A SEARCH FOR INFRARED EMISSION FROM CORE-COLLAPSE SUPERNOVAE AT THE TRANSITIONAL PHASE

To search for IR emission from SNe at the transitional phase, we target seven core-collapse SNe, namely, SNe 1909A, 1917A, 1951H, 1962M, 1968D, and 1978K, discovered in three very nearby galaxies NGC 1313, NGC 6946, and M101. These targets are selected based on their proximity to the host galaxy and the availability of NIR and MIR imaging data taken with the Infrared Camera (IRC; Onaka et al. 2007) onboard the AKARI satellite (Murakami et al. 2007). General information on our targets is summarized in Table 1.

Table 1. List of Targets

Host Galaxy	Distance	SN	SN Type	Position		Position Ref.
	(Mpc)			α (J2000.0)	δ (J2000.0)
NGC 1313	4.13 ± 0.11	SN 1962M	II	03h18m122	−66°31'38''	1
		SN 1978K	IIn	03h17m390	−66°33'04''	2
NGC 6946	5.6 ± 0.3	SN 1917A	II	20h34m4690	+60°07'2908	3
		SN 1968D	II	20h34m5841	+60°09'3448	4
M101	6.7 ± 0.3	SN 1909Aa	II peculiar	14h02m031	+54°27'58''	3
		SN 1951H	II	14h03m553	+54°21'41''	3

References. (1) Artyukhina et al. 1996; (2) Dopita & Ryder 1990; (3) Barbon et al. 1999; (4) van Dyk et al. 1994. ^aSN 1909A is out of field of view in the IRC images, and only in the Spitzer/MIPS image.

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Note that SN 1948B (Type II) in NGC 6946 is excluded because of the uncertainty in its position. SNe 1939C and 1969P in NGC 6946 are also excluded since SN 1939C is classified as merely Type I (either core-collapse SNe or Type Ia SNe) and the type of SN 1969P is unknown. SN 1980K is also located in NGC 6946. But since it is found that AKARI data are not as deep as the data presented by Sugerman et al. (2012), we do not include SN 1980K. SN 1970G (Type IIL) in M101 is excluded because the SN position is heavily contaminated by the H ii region NGC 5455 (Fesen 1993).

For the distance to NGC 1313, we adopt 4.13 ± 0.11 Mpc, which is estimated from the tip of the red giant branch (Méndez et al. 2002). We use this value since it was also used by Lenz & Schlegel (2007) and Smith et al. (2007) for the study of SN 1978K. For NGC 6946, there are several distance measurements. We adopt the average value of 5.6 ± 0.3 Mpc from the H i Tully–Fisher relation (Pierce 1994), the CO Tully–Fisher relation (Schoniger & Sofue 1994), and the expanding photosphere method of Type II SN 1980K (Schmidt et al. 1994) and SN 2004et (Sahu et al. 2006). The distance to M101 is assumed to be 6.7 ± 0.3 Mpc, which is estimated by the Cepheid calibration (Freedman et al. 2001).

The AKARI/IRC equips three channels, i.e., NIR (1.8–5.5 μm), MIR-S (4.6–13.4 μm), and MIR-L (12.6–26.5 μm). Each channel has about a 10' × 10' field of view. The NIR and MIR-S share the same field of view while the MIR-L observes a sky about 25' away. Thus, the data we use in this paper consist of two observational sequences. The pixel scales of the detectors are 146 × 146, 234 × 234, and 251 × 239 for the NIR, MIR-S, and MIR-L, respectively.

We use the archived imaging data taken in the AKARI mission program "ISM in our Galaxy and Nearby galaxies" (ISMGN; Kaneda et al. 2009). The data are taken with the two-filter mode (Astronomical Observation Template (AOT) IRC02; see Onaka et al. 2007). We use six band images in total, i.e., N3 (reference wavelength 3.2 μm), N4 (4.1 μm), S7 (7.0 μm), S11 (11.0 μm), L15 (15.0 μm), and L24 (24.0 μm). The wavelength coverage with the response larger than 1/e of the peak is 2.7–3.8 μm, 3.6–5.3 μm, 5.9–8.4 μm, 8.5–13.1 μm, 12.6–19.4 μm, and 20.3–26.5 μm for N3, N4, S7, S11, L15, and L24, respectively. For more details of the IRC instrument, see Onaka et al. (2007). The log of observations is summarized in Table 2.

