hotter by m_p/m_e), but then the temperature drops first due to free–free emission, then by IC scattering. The detailed evolution differs for the RS and FS, which we discuss in detail below.

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3.1.1. RS Region

In the RS region bremsstrahlung tends to be the dominant process for cooling electrons, owing to the high density and low shock velocity (Chevalier & Irwin 2012). The available energy density at the shock downstream is

$\begin{eqnarray}\begin{array}{rcl}\frac{3}{2}{n}_{i}{k}_{{\rm{B}}}{T}_{{\rm{p}},\mathrm{rs}} & \approx & \frac{9}{14\alpha }\frac{\dot{M}}{4\pi {v}_{{\rm{w}}}}{\left(\frac{{v}_{\mathrm{rs}}}{{r}_{\mathrm{sh}}}\right)}^{2}\\ & \sim & {10}^{3}\mathrm{erg}\,{\mathrm{cm}}^{-3}{\left(\frac{{ \mathcal T }}{10\mathrm{days}}\right)}^{-2}\left[\frac{\dot{M}/{v}_{{\rm{w}}}}{{\left(\dot{M}/{v}_{{\rm{w}}}\right)}_{* }}\right]\end{array}\end{eqnarray} \tag{ 45 }$

where α is a parameter that appears in the self-similar solution of shock dynamics, as can be seen in Equation (A4). This thermal energy density corresponds to 5 × 10⁶ K for the E1M1 model at 10 days. The cooling timescale is dominant at the immediate downstream when T_e is highest, and scales with time and model parameters as

$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{cool}} & \sim & \frac{(3/2){n}_{i}{k}_{{\rm{B}}}{T}_{{\rm{p}},\mathrm{rs}}}{1.4\times {10}^{-27}\mathrm{cgs}\,{\bar{g}}_{\mathrm{ff}}{T}_{{\rm{p}},\mathrm{rs}}^{1/2}{n}_{i}^{2}}\\ & \sim & 300\,{\rm{s}}\,{\bar{g}}_{\mathrm{ff}}^{-1}{\left(\frac{{ \mathcal T }}{10\mathrm{days}}\right)}^{13/8}\\ & & {\left(\frac{{E}_{\mathrm{ej}}}{{10}^{51}\mathrm{erg}}\right)}^{21/16}{\left[\frac{\dot{M}/{v}_{{\rm{w}}}}{{\left(\dot{M}/{v}_{{\rm{w}}}\right)}_{* }}\right]}^{-11/8},\end{array}\end{eqnarray} \tag{ 46 }$

where ${\bar{g}}_{\mathrm{ff}}\approx 1.2$ is the frequency-averaged value of g_ff (Rybicki & Lightman 1979). This timescale is consistent with the curves in the left panel of Figure 2. This t_cool is generally much shorter than the dynamical time t, and the plasma efficiently cools down and is eventually balanced with heating by free–free absorption. In this case the gas energy density is converted to blackbody radiation, whose temperature is given as

$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{BB}} & \sim & {\left[\frac{(3/2){n}_{i}{k}_{{\rm{B}}}{T}_{{\rm{p}},\mathrm{rs}}}{a}\right]}^{1/4}\\ & & \sim \,2\times {10}^{4}\,{\rm{K}}{\left(\frac{{ \mathcal T }}{10\mathrm{days}}\right)}^{-1/2}{\left[\frac{\dot{M}/{v}_{{\rm{w}}}}{{\left(\dot{M}/{v}_{{\rm{w}}}\right)}_{* }}\right]}^{1/4},\end{array}\end{eqnarray} \tag{ 47 }$

where a is the radiation density constant. There is also a contribution from the FS region, which can further increase T_e due to absorption and interaction of FS photons with the electrons. Compton heating is more important for models with higher ejecta energy, as it makes the seed photons harder. For some cases this balances free–free cooling and halts the decrease of T_e.

The escape of photons after a diffusion time contributes to the lowering of T_e, as the supply of hard photons emitted in the RS region or streaming from the FS region would be limited. This generally makes T_e < T_BB, and for some cases even reaching around 6000 K, where hydrogen recombines.

