INTERSTELLAR-MEDIUM MAPPING IN M82 THROUGH LIGHT ECHOES AROUND SUPERNOVA 2014J

Light echoes are the scattered light of a transient event that arise from dust clouds. Here we consider the case of an SN and CSM/ISM. Because of the high initial brightness of SNe, searches for late-time off-source flux excesses have been the main approaches to detect light echoes residing close to the SNe, i.e., the slowly fading light-curves of SN 1991T (Schmidt et al. 1994; Sparks et al. 1999), SN 1998bu (Cappellaro et al. 2001), and SN 2006X (Wang et al. 2008). Outside the solar system, spatially resolved light echoes have been rare events. The first one reported arose around Nova Persei 1901 (Kapteyn 1901; Ritchey 1901), followed by Nova Sagittarii 1936 (Swope 1940). Echoes were also found from the Galactic Cepheid RS Puppis (Havlen 1972) and, with Hubble Space Telescope (HST) angular sampling, from the eruptive star V838 Monocerotis (Bond et al. 2003). Vogt et al. (2012) reported the detection of an infrared echo near the Galactic supernova remnant Cassiopeia A. Additionally, spectroscopic observations of nearby light echoes provide unique opportunities to probe the progenitor properties of historical transients (Rest et al. 2008; Davidson & Humphreys 2012) and in some cases the three-dimenisonal structure of the explosion, for instance, an ancient eruption from η Carinae (Rest et al. 2012), or asymmetry in the outburst of SN 1987A (Sinnott et al. 2013) and Cassiopeia A (Grefenstette et al. 2014). In recent years, the number of light echoes from extragalactic SNe has grown rapidly, mostly thanks to HST. Table 1 provides an overview of the events recorded to date, updated from Table 1 of Van Dyk et al. (2015).

Photons from spatially resolved light echoes travel a slightly different path than the DLOS from the SN to Earth. Therefore, observations of a resolved light echo around a nearby SN provide a unique opportunity to measure the extinction properties of the dust along the DLOS and the scattering properties of the echo-producing dust independently and simultaneously. As the SN fades, outer echoes (echoes with larger angular diameter) associated with ISM at large distances to the SN will become less contaminated by its bright light, and any inner echoes associated with ISM at small distances to the SN, and even the CSM, will become detectable. The expansion with time of the light echoes maps out the 3D structure of ISM along and close to the line of sight.

Detailed introductions to the relation between two-dimensional light echoes and three-dimensional scattering dust distributions has been given in various studies (Chevalier 1986; Sparks 1994; Sugerman 2003; Tylenda 2004; Patat 2005). Here, we briefly define the geometry used throughout this paper, also shown in Figure 1, which considers the SN event as an instantaneous flash of radiation. The locus of constant light travel time is an ellipsoid with the supernova at one focus, which we refer to as an iso-delay surface. The ellipsoid grows with time as the light propagates in space.

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The angular radius of the light echo (α) can be easily measured in two-dimensional images. The SN is centered at the origin of the plane, the x and y give the coordinates of the scattering materials in the plane of the sky. The projected distance ($\rho =\sqrt{{x}^{2}+{y}^{2}}$) of scattering material to the SN perpendicular to the DLOS is related to the distance (D) to the SN as $\tan \alpha =\rho /D$, ϕ gives the position angle (PA). Because D is significantly larger than other geometric dimensions, the light echo can be very well approximated by a paraboloid, with the SN lying at its focus. ρ can be obtained by

$\begin{eqnarray}&&\rho =\sqrt{{ct}(2z+{ct})}.\end{eqnarray} \tag{ 1 }$

Here t is the time since the radiation burst, z gives the foreground distance of the scattering material along the line of sight, and c denotes the speed of light. The distance r of the scattering material from the SN is

$\begin{eqnarray}&&r=\displaystyle \frac{1}{2}\left(\displaystyle \frac{{\rho }^{2}}{{ct}}+{ct}\right).\end{eqnarray} \tag{ 2 }$

The scattering angle can be obtained from $\cos \ \theta (\rho ,t)=z/(z+{ct})$ or from $\tan \theta =\rho /z$.

