Time-Varying Extinction, Polarization, and Colors of Type Ia Supernovae due to Rotational Disruption of Dust Grains
Nguyen Chau Giang Thiem Hoang Le Ngoc Tram





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ApJL
ApJ
MNRAS
ARA& A
ApJL
Annu. Rev. Earth Planet. Sci.
PhRL
NIMPB
JSpRo

 cm
 K
 g
 μm
 erg
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    Grain Disruption Effects on SNe IaGiang, Hoang, and TramUniversity of Science and Technology of Hanoi, VAST, 18 Hoang Quoc Viet, VietnamKorea Astronomy and Space Science Institute, Daejeon 34055, South KoreaKorea University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, South KoreaSOFIA-USRA, NASA Ames Research Center, MS 232-11, Moffett Field, 94035 CA, USAUniversity of Science and Technology of Hanoi, VAST, 18 Hoang Quoc Viet, VietnamThiem Hoangthiemhoang@kasi.re.kr
    Photometric and polarimetric observations toward type Ia supernovae (SNe Ia) frequently report an unusually low total-to-selective extinction ratio (R_ V < 2) and small peak wavelength of polarization (λ_ max< 0.4). Recently, Hoang et al. proposed that the increase in the abundance of small grains relative to large grains near SNe Ia due to RAdiative Torque Disruption (RATD) can explain this puzzle. To test this scenario, we will perform detailed modeling of dust extinction and polarization of SNe Ia accounting for grain disruption by RATD and grain alignment by RAdiative Torques (RATs). For dust clouds at distance d< 4 pc from the source, we find that R_ V decreases rapidly from the standard value of 3.1 to ∼ 1.5 after a disruption time t_ disr< 40 days. We then calculate the observed SNe Ia light curve and find that the colors of SNe Ia would change with time due to time-varying extinction for dust clouds at distance d<4 pc. We also calculate the wavelength-dependence polarization produced by grains aligned with the magnetic fields by RATs. We find that λ_ max decreases rapidly from ∼ 0.55 to ∼ 0.15 over an alignment time of t_ align< 10 days due to the enhanced alignment of small grains. By fitting the theoretical polarization curve with the Serkowski law, we find that the parameter K from the Serkowski law increases when large grains are disrupted by RATD which can explain the K vs. λ_ max data observed for SNe Ia. Finally, we discover an anti-correlation between K and R_ V which might already be supported by SNe Ia observational data. Our results demonstrate the important effect of rotational disruption of dust grains by radiative torques on the time-dependent extinction, polarization, and colors of SNe Ia.
    
    
    § INTRODUCTION
    
    Type Ia supernovae (hereafter SNe Ia) is an explosion of the white dwarf star in the binary system when the accretion of material from the evolving companion star onto the white dwarf increases its mass to ∼ 1.4M_⊙, i.e., the Chandrasekhar limit (). Beyond this mass limit, the electron degeneracy pressure is insufficient to support against the gravity, and the white dwarf collapses, releasing a large amount of material and energy into the surrounding environment. SNe Ia have similar intrinsic luminosity because they explode at similar mass, for which SNe Ia are considered "standard candles" to measure cosmological distances (). SNe Ia are also useful to study the physical and chemical properties of the interstellar medium of external galaxies (; ; ).
    
    Dust along the line of sight toward SNe Ia absorbs, scatters, and polarizes the SNe radiation in the UV-NIR wavelength range (see, e.g., ). Photometric observations of SNe Ia show that the total-to-selective extinction ratio R_ V=A_ V/E(B-V), with A_ V the optical extinction and E(B-V)=A_ B-A_ V the color excess, is much lower than the typical value of interstellar dust in the Milky Way of R_ V∼ 3.1. For example, 80 SNe Ia studied by <cit.> have an average value of R_ V∼ 1.75. Using Hubble space telescope and Swift satellites, <cit.> and <cit.> estimated a range of R_ V∼ 1.4-3. These examples can be explained by the peculiar properties of dust in the SNe Ia's host galaxy. Other suggestions include the scattering of circumstellar (CS) material in the vicinity of SNe Ia (; ) and the enhancement of small grains (smaller than 0.05) relative to large grains (size above 0.1) in the local interstellar environment of SNe Ia (; ).
    
