PROBING SHOCK BREAKOUT AND PROGENITORS OF STRIPPED-ENVELOPE SUPERNOVAE THROUGH THEIR EARLY RADIO EMISSIONS

Core-collapse supernovae (CC-SNe) are explosions of a massive star with a zero-age main-sequence mass (M_ms) exceeding ∼8 M_☉. Their observational appearance is diverse, controlled by the nature of the progenitor star, the explosion, and the circumstellar environment. These are probably mutually connected through intrinsic controlling factors, including at least the progenitor mass, metallicity, and binarity. Clarifying these mutual relations is a key issue in the study of supernovae (SNe). For this reason, stripped-envelope SNe (SNe IIb/Ib/Ic; Filippenko 1997) have been a target of various studies. Being explosions of stars that have lost all or most of their hydrogen envelopes before the explosion (Nomoto et al. 1993), their origins are likely a mixture of different evolutionary paths (i.e., a single massive star with M_ms ≳ 25 M_☉ or a binary evolution with M_ms ≲ 25 M_☉). They show diverse properties in explosion energies (Nomoto et al. 2010), with the most energetic ones linked to gamma-ray bursts (GRBs; see, e.g., Woosley & Bloom 2006 for a review). In this paper, we address the following three issues about their nature: (1) the nature of the shock breakout, (2) the progenitor structure and its relation to the shock breakout signal, and (3) radio emissions and the circumstellar material (CSM) environment.

The shock breakout emission is the first electromagnetic signal from SNe (Falk & Arnett 1977; Klein & Chevalier 1978). When the shock wave launched in the deepest part of the progenitor emerges from the stellar surface, an intensive UV/X-ray flash bursts out of the shock wave. The typical spectral energy of the emission is predicted to be sensitive to the progenitor radius, and for a Wolf–Rayet (W-R) progenitor it is mostly in X-rays. The shock breakout phenomenon was predicted theoretically more than a few decades ago, but so far its detection has been rare (Soderberg et al. 2008; Schavinski et al. 2008; Gezari et al. 2008; Modjaz et al. 2009) and was first inferred from the ionization structure around SN 1987A (Lundqvist & Fransson 1996). The shock breakout is essentially a very brief transient event. It is now an important target of time-domain astronomy, including future optical survey proposals to catch the shock breakout signals from high-z SNe (e.g., Tominaga et al. 2011). However, such optical surveys are not optimized for SNe from a W-R progenitor (i.e., a leading progenitor scenario for SNe Ib/c), since it is an X-ray event lasting only for ∼R_*/c ∼ 10 s (where R_* is the progenitor radius and c is the speed of light).

The final stage of stellar evolution has not been clarified for stripped-envelope SNe. Estimating the radius of a progenitor star provides important insight here, highlighted by the direct detection of an unprecedented blue supergiant progenitor with R_* ∼ 50 R_☉ for SN II 1987A (see Arnett et al. 1989 for a review). So far, without direct progenitor detection (see Smartt 2009 for a recent review), the shock breakout signal and subsequent early optical emission have been used to estimate the size of the progenitors of SNe Ib/c (Soderberg et al. 2008; Chevalier & Fransson 2008; Modjaz et al. 2009; Rabinak & Waxman 2011). SN Ib 2008D is the only example where the shock breakout X-ray was convincingly detected (Soderberg et al. 2008; Modjaz et al. 2009), and its estimated radius is still under debate (Chevalier & Fransson 2008; Rabinak & Waxman 2011). Any other independent information on the as yet unresolved nature of the progenitors is highly valuable.

Radio emissions have been detected from a number of nearby SNe IIb/Ib/Ic (see Chevalier & Fransson 2006 for a review). The radio emission is created through SN–CSM interaction, thus it is a strong tool with which to study the CSM environment. A relation between the time of the peak (t_p) and the luminosity at the peak (L_p) can be used to estimate the size of the emitting region (thus the velocity of the shock wave at the SN–CSM interaction), and the CSM density (Chevalier 1998). Generally SNe of different spectral sub-classes occupy different regions in the t_p–L_p plot, and SNe Ib/c (plus a part of SNe IIb, i.e., "compact" SNe IIb; "cIIb" hereafter) are characterized by relatively low t_p and large L_p for a given t_p (Chevalier & Fransson 2006; Chevalier & Soderberg 2010). However, radio emission from SNe IIb/Ib/Ic is diverse in a sense that, even excluding outliers, they span more than an order of magnitude in both t_p and L_p (t_p ∼ 10–100 days, L_p ∼ 10²⁶–10²⁸ erg s⁻¹ Hz⁻¹). Among SNe IIb/Ib/Ic, SN Ic 2002ap is a suggested outlier characterized by small t_p and L_p. As we show in Section 2, SN Ic 2007gr has properties similar to SN 2002ap, and both of these SNe show features different from other SNe aside from just the t_p and L_p.

In this paper, we propose a new idea to explain the peculiarities of radio emission from SNe Ib/c with low radio luminosities and short rising times (i.e., SN 2002ap, 2007gr), and link our interpretation with the properties of the shock breakout and the progenitor. In Section 2, we summarize the radio properties of SNe 2002ap and 2007gr, suggesting that these two SNe belong to a sub-class of stripped-envelope SNe (in terms of radio properties). In Section 3, we explore the consequences of the hydrodynamic interaction between the SN ejecta and CSM in the early phase, and show that the evolution can be different for SNe exploding in low-density (SNe 2002ap and 2007gr) and high-density (other SNe IIb/Ib/c) CSM environments. In Section 4, we calculate multi-band radio light curves for SNe 2002ap and 2007gr, showing that their radio properties are naturally explained in terms of the hydrodynamic evolution discussed in Section 3. In Section 5, we discuss implications of our findings for the nature of the shock breakout and the progenitor. The paper is summarized in Section 6 with our conclusions and discussion. Our scenario requires (relatively) inefficient acceleration of electrons at a shock, and a discussion of this issue is given in the Appendix.