3.1.2. FS Region

The situation can be different for the FS region, due to the higher downstream energy density and, more importantly, the much lower density at the shock front. The free–free cooling time is orders of magnitude longer than in the RS region, with a scaling

$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{cool}} & \sim & 4\times {10}^{4}\,{\rm{s}}\,{\bar{g}}_{\mathrm{ff}}^{-1}{\left(\frac{{ \mathcal T }}{10\mathrm{days}}\right)}^{13/8}\\ & & \times {\left(\frac{{E}_{\mathrm{ej}}}{{10}^{51}\mathrm{erg}}\right)}^{21/16}{\left[\frac{\dot{M}/{v}_{{\rm{w}}}}{{\left(\dot{M}/{v}_{{\rm{w}}}\right)}_{* }}\right]}^{-11/8}.\end{array}\end{eqnarray} \tag{ 48 }$

This roughly explains the drop at later epochs, such as at day 103. However at earlier epochs, due to inefficient free–free cooling, IC can take over the cooling when there is enough supply of seed photons (Chevalier & Irwin 2012).

To see this more quantitatively, in the bottom panel of Figure 3 we plot the time evolution of the inverse of various electron heating and cooling timescales in the E1M1 model. The Coulomb heating timescale is t_e−p ∼ (3n_i k_B T_e/2)/P_e−p, the IC cooling timescale is t_IC ∼ (3n_i k_B T_e/2)/[∫d ν j_ν,IC], and the free–free absorption heating timescale is t_abs ∼ (3n_i k_B T_e/2)/[∫d ν α_ν,ff ce_rad,ν ]. The aforementioned transition of IC cooling being the dominant cooling process can be seen in the bottom panels, at ∼10⁴ s in day 10.5. For early phases, this is likely to be responsible for significantly reducing the electron temperature (see top panel of Figure 3) before free–free cooling takes back the role.

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In the early epochs we see a convergence of T_e, which is due to (i) a balance of free–free absorption and cooling given that the density is high enough, (ii) a balance of free–free cooling and Compton heating by photons from the RS, or (iii) a recombination of hydrogen for those that converge around 6000 K.

Figure 4 shows the resulting spectra of photons escaping the FS region for our choice of model parameters. We find that for the E1M1 and all the M10 models soft X-rays emitted from the shock are significantly absorbed by the unshocked CSM, and the observed spectrum is much different from the spectrum emergent from the shock front. A fraction of the absorbed X-ray photons are converted to optical and UV emission, with a large number of emission lines, including Hα at ≈1.9 eV. This matches the naive expectation that for low E_ej and/or high $\dot{M}/{v}_{{\rm{w}}}$, the shock expands more slowly and the optical depth of the CSM is kept large. On the other hand, the observed spectra for models E3M1 and E10M1 are largely unmodified from the spectrum from the shock. In addition to the CSM quickly becoming optically thin, higher explosion energies (and hence higher shock velocities) extend the FS spectra to hard X-rays, which avoid significant absorption by the CSM.

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An important aspect that determines the spectrum from the shock is the optical depth to Thomson scattering inside the FS region. For a wind-profile CSM this is equivalent to that in the unshocked CSM, and is given as

$\begin{eqnarray}&&{\tau }_{\mathrm{fs}}\sim {\kappa }_{\mathrm{scat}}{\rho }_{{\rm{w}}}{r}_{\mathrm{sh}}\sim 1.7\left[\displaystyle \frac{\dot{M}/{v}_{{\rm{w}}}}{{(\dot{M}/{v}_{{\rm{w}}})}_{* }}\right]{\left(\displaystyle \frac{{r}_{\mathrm{sh}}}{{10}^{15}\mathrm{cm}}\right)}^{-1},\end{eqnarray} \tag{ 49 }$

where we assume κ_scat = 0.34 cm² g⁻¹.

For an optically thick (τ_fs ≫ 1) case, Compton downscattering can drastically reduce the photon energy. The number of Thomson scatterings per unit time is κ_scat × 7ρ_w c, and photons stay in the FS region for a diffusion time ${t}_{\mathrm{diff},\mathrm{fs}}\sim {\kappa }_{\mathrm{scat}}{\rho }_{{\rm{w}}}{r}_{\mathrm{sh}}^{2}/7c$, where the factor 7 comes from the compression ratio for adiabatic index 4/3. Thus there are ${\tau }_{\mathrm{fs}}^{2}$ scatterings in the FS region before escaping, and as a result the energy of photons reaching the observer is reduced to $\approx 511\,\mathrm{keV}/{\tau }_{\mathrm{fs}}^{2}\sim 0.3\,\mathrm{keV}{\left({\tau }_{\mathrm{fs}}/40\right)}^{-2}$. This is consistent with the cutoff in the spectra at the FS for the E1M10 model on day 10.6, when τ_fs ≈ 38 (blue dotted line in the top-right panel of Figure 4).

