On properties of interstellar dust
responsible for steep extinction curves
measured toward Type Ia supernovae

Takaya Nozawa(National Astronomical Observatory of Japan)
E-mail: takaya.nozawa@nao.ac.jp

Abstract
The extinction laws along lines of sight to Type Ia supernovae (SNe Ia) are
powerful probes that provide us with key clues to the properties ofinterstellar dust in external galaxies. It has been known, however, that theextinction curves toward SNe Ia are very steep with unusually low total-toselective extinction ratios of Rv = 1.0-2.5. In order to reveal the properties ofinterstellar dust that causes such unusual extinction laws, we search forphysical dust models that lead to good fits to the extinction curves obtained
from the empirical formula by Cardelli et al. (1989) with Rv = 2.0, 1.5, and1.0. In the fitting calculations, we apply a graphite-silicate dust model andconsider two grain size distributions of power-law and lognormal functions.
We find that the steep extinction curves with Rv = 2.0, 1.5, and 1.0 can bereasonably explained even by the simple power-law dust models with a fixedindex of -3.5 by taking the maximum cut-off radii of 0.13 µm, 0.094 µm, and0.057 µm, respectively. These maximum cut-off radii are smaller than 0.24
µm considered valid in the Milky Way, clearly demonstrating that theinterstellar dust responsible for steep extinction curves is biased to smaller
sizes. We show that lognormal size distribution can also yield good fits to theextinction curves with Rv = 1.0-3.1 by adopting very small peak radius andlarge standard deviation of the distribution.

〇 Extinction curves and CCM formula・ Extinction curves (Aλ/Av)‐ wavelength dependence of UV-to-NIR
extinction caused by interstellar dust
➜ powerful tool to know compositionand size distribution of dust grains
・ CCM formula (Cardelli, Clayton, Mathis 1989)‐describes variety of extinction curvesin the Milky Way (MW) through
RV = Av / E(B - V)
➜ Rv = 2.2-6.0 depending on sightlines➜ Rv = 3.1 for average extinction curve

Figure 1;
UV-to-NIR extinction curves measured
along different lines of sight in the MW(gray curves, Fitzpatrick & Massa
2007). The black circles indicates theMW average extinction curves. The
four colored curves show the extinction
curves calculated with the CCM
formula for Rv = 2.4, 3.1, 4.4, and 5.5.

・ Classical dust model in the MW‐two components of graphite & silicate‐power-law size distribution: n(a) ∝ a-q
q = 3.5, amax~0.25 µm, amin~0.005 µm (Mathis, Rumple, & Nordsieck 1977)

〇 Extinction laws toward SNe Ia
・ Type Ia supernovae (SNe Ia)‐ thermonuclear explosion of a WD
‐ discovered in all types of galaxies(star-forming, elliptical, irregular etc…)‐ used as cosmic standard candles

Howell+2011

ideal targets to probe the extinction(dust) properties in external galaxies

However, the values of Rv measured
for SNe Ia are very low (Rv ~ 1.0-2.5),
which never appear toward any linesof sight in the MW

Figure 2; (taken from Howell 2011)
Rv values toward SNe Ia derived from avariety of samples and techniques. The
cyan and blue points indicates the averagevalues obtained from ensembles of SNe Ia,
while the red ones show the Rv estimatedon the basis of SEDs of individual SNe Ia.

〇 Aim of this study and dust model
What properties of dust cause such steep extinction curves?
We perform the fitting calculations to the extinction curves
with Rv = 1.0-2.0 by applying a graphite-silicate dust model➜ two function forms for grain size distribution(1) power-law distribution(2) lognormal distribution
where➜ goodness of fitting to the extinction data