Properties of radio emissions from SNe are characterized by a relation between the peak luminosity and the peak date (normalized at 5 GHz; Chevalier 1998; Chevalier et al. 2006). These are especially useful diagnostics for SNe Ib/c (Chevalier & Fransson 2006; Chevalier & Soderberg 2010): since the synchrotron self-absorption (SSA) is a dominant absorption process for these SNe exploding within a relatively low-density CSM environment, a place in this t_p–L_p plot provides an estimate of the size of the emitting region (thus the velocity of the shock wave). The velocity of the shock wave estimated in this way is typically ∼0.1c for SNe cIIb/Ib/Ic. At the same time, within the framework of the SSA, t_p is smaller for a less dense CSM.

SN 2002ap is a classical example of a broad-line SN Ic in optical wavelengths, which is suggested to be more energetic than canonical SNe (Mazzali et al. 2002). SN 2002ap also exhibits peculiar radio emission properties as compared to other SNe Ib/c (Berger et al. 2003; Björnsson & Fransson 2004). While L_p ∼ 10²⁷ erg s⁻¹ Hz⁻¹ and t_p(ν_p/5 GHz) ∼ 30 days for a majority of SNe Ib/c, in SN 2002ap, L_p ∼ 10²⁵ erg s⁻¹ Hz⁻¹ and t_p(ν_p/5 GHz) ∼ 3 days, characterized by an extremely low luminosity and a fast rise to the peak. According to the SSA scaling relation, the CSM density is estimated to be lower than that of other SNe Ib/c at least by an order of magnitude. In the optically thin phase, a majority of SNe Ib/c show a radio spectral slope of α ∼ −1 and a temporal slope of −1.5 ≲ β ≲ −1.3 (where L_ν∝ν^αt^β; Chevalier & Fransson 2006). However, SN 2002ap showed a shallow decay with β ∼ −0.9 while the spectral index was similar to that of other SNe (α ∼ −0.9; Berger et al. 2003; see also Björnsson & Fransson 2004; Chevalier & Fransson 2006). This peculiarity negates a standard interpretation for SN 2002ap. There has been only one theoretical interpretation suggested so far: the slope of the relativistic electrons' energy distribution might have been different and flatter than in other SNe Ib/c (Björnsson & Fransson 2004; see also Chevalier & Fransson 2006).

We point out that SN Ic 2007gr has radio properties similar to SN 2002ap, despite its optical properties belonging to a "normal" (or non-broad-line) class (Hunter et al. 2008; Valenti et al. 2008). In Figure 1, we compare the multi-band radio light curves of SNe 2002ap (Berger et al. 2003) and 2007gr (Soderberg et al. 2010a). SN 2007gr showed L_p ∼ 10²⁶ erg s⁻¹ Hz⁻¹ and t_p(ν_p/5 GHz) ∼ 5 days, both about an order of magnitude smaller than typically found for SNe Ib/c, placing this SN close to SN 2002ap in these properties. Furthermore, the radio spectral index and decay slope in optically thin phases are similar to those of SN 2002ap. Not only the t_p and L_p, but also the temporal slopes of SNe 2002ap and 2007gr (β ≳ −1) are different from other SNe Ib/c (β ≲ −1.3), while their spectral indices are similar (α ∼ −1) to those of other SNe Ib/c.

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The previous theoretical interpretation of the radio emission from SN 2002ap was based on the slow decay rate as mentioned above (Björnsson & Fransson 2004), and this is our motivation for investigating an alternative explanation for this behavior. However, it should be noted that the quality of the radio data of SN 2002ap, as well as that of SN 2007gr, does not allow for a very accurate determination of the decay rate. Fitting the radio light curve of SN 2002ap (Berger et al. 2003) during 4–50 days (eight points) by a function f_ν∝t^β, we obtain β = −0.87 ± 0.17 at 8.46 GHz.¹ The error indicated here is 1σ. For SN 2007gr (Soderberg et al. 2010a), we obtain β = −0.85  ±  0.12 (4–18 days, five points). For comparison, the "radio-normal" SN cIIb 2011dh (Soderberg et al. 2012; Krauss et al. 2012) shows β = −1.13  ±  0.16 at 29.0 GHz (30–100 days, four points). So, the preferred value of the decay rate (β) is flatter by ∼0.3 in SNe 2002ap and 2007gr than in the canonical case. The deviation of SN 2002ap in the decay rate from the canonical value (β = −1.3; Chevalier & Fransson 2006) is therefore at the 2.5σ level, and for SN 2007gr it is above 3σ. Given the small number of data points, however, this nominal significance should be regarded as merely indicative. In any case, in this paper we will investigate the implications provided by this slow decay.

As summarized in this section, SNe 2002ap and 2007gr (seem to) share common properties, and we suggest that they form a sub-class of stripped-envelope SNe based on their radio properties. Observationally this class of objects is rare (about 10% of radio-detected stripped-envelope SNe), but may well intrinsically include a large fraction of stripped-envelope SNe: SNe 2002ap and 2007gr are among the nearest examples of stripped-envelope SNe, and they would not have been detectable at ∼30 Mpc, i.e., the typical distance of most radio-detected SNe (Soderberg et al. 2010a). Future observations of nearby SNe in radio wavelengths will be critical to establish if the shallow decay is a common property of these radio low-luminosity stripped-envelope SNe.