The data were reduced by using the IRC imaging pipeline software version 20091022 (see Lorente et al. 2007 for details). The pipeline performs the basic reduction, e.g., correction of bad pixels, subtraction of dark current, rejection of cosmic rays, linearity correction, flat fielding, and co-adding of the frames.

To determine the accurate position, we match the sources in the N3, N4, and S7 band images with those in Two Micron All Sky Survey catalog (Skrutskie et al. 2006). The uncertainty in the position is <2'' for the N3 and N4 band images, and <3'' for the S7 band images. Then, the S11, L15, and L24 images are matched with S7 images. In these images, the uncertainty in position is estimated as ∼3'', which is larger than that in the N3, N4, and S7 images because the matching is based on a small number of common sources in the field of view.

To obtain the flux densities of the sources, we perform aperture photometry with the aperture of 10 pixels and 7.5 pixels for the NIR and MIR-S/L bands, respectively, As the background, we use the annulus of 5 pixel width just outside of the aperture radius. When the SN is located at the edge of the image, the aperture correction is applied. The flux is calibrated by following Lorente et al. (2007). Color correction is not performed to the derived photometry because the intrinsic spectral shape of the source is not known. Instead, we apply correction to the model when it is compared with the observations (see Section 3).

When the SN is not detected, we derive an upper limit of the flux by putting an artificial source and measuring the flux of the source. For this purpose, we need a point-spread function (PSF) of each image. For the N3 and N4 images, PSF is constructed by the point sources in the same image. Since there are not many sources in the S7, S11, L15, and L24 images, we use an average PSF constructed from sources in different fields (Arimatsu et al. 2011).

In addition to the AKARI/IRC data, we use archive data taken with the Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004) onboard Spitzer (Werner et al. 2004).

All the six SNe are within the field of view of some observations with the MIPS 24 μm band. NGC 1313 was observed with the scan mode in the Spitzer Local Volume Legacy program (Dale et al. 2009). M101 was also observed with the scan mode by Gordon et al. (2008). SNe 1917A and 1968D in NGC 6946 are within the field of SN 2002hh (Meikle et al. 2006). A summary of the data, including the Astronomical Observation Request (AOR) keys, program IDs, and PI names, is shown in Table 3.

We use the post-basic calibrated data (PBCD) products, which are produced by the Spitzer pipeline. The pixel scale of the final image is 245 × 245. The reference wavelength of the MIPS 24 μm band is 23.68 μm and the wavelength coverage is 20.5–28.5 μm (the range where the response is >10% of the peak). The positional uncertainty is 14.

For photometry, we perform aperture photometry. When the SN is not detected, we derive an upper limit of the flux by putting an artificial source around the SN position. We constructed a PSF from point sources in the scan mode images of NGC 1313 and M101 and use it to derive the upper limits.

The Spitzer/MIPS data are found to be slightly deeper than the AKARI/IRC data at 24 μm (Section 2.4). Thus, we use MIPS data for the flux of 24 μm.

We detect IR emission associated with SN 1978K in all the bands (Table 4). Figure 1 shows the AKARI/IRC images around SN 1978K (180'' × 180'' section). The source has an extremely red color with a flux ratio of F_ν(S7)/F_ν(S11) = 0.1, which is out of a typical range in the spiral arm and the inter-arm of the galaxy (∼0.8–1.4; Sakon et al. 2007). The origin of this emission is discussed in Section 4.

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For the other five objects, we do not detect a significant signal from SNe. Upper limits of the flux are shown in Table 4.