    Moreover, polarimetric observations also reveal unusually low values of the wavelength at the maximum polarization curve of SNe Ia, i.e., λ_ max< 0.4. For instance, four SNe SN 1986G, 2006X, 2008fp and 2014J shows λ_ max from 0.05 to 0.43 (; ; ), and nine SNe Ia listed in <cit.> have λ_ max< 0.4, much lower than the standard value of the ISM of λ_ max=0.55 (). Based on simultaneous fitting to extinction and polarization of SNe Ia, <cit.> found that the unusual values of λ_ max can be reproduced if small grains  can be aligned efficiently as large grains. 
     
    The enhancement in the abundance of small grains relative to large grains around SNe Ia is the popular explanation for the anomalous values of R_ V and λ_ max<cit.>. Several mechanisms were proposed to explain the disruption of large grain such as grain-grain collision, thermal sublimation. However, the thermal sublimation mechanism only destroys grains within radius of R_ sub≃ 0.015(L_ UV/10^9L_⊙)^1/2(T_ sub/1800)^-2.8 pc where L_ UV is the total luminosity of SNe Ia in the NUV-optical band and T_ sub is the grain sublimation temperature (; ). The grain-grain collision mechanism destroys dust by the strong interaction between dust and gas in the interstellar medium (ISM). This process takes near 100 years for gas release from SNe Ia reach the dust in the ISM at 1 pc if we assume gas moves at 0.028 c. However, the values of R_ V and λ_ max are measured during the early-time observations (less than a few weeks after the explosion). Thus, the question of why small grains are predominant around SNe Ia remains mysterious.
     
    The mystery in the predominance of small grains around SNe Ia might be resolved when <cit.> introduced a new mechanism of grain disruption, so-called Radiative Torque Disruption (RATD), which is based on centrifugal force within rapidly spinning grains spun-up by radiative torques. This RATD mechanism can disrupt large grains into small fragments on a characteristic timescale of tens of days, which dramatically changes the grain size distribution and observable properties of dust near SNe Ia. Therefore, the first goal of this paper is to model the time-dependent extinction and polarization curves due to dust grains which are being disrupted by RATD. We will demonstrate that RATD can indeed reproduce the low values of R_ V and λ_ max as observed toward SNe Ia.
    
    Intrinsic colors and light curves of SNe Ia are critically crucial for achieving an accurate measurement of cosmological constant and better constraint of dark energy, which are the next frontiers of SNe Ia cosmology. Yet, the major systematic uncertainty arises from uncertainty in dust extinction toward SNe Ia (see, e.g., ). Moreover, intrinsic lightcurve is essential for understanding the trigger mechanism of SNe Ia. The second goal of this paper is to quantify the time-variation of the SNe Ia colors due to time-varying extinction caused by RATD. 
     
    The structure of this paper is as follows. In Section <ref>, we review the RATD mechanism and calculate the disruption size of dust grains as functions of time and cloud distance toward SNe Ia. In Sections <ref> and <ref>, we will model dust extinction and polarization curves induced by grains aligned with the magnetic fields using the grain size distribution determined by RATD and grain alignment function by RAT alignment theory. In Section <ref>, we will predict the observed light curve of SNe Ia, which vary over time due to RATD. Discussion and summary of our main findings are given in Section <ref> and <ref>, respectively.
    
    
    
    § ROTATIONAL DISRUPTION OF DUST GRAINS BY SNE IA
    
    
     §.§ The RATD mechanism
    
    
    When dust grains of irregular shapes are located in an intense radiation field, such as around SNe Ia, radiative torques induced by the interaction of anisotropic radiation with the irregular grain can spin the grain up to extremely fast rotation (; ; ). Such suprathermal rotation of grains produces a strong centrifugal force, which can exceed the tensile strength of grain material, so that grains are disrupted into small fragments. Note that the grain shape is assumed to be irregular, but we can approximate it as an equivalent spherical grain of the effective size a which has the same volume as the irregular one (see, e.g., ; ). This RATD mechanism was described in detail in <cit.>. Therefore, here we briefly describe the RATD mechanism for reference.
    