One of the striking features of radio emission from SNe 2002ap and 2007gr is the short rise time, with timescales less than 10 days. In this section we discuss the hydrodynamic evolution of the shocked region in this early phase. In previous studies, the self-similar solution for the interaction between the expanding SN ejecta and CSM has been assumed (Chevalier 1982). However, the basic assumption in the formalism would not apply in the very early phase of the expansion and/or low CSM density. The effect of the interaction on the hydrodynamic evolution becomes important only after a sufficient amount of CSM is swept up by the forward shock, and then the ejecta begin to decelerate, following the self-similar solution that is eventually established. Before this phase, the ejecta feel the CSM almost as a vacuum, thus they are in the free-expansion phase.²

The density structure at the outermost ejecta can be approximated by

$$\begin{eqnarray} \rho _{\rm SN} & \sim & 8.3 \times 10^{-18} E_{51}^{3.59} \left(\frac{M_{\rm SN}}{M_{\odot }}\right)^{-2.59} \nonumber\\ &&\times \left(\frac{V}{0.3 c}\right)^{-10.18} t_{\rm d}^{-3} \ {\rm g} \ \rm {cm}^{-3}, \end{eqnarray} \tag{ 1 }$$

where ρ_SN is the density of the SN ejecta at velocity V, E₅₁ is the kinetic energy of the SN ejecta (in unit of 10⁵¹ erg), M_SN is the ejecta mass, and t_d is the time since the explosion in days (Matzner & McKee 1999; Chevalier & Fransson 2008).

The maximum velocity of the SN ejecta is determined by the shock breakout set by the radiative losses due to the shock breakout flash, unless a process exists that could alter the dynamics at the highest velocity ejecta after the breakout. Although detailed radiation hydrodynamic modeling is required to obtain the exact value and the result depends on details in the treatment of the physics involved, the first-order estimate can be obtained by analytic considerations (Matzner & McKee 1999):

$$\begin{eqnarray} V_{\rm SN} & \sim & 0.48 c \left(\frac{\kappa }{0.34\, {\rm cm}^2\, {\rm g}^{-1}}\right)^{0.16} E_{51}^{0.58} \left(\frac{M_{\rm SN}}{M_{\odot }}\right)^{-0.42} \nonumber\\ &&\times \left(\frac{R_{\rm *}}{10\, R_{\odot }}\right)^{-0.32}, \end{eqnarray} \tag{ 2 }$$

where R_* is the radius of the progenitor (for that with a radiative envelope).

Let us assume that the density distribution of the CSM is expressed by the steady wind solution, $\rho _{\rm CSM} = \dot{M} / 4 \pi r^{2} v_{\rm w}$ (where $\dot{M}$ and v_w are the mass-loss rate and the wind velocity). Thus,

$$\begin{equation} \rho _{\rm CSM} = 5 \times 10^{11} A_{\rm *} r^{-2} \ {\rm g} \ {\rm cm}^{-3}, \end{equation} \tag{ 3 }$$

where A_* ∼ 1 is a reference value corresponding to typical W-R mass-loss properties (i.e., $\dot{M} \sim 10^{-5}\, M_{\odot }$ yr⁻¹ and v_w ∼ 1000 km s⁻¹). Once the self-similar solution is established, the evolution of the velocity at the contact discontinuity follows the following form (Chevalier 1982; Chevalier & Fransson 2006):

$$\begin{equation} V_{\rm c} \sim 8 \times 10^9 E_{51}^{0.43} \left(\frac{M_{\rm SN}}{M_{\odot }}\right)^{-0.32} A_{\rm *}^{-0.12} t_{\rm d}^{-0.12} \ {\rm cm} \ {\rm s}^{-1}. \end{equation} \tag{ 4 }$$

Since the self-similar solution describes the deceleration of the ejecta, the velocity here should be smaller than V_SN. Thus, the self-similar solution does not apply if

$$\begin{equation} t \lesssim t_{\rm dec} = 0.4 E_{51}^{3.58} \left(\frac{M_{\rm SN}}{M_{\odot }}\right)^{-2.67} A_{\rm *}^{-1} \left(\frac{V_{\rm SN}}{0.3 c}\right)^{-8.33} \ {\rm day{\rm s}}. \end{equation} \tag{ 5 }$$

Assuming V_SN ≫ 0.1c and the typical value of a few M_☉ for the ejecta mass and A_* ∼ 1, the strong interaction starts at the latest a few days after the explosion, and thus the free-expansion phase is negligible. This justifies the use of the self-similar solution for most SNe. However, if V_SN ⩽ 0.1c, then even a typical W-R wind density is not enough to decelerate the ejecta to reach the self-similar solution within the timescale of ∼100 days. Furthermore, a low CSM density prevents the SN ejecta from being decelerated, and with A_* ∼ 0.01 as inferred for the sub-class of SNe Ib/c (for example, the scaling relation of the SSA peaks implies that A_* ∼ 0.04 for SN 2002ap), the ejecta do not experience significant deceleration for ∼100 days even with V_SN ∼ 0.2c.

To confirm the above arguments, we have performed a series of hydrodynamic simulations for the early phase of the interaction. The ejecta structure is assumed to follow a power law up to the maximum ejecta velocity V_SN (Equation (1)), with E₅₁ = 1 and M_ej = 3 M_☉. The CSM density distribution is assumed to be a power law as described by Equation (3). We have varied V_SN and A_* as parameters, choosing V_SN = 0.1c, 0.2c, 0.3c and A_* = 0.01, 1, 100 (thus a total of nine models were investigated). The interaction starts at day 1 since explosion. Adiabatic Euler equations under spherical symmetry have been solved with the adiabatic index of 5/3, using the Harten–Lax–van-Leer Contact solver (Toro 1999) to treat the discontinuity and the shock waves (Maeda et al. 2002). The number of meshes is 8500, linearly covering up to 7.8 × 10¹⁶ cm, so the spatial resolution is about 10¹³ cm which is sufficient to resolve the detailed structure of the interaction region throughout the computation.