Among our seven targets, we detect IR emission only from SN 1978K. SN 1978K is an extraordinarily strong Type IIn SN, which is visible both in radio and X-ray even at ∼30 yr after the explosion (Ryder et al. 1993; Petre et al. 1994; Chugai et al. 1995; Schlegel et al. 1996, 1999, 2000, 2004; Montes et al. 1997; Chu et al. 1999; Lenz & Schlegel 2007; Smith et al. 2007). Such long-lasting radio and X-ray emissions are clear evidence for the presence of the dense CSM, with which the SN ejecta has been interacting (Chevalier 1982; Chevalier & Fransson 1994; Chugai et al. 1995).

Figure 2 shows the spectral energy distribution (SED) of SN 1978K from IR to radio wavelengths (filled circles). The SED of Cassiopeia A (Cas A) at the distance of NGC 1313 is also shown for comparison (open squares). These two objects show similar overall SEDs from IR to radio wavelengths.

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Since SN 1978K is powerful in radio wavelengths (Smith et al. 2007), we must take into account the influence of the synchrotron emission on the IR bands (see, e.g., Mezger et al. 1986; Rho et al. 2003; Dwek 2004; Hines et al. 2004; Barlow et al. 2010, for the case of Cas A). The radio emission of SN 1978K is well fitted by a power-law spectrum F_ν∝ν^α (Montes et al. 1997; Smith et al. 2007). When we fit the flux of four radio bands of SN 1978K (Smith et al. 2007), we derive α = −0.67 ± 0.05 (blue dashed line in Figure 2). The flux densities at the N3 and N4 bands are almost in line with the power-law spectrum. If we include the flux of these two bands for the fitting, we obtain α = −0.60 ± 0.02 (red dashed line). Since these two values are consistent within the error, we assume that the emissions at the N3 and N4 bands are of synchrotron origin and use α = −0.60 in the following discussions.

The flux density of the S7, S11, L15, and L24 (MIPS 24 μm) bands is clearly above the synchrotron component. Therefore, we conclude that this excess is caused by the dust emission associated with SN 1978K.

We fit the observed flux with the model spectra of the four dust species with the dust temperature and mass as parameters. The quality of the fit is judged by χ² for the four observational points, i.e., the IRC S7, S11, L15, and MIPS 24 μm bands. For each dust species, we choose the best model that gives the smallest χ². Since the dependence on the temperature and mass is simple, the fit is unique, i.e., there is no other solution with a different set of temperature and mass.

Among the four dust species, the smallest χ² is obtained for astronomical silicate with T_dust = 230 K and M_dust = 1.3 × 10⁻³ M_☉ (blue line in Figure 3). The total IR flux by the dust emission is F_IR = 7.0 × 10⁻¹³ erg cm⁻² s⁻¹ and the total IR luminosity is L_IR = 1.5 × 10³⁹ erg s⁻¹. Silicate dust grains are favored over carbon grains because the observation shows excess at the S11 band, which can be attributed to the Si–O stretching mode of silicate dust at 10 μm (Draine & Lee 1984; Draine 2003a). As shown in the red line in Figure 3, amorphous carbon dust grains are not able to explain the excess. Forsterite fits the observations quite well, but gives a band feature at 10 μm slightly narrower than astronomical silicate. Hereafter, we simply call the dust species found in SN 1978K "silicate dust."

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SN 1978K is associated with 1.3 × 10⁻³ M_☉ of silicate dust. The derived temperature is relatively high (230 K). In this subsection, we discuss possible origins of the IR emission. Since SN 1978K is a CSM-rich SN, we first consider CS dust. CS dust can radiate IR emission by light echo (Section 4.2.1). Then, the echoing CS dust is located at the distance larger than D = ct/2 = 1.4 × 10¹⁹(t/30 yr) cm, where the SN forward shock cannot reach. On the other hand, the CS dust swept up by the SN forward shock can be heated via the collisions with energetic electrons and emits thermal radiation (Section 4.2.2). Then, the typical location is at ∼5 × 10¹⁷(t/30 yr)(v_SN/5000 km s⁻¹) cm. Another possibility is the SN dust (Section 4.2.3). We show that, among these possibilities, the shocked CS dust is the most probable origin for the IR emission from SN 1978K.