The degree of absorption of soft X-rays depends on this cutoff frequency and the luminosity of ionizing photons L_ion, as discussed in Chevalier & Irwin (2012). Qualitatively, in order for soft X-rays to reach the observer the following two conditions must be satisfied: (i) τ_fs should be at most a few tens, so that the cutoff is in or beyond the soft X-ray band, and (ii) the flux of ionizing photons is high enough that the main elements that contribute to absorption of soft X-rays (oxygen and iron) are completely ionized. The latter is quantified as the ionization parameter $\xi \equiv {L}_{\mathrm{ion}}/({{nr}}^{2})\approx 4\pi {m}_{{\rm{p}}}{L}_{\mathrm{ion}}/(\dot{M}/{v}_{{\rm{w}}})$ in cgs units. Chevalier & Irwin (2012) find that ξ greater than 5000–10⁴ will ionize all the elements, a precise value depending on the characteristic temperature of the spectrum. For our parameters

$\begin{eqnarray}&&\xi \sim 3\times {10}^{3}\left(\displaystyle \frac{{L}_{\mathrm{ion}}}{{10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left[\displaystyle \frac{\dot{M}/{v}_{{\rm{w}}}}{{(\dot{M}/{v}_{{\rm{w}}})}_{* }}\right]}^{-1},\end{eqnarray} \tag{ 50 }$

which is consistent with our results within an order of magnitude, and explains the tendency seen in our spectra that models with higher E_ej (i.e., higher L_ion) and smaller $\dot{M}/{v}_{{\rm{w}}}$ are visible in soft X-rays.

Another notable feature is that the spectrum νL_ν below the cutoff is nearly flat for the E1M1 and E3M1 cases, while it becomes steeper up to νL_ν ∝ ν for the E10M1 case. The latter is simply due to the fact that the free–free emissivity j_ν,ff is flat at h ν ≪ k_B T_e, and the Compton and IC scatterings are ineffective. Repeated scattering between electrons and photons can convert X-ray photons to lower energies, which flattens the spectrum from the free–free one.

We calculate the optical and X-ray light curves to compare them with observations. For the optical light curves, we use the νL_ν from our model and adopt the Johnson–Cousins filter response function to obtain the (absolute) B- and V-band magnitudes with

$\begin{eqnarray}&&M=-2.5{\mathrm{log}}_{10}\left\{\frac{\int [{L}_{\nu }/4\pi {\left(10\mathrm{pc}\right)}^{2}](c/{\lambda }^{2})(\lambda /{hc})e(\lambda )d\lambda }{\int {f}_{\nu ,\mathrm{ref}}(c/{\lambda }^{2})(\lambda /{hc})e(\lambda )d\lambda }\right\},\end{eqnarray} \tag{ 51 }$

where e(λ) is the filter response function for each band, ⁶ f_ν,ref is the reference flux that defines the zero-point, and we use the conversion f_λ = (c/λ²)f_ν . For f_ν,ref we adopt the values 4260 Jy and 3640 Jy for the B and V bands, respectively (see Table 4 of Bessell 1979). The resulting light curves in the B and V bands are shown in Figure 5. The peak magnitude of average Type IIn SNe is estimated to be M_V = −18.4 ± 1.0 (Kiewe et al. 2012), with more recent estimates (Richardson et al. 2014; Nyholm et al. 2020) giving consistent results. Our results that roughly reproduce this magnitude favor mass-loss rates of the order of 0.1 M_⊙ yr⁻¹. This is at odds with the much lower $\dot{M}$ derived for some SNe IIn in previous works (Kiewe et al. 2012; Taddia et al. 2013). This should be partly due to the shortcomings of our model with incomplete opacity in the shocked region and an assumption of a one-zone density profile, which are elaborated in Section 5. The discrepancy can also be from the previous works' assumption of a high conversion efficiency from the kinetic energy dissipation rate to the Hα luminosity, _Hα = 0.1. Inspecting Figure 4, some spectra show _Hα ≪ 0.1, especially for the M1 models. A low _Hα is also implied from a more recent SN IIn sample (Kokubo et al. 2019). As the value of $\dot{M}$ derived from this method scales as the inverse of _Hα , this implies that the actual $\dot{M}$ can be much higher than previously estimated. A more sophisticated modeling of _Hα may give a better understanding of $\dot{M}$, although it is beyond the scope of this work.