Figures 2 and 3 show the evolution of the forward shock velocity and radius as a function of time. It is seen that the above analytical estimate roughly explains the behavior of the shock wave velocity: models with low V_SN and/or low A_* do not follow self-similar deceleration, but rather show an almost constant forward shock velocity. According to Equation (5), we expect t_dec as follows: for A_* = 100, t_dec < 2 days for all values of V_SN, for A_* = 1, t_dec < 1 day for V_SN > 0.2c but ∼200 days for V_SN = 0.1c, and for A_* = 0.01, t_dec ∼ 2, 60, 20, 000 days for V_SN = 0.3c, 0.2c, 0.1c, respectively. In the simulations, for A_* = 100 the self-similar solution is approximately reached for all the cases. For A_* = 1 the high-velocity models with V_SN ≳ 0.2c quickly follow the self-similar solution but the model with V_SN = 0.1c starts to decelerate at t ∼ 100 days. For A_* = 0.01, the model with V_SN = 0.1c never experiences deceleration during the simulated period of time, and the model with V_SN = 0.2c does not show signs of deceleration before t ∼ 40 days.

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Figure 4 shows the evolution of the velocity profiles for SN models expanding into the low-density CSM with A_* = 0.01, with the SN ejecta velocity V_SN = 0.1c, 0.2c, and 0.3c. One immediately notices that all three models do not follow the self-similar solution in the early phase, since in such a case there must be no difference according to different V_SN. In our simulations, the velocity of the leading edge of the ejecta is not identical to the initial value (i.e., V_SN). At 6 days after the explosion (which may depend on the initial setup), the forward shock begins to develop, and the maximum velocity of the ejecta is ∼75, 000 km s⁻¹ for V_SN = 0.3c, 65, 000 km s⁻¹ for V_SN = 0.2c, and 45, 000 km s⁻¹ for V_SN = 0.1c. Thus, the kinetic energy is redistributed in such a way that it is decelerated for a large value of V_SN and accelerated for a small value of V_SN. This kinetic energy redistribution may partly stem from a specific simulation setup, and details of this process could be dependent on the initial conditions. In any case, once the velocity profile is quickly adjusted, there is no further significant evolution, and the following arguments should not be sensitive to details in the numerical setup. The forward shock is formed with the velocity at roughly 80% of the (redistributed) maximum velocity of the ejecta, and the forward shock expands into the CSM at almost constant velocity without deceleration.

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It has been argued that the free-expansion phase is negligible for SNe Ib/c, with the shock breakout velocity V_SN > 0.3c estimated with Equation (2) or similar expressions, under the assumption that R_* ∼ R_☉ (Chevalier & Fransson 2006). However, it is not yet clear to what extent the analytic expression of the shock breakout physics is accurate: the expression is an order-of-magnitude estimate with various assumptions. The progenitor structure just before the explosion is under debate, without direct detection of the SNe Ib/c progenitors. For example, a peculiar SN Ib 2006jc showed a luminous-blue-variable-like eruption about two years before the SN explosion (Pastorello et al. 2007). The mechanism of such an outburst of the proposed WCO W-R progenitor (Tominaga et al. 2008) has not yet been clarified, highlighting our ignorance of the final evolution of W-R stars toward an SN explosion.

Considering all of these uncertainties in the shock breakout dynamics and progenitor scenarios, our motivation in this paper is to tackle these issues from the opposite direction: we first provide possible, observationally based information on these issues, then compare this with theoretical expectations. Our strategy is as follows. Rather than starting with the shock velocity based on Equation (2) (or similar expressions), we first compare the velocity of the forward shock estimated from the radio data and that predicted by the self-similar decelerating solution. The two should agree if the dynamics are in the self-similar phase, while the radio-derived velocity will be lower than the self-similar expectation if the ejecta are in the free-expansion phase. We then examine the radio spectra and light curve properties of these SNe to see if the free-expansion solution provides a consistent view. Once we obtain information on the free-expansion velocity through these analyses, then we could translate this information to the velocity of the shock breakout. This can then be used to calibrate the shock breakout physics (e.g., Equation (2)) and/or the nature of progenitors.

Useful diagnostics of the expansion of the forward shock, especially for stripped-envelope SNe, are provided by the observed t_p–L_p relation. Figure 5 is a reproduction of the diagram from Soderberg et al. (2012; see also Chevalier 1998; Chevalier et al. 2006). The overplotted lines in the figure are the estimates that divide the (truly) free-expansion phase and the self-similar deceleration phase using Equation (5), for different choices of ejecta mass. Below these lines, the observationally derived velocity (through the SSA model) is smaller than the self-similar prediction; thus SNe in this range are very likely in the free-expansion phase.

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Figure 5 shows that SNe Ib/c as well as cIIb generally follow the self-similar solution with E₅₁ ∼ 1 and M_ej ∼ 1–3 M_☉. An exception is SN 2002ap (as well as 1987A) which falls well below the self-similar prediction. Since different SNe have different properties in E₅₁ and M_ej, one has to compare the self-similar evolution and the radio properties on a case-by-case basis. With E₅₁ ∼ 5 and M_ej ∼ 3 M_☉ for SN 2002ap estimated through optical modeling (Mazzali et al. 2002), the discrepancy between the self-similar solution and the observed radio properties is even larger. The same argument applies to SNe 2007gr (E₅₁ ∼ 2, M_ej ∼ 2 M_☉; Hunter et al. 2008; Valenti et al. 2008) and 1987A (E₅₁ ∼ 1.4, M_ej ∼ 14 M_☉; e.g., Blinnikov et al. 2000; Maeda et al. 2002), which showed that their radio properties are also below the self-similar case. Thus, we suggest that the shock wave evolution of SNe 2002ap and 2007gr (and 1987A) was in the free-expansion phase in these early epochs responsible for the early radio emission covering the SSA peak.