The nearby SN Ia 2014J in M82 (3.53 ± 0.04 Mpc, Dalcanton et al. 2009) offers the rare opportunity to study the physical properties and spatial distribution of dust particles along and close to the DLOS and also in the vicinity of the SN. SN 2014J suffers from heavy extinction (A_V = 2.07 ± 0.18, Foley et al. 2014) and is located behind a large amount of interstellar dust (Amanullah et al. 2014). Additionally, the absorption profiles of Na and K lines from high-resolution spectroscopy exhibit more than ten extragalactic absorption components, indicating that the extinction along the DLOS is caused by the combined presence of a large number of distinct interstellar dust clouds along the DLOS (Patat et al. 2015). SN 2014J was discovered on January 21.805 UT by Fossey et al. (2014). Later observations constrained the first light of the SN to January 14.75 UT (Goobar et al. 2014; Zheng et al. 2014).

SN 2014J reached its B-band maximum on February 2.0 UT (JD 2,456,690.5) at a mgnitude of 11.85 ± 0.02 (Foley et al. 2014). Continuous photometric and spectroscopic observations through late phases have been made by various groups (Johansson et al. 2014; Lundqvist et al. 2015; Bonanos & Boumis 2016; Porter et al. 2016; Sand et al. 2016; Srivastav et al. 2016).

There is clear evidence that the strong extinction measured from SN 2014J is caused primarily by interstellar dust (Brown et al. 2015; Patat et al. 2015), although a mix of interstellar and circumstellar dust is also possible (Foley et al. 2014; Bulla et al. 2016). Several independent studies, including photometric color fitting from Swift/UVOT and HST (Amanullah et al. 2014), near-UV/optical grism spectroscopy from Swift UVOT (Brown et al. 2015), HST STIS spectroscopy and WFC3 photometry (Foley et al. 2014), reddening curve fitting near the SN maximum using the silicate-graphite model (Gao et al. 2015), as well as optical spectroscopy from Goobar et al. (2014) found an R_V ∼ 1.4 toward SN 2014J. Moreover, ground-based broadband imaging polarimetry (Kawabata et al. 2014; Srivastav et al. 2016) and spectropolarimetry (Patat et al. 2015; Porter et al. 2016) have shown that the polarization peak that is due to interstellar dust extinction is shortward of ∼0.4 μm, which indicates that this line of sight has peculiar Serkowski parameters (see Patat et al. 2015). This polarization wavelength dependence can be interpreted in terms of a significantly enhanced abundance of small grains (Patat et al. 2015). Models considering both interstellar dust and circumstellar dust simultaneously and fitted to observed extinction and polarization (Hoang 2015) find that a significant enhancement (w.r.t. the Milky Way) in the total mass of small grains (<0.1 μm) is required to reproduce low values of R_V. Multiple time-invariant Na i D and Ca ii H&K absorption features as well as several diffuse interstellar bands have also been identified (Graham et al. 2015; Jack et al. 2015). These are most likely associated with multiple dust components of interstellar material along the DLOS.

The nature (amount and distribution) of CSM is of interest when probing the possible diversity of progenitors of SNe Ia and for accurately correcting the extinction when using SNe Ia as standard candles. Johansson et al. (2014) found no evidence of heated dust in the CSM of SN 2014J with r < 10¹⁷ cm (∼39 light days). Graham et al. (2015) reported variable interstellar K i lines in high-resolution spectra, which may form about 10 light years (∼10¹⁹ cm) in front of the SN.