    The luminosity of SNe Ia varies over time, which can be approximately given by an analytical formula <cit.>:
    
        L_ SNIa(t)= L_0(t-t_0/t_p)^α_r[1+(t-t_ 0/t_p)^sα_d]^-2/s,
    
    where L_ 0 is the scaling parameter, and we take L_ 0=2 × 10^10 L_⊙. The first term describes the rising luminosity of SNe Ia () while the second term is the decrease luminosity function after the peak around 20 days. We take α_ r= 2, α_ d= 2.5, s= 1 , t_0= 0 days and t_ p= 23 days ().
    
    Since the luminosity of SNe Ia is time-dependent, the grain angular velocity can only be obtained by numerically solving the equation of motion (see ): 
    
        I dω/dt = Γ_ RAT-Iω/τ_ damp,
    
    where I is the grain inertia moment with I=8 πρ a^5/15, Γ_ RAT is radiative torques, and τ_ damp is the characteristic timescale of grain rotational damping. 
    
    Usually, Γ_ RAT is the radiative torque averaged over the radiation spectrum as given by: 
    Γ_ RAT = π a^2γ u_ rad(λ̅/2π) Q_Γ,
    
    where π a^2 is geometric cross-section, γ is the anisotropy degree of the radiation field (0≤γ≤ 1), λ̅ is the mean wavelength of the radiation spectrum, Q_Γ is the averaged RAT efficiency. Here u_ rad is the radiation energy density of the radiation field which is equal to:
    
        u_ rad =  L_ bole^ -τ/4 π c d^ 2,
    
    where τ the effective optical depth defined by e^-τ=∫ u_λ× e^-τ_λ dλ/u_ rad. Let U=u_ rad/u_ ISRF where u_ ISRF=8.64× 10^-13^-3 is the energy density of the average radiation field in the solar neighborhood (see ).
    
    
    In general, Q_Γ depends on the grain size and shape, the wavelength and the angle between the direction of the grain and incoming radiation (; ). Spectacularly, when the anisotropic radiation direction is parallel to the axis of maximum moment of the inertia of grain, <cit.>, <cit.> and <cit.> showed that Q_Γ could be approximated by a power law:
    Q_Γ={[     ≈ 2(λ̅/a)^-2.7  for  a ≲ a_ trans;              ∼ 0.4 for  a > a_ trans, ]}
    where a_ trans=λ̅/1.8.
    
    The damping of grain rotation can arise from collisions with gas atoms followed by evaporation and re-emission of infrared radiation. The total damping rate can be written as ():
    τ_ damp = τ_ gas/1+F_ IR,
    
    where τ_ gas is the rotational damping timescale due to collisions given by
    τ_ gas≃ 8.74× 10^4a_-5ρ̂(30 ^-3/n_ H) (100/T_ gas)^1/2 yr
    
    with a_-5 = a/(10^-5) and ρ̂=ρ/3^-3 with ρ the dust mass density, n_ H the gas density and the gas temperature T_ gas (see ; ): 
    
    Assuming that grains are in thermal equilibrium between heating by stellar radiation and cooling by IR re-emission, the IR damping coefficient can be estimated as (see ):
    
    F_ IR≃0.4U^2/3/a_-530^-3/n_ H(100/T_ gas)^1/2.
    
    
    A spinning grain of angular velocity ω is disrupted when the centrifugal stress S=ρ a^2ω^2/4 exceeds the maximum tensile strength of grain material S_ max. The critical velocity at which the disruption occurs is determined by S≡ S_ max, which yields
    ω_ disr = 2/a(S_ max/ρ)^1/2,
    
    where S_ max depends on the grain material, internal structure, and the grain size (see ). 
    
    The disruption time is roughly estimated by
    
    t_ disr   =   Iω_ disr/dJ/dt=Iω_ disr/Γ_ RAT
       ≃    38(γ U_6)^-1λ̅_0.5^1.7ρ̂^1/2S_max,7^1/2a_-5^-0.7  days
    for a_ disr<a ≲ a_ trans, and
    
    t_ disr≃    2.5(γ U_6)^-1λ̅_0.5^-1ρ̂^1/2S_max,7^1/2a_-5^2  days
    for a>a_ trans where U_6=U/10^6.
    
