Dust Extinction towards the Type Ia Supernova 2012cu in NGC 4772

Using photometric and spectroscopic data of Supernova (SN) 2012cu, a fairly reddened type Ia supernova, we derived its color excess curves and probed the dust extinction in its host galaxy, NGC 4772. In order to derive the extinction as a function of wavelength (i.e., $A_\lambda$), we model the color excess curves of SN 2012cu in terms of dust models consisting of silicate and carbonaceous (graphite or amorphous carbon) dust. The modeled extinction law towards SN 2012cu extends flatly to the far-ultraviolet (UV) bands, which is much flatter than those of the Milky Way and Magellanic Clouds, and the 2175 $\AA$ feature is very weak or absent. The flatness of the modeled extinction curve in the UV bands suggests a"grey"extinction law of the active galactic nucleus in the vicinity of the SN 2012cu-Earth line of sight. Our results indicate that the sizes of the dust in the ISM towards SN 2012cu in NGC 4772 are larger than those of the Milky Way and the Large Magellanic Cloud, and much larger than that of the Small Magellanic Cloud. The best fitting gives an observed visual extinction towards SN 2012cu of $A_V \approx 2.6$ mag, a reddening of $E(B-V) \approx 1.0$ mag, with a total-to-selective extinction ratio $R_V \approx 2.7$, consistent with previous results.


Introduction
Attributed to their high luminosity with small dispersion at the maximum of light curves, type Ia supernovae (SNe Ia) are one of the most powerful "standard candles" for measuring cosmological distances (Riess et al. 1998(Riess et al. , 2016Perlmutter et al. 1999), leading to the discovery of accelerating universe and hence the presence of mysterious dark energy. Incomprehensive knowledge on the extinction towards SNe Ia will, however, result in a systematic uncertainty in the intrinsic luminosity and distances to SNe Ia, which leads to one of the major issues in SNe Ia cosmology as the correction for the line-of-sight extinction. In addition, due to the undetectable extragalactic environment, SNe Ia are often the only approach to study the extinction of extragalactic dust.
The wavelength dependence of the interstellar extinction [known as the "interstellar extinction law (or curve)"] is defined as A λ at wavelength λ.
Since A λ is hard to obtain directly, astronomers often measure the color excesses E(λ−V ) ≡ A λ −A V , where A V is the extinction in the visual band. The color excesses of SNe Ia are often determined by comparing spectrophotometry of two sources with the same spectral shape, one of which has minor foreground reddening (i.e., Amanullah et al. 2014Amanullah et al. , 2015. The total-to-selective extinction ratio R V ≡ A V /E(B − V ) provides an adequate description of the extinction laws of the Milky Way (MW) dust (e.g., Cardelli et al. 1989, hereafter CCM89;Fitzpatrick 1999, hereafter F99), in which R V ranges from 2.2 to 5.5 with the average value of 3.1 (e.g., Fitzpatrick & Massa 2007).
However, CCM89 and F99 are not necessarily valid for the extinction laws for external galaxies, even for the Large and Small Magellanic Clouds (LMC, SMC; Gordon et al. 2003).
There is increasing evidence that extinction curves towards SNe Ia systematically favor a steeper law (R V < 3, see, for instance, Nobili & Goobar 2008;Folatelli et al. 2010;Cikota et al. 2016) compared to the Galactic average value (R V = 3.1, see CCM89 or F99). This suggests the properties of extragalactic dust may be incompatible with the Milky Way dust. Additionally, scattering by circumstellar dust also tends to reduce R V in the optical band (Wang 2005;Patat et al. 2007;Goobar 2008). This discrepancy leads to another one of the major issues in SNe Ia cosmology, which is to understand whether the systematically low R V values towards SNe Ia are caused by 1) systematic differences from the optical properties of extragalactic dust grains, or 2) modifications by the circumstellar matter scattering.