4.2.1. Light Echo

The first possibility for the IR emission is echo emission of CS dust. The dust can absorb the ultraviolet or optical photons of an SN, and radiate IR emission (Bode & Evans 1980; Wright 1980; Dwek 1983, 1985; Graham & Meikle 1986). In fact, the presence of CS dust echo has been suggested by early phase observations of SNe (Meikle et al. 2006, 2007; Andrews et al. 2011) and SN 1980K at the transitional phase (Sugerman et al. 2012).

We calculate the luminosity of the light echo by following Dwek (1983). The gas mass-loss rate of the progenitor of SN 1978K is estimated to be about $\dot{M}= 10^{-4}\, M_\odot \; {\rm yr^{-1}}$ from optical, X-ray, and radio observations (Ryder et al. 1993; Chugai et al. 1995; Smith et al. 2007). The duration of mass loss is assumed to be t_wind = 10⁵ yr. This is a rather extreme assumption since the progenitor loses mass as large as 10  M_☉ prior to the explosion. We assume that the wind velocity is v_wind = 10 km s⁻¹, which gives outer radius of CSM about R₂ = 3 × 10¹⁸ cm. The mass-loss rate is assumed to be constant, and thus, the CSM density has a power-law radial profile ∝ r⁻².

For the SN luminosity, we use an exponential form L = L₀exp (t/t_SN). We assume a normal peak luminosity L₀ = 6 × 10⁴² erg s⁻¹ and t_SN = 25 days (Sahu et al. 2006; Kotak et al. 2009). The other parameters are set as follows: the gas-to-dust mass ratio f = 200, the grain radius a = 0.1 μm, the mean absorption efficiency $\bar{Q} = 1$, and the evaporation temperature of dust T = 1500 K. The SN outburst luminosity will make a dust cavity with a radius of R₁ = 7 × 10¹⁶ cm.

The left panel of Figure 4 shows the computed IR luminosity of the CS dust echo (solid black line). Optical data of SN 1978K are also plotted for comparison under a simple assumption of zero bolometric correction (blue and red points). The IR echo luminosity at t = 10–100 yr depends largely on the outer radius (R₂) of the CSM since the luminosity from the CS echo declines rapidly at t = 2R₂/c ∼ 7 yr (Dwek 1983). To illustrate this dependence, we also show the model with a longer mass-loss duration (dashed line for t_wind = 10⁶ yr). With t_wind ≳ 5 × 10⁵ yr, with which the progenitor loses more than 50  M_☉, the IR echo does not drop dramatically at t < 30 yr. But, even with this extreme case, the IR luminosity is only an order of 10³⁷ erg s⁻¹ at t = 30 yr, which is much less than the observed IR luminosity. Thus, we conclude that the CS dust echo of SN outburst is unlikely to be the origin of the IR emission.

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We also calculate the IS dust echo by the same method as that for the CS dust echo. In fact, IR echo by the IS dust has been detected for several extragalactic SNe (Meikle et al. 2007, 2011; Kotak et al. 2009) and Cas A (Krause et al. 2005; Dwek & Arendt 2008). The difference from the CS dust is that the density is set to be uniform and that the much larger outer radius is allowed. Here, the distribution of IS dust is approximated to be a sphere with the radius of R₂ = 100 pc (t = 2R₂/c ∼ 700 yr). The black solid and dashed lines in the right panel of Figure 4 show the luminosity evolution of the IS dust echo for the ISM gas density of n = 0.1 and 1 cm⁻³, respectively.

With the normal SN luminosity, the expected IR luminosity at t = 30 yr is not high enough to explain the observed luminosity of SN 1978K. To explain the observation, the SN had to be very luminous, L₀ ∼ 6 × 10⁴³ erg s⁻¹, or the absolute optical magnitude of about −20 mag. It would imply that the brightness of SN 1978K was of the 8th magnitude. This is in contrast to the suggestion by Ryder et al. (1993) that SN 1978K was subluminous. Although a very large luminosity of SN 1978K is not completely excluded given the sparse observations in 1978, we conclude that IS dust echo is also unlikely.