    Assuming the mass density ρ=3.5^-3 and 2.2^-3 for silicate and graphite grains, one can see that the disruption time of graphite grains is shorter than that of silicate grains due to its lower mass density. 
    
    
    
     §.§ Grain disruption sizes
    
    The grain disruption size by the RATD mechanism depends on the tensile strength of grain material S_ max, the strength of the radiation field which is determined by the cloud distance to SNe Ia, the spectrum of the radiation field, and the irradiation time. As in <cit.>, SNe Ia are considered a black-body radiation source and have a constant effective temperature of SNe Ia of 15000. This yields the mean wavelength of λ̅≈ 0.35 . We assume that the radiation field is unidirectional, i.e., γ =1.  
    
    We solve Equation (<ref>) numerically to obtain ω_ RAT(t) for a grid of grain sizes a and compare it with ω_ disr(a) (Eq. <ref>). The grain disruption size a_ disr and disruption time t_ disr are then determined (see  for details). As discussed in <cit.>, small grains acquire lower radiative torques than large grains such that they can be spun-up to lower rotation rates. In addition, Equation (<ref>) reveals that small grains have the higher critical angular velocity to be disrupted. As a result, large grains can be disrupted into smaller ones, but small grains may survive RATD.
    
    
    
    Figure <ref> (upper panel) shows the variation of grain disruption size with cloud distance at t = 15, 30, 45 and 60 days, assuming S_ max=10^7^-3 and the typical gas density n_ H=30^-3. For distant clouds (d>3.5 pc), grain disruption cannot occur because of insufficient radiation intensity, and we set a_ disr to the popular upper cutoff of the grain size distribution of a_ max=0.25 (). As the cloud distance decreases (i.e., radiation energy density increases as d^-2), the grain disruption size increases rapidly. Moreover, within 15 days, only grains within d∼ 1.5 pc can be disrupted, and after 60 days, even grains located at d∼ 3 pc can be disrupted.
    
    Figure <ref> (lower panel) shows the variation of the grain disruption size with time for the different cloud distances where the initial value is equal to 0.25. For a given distance, the disruption size starts to decrease rapidly with time from the original value after some disruption time t_ disr, and it achieves saturated values when the disruption ends. Dust grains closer to SNe Ia receive higher radiation flux and have shorter t_ disr. As a result, one sees the decrease of a_ disr begins earlier, but its saturation also ends earlier than grains at farther distances. For example, from Figure <ref> (lower panel), one can obtain t_ disr∼ 6, 10, 19 and 34 days for silicate grains at d=0.5, 1, 2 and 3 pc, respectively (see more details in Figure 3 in <cit.>. The decrease of a_ disr starts from t∼ t_ disr and achieves its saturation of a_ disr∼ 0.01-0.03. We note that for all considered cloud distances in Figure <ref>, the grain disruption ceases after about about 60 days because the SNe luminosity already fade away substantially after its peak.
    
    
    
    Figure <ref> (upper panel) presents the disruption size of silicate grains as a function of the cloud distance for different maximum tensile strengths. Grains with lower tensile strength can be disrupted out to a larger distance than those with higher S_ max. This can be seen from Equation (<ref>) that grains with higher S_ max have a higher critical angular velocity and require a higher radiation energy (i.e., smaller distance) to be disrupted. For example, at a given cloud distance of 1 pc, the disruption size a_ disr∼ 0.009 for S_ max=10^7^-3, it increases to a_ disr∼ 0.07 for S_ max= 10^9^-3, and there is no disruption for very strong material of S_ max=10^10^-3. Moreover, the active region D_ disr of RATD, i.e., disruption region, is smaller when grains have higher maximum tensile strength. For example, the disruption distance decreases from D_ disr=4 pc for S_ max=10^7^-3 to D_ disr=0.5 pc for S_ max=10^10^-3.
    
    Figure <ref> (lower panel) shows the disruption size of silicate as a function of time for a dust cloud at 1 pc for different S_ max. The disruption time of grains with higher S_ max is longer and the grain disruption stops earlier at a higher saturated grain size. For example, the disruption time is t_ disr∼ 10, 17 and 38 days for S_ max=10^7, 10^8 and 10^9^-3, respectively. The final disruption sizes are a_ disr∼ 0.01, 0.03 and 0.1 for these tensile strengths.
    