In our previous work, Gao et al. (2015) found that the peculiar extinction law in the sightline of SN 2014J, one SN Ia with R V < 2 (e.g., R V = 1.6±0.2, Foley et al. 2014), cannot be explained by CCM89. For comparison, the dust models of a mixture of silicate and graphite/amorphous carbon dust grains provide an excellent fitting to the observed color excess curves towards SN 2014J. The reddening curve fitting around the peak luminosity of SN 2014J gives R V ∼ 1.7 towards the SN 2014J line of sight, which is also generally consistent with many other studies (Amanullah et al. 2014;Foley et al. 2014;Goobar et al. 2014;Brown et al. 2015;Yang et al. 2017).
SN 2012cu was first discovered on June 11.2 UT, 2012 by Itagaki et al. (2012) and later classified as a SN Ia on June 15 UT, 2012 by Marion et al. (2012) (Haynes et al. 2000), and also classified as a lowluminosity "dwarf" Seyfert nuclei (or low-ionization nuclear emission-line region, LINER) according to its spectral lines (Ho et al. 1997 In this work, using the photometric data from A15 and spectroscopic data from H17 (see §2), we measure the color excesses E(λ − V ) of SN 2012cu. The results are presented in §4, discussed in §5, and summarized in §6.

Observational Data
We utilize the photometric data of SN 2012cu published by A15 where X (V ) is the rest-frame magnitude in the filter X (V); m X and m V are the observed magnitudes; K X is the combined K and S corrections; A MW and K X are given by A15, who adopted the values of Schlafly & Finkbeiner (2011). Then the color excesses of SN 2012cu can be derived by using the following equation where (X − V ) 0 is the intrinsic color of the unreddened objects listed by A15, who combined the spectroscopic data of SN 2011fe and created a series of daily sampled unreddened SED templates to derive the intrinsic colors of SNe Ia. It will be described later in this section.
The spectroscopic data of SN 2012cu are taken from H17 5 (see Figure 13 therein), which span 17 epochs from -6.8 days to 46.2 days. In this work, we use the spectra which span 15 epochs from -6.8 days to 31.2 days 6 . The spectra of SN 2012cu cover a wavelength range from 3300 to 9200Å, and have already been flux calibrated (Buton et al. 2013), host-galaxy subtracted and corrected for the MW foreground extinctions (Schlafly & Finkbeiner 2011).
In order to determine the intrinsic colors of SN 2012cu, we use the SED templates of SN 2011fe privately provided by Amanullah R. (A15). SN 2011fe, one of the best studied SNe Ia, is the rare one with high-quality UV time series spectra obtained with HST (Foley et al. 2014;Mazzali et al. 2014 0.026 ± 0.036 mag, Nugent et al. 2011;Pereira et al. 2013;Zhang et al. 2016) and the absence of complex absorption profiles (Patat et al. 2013) towards SN 2011fe suggest negligible extinction from its host galaxy. Hence, with the wide wavelength and comprehensive phase coverage dataset, the nearby, reddening-free SN 2011fe provides an excellent calibration for extinction comparison.
Assuming that SN 2012cu has the same SED as the lightly reddened SN 2011fe, the color excess curves of SN 2012cu can be directly derived by comparing the spectra of SN 2012cu with the SED templates of SN 2011fe at similar epochs (A15), as where at wavelength λ for phase p, f (λ; p) and S 0 (λ; p) are flux densities from the spectra of SN 2012cu and the SED templates of SN 2011fe, respectively. Figure 1 shows the color excess curves derived from the spectroscopic data.
These color excess curves are smoothed by a simple moving average method.
The substantially fluctuant parts of curves beyond wavenumber ≥ 2.6µm −1 are also excluded.