The extremely dusty environment in M82 and its relative proximity to Earth lead to the expectation of complex and evolving light echoes if SN 2014J exploded inside the galactic disk. In fact, Crotts (2015) discovered the first light echoes surrounding SN 2014J in HST images from 2014 September 5, 215.8 days past B-band maximum light (referred to as +216 d hereafter) on JD = 2456690.5 (Foley et al. 2014). The echo signal tends to be associated with pre-explosion nebular structures in M82 (Crotts 2015).

In the following, we present the evolution of multiple light echoes of SN 2014J as revealed by new HST ACS/WFC multiband and multi-epoch imaging around ∼277 days and ∼416 days past B-band maximum (referred to as +277 d and +416 d below). We also qualitatively discuss similar archival WFC3/UVIS images obtained on +216 d and +365 d.

Photometry of SN 2014J at four epochs was performed in the background-subtracted images described above, and shown in Table 3. Measurements were made with a circular aperture of 04 (8 pixels in the ACS/WFC FOV and 10 pixels in the WFC3/UVIS FOV) in the WFC3/UVIS F438W, F555W, F814W images from +216 d, and the F438W and F555W images from +365 d. We applied aperture corrections according to Hartig (2009) and Sirianni et al. (2005) to estimate the total flux from SN 2014J. The photometric uncertainties in Table 3 include the Poisson noise of the signal, the photon noise of the background, the readout noise contribution (3.75 electrons/pixel for ACS/WFC), and the uncertainties in aperture corrections. These quantities were added in quadrature. The magnitudes are presented in the Vega system with zero points from the CALSPEC⁹ archive. The total flux of the source within the aperture equals the product Total Counts × PHOTFLAM, where PHOTFLAM is the inverse sensitivity (in erg cm⁻² s⁻¹ Å⁻¹ and representing a signal of 1 electron per second). For WFC3/UVIS images, we adopted the values of the PHOTFLAM keyword in the image headers. However, for the ACS/WFC polarizer images, which were corrected for the throughputs of the polarizers to generate the intensity maps, we discarded the default PHOTFLAM values. Instead, we adopted the most up-to-date PHOTFLAM values in the ACS filter bands for images obtained without polarizers (Bohlin 2012). This is required by the mismatch between (i) the polarizer throughput curves used by SYNPHOT¹⁰ for unpolarized sources and (ii) the values found by comparing unpolarized sources in both the polarizing and nonpolarizing filters (Cracraft & Sparks 2007). Therefore, the PHOTFLAM keywords in ACS/WFC polarized images are not applicable to intensity maps derived from polarized images. Polarization properties of SN 2014J will be discussed in a separate paper (Y. Yang et al. 2016, in preparation).

Two main echo components are evident. In Figure 4 we show a luminous quarter-circle arc and a diffuse ring at angular distance larger than 03 from the SN. Closer to the SN, uncertainties in the PSF correction prevent reliable detections. On +277 d, the most notable features of the light echoes in F475W are three luminous clumps at angular radius α = 060 and PAs 80°, 120°, and 150°, measured from north (0°) through east (90°). These clumpy structures are already present on +216 d at the same PAs, but appear smoother and more extended. They eventually evolve into a fairly continuous luminous quarter-circle arc seen on both +365 d and +416 d extending from PA = 60°–170°. Images obtained on +216 d with F438W and F555W show the luminous arc at angular radii α = 054 and α = 069, over roughly the same range in PA, in agreement with Crotts (2015). However, for the arc we find a foreground distance of the scattering material that ranges from 226 to 235 pc in the four epochs (Table 4) and has a mean value of 228 ± 7 pc. This is different from the foreground distance of ∼330 pc discussed for this prominent echo component by Crotts (2015). This discrepancy may be due to the difficulties and uncertainties in subtracting the PSF in earlier epoch when the SN is still bright, or in distinguishing the multiple light-echo components identified in our multi-epoch data.