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For the X-ray luminosity, most observations of SNe IIn, using, e.g., Chandra and XMM-Newton, have been done in the soft X-ray band covering energies up to 10 keV. Here we define the X-ray luminosity as

$\begin{eqnarray}&&{L}_{{\rm{X}}}={\int }_{2\mathrm{keV}}^{10\mathrm{keV}}{L}_{\nu }d\nu .\end{eqnarray} \tag{ 52 }$

This is to be compared with the bolometric luminosity integrated over the frequency range in our model:

$\begin{eqnarray}&&{L}_{\mathrm{bol}}={\int }_{0.1\mathrm{eV}}^{{10}^{7}\mathrm{eV}}{L}_{\nu }d\nu .\end{eqnarray} \tag{ 53 }$

We show the resulting light curves in Figure 6. For the E1M1, E1M10, and E3M10 models, CSM absorption is significant enough to make X-rays observable only at later times. The E1M10 model is extreme, as soft X-ray photons are completely absorbed until the end of our simulation of ∼200 days. For most SNe IIn X-rays have been observed only around a year after explosion (Chandra 2018). This observational fact may favor mass-loss rates of the order of 0.1 M_⊙ yr⁻¹ for SNe IIn, independent of the previous argument on the optical luminosity. For models E3M1 and E10M1, where the CSM quickly becomes transparent to X-rays, the X-ray luminosity can become comparable to the bolometric luminosity. In fact, they show very bright X-ray emission of luminosity 10⁴²–10⁴⁴ erg s⁻¹ for ∼100 days. While the rate of such energetic explosions inside a moderate CSM is uncertain, they are good targets for all-sky X-ray surveys carried out with, e.g., eROSITA (see Figure 5.7.2 of Merloni et al. 2012).

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SN 2010jl was detected in a nearby star-forming galaxy, UGC 5189A (Newton & Puckett 2010), whose distance is measured to be 49 Mpc. Observations from the UV to the near-IR have revealed it to be a luminous Type IIn SN with bolometric luminosity of at least 3 × 10⁴³ erg s⁻¹ (Zhang et al. 2012; Fransson et al. 2014; Ofek et al. 2014). Various studies converge on the CSM being massive, with estimates of the mass-loss rate ranging from 4 × 10⁻² to 1 M_⊙ yr⁻¹ (Zhang et al. 2012; Moriya et al. 2013; Fransson et al. 2014; Ofek et al. 2014; Chandra et al. 2015).

X-ray emission was detected from around 40 days after the explosion (Chandra et al. 2012b). Long-term X-ray observations by Chandra et al. (2015) found an evolving column density from N_H ∼ 10²⁴ cm⁻² to 10²¹ cm⁻² over a few years, and radio emission was first detected on day 570. This implies that the shock has been running through a dense CSM that efficiently absorbs X-ray and radio emission.

We compare our model to the bolometric and X-ray light curves of SN 2010jl. We adopt parameters close to those estimated by previous works: ejecta with energy 1.6 × 10⁵¹ erg for mass of 10 M_⊙, an outer ejecta slope of n = 7, and a CSM mass-loss rate of 0.13 M_⊙ yr⁻¹. As an input of CLOUDY we adopt a CSM metallicity of 0.35 solar, which is around the upper limit inferred from observations of the host galaxy (Stoll et al. 2011). We compare the light curves only up to around day 200. The bolometric light curve shows a break between days 200 and 400 (Fransson et al. 2014), which can be because either the FS has reached the outer edge of the dense CSM or the RS has reached the inner ejecta. In either case, our assumption of self-similarity in deriving the shock dynamics is inapplicable beyond this time.

The result is shown in Figure 7. We find the bolometric light curve matches well the best fit obtained from observations in Fransson et al. (2014). Both the FS and RS are radiative throughout the time range we consider, which is consistent with the analytical estimate by Fransson et al. (2014).

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While the bolometric light curve is consistent, the X-ray luminosity in the right panel is not reproduced. Our predicted X-ray luminosity (orange dashed line) falls far below the observed one (black points). Furthermore, the X-ray luminosity at the FS front obtained from our model (solid line) overshoots the unabsorbed luminosity, calculated from spectral analysis of observational data (gray points) by Chandra et al. (2015).

These inconsistencies can be reconciled by abandoning the simplest assumption of the CSM being spherical, and by introducing, e.g., the existence of clumps or asphericity as mentioned in previous works (Fransson et al. 2014; Chandra et al. 2015; Katsuda et al. 2016). To show this, we consider a case where 1% of the emission from the shock front can escape, possibly due to incomplete coverage of the CSM. The bolometric light curve in the left panel of Figure 7 is hardly affected with this small solid angle.