4.2.2. Shocked CS Dust

The second source for the IR emission is the CS dust heated by the forward shock of the SN. The mid-IR emission from the ring in SN 1987A is thought to arise from this mechanism (e.g., Dwek et al. 2008). It is interesting to note that the emission properties of SN 1987A ring and SN 1978K look similar. The dust temperature in the SN 1987A ring is ∼165–185 K at t = 17.9–19.7 yr (Bouchet et al. 2006; Dwek et al. 2008; Seok et al. 2008), while that in SN 1978K is 230 K at t = 29 yr. In addition, the IR to X-ray flux ratio is of the order of unity in both objects (For SN 1978K, F_X ∼ 1 × 10⁻¹² erg cm⁻² s⁻¹; Smith et al. 2007). The similarity seems to support the shock-heated CS dust as an origin of IR emission from SN 1978K.

We estimate the mass of the emitting CS dust swept by the forward shock. CS dust is subject to the destruction by the SN forward shock. The timescale of sputtering is given by

$$\begin{equation} \tau _{\rm sput} \equiv \left(\frac{1}{a}\frac{da}{dt} \right)^{-1} \simeq 10^6 \; {\rm yr} \ \left(\frac{a}{1\ \mu {\rm m}} \right) \left(\frac{n_{\rm g}}{1 \, {\rm cm^{-3}}} \right)^{-1} \end{equation} \tag{ 4 }$$

(Dwek & Arendt 1992), where n_g is the number density of shocked gas. Then, the evolution of the dust radius is estimated as a(t) = a₀(1 − t/τ'_sput), where τ'_sput = 10⁶ yr(a₀/1 μm)(n_g/1 cm⁻³)⁻¹. Hereafter, the subscript "0" denotes the value before the dust destruction. The gas mass-loss rate and the CS dust density are related as follows: $\dot{M} = 4 \pi R^2 \rho _{\rm wind} v_{\rm wind}= 4 \pi R^2 f \rho _{\rm dust, 0} v_{\rm wind}$, where ρ_wind is the mass density of the wind gas. We also define the mass density of dust before the destruction; ρ_{dust, 0} ≡ (4/3)πρa³₀n_{dust, 0}, where n_{dust, 0} is the number density of dust before the destruction. The CS dust density after the destruction is given by ρ_dust(t) = (4/3)πn_{dust, 0}ρa³ = (4/3)πn_{dust, 0}ρa³₀(1 − t/τ'_sput)³ = ρ_{dust, 0}(1 − t/τ'_sput)³.

Using these relations, the mass of the emitting, shocked CS dust is described as follows:

$$\begin{eqnarray} M_{\rm CS dust} (t) &=& \int ^{t}_{0} 4 \pi R^2 \rho _{\rm dust}(t^{\prime }) v_{\rm SN}dt^{\prime } \nonumber \\ &=& \int ^{t}_{0} 4 \pi R^2 \rho _{\rm dust, 0} v_{\rm SN}(1 - t^{\prime } /\tau _{\rm sput})^3 dt^{\prime } \nonumber \\ &=& 8 \times 10^{-4} \,M_{\odot }\left(\frac{\dot{M}_{\rm wind}}{10^{-5}\ \,M_{\odot }\ {\rm yr^{-1}}} \right) \left(\frac{v_{\rm SN}}{5000\, {\rm km\, s^{-1}}} \right) \nonumber \\ &&\!\!\!\times \! \left(\!\frac{v_{\rm wind}}{10\, {\rm km\, s^{-1}}}\! \right)^{-1} \left(\!\frac{f}{200} \!\right)^{-1}\! \left(\!\frac{\int ^t_0 (1 - t^{\prime } /\tau _{\rm sput}^{\prime })^3 dt^{\prime }}{30\ {\rm yr}}\! \right).\nonumber\\ \end{eqnarray} \tag{ 5 }$$

At the limit of no dust destruction, the integral in the last parentheses becomes t, and the shock-heated CS dust is proportional to the mass-loss rate. But τ'_sput is a function of the shocked gas density, and thus, the mass-loss rate. With a larger mass-loss rate, the sputtering timescale becomes shorter, and the integral in the last parentheses becomes smaller than t.