To enable a more quantitative description of the light echoes and their evolution, we performed photometry on them in backgound-subtracted images (Figure 4). We measured the surface brightness of the light-echo profile at different radii and over different ranges in PA. Fan-shaped apertures centered on the SN were used to sample the intensity. The width in PA of each aperture is 45°. In contrast to the luminous arc, the diffuse echo can be seen over the full range in PA from 0° to 360°, but it does not exhibit a common radial profile (Figures 5 and 6).

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In the following subsections, we use these measurements to investigate the evolving profile of the light echoes, conduct geometric and photometric analyses, and estimate the dust distribution and scattering properties responsible for the observed light echoes along and close to the DLOS. A function characterizing the properties of the scattering material is constructed to represent the brightness evolution of the observed light echoes on +277 d and +416 d.

A comprehensive discussion of the formation of light-echo arcs is available from Tylenda (2004). In the context of this paper, it is sufficient to recall that a circular light echo is created from the intersection of the dust slab with the iso-delay parabaloid. Any uneven distribution of material in the slab results in an uneven flux distribution along the circle, and the light echo may be composed of incomplete arcs. A dust slab always produces a (complete or incomplete) circular light echo, regardless of its inclination with respect to the line of sight. When a dust slab is not perpendicular to the line of sight, the center of the light-echo circle will not coincide with the SN position, and it moves with time.

The luminous arc echo is unresolved with a full width at half maximum of the radial profile approximately that of the SN measured in the same images, i.e., ∼01 (2 pixels). Therefore, we consider that the luminous arc was formed by a thin dust slab intersecting the line of sight. We have fitted circles to the positions of the luminous arc at all available epochs. None of them are significantly decentered from the SN. This implies that the dust slab producing the arc echo is fairly perpendicular to the line of sight. Table 4 summarizes the geometric properties measured from the luminous arc.

In addition to the luminous arc, a radially extended and diffuse structure is identified, which on +277 d is present in F475W and F606W and spread over α = 040 to α = 090. This structure can also be noted on +365 d in F438W and F555W (from α = 047 to α = 103). It appears more clearly on +416/417 d in F475W and F606W (from α = 050 to α = 108) because for these observations longer exposure times were used. The epochs of observation and the exclusion of the inner 03 limit the explored foreground distances from z = 100 pc to z = 500 pc. On +216 d, the diffuse component cannot be identified in F438W, but is marginally seen in F555W. However, the inner and outer radii of the diffuse structure cannot be well determined because of uncertainties in the PSF subtraction. The diffuse light echo observed on +277 d can be produced by a dust cloud intersecting the iso-delay surface over a wide range in foreground distance. The line-of-sight extent of this diffuse dust cloud is indicated by the filled profile of the echoes. This shows that a continuous dust distribution over a certain range of foreground distances along the line of sight is required.

In each panel of the radial profiles in Figures 5 and 6, the radially resolved positive flux excesses (on +277 d and +416 d), and also the radially extended negative flux that is due to the subtraction of the light echo on +796 d, suggest the presence of an extended and inhomogeneous foreground dust distribution. Outside the ∼03 region, as discussed earlier, the imperfect PSF subtraction makes the detection of echoes unreliable. The most prominent structure with an intensity peak at the second and third curve near the top in Figure 5 can be clearly seen on +277 d with an angular radius of ∼060, which at the distance of M82 (3.53 ± 0.04 Mpc, Dalcanton et al. 2009) is at a radius ρ = 10.3 pc from the SN in the plane of the sky. By +416 d, the radius has increased to ∼0735 or ρ = 12.6 pc from the SN. The scattering angles are 26 and 32, respectively.

To our knowledge, and with the exception of SN 1987A in the Large Magellanic Cloud (Crotts 1988; Suntzeff et al. 1988), this is the first radially extended light echo detected from any SN. For epochs discussed in this paper, the diffuse echo component around SN 2014J reveals the SN-backlit ISM over ∼40 pc × 40 pc around the DLOS. Standard methods for estimating the optical properties of the ISM toward the supernova only consider the extinction along the DLOS. They include the spectrophotometric comparison between the observed SN and an unreddened SN or template, and comparing the integrated echo flux with the surface brightness calculated from the scattering properties of various dust models. The resolved dust echoes of SN 2014J and their temporal evolution in the gas-rich and very nearby galaxy M82, however, provide an unprecedented opportunity to do better. In the following, we take advantage of this to measure the scattering properties of the ISM at different foreground distances and PAs relative to SN 2014J.