The X-ray light curve in this case is shown as green dotted lines, which roughly matches the observed X-ray luminosity up to day 200. With this 1% escape fraction, the X-ray spectrum at around 60 days (Figure 8) extends to about 10 keV, consistent with Chandra observations at a similar epoch (Chandra et al. 2015; Katsuda et al. 2016). Intriguingly, an order of 1% escape fraction is also inferred independently from the modeling of the radio emission (Murase et al. 2019), though the radio data were taken long after the time range we consider (at around day 600). Nonetheless, SN 2010jl demonstrates that observations in both the optical and X-ray bands would be an effective way of diagnosing the presence of asymmetry in the CSM.

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SN 2014C was first detected by the Lick Observatory Supernova Search (Kim et al. 2014). It was originally classified as an SN Type Ib in a host galaxy, NGC 7331, at a distance of 15 Mpc. However, from ∼300 days after explosion the source showed strong hydrogen emission lines analogous to those of Type IIn (Milisavljevic et al. 2015), and became bright in X-rays (Margutti et al. 2017). This transition from a free expansion to a strong interaction indicates that the progenitor exploded inside a shell-like massive CSM embedding a low-density cavity.

We compare our model with X-ray observations of SN 2014C. However our dynamical modeling, which assumes an extended CSM with a power-law density profile, cannot be applied to the shell-like CSM argued for SN 2014C. Instead, we focus on one epoch, at day 400, and construct an X-ray spectrum at this epoch.

The X-ray spectrum at this epoch obtained by Margutti et al. (2017) shows a clean bremsstrahlung emission with a characteristic temperature of 18 keV. From this spectrum Margutti et al. (2017) deduced that the X-ray emission mainly originates from the optically thin FS region, which is close to adiabatic and is moving at a velocity of V_sh ∼ 4000 km s⁻¹. The shock radius at this epoch is estimated to be 6.4 × 10¹⁶ cm from very long baseline interferometry imaging (Bietenholz et al. 2018). We adopt these values and use our model with only the FS component to obtain the spectrum, as the RS component is argued to be negligible (Margutti et al. 2017). We vary the downstream number density n_down as a parameter. We put a CSM of solar metallicity (Milisavljevic et al. 2015) with a flat density profile with number density n_down/4. The 4 in the denominator is the compression ratio for γ = 5/3, as this shock is close to adiabatic. The width of the CSM is determined so that the hydrogen column density is 3 × 10²² cm⁻², as obtained from X-ray spectral fitting. For this case the advection time r_sh/V_sh ∼ 5 yr is much longer than 400 days. Thus we only calculate with our model up to 400 days, and obtain L_ν as

$\begin{eqnarray}&&{L}_{\nu }=4\pi {r}_{\mathrm{sh}}^{2}\frac{{r}_{\mathrm{sh}}}{4}\frac{{e}_{\mathrm{esc},\nu }(t=400\,\mathrm{days})}{400\,\mathrm{days}}.\end{eqnarray} \tag{ 54 }$

This time we do not include the factor 2 accounting for direction, as we do not consider the reprocessing by the (presumably optically thin) RS region. We also neglect the effect of diffusion since the diffusion timescale is negligible at this epoch.

We compare our spectrum (solid lines) and the observed spectrum by Margutti et al. (2017) (blue points) in Figure 9. Our spectrum, with a downstream density of n_down = 1.4 × 10⁶ cm⁻³, reasonably matches the bremsstrahlung component in Margutti et al. (2017). The prominent Hα line observed in Milisavljevic et al. (2015) is also seen in our spectrum, although a detailed comparison is subject to uncertainties in the profile and geometry of the CSM, and is beyond the scope of this work. One caveat is that the strong excess at 6.7–6.9 keV that Margutti et al. (2017) attribute to Kα transitions of H- and He-like ions of iron is not seen in our model. This excess is seen in SN 2006jd as well, and may be explained by an enhancement of iron as compared to the solar metallicity, or by the coexistence of a cooler component, e.g., from the RS region or from clumps (see Chandra et al. 2012a; Margutti et al. 2017 for a discussion).