Figure 5 shows the shocked (but surviving) mass of the CS dust at t = 30 yr as a function of mass-loss rate. The dashed, solid, and dotted lines in black show the models with a₀ = 0.3, 0.1, and 0.01 μm, respectively. The red solid line is the model without dust destruction with a₀ = 0.1 μm. Other parameters are assumed as follows: v_SN = 4000 km s⁻¹, v_wind = 10 km s⁻¹, and f = 200. The SN shock velocity of SN 1978K is estimated to be v_SN < 4000 km s⁻¹ by the very long baseline interferometry image, which marginally resolves it (Smith et al. 2007). Thus, we assume v_SN = 4000 km s⁻¹.

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For small mass-loss rates, the destruction is not so effective that the shocked CS dust mass is roughly proportional to the mass-loss rate. For large mass-loss rates, the surviving dust mass is not sensitive to the mass-loss rate. Because of the high gas density, the destruction is effective enough that the mass of dust swept up by the shock compensates for the mass destroyed by the shock.⁸ In the calculation above, we assume there is no dust-free cavity created by the SN outburst. Since most of surviving dust is located near the SN shock (at R ∼ 5 × 10¹⁷(t/30 yr)(v_SN/5000 km s⁻¹) cm), which is larger than the cavity size (∼7 × 10¹⁶ cm). Thus, the results presented here are not largely affected by the presence of the dust-free cavity.

From the IR SED of SN 1978K, we estimate the mass of emitting dust to be 1.3 × 10⁻³ M_☉. The model of the shocked CS dust with $\dot{M} = 10^{-4}\ \,M_{\odot }\ {\rm yr^{-1}}$ and v_SN = 4000 km s⁻¹ is consistent with the observed dust mass if the initial dust radius is a₀ = 0.3 μm. A similar conclusion has been reached by Dwek et al. (2008) for SN 1987A.

We also check the expected temperature for the shocked CS dust. With $\dot{M}= 10^{-4}\ \,M_{\odot }\ {\rm yr^{-1}}$, the shocked gas density is about 6 × 10³ cm⁻³. For the electron temperature about 10⁷ K, the expected dust temperature is T_dust ∼ 270 and 220 K for the dust radius a = 0.1 and 0.3 μm, respectively (Dwek et al. 2008). Thus, the expected temperature of the shocked CS dust is also in good agreement with the observations. The surviving, shocked CS dust is distributed in a thin shell whose inner radius is about 95 % of the radius at the forward shock. Using the mass absorption coefficient of dust κ_ν for astronomical silicate at 24 μm (Draine 2003b), the optical depth of dust is about 4 × 10⁻⁴(κ_ν/562 cm² g⁻¹), which is consistent with our assumption of optically thin dust (Section 3). In summary, the IR emission from the shocked CS dust can naturally explain the observed IR emission from SN 1978K.

4.2.3. SN Dust

For CSM-rich SN 1978K, we have discussed IR emission expected by (1) light echo from CS and IS dust, (2) shocked CS dust, and (3) SN dust. In this section, we summarize the IR emissions from normal SNe by these processes, presenting the expected IR luminosity and typical dust temperature.

We have shown the light curve of the light echo by CS dust for $\dot{M}= 10^{-4}\ \,M_{\odot }\ {\rm yr^{-1}}$ in Figure 4. Using the same method, we calculate the echo luminosity with a normal mass-loss rate $\dot{M}= 10^{-5}\ \,M_{\odot }\ {\rm yr^{-1}}$, and a normal SN luminosity L₀ ∼ 6 × 10⁴² erg s⁻¹. The solid blue line in Figure 6 shows the IR luminosity with the wind duration t_wind = 10⁶ yr. The echo luminosity is about L = 3 × 10³⁶ erg s⁻¹ at t = 30 yr.