Dust Models
We adopt a two-component grain model consisting of astronomical silicate and graphite (GRA, Draine & Lee 1984) or amorphous carbon (AMC, Rouleau & Martin 1991). We assume that both the silicate and carbonaceous grains have the same size distribution, i.e., a power-law function with  an exponential cutoff (Kim et al. 1994;Wang et al. 2014;Gao et al. 2015): where α and a b are the power index and exponential cutoff radius, respectively; dn i is the number density of dust species i (silicate or carbonaceous component) with radii between [a, a + da]; n H is the number density in unit of hydrogen nucleon H; B i is the normalization constant. A classical MRN grain model (Mathis et al. 1977, hereafter MRN) has a power-law size distribution, i.e., dn(a)/da ∝ a −3.5 (Clayton et al. 2003). When a b ≫ a, the exponential cutoff will have exp(−a/a b ) ≈ 1, which decays Equation (4) for silicate dust, and and 2.24 g cm −3 , respectively. The density of amorphous carbon (ρ AMC ) is constrained in the silicate dust, and the fraction of gas-phase carbon is 0% (Gao et al. 2013(Gao et al. , 2015Wang et al. 2014). In addition, f cs will be 0.3 when the fraction of gas-phase carbon is 50%.
Then the modeled extinction A λ at wavelength λ is calculated by where N H and n H are the column density and volume density in unit of hydrogen nucleon H, respectively. All dust grains are assumed to be spherical with the radius of 0.005 µm = a min ≤ a ≤ a max = 5 µm. The extinction cross section C ext,i (a, λ) (cm −2 ), for the grain of species i with size a at wavelength λ, is calculated with the Mie theory code derived from BH-MIE (Bohren & Huffman 1983). The optical constants of dust are taken from Draine & Lee (1984) for astronomical silicate and graphite, and from Rouleau & Martin (1991) for amorphous carbon.
The modeled extinction A X,p in X band for phase p is subsequently derived from the similar formula by A15: where the extinction A λ is initially calculated by Equation (7); T X (λ) is the filter transmission, and S 0 (λ; p) is the unreddened flux density of the SED at wavelength λ for phase p (see Equation (3)).
In order to reproduce the observed extinction curves with dust models, where at wavelength λ for phase p, the modeled color excesses is derived from the observed data by Equation (2) and (3), for the photometric data and the spectroscopic data, respectively; 1/σ 2 is the weight, i.e., the uncertainties of data; N data is the number of observed data points used for fitting, and N para is the number of adjustable parameters; d.o.f ≡ N data − N para is the degree of freedom. Since we assume that the silicate and carbonaceous grains have the same dust size distribution, there are three parameters (N para = 3) in our dust models: the size distribution power index α, the exponential cutoff size a b , and the H column density N H . The grid of α ranges from 0.4 to 5.0 with a step of 0.1, while a b ranges from 0.01 to 0.30 with a step of 0.01. 7 A mod λ,p is calculated by Equation (8) when fitting the photometric data, instead, by Equation (7) when fitting the spectroscopic data.

Results
In this work, we calculate the color excesses, E(X −V ) of SN 2012cu in the UV-to-NIR bands (2346−21295Å, see §2) and the color excess curves, E(λ − V ) in the optical band (3850−9150Å). We fit the combination of the color excesses and the color excess curves using the silicate+graphite/amorphous carbon dust models. The color excesses derived from the photometric data contain 31 points and cover eight epochs (i.e., N data = 31 and phase p = 8).
The color excess curves derived from the spectroscopic data cover 15 epochs (p = 15, see black lines in Figure 1). When fitting with the dust models, each of these 15 color excess curves is evenly divided into 54 points with a step of 100Å, which yields 810 points (N data = 810) in total. Thus, in Equation (9), we finally take p = 23 and N data = 841 for summation and search for the best fitting parameters by minimizing χ 2 /d.o.f.
The best-fit results to our dust models are shown in Table 1. The five rows present the dust models with different types of carbonaceous component (GRA or AMC) and mass ratio f cs between carbonaceous and silicate dust.
In   We also adopt the MRN size distribution to fit the color excess curves.