: data of extinction Aλi/Av derived from the CCM formula at photometric bands

〇 Results for power-law size distribution
・ Best-fits for Model 1 with q = 3.5 and amax,gra = amax,sil➜ amax = 0.13, 0.094, 0.057 µm for Rv = 2.0, 1.5, 1.0(i.e., smaller amax for steeper extinction curve)➜ Fits for Rv = 2.0 and 1.5 are good (solid lines in Fig. 3)
・ Best-fits for Model 2 with amax = 0.25 µm and qgra = qsil
➜ q = 4.05, 4.40, 5.08 for Rv = 2.0, 1.5, 1.0(i.e., larger q for steeper extinction curves)➜ Fits for Rv = 1.5 and 1.0 are poor (dashed lines in Fig. 3)

The simplest power-law size distributions with q = 3.5
account for the entire shapes of steep extinction curves
with Rv = 1.0-2.0 by taking amax = 0.06-0.13 µm.

〇 Extinction curves from power-law model(a) Rv = 2.0(c) Rv = 1.0(b) Rv = 1.5

Figure 3;
Extinction curves calculated from the best-fit
parameters in Model 1 (red solid), Model 2(green dashed), and Model 5 (black solid) withpower-law size distributions for (a) Rv = 2.0,(b) Rv = 1.5, and (c) Rv = 1.0. The filled circles
are the data of extinction at wavelengths ofrepresentative UV-to-NIR photometric bands,
derived from the CCM formula for each Rv
value. For reference, the extinction curve forRv = 3.1 is shown by the blue dot-dashed line.

〇 Best-fit parameters for power-law model
Table 1 :Table 2 :

〇 Results for lognormal size distribution
・ Best-fits for Model 1s with γj = 0.5➜ a0,j = 0.013-0.024 µm for Rv = 1.0-2.0(i.e., smaller a0,j for steeper extinction curve)➜ Fits for Rv = 2.0 and 1.5 are poor (dashed lines in Fig. 4)・ Best-fits for Model 2s with no fixed parameter➜ a0,j < ~3x10-3 µm and γj ~ 1.0 for Rv = 1.5 and 2.0
which appears an exponential-like distribution (Fig. 5)➜ Fits for Rv = 2.0 and 1.5 are good (solid lines in Fig. 4)

The lognormal size distribution can lead to good fits to
the CCM extinction curves with Rv = 1.0-2.0 by taking
the appropriate sets of a0,j and γj.

〇 Extinction curves from lognormal model(a) Rv = 2.0(c) Rv = 1.0(b) Rv = 1.5

Figure 4;Extinction curves calculated from the best-fit
parameters in Model 1s (red dashed) andModel 2s (black solid) with lognormal size
distributions for (a) Rv = 2.0, (b) Rv = 1.5, and(c) Rv = 1.0. The filled circles are the data ofextinction at wavelengths of representative
UV-to-NIR photometric bands, derived fromthe CCM formula for each Rv value. For
reference, the extinction curve for Rv = 3.1 isshown by the blue dot-dashed line.

〇 Best-fit parameters for lognormal modelTable 3 :Table 4 :

〇 Examples of grain size distributions(a) Rv = 2.0(c) Rv = 1.0(b) Rv = 1.5

Figure 5;
Grain size distribution that lead to the best-fit
to the extinction curves with (a) Rv = 2.0, (b)Rv = 1.5, and (c) Rv = 1.0. The solid lines show
the power-law size distributions obtained from
Model 5, while the dashed lines indicates thelognormal size distributions obtained from
Model 2s. Graphite and silicate are drawn in
red and blue, respectively.

Summary1) Many observational studies of Type Ia supernovae suggestthat the Rv values measured toward SNe Ia are significantly
low (Rv ~ 1.0-2.5), compared with Rv = 3.1 in the MW2) The steeper extinction curves described by Rv = 1.0-2.5 are
reproduced by the dust models whose size distributions are
skewed to smaller radii than that in the MW.
a) Power-law size distribution➜ Maximum cut-off radius (amax) is an important quantity todescribe the variety of extinction curves, and if the powerindex is fixed to be q = 3.5, amax = 0.06-0.13 µm is required.
b) Lognormal size distribution➜ Very small peak radius a0,j and large standard deviation γj
are needed to yield good fits to the steep extinction curves.