    We also calculated the grain disruption size for the different gas density n_ H∼ 10-10^4^-3 and found that the disruption size is almost the same for clouds within 4 pc from the source. This arises from the fact that the strong radiation field of SNe Ia induces F_ IR 1, IR damping dominates over gas damping, so that n_ H from τ_ gas and F_ IR are canceled out in Equation <ref>. As a result, the rotational damping rate τ_ damp only depends on the radiation strength.
    
    
    
    § EXTINCTION OF SNE LIGHT IN THE PRESENCE OF RATD EFFECT
    
    
     §.§ Grain size distribution
    
    For simplicity, we assume that dust grains follow a power law size distribution:
    
        dn^j/da = C_j n_ H a^α,
     
    where j denotes the grain composition of silicate and graphite, C_j is the constant, and α is the power slope. 
    
    For the standard ISM in our galaxy, <cit.>) derives the slope α=-3.5, and C_ sil=10^-25.11^2.5 for silicate grains and C_ gra=10^-25.14^2.5 for graphite grains. The size distribution has a lower cutoff a_ min=3.5 Å determined by thermal sublimation due to temperature fluctuations of very small grains (see e.g., ), and an upper cutoff of a_ max=0.25 (). 
    
    In the presence of RATD, largest grains are disrupted into smaller ones, resulting in the decrease of a_ max, and we can set a_ max=a_ disr for the diffuse ISM. Since the dust mass is conserved in the RATD picture, obviously, the grain size distribution dn/da must be modified. 
    
    Due to the lack of experimental study on the size distribution of resulting fragments from rotational disruption, we will assume that the new grain size distribution also follows a power law. One can imagine there are three possible ways to model the size distribution of the new size distribution. First, the power slope should vary, but the normalization constant C is assumed to be fixed (model 1). Second, the slope is assumed to be fixed, but the normalization constant C varies (model 2). Third, both C and the slope should vary (model 3). Model 1 corresponds to the case in which the original large grain is made up of small constituents that have various sizes. The model 2 corresponds to the case where an original large grain is disrupted into smaller fragments which follows the same size distribution as the original one. Model 3 would require a detailed understanding of the internal structure of dust grains which is uncertain and thus beyond the scope of this paper. So we consider here model 1 and will discuss its difference with model 2 in Appendix <ref>. 
    
    For a given grain disruption size a_ disr, the new value of the size distribution slope is determined by the conservation of the dust mass:
    ∫_a_ min^a_ max a^3 C a^-3.5 da =∫_a_ min^a_ disr a^3 C_ new a^α da,
    
    which yields
    
        a_ disr^4+α - a_ min^4+α/4+α=a_ max^0.5 - a_ min^0.5/0.5.
    
    
    The new slope α is obtained by solving numerically Equation (<ref>). Note that the unit of the constant C_ new must be changed accordingly to conform with the new slope α.
    
    
    
     §.§ Extinction curve
    
    
    The radiation intensity of SNe Ia is reduced due to the absorption and scattering (i.e., extinction) of dust and gas along the line of sight. The extinction efficiency of light by a dust grain is defined by:
    
    
        Q_ ext = C_ ext/S = C_ ext/π a^2,
    
    
    where C_ ext is the extinction cross-section. We use a mixed-dust model comprising silicate and graphite materials (; ) and take C_ ext calculated for oblate spheroidal grains of axial ratio a/b=2 from <cit.>.  
    
    The extinction of supernova light at wavelength λ in a unit of magnitude per atom is given by (see e.g., ):
    
    
        A(λ)/N_ H = ∑_j=sil,gra 1.086 ∫_a_ min^a_ max  C_ ext^j(a)(1/n_ Hdn^j/da)da,
    
    
    where N_ H=∫ n_ Hdz=n_ HL with L the path length is the column density, dn^j/da is the grain size distribution of dust component j, and a_ max=a_ disr.
    
    For a given a_ disr, the new grain size distribution function is obtained, and we can calculate the wavelength-dependence extinction by dust being modified by RATD using Equation (<ref>).
    