We assume that dust scattering follows the Henyey–Greenstein phase function (Henyey & Greenstein 1941):

$\begin{eqnarray}&&{\rm{\Phi }}(\theta )=\displaystyle \frac{1-{g}^{2}}{{(1+{g}^{2}-2g\cos \theta )}^{3/2}},\end{eqnarray} \tag{ 4 }$

where $g=\overline{\cos \theta }$ is a measure of the degree of forward scattering. With ${L}_{\lambda }(t)$ as the number of photons emitted per unit time by the SN at a given wavelength, ${F}_{\lambda }(t)={L}_{\lambda }(t)/4\pi {D}^{2}$ is the number of photons observed at time t. D is the distance to the SN. For the modeling of our observations, t is the time of the light-echo observation, t_e denotes the time when photons emitted by the SN would be directly observed along the DLOS, and ${F}_{\lambda }(t-{t}_{e})$ is the brightness of the SN at $(t-{t}_{e})$. At t, the photons emitted at the same time as t_e, but experiencing scattering leading to a light echo, arrive at the observer with a time delay $(t-{t}_{e})$.

For a single short flash of light of duration Δt_e emitted by the SN at t_e, ${F}_{\nu }(t-{t}_{e})=0$ for $t\ne {t}_{e}$ and ${\int }_{0}^{t}{F}_{\nu }(t-{t}_{e}){{dt}}_{e}={F}_{\nu }(t-{t}_{e}){| }_{t={t}_{e}}{\rm{\Delta }}{t}_{e}$. Then, the surface brightness, Σ, of a scattered light-echo at frequency ν and arising from an infinitely short (δ function) light pulse is given by

$\begin{eqnarray}{{\rm{\Sigma }}}_{\nu }^{\delta }(\rho ,\phi ,t) & = & {n}_{d}{Q}_{s}{\sigma }_{d}\ \displaystyle \frac{{F}_{\nu }(t-{t}_{e}){| }_{t={t}_{e}}{\rm{\Delta }}{t}_{e}}{4\pi {r}^{2}}\ \left|\displaystyle \frac{{dz}}{{dt}}\right|{\rm{\Phi }}(\theta )\\ & = & {n}_{d}{Q}_{s}{\sigma }_{d}\displaystyle \frac{{\int }_{0}^{t}{F}_{\nu }(t-{t}_{e}){{dt}}_{e}}{4\pi {r}^{2}}\left|\displaystyle \frac{{dz}}{{dt}}\right|{\rm{\Phi }}(\theta ).\end{eqnarray} \tag{ 5 }$

Here n_d is the volume number density of the scattering material in units of cm⁻³; Q_s is a dimensionless number describing the scattering efficiency of the dust grains; σ_d is the geometric cross section of a dust grain, Φ(θ) is the unitless scattering phase function. This means that the surface brightness at a certain instance of the light echo at $t={t}_{e}+(t-{t}_{e})$ is determined by the flux emitted from the SN at t_e, together with the local geometric properties of the iso-delay surface at t − t_e.