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As mentioned in Section 2.2 we have included only free–free processes as the emission/absorption mechanism of photons in the shocked region. Although this may be a good approximation at low-metallicity environments, there are additional contributions from bound–free (and bound–bound) processes that can be important at metallicities around and beyond the solar metallicity. As seen in Figure 2 the plasma temperature in the shocked region can drop down to ∼10⁴–10⁵ K. At this temperature range metals are not fully ionized and thus can efficiently convert soft X-rays to UV and optical radiation. ⁷

We defer the precise modeling of this to future work, but the qualitative effect of this may be seen by enhancing the opacity by a factor η = [1 + (κ_bb + κ_bf)/κ_ff], where κ_bb, κ_bf, and κ_ff are the bound–bound, bound–free, and free–free opacities (Svirski et al. 2012). We crudely assume that η is independent of frequency and temperature, i.e., the free–free emissivity and absorption coefficient (Equations (22) and (23)) are multiplied by η, with everything else unchanged. We compare our results (assuming η = 1) for the E1M1 model to the case where η = 10.

The comparison can be seen in Figure 10. The left panel shows the spectrum at the shock radius, and the right panel shows the spectrum that is reprocessed through the CSM. We generally see that for a larger η there is enhancement of the optical component at the cost of reducing the X-ray component. The difference is most clearly seen at late epochs. Thus we conclude that inclusion of a more realistic bound–free opacity would be important for characterizing emission in the optical and soft X-rays, although this is included in the radiation transfer through the unshocked CSM. However, we should note that the hard X-ray spectrum should be close to the η = 1 one, as plasma reaching temperatures hotter than ∼10 keV is fully ionized and free–free emission should dominate there.

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Another caveat of our work is that we neglect the effect of radiative cooling on the dynamics. The effect of this can be summarized in two aspects.

One is the shock propagation. In this work we have adopted the self-similar model of shock evolution by Chevalier (1982), which assumes the shock downstream is adiabatic. We observe in some cases the shock becoming radiative, where solving the momentum equation assuming a thin shell (Chevalier & Fransson 2003) is more appropriate. In this case we obtain the time evolution of the shock radius as ${r}_{\mathrm{sh}}={[8\pi {v}_{{\rm{w}}}{g}^{n}/(n-4)(n-3)\dot{M}]}^{1/(n-2)}{t}^{(n-3)/(n-2)}$, which differs from Equation (A4) by a factor of [α(n − 4)(n − 3)/2]^1/(n−2). In the case of our model (α = 0.0595, n = 10) this is merely a ∼3% difference, which affects the bolometric light curve ($\propto {r}_{\mathrm{sh}}^{2}{\rho }_{{\rm{w}}}{v}_{\mathrm{sh}}^{3}$ for a radiative case) by ≲10%. We conclude that refinement of the shock dynamics will only have a small impact on our results.

The other point, perhaps more important, is that we neglect the density gradient in the shocked region for simplicity. The radiative processes adopted in our model depend on the plasma density, and thus cooling and heating will accelerate if this compression is taken into account. This can be partially taken into account by adopting a lower adiabatic index close to 1, which enhances the downstream density.

To see the effects of lowering γ we compare our E1M1 model (assuming γ = 4/3) with a result assuming γ = 1.2. With this low γ the downstream density is enhanced due to the higher compression ratio (factor 11/7), and slightly more due to the lower value of α by order 10%. Figure 11 shows the comparison of these two values of γ at the same epochs. In the early phase, the density enhancement helps convert more UV/X-ray photons to optical, as shown in the left panel. This can have a large impact on the absorption of soft X-rays around 1 keV, as shown in the right panel for day 24.7. In the late phase, the density has sufficiently dropped and the resulting spectra become almost the same.

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We have considered Coulomb collision as a relaxation process for protons and electrons. However it is known that for interactions of the ejecta and CSM, the FS turns collisionless around breakout (Katz et al. 2011; Murase et al. 2011). As a result collisionless relaxation can occur, which can modify the relaxation timescale of the plasma. Furthermore, a non-negligible fraction of the post-shock energy may be used to accelerate electrons and protons to relativistic energies through the diffusive shock acceleration mechanism (e.g., Blandford & Eichler 1987).

The latter is proposed to give rise to various emissions in both photons and neutrinos (Katz et al. 2011; Murase et al. 2011, 2019, 2014; Petropoulou et al. 2016; Zirakashvili & Ptuskin 2016; Murase 2018). The particles and their emissions also interact with optical/X-ray photons, e.g., through IC processes and two-photon annihilation processes. Detailed modeling of the spectrum such as done in this work may thus be important for deriving nonthermal emission from these relativistic particles. In future work we plan to apply our current emission model to calculate the nonthermal emission of interaction-powered transients. We note that these photons are expected to be prominent in radio and gamma rays, and we expect inclusion of this will not greatly modify our present results, which cover the optical to X-ray bands.