Results from previous studies on the radio properties of SN 2002ap have been controversial. In the following, we show that a self-consistent picture can be obtained by considering the "free expansion" which was not taken into account in previous studies. We further show that the same solution could apply to SN 2007gr as well. In our scenario, the peculiar features of SNe 2002ap and 2007gr are natural consequences of SNe exploding within a relatively low-density CSM.

We calculate radio emissions from SNe Ib/c as follows. The basic formalisms have been developed by Fransson & Björnsson (1998) and Björnsson & Fransson (2004), and specific prescriptions used here are given by Maeda (2012; see also Chevalier 1998; Soderberg et al. 2005, 2010a; Chevalier et al. 2006; Chevalier & Fransson 2006). The synchrotron radio luminosity νL_ν in the optically thin phase is given as

$$\begin{eqnarray} &&\nu L_{\nu } \sim \pi R_{\rm sh}^2 V_{\rm sh} n_{\rm rel} \gamma _{\nu }^{2-p} m_{\rm e} c^2 \left[1 + \frac{t_{\rm synch (\nu)}}{t} + \frac{t_{\rm synch (\nu)}}{t_{\rm other} (\nu)}\right]^{-1}. \nonumber\\ \end{eqnarray} \tag{ 6 }$$

Here R_sh and V_sh are the position and the velocity of the forward shock, and n_rel is the number density of the relativistic electrons. The relativistic electrons are assumed to follow a power-law distribution with index p as a function of energy (note that throughout this paper we use p as the intrinsic power-law index, before being altered by cooling effects). t_synch is the synchrotron cooling timescale. t_other is the timescale for other energy loss processes that do not emit at the radio frequency (i.e., inverse Compton (IC) scattering in the situation investigated here). The Lorentz factor of the electrons emitting at frequency ν is γ_ν ∼ 80ν^0.5₁₀B^−0.5 (here ν₁₀ = ν/10¹⁰ Hz and B is in Gauss).

In SNe IIb/Ib/Ic, the assumption of equipartition has been proven to work well (Fransson & Björnsson 1998; Soderberg et al. 2005; Chevalier & Fransson 2006). The energy densities of the relativistic electrons and the magnetic field behind the shock wave are proportional to the thermal energy density created by the shock wave, while the proportional coefficients (_e and _B) are generally found to be lower than the full equipartition: typically _e = _B = 0.1 is assumed for simplicity, while Fransson & Björnsson (1998) and Maeda (2012) have suggested _e ≲ 0.01 and _B ∼ 0.01–0.15. Our general arguments are independent of the values of _e and _B, a further discussion of which is given in the Appendix. The amplified magnetic field strength and the relativistic electron density are expressed as follows:

$$\begin{eqnarray} B & \sim & 2.2 \times 10^{6} \epsilon _{B, -1}^{0.5} A_{\rm *}^{0.5} \frac{V_{\rm sh}}{R_{\rm sh}}\ {\rm gauss}, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} n_{\rm rel} & \sim & 2.4 \times 10^{17} \frac{p-2}{p-1} \epsilon _{e, -1} A_{\rm *} \left(\frac{V_{\rm sh}}{R_{\rm sh}}\right)^2 {\rm cm}^{-3}. \end{eqnarray} \tag{ 8 }$$

Substituting these expressions into Equation (6), the synchrotron emission is scaled as

$$\begin{equation} L_{\nu } \propto V_{\rm sh}^3 \nu ^{-p/2} B^{(p-2)/2} \left[1 + \frac{t_{\rm synch (\nu)}}{t} + \frac{t_{\rm synch (\nu)}}{t_{\rm other} (\nu)}\right]^{-1}. \end{equation} \tag{ 9 }$$

Note that so far no assumption has been made on the time dependence of the expansion of the interaction region.

In these SNe, the main cooling agencies are synchrotron cooling and IC scattering. These cooling timescales are estimated as

$$\begin{eqnarray} t_{\rm synch} (\nu) & \sim & 110 \nu _{10}^{-0.5} B^{-1.5} \ {\rm days}, \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} t_{\rm IC} (\nu) & \sim & 1.7 \nu _{10}^{-0.5} B^{0.5} \left(\frac{L_{\rm SN}}{10^{42}\, {\rm erg\, s}^{-1}}\right)^{-1} \left(\frac{R_{\rm sh}}{10^{15}}\, {\rm cm}\right)^{2}\ {\rm days}. \nonumber\\ \end{eqnarray} \tag{ 11 }$$

Now, the spectral index (α) and the temporal slope (β) of the synchrotron emission can be computed (here L_ν∝ν^αt^β). Describing the expansion dynamics as R_sh∝t^m (therefore V_sh∝t^{m − 1}), the result is summarized in Table 1.

Table 1. Characteristics of the Synchrotron Emission^a

Indices	Adiabatic	Syn.	IC
α	$\frac{1-p}{2}$	$-\frac{p}{2}$	$-\frac{p}{2}$
β	$(3m-3)+\frac{1-p}{2}$	$(3m-3)+\frac{2-p}{2}$	$(5m-5) +\frac{2-p}{2}+\delta$
α(p = 2)	$-\frac{1}{2}$	−1	−1
β(p = 2)	$(3m-3)-\frac{1}{2}$	(3m − 3)	(5m − 5) + δ
α(p = 3)	−1	$-\frac{3}{2}$	$-\frac{3}{2}$
β(p = 3)	(3m − 3) − 1	$(3m-3)-\frac{1}{2}$	$(5m-5)-\frac{1}{2}+\delta$

Note. ^aThe spectral index (α) and temporal slope (β) are shown for different cooling regimes (L_ν∝ν^αt^β). These are characterized by the electron distribution power-law index (p), the evolution of the forward shock (m, where R_sh∝t^m), and the evolution of the bolometric luminosity (δ, where L_bol∝t^δ).