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Figure 6 also shows the luminosity by IS dust echo as calculated in Section 4.2.1. The light curve of the IS dust echo has a long timescale, and will exceed that of the CS dust echo after t = 5–10 yr for the ISM density of n = 0.1–1.0 cm⁻³. Note that a typical IS density is not well constrained. Gaustad & van Buren (1993) derived a filling factor of IS gas with density >0.5 cm⁻³ to be only about 15%. Dring et al. (1996) derived an even smaller filling factor: 8.5% for the gas density >0.01 cm⁻³. When the density is lower than 0.1 cm⁻³, the IR luminosity by the echo is roughly proportional to the gas density since the IS dust is optically thin.

For both CS and IS dust cases, the typical temperature of IR emission can be roughly estimated at the closest point from the SN, i.e., at D = ct/2 = 1.4 × 10¹⁹(t/30 yr) cm, which has the largest contribution to the total luminosity. Suppose that a dust grain is irradiated by an SN with the luminosity L_SN at a distance D. For simplicity, the spectrum of the luminous source is assumed to be blackbody with a temperature T_s. Then, the absorbed luminosity is given by L_abs = (L_SN/4πD²)πa²〈Q^abs(T_s)〉, where 〈Q^abs(T)〉 is the Planck mean of the absorption efficiency factor. When the dust particle emits radiation with a temperature T_dust, the emitted luminosity is L_rad = 4πa²σT⁴_dust〈Q^abs(T_dust)〉. Under the equilibrium, i.e., L_abs = L_rad, we can derive the temperature as follows:

$$\begin{eqnarray} T_{\rm dust}&\simeq & 140\ {\rm K} \left(\frac{D}{1.4 \times 10^{19}\, {\rm cm}} \right)^{-1/3} \left(\frac{L_{\rm SN}}{6 \times 10^{42}\, {\rm erg\, s^{-1}}} \right)^{1/6} \nonumber \\ && \times \left(\frac{\langle Q^{\rm abs}(T_s)\rangle }{1} \right)^{1/6} \left(\frac{a}{0.1\ {\rm \mu m}} \right)^{-1/6}. \end{eqnarray} \tag{ 6 }$$

Here, we used 〈Q^abs(T_dust)〉/a = 0.127T²_dust, which is derived for T ≲ 100 K by Dwek (1987). We find that this formula is applicable up to T ∼ 250 K for our purpose. The expected temperature for the light echo is ∼140 K at t = 30 yr.

Next we consider the shocked CS dust. Figure 5 shows that the typical mass of the surviving, shock-heated dust is about 3 × 10⁻⁴ M_☉ at t = 30 yr for $\dot{M}= 10^{-5}\ \,M_{\odot }\ {\rm yr^{-1}}$ and v_SN = 5000 km s⁻¹. With this mass-loss rate, the shocked gas density is about 6 × 10² cm⁻³. For the electron temperature about 10⁷ K, the expected dust temperature is T_dust ∼ 160 K for the grain radius of a = 0.1 μm (Dwek et al. 2008). The total IR luminosity can be derived by integrating Equation (1) over wavelength:

$$\begin{eqnarray} L_{\rm IR} &=& M_{\rm dust}\left(\!\frac{4\pi }{3}\rho a^3 \!\right)^{-1} 4\pi a^2 \sigma T_{\rm dust}^4 \langle Q^{\rm abs}(T_{\rm dust})\rangle \nonumber \\ &\simeq & 1.8 \times 10^{37}\left(\frac{M_{\rm dust}}{0.1\ \,M_{\odot }} \right) \nonumber \\ && \times \left(\frac{\rho }{3.8\, {\rm g\, cm^{-3}}} \right)^{-1} \left(\frac{T_{\rm dust}}{50\ {\rm K}} \right)^6\, \rm erg\, s^{-1}, \end{eqnarray} \tag{ 7 }$$

where, as in Equation (6), we adopt 〈Q^abs(T_dust)〉/a = 0.127T²_dust. Using this equation, the total luminosity of the shocked CS dust is estimated as ∼6 × 10³⁷ erg s⁻¹. For a higher mass-loss rate $\dot{M}= 10^{-4}\ \,M_{\odot }\ {\rm yr^{-1}}$, the shocked CS dust mass becomes M_dust ∼ 1 × 10⁻³ M_☉, and the temperature is T_dust ∼ 270 K. The total IR luminosity becomes 4 × 10³⁹ erg s⁻¹.