In reality, the SN emission has a finite duration. ${F}_{\nu }(t-{t}_{e})$ is no longer a δ function, and the surface brightness of the light echo unit at a certain frequency Σ_ν is the time integral of ${F}_{\nu }(t-{t}_{e})$ from 0 to t:

$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\nu }(\rho ,\phi ,t)=\displaystyle \frac{{Q}_{s}{\sigma }_{d}}{4\pi }{\int }_{0}^{t}\displaystyle \frac{{n}_{d}{F}_{\nu }(t-{t}_{e}){{dt}}_{e}}{{r}^{2}}\ \left|\displaystyle \frac{{dz}}{{dt}}\right|{\rm{\Phi }}(\theta ).\end{eqnarray} \tag{ 6 }$

By recalling that

$\begin{eqnarray}&&z=\displaystyle \frac{{\rho }^{2}}{2{ct}}-\displaystyle \frac{{ct}}{2},\end{eqnarray} \tag{ 7 }$

one can easily find

$\begin{eqnarray}&&\displaystyle \frac{{dz}}{{dt}}=-\displaystyle \frac{c}{2}\left(\tfrac{{\rho }^{2}}{{c}^{2}{t}^{2}}+1\right),r=z+{ct}=\displaystyle \frac{{ct}}{2}\left(\tfrac{{\rho }^{2}}{{c}^{2}{t}^{2}}+1\right).\end{eqnarray} \tag{ 8 }$

Therefore,

$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\nu }(\rho ,\phi ,t)=\displaystyle \frac{{Q}_{s}{\sigma }_{d}c}{2\pi }{\int }_{0}^{t}\displaystyle \frac{{n}_{d}(\rho ,\phi ,t)}{{c}^{2}{t}^{2}+{\rho }^{2}}{\rm{\Phi }}(\theta )\ {F}_{\nu }(t-{t}_{e}){{dt}}_{e}.\end{eqnarray} \tag{ 9 }$

Because of the relative proximity of M82, some light echoes around SN 2014J are resolved by HST at late phases, and each pixel represents the surface brightness of the light echo multiplied by the physical area covered by the pixel in the sky.

Therefore, in order to compare the model flux distribution with the flux in a two-dimensional image, one needs to integrate the model flux over the physical depth covered by the pixel. Since each pixel has size ΔxΔy, and Δx = Δy, this implies

$\begin{eqnarray}&&{\mathrm{Im}}_{\nu }(x,y,t)={\int }_{x-\tfrac{{\rm{\Delta }}x}{2}}^{x+\displaystyle \frac{{\rm{\Delta }}x}{2}}{\int }_{y-\tfrac{{\rm{\Delta }}y}{2}}^{y+\displaystyle \frac{{\rm{\Delta }}y}{2}}{\rm{\Sigma }}(x,y,t){dxdy}.\end{eqnarray} \tag{ 10 }$

The geometric factor is determined by the radial distance to the SN, $\rho =\sqrt{{x}^{2}+{y}^{2}}$. Therefore, in the tangential direction inside each pixel, we approximate the integration by assuming that n_d (x, y, t) is invariant over the angle Δϕ subtended by a single pixel. Furthermore, the angular size of each ACS/WFC pixel is 005. At the distance of D = 3.53 ± 0.04 Mpc, the corresponding physical pixel size in the sky is

$\begin{eqnarray}\mathrm{pixscale} & = & (3.53\pm 0.04)\ \mathrm{Mpc}\times \tan (0\buildrel{\prime\prime}\over{.} 05)\\ & = & (0.86\pm 0.01)\ \mathrm{pc}={\rm{\Delta }}x={\rm{\Delta }}y.\end{eqnarray} \tag{ 11 }$

We recall the geometric configuration of the iso-delay light surface at +277 d presented by Figure 1. In Figure 7 we modify this schematic diagram to demonstrate how we use a two-dimensional image to map the ISM in three dimensions. The gray-shaded fields on the vertical axis show the pixelation of the sky view by the camera, with each pixel measuring 0.86 pc on both sides. Δz is the position-dependent line-of-sight extent of the foreground column covered by each pixel. Gray-shaded rectangles superimposed on the iso-delay light surface mark columns of ISM that would be responsible for the respective light echoes as projected onto the sky. The fixed size of the sky pixels leads to varied lengths of the foreground columns of ISM. If the ISM is homogeneously distributed in the x/y plane, then the total per-sky-pixel extinction of the scattering materials as revealed by the light echo can be estimated by summing the extinction along each rectangular column of ISM intersecting the iso-delay light paraboloid. Comparison of the extinction by the scattering materials to the extinction along the DLOS (marked by the gray line on the z-axis in Figure 7) may reveal whether they are caused by the same dust mixture and perhaps even the same dust cloud.