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A majority of SNe Ib/c show a spectral index of α ∼ −1 and a temporal slope of −1.5 ≲ β ≲ −1.3 (Chevalier & Fransson 2006), consistent with the adiabatic expansion following the self-similar deceleration (m ∼ 0.9) with p ∼ 3. The "low-density CSM" SNe Ic, 2002ap and 2007gr, differ from the other cases in temporal slope: they show a rather shallow decay, β ∼ −1, while the spectral index is similar to that of others (α ∼ −1). One solution is to assume that the intrinsic slope of the relativistic electron distribution is flatter (p ∼ 2) than other SNe Ib/c (p ∼ 3) and that the electrons emitting in the radio frequency are in the IC cooling regime (Björnsson & Fransson 2004), while assuming self-similar dynamics (m ∼ 0.9). In this case, the spectral slope is reproduced (see Table 1). The temporal evolution of the (optical) bolometric luminosity of SN 2002ap was roughly ∝t^0.8 before 10 days after the explosion, and ∝t^−0.5 after that until 20 days after the explosion (Yoshii et al. 2003; Tomita et al. 2006). Thus, if the IC cooling dominates, then the radio temporal evolution will follow L_ν∝t^0.3 and ∝t⁻¹ before and after the optical peak (if the frequency of interest is optically thin). Well after the optical peak, the cooling will become less important as time goes by, so the radio light curve will be flattened until it reaches the temporal behavior of L_ν∝t^−0.8 (for p = 2). Thus, this scenario roughly explains the radio flux evolution, β ∼ −0.9 as observed.

Although the scenario is consistent with the available observational data and can also explain the X-ray emission together with the radio (Björnsson & Fransson 2004), a drawback of the scenario is that it requires an electron distribution quite different from that of other SNe Ib/c. Note that for a magnetic field strength of B ∼ 0.3 Gauss (Björnsson & Fransson 2004), the radio emission is produced by electrons with an energy of γ ∼ 50–150, similar to those responsible for radio emission from other SNe cIIb/Ib/Ic (e.g., Maeda 2012). Thus, the different slopes cannot be attributed to the different energy regimes probed for different objects. If this is true, it might mean that being a broad-line SN Ic, SN 2002ap might have much more efficient electron acceleration (Chevalier & Fransson 2006), but as we show in Section 2 the same argument should apply to SN 2007gr, which is a non-broad-line canonical SN Ic, making this interpretation less appealing.

In this paper, we suggest another solution for radio emission from SN Ic 2002ap, which also naturally explains why the "canonical" SN Ic 2007gr shares similar properties. The analysis in the previous section suggests that the ejecta expansion of SNe 2002ap and 2007gr is approximated by the free expansion without any deceleration, rather than by the decelerating self-similar solution. In this case, α ∼ −1 and β ∼ −1 are obtained with p = 3, if the cooling is not efficient. This is the same situation as has been inferred for other SNe cIIb/Ib/Ic (e.g., Chevalier & Fransson 2006; Soderberg et al. 2012), with the only difference being the dynamic evolution: assuming a general value of p = 3, we predict that SNe in the low-density environment peak early in radio with a temporal index of β ∼ −1, while in the higher density environment SNe peak later and show a temporal evolution of β ∼ −1.3.

The radio light curves computed for the free-expansion case as compared to those of SN 2002ap are shown in Figure 6. The microphysics parameters are set as _B = _e = 0.1, as typically assumed for radio SNe. As expected from Figure 5, we require a relatively high forward shock velocity (V_sh = 60, 000 km s⁻¹) and low-density CSM (A_* = 0.007). The synchrotron cooling is included in the model, but its effect is negligible. Thus, the light curves show the temporal slope β ∼ −1 as observed (red solid line in the figure), if the other cooling mechanism, i.e., the IC cooling, is ignored.

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Although it looks like a simple and straightforward interpretation, the story is complicated by the effect of the IC cooling. In Figure 6, we also show the light curves with the IC cooling included. With the set of parameters adopted, the IC cooling is indeed important, confirming the claim made in previous works (Björnsson & Fransson 2004; Chevalier & Fransson 2006). With the standard p ∼ 3, the model now fails to reproduce the spectral index.

Since the match between the observed radio behaviors and the prediction without the IC cooling is striking, we investigate what conditions are necessary to remedy this problem. The Compton cooling timescale is larger if _e is smaller to reproduce a given luminosity. Figure 7 shows an example where we adopt a low value for _e. The model parameters are V_sh = 80, 000 km s⁻¹, A_* = 0.05, _B = 0.1, and _e = 10⁻³. The required mass-loss parameter is now increased roughly following the SSA scaling, _BA_*(_e/_B)^8/19, but it is still very low compared to other SNe Ib/c. The required velocity is also increased for the smaller value of _e roughly following the SSA scaling V_sh∝(_e/_B)^−1/19. Not only the decrease in the relativistic electron density, but also the increase in the velocity, thus radius, has the effect of reducing the IC cooling effect. The light curve with low _e is similar to the standard case. The difference is that in this model, the IC cooling is now negligible. With the parameters for this "inefficient electron acceleration" model and E₅₁ = 5 and M_ej = 3 M_☉, Equation (5) predicts that t_dec ∼ 360 days, justifying our assumption of free expansion.