Finally, let us consider the SN dust. For the SN dust, we assume L_IR < 10³⁷ erg s⁻¹, which is expected by the radioactive decay of ⁴⁴Ti. If the mass of SN dust is M_dust = 0.1 M_☉, the temperature can be T_dust < 50 K. Even if only M_dust = 10⁻³ M_☉ of dust is formed, the temperature is about 100 K, which is lower than the expected temperatures for the light echo and the shocked CS dust.

In summary, IR emission of normal SNe at the transitional phase can result from three types of mechanisms, i.e., IS dust echo, shocked CS dust, and SN dust. IS dust echo and shocked CS dust have a relatively high temperature (T_dust = 150–270 K) while the SN dust has a lower temperature (T_dust < 50–100 K). The total luminosity of the IS echo can be L_IR > 10³⁷ erg s⁻¹ for the IS density of n > 0.1 cm⁻³ (Figure 6). The shocked CS dust could also have L_IR ∼ 6 × 10³⁷–4 × 10³⁹ erg s⁻¹ for $\dot{M}=$ 10⁻⁵–10⁻⁴  M_☉ yr⁻¹ (see the arrow in Figure 6). The luminosity of the SN dust must be L_IR < 10³⁷ erg s⁻¹.

Figure 7 shows SEDs of five SNe in the three different host galaxies. These upper limits are compared with the model spectra of amorphous carbon (red) and astronomical silicate (blue) with T_dust = 100 and 200 K (solid and dashed lines, respectively). The assumed dust mass is shown in the panels. In Table 5, we show the upper limit of the dust mass for the case of amorphous carbon and astronomical silicate for the temperature T_dust = 100 and 200 K.

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The upper limit of the dust mass strongly depends on the dust temperature. The blue dashed line in Figure 8 show the upper limits of the dust mass for SN 1962M as a function of the dust temperature for the case of astronomical silicate. Constraints on the other four SNe are weaker than those for SN 1962M by a factor of 2–20. For comparison, Figure 8 also shows the estimated dust mass and temperature of SN 1978K (this paper) and both warm and cold component of SN 1987A (Bouchet et al. 2004; Matsuura et al. 2011).

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The solid lines in Figure 8 show the dust mass as a function of dust temperature for L_IR = 10³⁶, 10³⁷, 10³⁸, and 10³⁹ erg s⁻¹ (from bottom to top). This can be analytically derived from Equation (7):

$$\begin{eqnarray} M_{\rm dust}&=& 5.6 \times 10^{-2} \left(\frac{L_{\rm IR}}{10^{37}\, {\rm erg\, s^{-1}}} \right) \nonumber \\ && \times \left(\frac{T_{\rm dust}}{50\ {\rm K}} \right)^{-6} \left(\frac{\rho }{3.8\, {\rm g\, cm^{-3}}} \right) \,M_{\odot }. \end{eqnarray} \tag{ 8 }$$

For these lines in Figure 8, we do not use the analytic formulae, but numerically integrate Equation (1) over wavelength.

As shown in Figure 8, our current observations are sensitive only to the IR luminosity as high as >10³⁸ erg s⁻¹. This limiting luminosity is much higher than the predictions for the SN dust. The luminosity of the IS dust echo can be 1 × 10³⁸ erg s⁻¹ with 1 cm⁻³. Our current observations can only exclude a homogeneous IS density of n ∼ 10 cm⁻³.

For the shocked CS dust, the expected ranges of the dust mass and temperature are shown by two gray boxes in Figure 8. The left lower box shows the range for $\dot{M}= 10^{-5}\ \,M_{\odot }\ {\rm yr^{-1}}$, while the right top for $\dot{M}= 10^{-4}\ \,M_{\odot }\ {\rm yr^{-1}}$, Our current best observational limit excludes the existence of the shocked CS dust with $\dot{M}= 10^{-4}\ \,M_{\odot }\ {\rm yr^{-1}}$.