    
    
    Figure <ref> shows the extinction curves evaluated at different times for dust grains located at distance d= 1 pc from the source and the maximum tensile strength S_ max=10^7^-3 (upper panel) and S_ max=10^8^-3 (lower panel). In the upper panel, the extinction curve at t = 5 days (red dashed line) is the same as the extinction at t=1 days because t<t_ disr (see Eq. <ref>). For t> t_ disr∼ 10 days (see Figure <ref>, lower panel), the optical-NIR extinction decreases rapidly with time due to the removal of large grains by RATD. On the other hand, the UV extinction is increased due to the enhancement in the abundance of small grains by RATD. The extinction at λ > 7 is essentially unchanged because of λ≫ (2π a).
    
    The lower panel of Figure <ref> shows the extinction curve for grains with S_ max=10^8^-3. The variation in the extinction curve with time is similar to the upper panel, but it starts later because of larger disruption time with t_ disr∼ 15 days (see Figure <ref> (lower panel)). 
    
    
    
    Figure <ref> shows the extinction curve for the different cloud distances at t=45 days, assuming S_ max=10^7^-3 (upper panel) and S_ max=10^8^-3 (lower panel). The optical-NIR extinction decreases while the NUV extinction increases with decreasing the cloud distance. This is a direct consequence of the dependence of the grain disruption size on the cloud distance as shown in Figure <ref>.
    
    
    
    Figure <ref> shows the time-dependence of the ratio A(λ,t)/A(λ,0) for the different photometric bands (FUV to R bands) and different cloud distances. Here we choose λ=0.15 for the far-UV band (FUV), λ=0.25 for the mid-UV band (MUV) and λ=0.3 for the near-UV band (NUV). As shown, the ratio A(λ,t)/A(λ,0) is constant during the initial stage of t<t_ disr before grain disruption, and it starts to rapidly change when RATD begins at t_ disr. For example, optical-NIR extinction (e.g., R-band, black line) decreases rapidly with time, but the UV extinction (FUV-U bands) increases first and then decreases with time. The reason is that the optical extinction is produced by large grains which have short disruption time by RATD (see ), and the production of small grains by RATD results in higher UV extinction. We also see that the amplitude of the extinction variation is smaller for more distant clouds.
    
    
    
     §.§ Time variation of color excess E(B-V) and R_ V
    
    From A(λ,t) we can calculate the color excess E( B-V)=A_ B-A_ V and the total-to-selective extinction ratio R_ V=A_ V/E(B-V) to study how these quantities vary with time due to RATD.
    
    
    
    Figure <ref> (upper panel) shows the variation of E(B-V,t)/E(B-V,0) with time for the different cloud distances. For a given cloud distance, the color excess remains constant until grain disruption begins at t∼ t_ disr. Subsequently, the ratio increases rapidly and then decreases to a saturated level when RATD ceases. For example, at distance d=2 pc, the color excess starts to rise at t∼ 20 days and declines again to the saturated value at t∼ 40 days.
    
    Figure <ref> (lower panel) shows the variation of R_ V with time. The value R_ V starts to rapidly decrease from the initial value of R_ V=3.1, corresponding with the average R_ V in the diffuse medium of the Milky Way, to R_V∼ 1-1.5 after less than 40 days. The moment where R_V starts to decline coincides with that of E(B-V), which is determined by the grain disruption time t_ disr (see, e.g., Figure <ref>). We note that although E(B-V) has some fluctuations over time, R_ V tends to decrease smoothly because A_V always decreases with time due to the disruption of large grains by RATD (see Figure <ref>). The decrease of R_ V starts earlier for grains closer to the source due to shorter t_ disr and also achieve a smaller terminal value. For example, a dust cloud located at 0.5 pc will have R_ V to decreases after 5 days and gives the minimum value of 0.5 after 15 days, while at 2 pc, it takes 20 days to drop R_ V down and the final value ∼ 1.4.
     