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For SN 2007gr, we have found a solution similar to that for SN 2002ap (Figures 8 and 9). The radio emission for the free-expansion evolution explains the multi-band light curves fairly well without the IC cooling. With _e = _B = 0.1, again the IC cooling alters the spectral index, and we require a low value of _e to fit the radio properties. For our fiducial model (V_sh = 70, 000 km s⁻¹, A_* = 0.15, E₅₁ = 2, and M_ej = 2 M_☉), t_dec ∼ 40 days, which is long and roughly consistent with the observation.

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We note that the low value of _e is not physically unreasonable. Indeed, detailed modeling of radio properties of SN eIIb 1993J (Fransson & Björnsson 1998) and SN cIIb 2011dh (Maeda 2012) points to _e ≲ 0.01, lower than generally assumed. No strong constraints have been placed on other SNe (e.g., Chevalier et al. 2006). We discuss a few issues from the previous works on _e in the Appendix. The low efficiency of the electron acceleration has an important consequence in interpreting the X-ray emission, which is also discussed in the Appendix. We conclude that, so far, there is no strong observational indication against the low value of _e in stripped-envelope SNe (and we suggest that the low value of _e may be a generic feature).

5.1. SNe Ic 2002ap and 2007gr: Structure of W-R Stars at the End of Their Lives

5.2. Implications for Other Classes of SNe

In this paper, we have investigated a consequence of SNe Ib/c exploding within low-density CSM (i.e., A_* ≲ 0.1, or $\dot{M} \lesssim 10^{-6}\, M_{\odot }$ yr⁻¹ for a typical W-R wind velocity). Such an SN should show a "free-expansion" phase before entering into the self-similar deceleration phase if the post-shock breakout velocity is V_SN ≲ 0.3c. The predicted radio properties of such SNe are different from those of SNe in the self-similar phase, characterized by a shallow decline in the temporal evolution while the spectral index is the same, if all the other properties (i.e., the acceleration mechanism of electrons) are unchanged.

SNe exploding in a low-density CSM environment should be characterized in radio frequency by a fast rise to peak (within 10 days after the explosion) and a low luminosity at the peak. We have shown that all stripped-envelope SNe with these properties observed so far (SNe 2002ap and 2007gr, as well as SN 1987A from a blue giant progenitor) indeed have the expected properties in the temporal and spectral indices. The synthesized multi-band light curves show a good match to those observed for SNe 2002ap and 2007gr. We note that the other example, SN 1987A, was also well modeled by free expansion dynamics (Chevalier 1998). In our interpretation, the efficiency of the acceleration of electrons must be low in order to avoid the IC cooling in radio frequencies. This is the same conclusion we obtained for SN 2011dh (Maeda 2012), and we suggest that this may be a generic property of the SN–CSM interaction.

Understanding the radio properties from SNe in the t_p–L_p plot in terms of the dynamics of the shock propagation, we propose new diagnostics on the shock breakout and progenitor properties through early radio emission: the forward shock velocity in the free-expansion phase (V_sh) is closely related to the maximum velocity obtained at/after the shock breakout (V_SN). We suggest that the relatively low post-shock breakout velocity (∼0.3c) we have derived for SNe Ic 2002ap and 2007gr could indicate the existence of an envelope driven by a near-Eddington luminosity near the end of the W-R evolution (Gräfener et al. 2012). This highlights the usefulness/uniqueness of the proposed strategy as compared to other methods (e.g., breakout flash and/or subsequent optical emission) of probing the shock breakout and the progenitor: velocity information cannot be obtained using other methods. Also, for a majority of SNe cIIb/Ib/Ic, we reject an RSG progenitor through the radio properties.

The idea can, in principle, provide slightly different approaches to estimating the post-shock breakout maximum velocity and placing constraints on progenitor structures. Once one identifies the free-expansion phase (i.e., the transition from the free expansion to the self-similar dynamics in the decay slope), then one can estimate the post-shock breakout velocity using Equation (5) or a similar expression, adopting ejecta properties and the CSM density estimated independently. The expression, however, requires further calibrations, and does not provide a cross-check of the decay slope. For these reasons, we adopted a more detailed approach, in which we fitted the multi-band light curves and checked the assumed dynamic evolution with hydrodynamic simulations. Also, a stronger constraint than that used here could be placed on the progenitor radius when SNe do not show the free-expansion for most radio-detected SNe IIb/Ib/Ic (using the information at the radio peak), by using the earliest data points in which SNe are in the self-similar phase. A complication is that either the absorption (in the low frequency) or cooling (in the high frequency) can change the temporal slope in the early phase, thus distinguishing the different dynamics is generally difficult well before the radio peak. For this reason, we have placed a conservative upper limit on the progenitor radius using the radio information around the peak. Further study and calibration of these methods could be possible applications of the idea presented in this paper.

Future, large observational data sets in the radio frequency will be highly valuable. Such data have been increasingly accumulated recently thanks to great efforts of researchers working in the field (e.g., Soderberg et al. 2012). SNe in a low-density CSM environment, such as SNe 2002ap and 2007gr, will provide a new possibility for tackling properties of the shock breakout and the progenitor as mentioned above. A majority of radio-detected SNe IIb/Ib/Ic follow self-similar evolution, and thus their properties and distribution in the t_p–L_p plot will reveal the general distribution of the progenitor mass and the energetics independently from optical wavelengths: so far, their distribution suggests that a majority of them (i.e., radio-detected stripped-envelope SNe) come from relatively low mass progenitors (≲ 25 M_☉), indicating that a binary interaction path could be a dominant one, in line with other recent studies (e.g., Sana et al. 2012).