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Now we can compare the intensity map obtained from the observations with the light echo modeled at each physical position for a given time t of the observation as follows:

$\begin{eqnarray}{\mathrm{Im}}_{\nu }(x,y,t) & = & \displaystyle \frac{\omega {C}_{\mathrm{ext}}c}{2\pi }{\rm{\Delta }}x{\displaystyle \int }_{x-\tfrac{{\rm{\Delta }}x}{2}}^{x+\tfrac{{\rm{\Delta }}x}{2}}{dx}{\displaystyle \int }_{0}^{t}\displaystyle \frac{{n}_{d}(x,y,t)}{{c}^{2}{t}^{2}+{\rho }^{2}}\\ & & \times \,{\rm{\Phi }}(\theta ){F}_{\nu }(t-{t}_{e}){{dt}}_{e}.\end{eqnarray} \tag{ 12 }$

The optical properties of the dust grains that are responsible for the light echoes around SN 2014J can be deduced within each observed pixel. We estimate the extinction properties of the scattering materials based on an approach that combines single scattering with attenuation (see Section 5 of Patat 2005 for more details). Conversions from the intensity map to the number-density map ("nd") are presented by Figure 8 based on Equation (12). We follow the sampling in Figures 5 and 6 and present the deduced optical properties of the dust grains for the PA sector 45°–90°, which includes the brightest part of the luminous arc, and PA sector 315°–360°, which covers the diffuse echo ring observed with the highest S/N. They are shown in Figure 9 for F475W and Figure 10 for F606W, both on +277 d. In these diagrams, the rectangular coordinates x and y are replaced with the polar coordinates ρ and ϕ, and the abscissa corresponds to the physical distances in the plane of the sky. The left ordinate represents the quantity ωC_extn_d(ρ, ϕ, t), which is determined by the optical properties of the dust grains. The right ordinate shows $\omega {C}_{\mathrm{ext}}{n}_{d}{dz}=\omega \tau$, where τ is the optical depth of the dust mapped onto a single pixel. By looking at the entire echo profile, we found that a major part of the luminous-arc echo spreads over 45°–180° in PA, and the diffuse echo ring attained the highest S/N over 270°–360° in PA.

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View extended figure (2 panels)

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View extended figure (2 panels)

We applied a Galactic extinction model with R_V = 3.1 to the scattering materials and compare the reproduced extinction properties with the extinction along the DLOS. Discrepancies between the derived quantities and the assumed model will indicate that the extinction properties of the scattering dust are different from the Milky Way dust with R_V = 3.1. For each photometric bandpass its pivot wavelength was used in interpreting the parameters from dust models. The extinction curve is obtained from Weingartner & Draine (2001) and Draine (2003a, 2003b).¹¹ For C_ext, the extinction cross section per hydrogen nucleon H, we adopted 5.8 × 10⁻²² cm²/H for F475W, and 4.4 × 10⁻²² cm²/H for F606W; for the scattering phase function, we adopted g = 0.555 for F475W, and g = 0.522 for F606W, and n_d is the H volume number density in units of cm⁻³.