However, the typical values of χ 2 /d.o.f give 0.5 to 0.6, almost twice as the corresponding values given by our silicate+graphite/amorphous carbon dust model shown in Table 1. Moreover, the MRN size distribution has even larger d.o.f than that of the power-law function with an exponential cutoff (Kim et al. 1994), due to the number of adjustable parameters N para = 2, i.e., the power index α and H column density N H . Therefore, we suggest that the reddening towards SN 2012cu disfavors the MRN dust size distribution.     We also show the extinction curve of the SMC bar (magenta short dashed line), the mean reddening curve of AGNs (black dashed line, Gaskell & Benker 2007), and the Calzetti attenuation law for starburst galaxies (cyan dotted line, Calzetti et al. 1994).   tinction curves exhibit the 2175Å bump produced by small graphite dust and display discernable deviation from the observed curve in the UV bands.
On the contrary, the silicate+amorphous carbon model (f AMC cs = 0.3) has no 2175Å feature and it well reproduces the extinction curve in F 225W and F 275W passbands.
In Figure 4a, the modeled extinction curves are much flatter in the UV bands than those of the MW and the SMC bar. It suggests that the size of the dust in the ISM towards SN 2012cu in NGC 4772 tends to be larger than that of the MW, and even much larger than that of the SMC bar. Our modeled extinction curves in the UV bands are nearly similar with the mean reddening curve of AGNs (Gaskell & Benker 2007) A15 used all measured colors (UV-NIR) to derive the extinction curves of SN 2012cu with F99 extinction law. Their best-fit result suggests R V = 2.8± 0.1, which is shown by green dashed line in Figure 4. H17 similarly utilized F99 law to deredden SN 2012cu spectra with the best fitting parameters E(X − V ) = 1.0 mag and R V = 2.95. The extinction curves and R V of SN 2012cu derived in this work are, however, based on the detailed dust models for the observed color excesses. The best-fit values of R V for all the cases listed in Table 1 are generally consistent with the studies by A15 and H17.
In Figure 4b, the presence of a weak 2175Å bump suggests the existence of bands without the 2175Å bump. Calzetti et al. (1994) derived the attenuation curves of starburst galaxies, which similarly show the flattening and no appreciable 2175Å bump.   Figure 4, we therefore infer that the size distribution of NGC 4772 is biased to large sizes compared with those of the MW, the LMC and the SMC.
In Table 2, we present the modeled extinction correction A λ towards SN 2012cu in NGC 4772 with the dust models with f AMC cs = 0.3 and f GRA cs = 0.3. These modeled extinction corrections cover most of the commonly used astronomical filters.
10 Nozawa (2016) calculated the average radii weighted by the size distribution of dust [see Equation (8) therein], for distinguishing the similarity between an exponential-like distribution and a lognormal distribution. We adopt the same equation to derive the average radii for comparing the typical sizes of dust grains in the MW. Note that the extinction law depends more probably on the size distribution of dust rather than average radii of dust grains.

Conclusion
In this work, we use the broadband photometric data provided by A15 and optical spectroscopic data obtained from H17 to study the extinction of SN 2012cu. We adopt silicate+graphite/amorphous carbon dust models to fit the observed color excess curves of SN 2012cu and derive the extinction as a function of wavelength (i.e., A λ ).
The best-fit results give the visual extinction towards SN 2012cu of A V ≈ 2.6 mag, the reddening of E(B − V ) ≈ 1.0 mag, and the total-to-selective extinction ratio R V ≈ 2.7. The reasonable modeled A λ towards SN 2012cu in NGC 4772 are also presented in Table 2 for extinction corrections. The modeled extinction curves towards the SN 2012cu sightline extend flatly to the far-UV bands with or without a very weak 2175Å feature, and are much flatter than those of the MW, the LMC, and the SMC. The flatness of the UV extinction curves suggests a "grey" line-of-sight extinction law towards the host AGN galaxy of NGC 4772, and indicates that the size distribution of dust in this sightline is skewed to large grains. The extragalactic extinction laws are inadequately based on R V and probably dependent on the local interstellar environment.