The low-density CSM around SNe Ic 2002ap and 2007gr may be related to the W-R progenitor structure just before the explosion. If the progenitor luminosity is larger, it would likely produce a higher-velocity wind, resulting in a lower value of A_*. This probably favors a massive WO star as a progenitor of these SNe Ic (e.g., Nugis & Lamers 2000). If this is true, we expect no SNe IIb/Ib would belong to the "rapid and faint" radio stripped-envelope SNe (i.e., SNe 2002ap and 2007gr are both of Type Ic), and future larger samples should be able to address this question.⁶ Searching for and observing these radio-faint stripped-envelope SNe will provide new clues about the progenitor scenario from this angle as well. Specifically, this will be best done by future observatories with better sensitivity than what is currently available. Because of the low radio luminosity of these objects, there may well be an observational bias in which we underestimate the frequency of these radio-weak SNe (Soderberg et al. 2010a). Once a volume-limited sample is constructed, it will hopefully connect the radio properties with different progenitor evolutionary paths.

K.M. thanks the anonymous referee for many valuable comments and suggestions. K.M. also thanks Claes Fransson for his comments on the manuscript and stimulating discussion. This research is supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan and by a Grant-in-Aid for Scientific Research for Young Scientists (23740141). K.M. acknowledges support from Stockholm University as a short-term visitor and thanks Jesper Sollerman and the staff there for their hospitality.

In our scenario, the acceleration of the relativistic electrons is required to be (relatively) inefficient (_e ≲ 0.01), as compared to frequently assumed (_e ∼ 0.1). As mentioned in the main text, so far there is no strong observational constraint against the low value of _e. Indeed, Fransson & Björnsson (1998) and Maeda (2012) suggested _e ≲ 0.01 for SN eIIb 1993J and SN cIIb 2011dh, respectively, based on detailed modeling of their radio properties. We discuss this issue in this appendix.

A.1. Previous Works

A.2. X-Ray Production through the Inverse Compton Mechanism

The IC upscattering of SN optical photons has been proposed as an X-ray emission production mechanism, especially favored for stripped-envelope SNe with A_* ∼ 1 (e.g., Björnsson & Fransson 2004; Chevalier & Fransson 2006; Soderberg et al. 2012). The scenario requires a large population of relativistic electrons with γ ≲ 50, and thus a large value of _e (≳ 0.1) if these electrons' energy distribution follows a power law extrapolated from the radio synchrotron-emitting electrons (typically with γ ∼ 50–200; see Maeda 2012 for a detailed discussion). In this section, we stress that having a large value of _e (≳ 0.1) is not a single solution for producing the IC X-rays, but there is an alternative interpretation in which the energy distribution of the IC-emitting electrons does not follow the extrapolation from the synchrotron-emitting electrons as suggested by Maeda (2012).

The low efficiency of the electron acceleration thus has an important consequence in interpreting the X-ray emission. Among several models suggested for the X-ray behavior of SN 2002ap, the IC scattering scenario seems the most plausible (Björnsson & Fransson 2004; Chevalier & Fransson 2006). However, if we adopt p ∼ 3, then we want to avoid the Compton cooling effect in the radio frequency (Section 4). We can estimate the IC X-ray luminosity in a form similar to the radio synchrotron emission:

$$\begin{equation} \nu L_{\nu } \sim \pi R^{2} V n_{\rm e} \gamma ^{2-p} m_{\rm e} c^2 \left(1+\frac{t_{\rm d}}{t_{\rm ic} (\gamma)}\right), \end{equation} \tag{ A1 }$$

where t_ic is the IC cooling timescale. The typical electron energy (γ_ic) where the Compton effect is significant is not sensitive to the value of _e since γ_ic∝L⁻¹_SNR_sh (not dependent on the microphysics parameters): at t_d = 5, γ_ic ∼ 140 and 250 for our models with _e = 0.1 and 10⁻³, respectively. On the other hand, the electron energy responsible for the X-ray emission through the IC is γ ∼ 30. Thus it is in the adiabatic phase, and the predicted X-ray luminosity is νL_ν(1keV) ∼ 6 × 10³⁶ erg s⁻¹ (for _e = 0.1) and ∼6 × 10³⁵ erg s⁻¹ (for _e = 10⁻³) at day 5. The former is lower than the observed value only by a factor of two, but the latter is more than an order of magnitude lower than observed. The estimate is consistent with the previous work by Björnsson & Fransson (2004).

We note, however, that the similar situation is found in SN cIIb 2011dh, for which the intensive radio and X-ray data allowed detailed modeling of its properties (Soderberg et al. 2012; Krauss et al. 2012; Bietenholz et al. 2012; Horesh et al. 2012). For SN 2011dh, there is almost no doubt that the power-law index of the relativistic electrons emitting in the radio frequencies is p ∼ 3, constrained by the late, optically thin light curves in multiple bands (Soderberg et al. 2012). While Soderberg et al. (2012) suggested a high value of _e and Horesh et al. (2012) suggested even more efficient electron acceleration with _e/_B ∼ 1000, Maeda (2012) pointed out that little change in the spectral index was observed once the radio emission became optically thin, and suggested that the IC cooling effect must be negligible in radio and that the acceleration of the relativistic electrons must be inefficient (_e ≲ 0.01). In this interpretation of Maeda (2012), the predicted IC emission in the X-ray was about one or two orders of magnitudes smaller than observed—this is the same situation we found for SNe 2002ap and 2007gr, but with the constraints on the spectral energy index of the electrons (p) and on the effect of the IC cooling being much stronger than for SNe 2002ap and 2007gr. To remedy this problem, Maeda (2012) suggested there is a distinct population of relativistic electrons below γ ∼ 50, with a number density (per γ) more than an order of magnitude larger than the extrapolation from the power-law distribution for the radio-emitting, higher-energy electrons, in order to explain the X-rays from SN 2011dh by the IC mechanism while still being consistent with the radio behaviors. We expect that the acceleration mechanism is essentially the same in SN 2002ap, thus we suggest that the same argument for SN 2011dh applies to SN 2002ap as well and the X-ray properties of SN 2002ap could be explained in the same manner.