For a uniform dust distribution in the x/y direction (in the plane of the sky), integrating ωτ over each position angle will provide a rough estimate of the product of the total optical depth and the scattering albedo, which is the main value added by the separate analysis of light echoes. We applied the same extinction measured along the DLOS to the scattered light-echoes and calculated the optical depth of the materials from scattering. This is labeled by the red text in the upper right of each panel of Figures 9 and 10. The inhomogeneity of the ISM in M82 has small scales, as is indicated by the rapid variability of the strength of the echo with PA along the rings as well as with time. The optical depth along the DLOS has been measured by Foley et al. (2014) around maximum light as τ_B = 3.11(0.18) and τ_V = 1.91(0.17) based on A_B = 3.38(20), A_V = 2.07(18) and using the relation ${A}_{\lambda }=-2.5{\mathrm{log}}_{10}(e){\tau }_{\lambda }^{\mathrm{ext}}=1.086{\tau }_{\lambda }^{\mathrm{ext}}$.

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View extended figure (2 panels)

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The hydrogen column number density along the line of sight is ${n}_{H}={\int }_{\mathrm{LOS}}{n}_{d}(z){dz}$. Therefore, n_H can be obtained by dividing the total optical depth per bin in position angle by $\omega {C}_{\mathrm{ext}}$ (Figure 9 for F475W and Figure 10 for F606W). For example, for F475W and +277 d, the maximum value of ωτ (ρ, ϕ, t) in the luminous arc was observed to be around 0.58. Using ω ∼ 0.65 for the Milky Way dust model with R_V = 3.1 given by Weingartner & Draine (2001), n_H can be estimated to be ${n}_{H}=0.58/\omega {C}_{\mathrm{ext}}=0.58/$ $(0.65\times 5.8\times {10}^{-22}$ cm²/H) ∼ 1.5 × 10²¹ H cm⁻² in the bin that shows the densest part of the dust slab producing the luminous arc echo. This is ∼15 times denser than the scattering material in the foreground of the Type II plateau SN 2008bk (Van Dyk 2013), for which the visual extinction of the dust responsible for the echo is A_V ≈ 0.05. It is also ∼4 times denser than the ISM in the foreground of the Type II plateau SN 2012aw (Van Dyk et al. 2015), for which the dust extinction in the SN environment responsible for the echo is consistent with the value that was estimated from observations of the SN itself at early times, i.e., A_V = 0.24.

From the scattering properties of the dust, its optical properties can be estimated by comparing the quantity ωC_extn_d derived for F475W and F606W. Figure 11 presents the division of the profiles of Figure 9 by Figure 10. This yields the wavelength dependence of the extinction cross section. As the ordinate of Figure 11 we use $\omega {\tau }_{F475W}/\omega {\tau }_{F606W}$. Overplotted histograms show (in red) the number density of the scattering material derived from the strength of the echoes in F475W. The horizontal gray dashed lines mark the value of ${\tau }_{F475W}/{\tau }_{F606W}={A}_{F475W}/{A}_{F606W}\,=$ 1.66, 1.30, and 1.19 for Milky Way-like dust with R_V = 1.4, 3.1, and 5.5, respectively, according to the algorithm determined by Cardelli et al. (1989). For completeness, extended versions of Figures 9, 10, and 11 over the entire eight bins of PA are available in the online journal.

Plausible estimates of $\omega {\tau }_{F475W}/\omega {\tau }_{F606W}$ can only be made in regions of the echoes with high S/N. In the left panel of Figure 11, the luminous arc at ρ = 10 ∼ 11 pc has an average value $\omega {\tau }_{F475W}/\omega {\tau }_{F606W}\sim 1.7$ (dimensionless), shown by the black histograms. For the diffuse structure, the right panel indicates an average value ∼1.3. This difference in the wavelength dependence measured from the scattering optical depth indicates that the size of the grains in the thin dust slab producing the luminous arc is different from the grain sizes in the foreground extended dust cloud producing the diffuse echo. While this difference is significant, one should be cautious about the inferred absolute values of R_V in this approach, considering the low S/N and the large uncertainties.

Figure 12 presents the three-dimensional dust distribution estimated for SN 2014J. Data points show the number densities as derived from two iso-delay paraboloids. Scattering materials producing the luminous arc and the diffuse echo were mapped out at epochs +277 d (inner layer) and +416 d (outer layer), respectively.