    
    
    Figure <ref> (upper panel) shows the variation of R_ V evaluated at t=45 days with cloud distance for the different tensile strength S_ max. Dust grains with a lower S_ max can be disrupted out to a larger distance (see Figure <ref>), so that R_ V starts to decrease from a larger distance. On the other hand, grains with very high S_ max≥ 10^9^-3 can be disrupted within a small distance of d∼ 1 pc, thus one only can observe the smaller value of R_ V if the cloud is close to the source.
    
    Figure <ref> (lower panel) shows the variation of R_ V with time for a dust cloud at distance d=1 pc. Weaker grains have R_ V decreasing sooner and achieve a lower terminal value than stronger ones due to the dependence of RATD on the material strength (see Figure <ref>). For example, dust in the cloud at 1.5 pc after 45 days will give R_ V=1.1 for S_ max=10^7^-3, 1.4 for S_ max=10^8^-3 and R_ V=3.1 with all grains of S_ max≥ 10^9^-3. 
    
    
    
    § POLARIZATION OF SNE LIGHT IN THE PRESENCE OF RAT ALIGNMENT AND RATD
    
    
     §.§ Grain alignment by RATs
    
    
    An anisotropic radiation field can align dust grains via the RAT mechanism (see  and  for reviews). In the unified theory of RAT alignment, grains are first spun-up to suprathermal rotation and then driven to be aligned with the ambient magnetic fields by superparamagnetic relaxation within grains having iron inclusions <cit.>. Therefore, grains are only efficiently aligned when they can rotate suprathermally. One can adopt the suprathermal condition as follows ():
    
        ω_ RAT(t) ≳ 3 ω_T,
    
    where ω_ T is the thermal angular velocity of dust grains at gas temperature T_ gas as given by
    
    
        ω_ T = √(2 k T_ gas/I)≃ 2.3× 10^5ρ̂^-1/2T_2^1/2a_-5^-5/2 rad  s^-1,
    
    where T_2=T_ gas/100. 
    
    
    
    Let a_ align be the critical size of aligned grains. Then, a_ align can be determined by using the condition ω_ RAT(t)=3ω_T.
    
    The grain alignment timescale is given the timescale required for RATs to spin-up grains to the suprathermal rotation:
    τ_ align≡3Iω_ T/Γ_ RAT≃ 0.6 (d_ pc^2/L_8e^-τ)λ̅_0.5^1.7a_-5^-2.2T_2^1/2  days,   
    where Γ_ RAT is given by Equation (<ref>) with a≲λ̅/1.8 (see also ).
    
    In Figure <ref>, we show the variation of alignment size with time for the different cloud distances. During the first stage, the alignment size is constant, which is equal to the typical value a_ align∼ 0.055 induced by the average ISRF. Due to intense SNe radiation, the alignment size starts to decrease after some alignment time (t_ align) from the original value to smaller values. One can see that the alignment time by SNe Ia is t_ align∼ 3-7 days for d=0.5-3 pc. Grains in a more distant cloud have larger alignment sizes, such as small grains at 0.5 pc can be aligned up to 0.005  after 10 days but it is still 0.05 with the cloud father than 1 pc (see also Eq. <ref>).
     
    
    
     §.§ Modeling SNe Polarization
    
    
    We assume that only silicate grains can be aligned with the magnetic field, whereas graphite grains are not efficiently aligned (; see  for explanation).[Although carbonaceous grains are expected to be aligned via k-RAT mechanism (see ), their degree of alignment is not yet quantified, in contrast for silicate grains which have alignment degree quantified in <cit.> using numerical simulations.] For the magnetic field in the plane of the sky, the degree of starlight polarization per H atom due to aligned grains in the unit of % is given by ():
    
        P(λ)/N_ H = 100 ∫_a_ align^a_ max1/2 C_ pol(a) f(a) (1/n_ Hdn/da)da,
    
    where C_ pol = Q_ polπ a^2 is the polarization cross-section with Q_ pol the polarization efficiency, and f(a) is the alignment function describing the grain-size dependence of the grain alignment degree, and a_ max≡ a_ disr. For our modeling, we consider the oblate grain shape and take data of Q_ pol computed by <cit.>. 
    
    The alignment function can be modeled by the following function:
    
        f(a) = 1-exp[-(0.5 a/a_ align)^3],
    
    which yields the perfect alignment f(a)=1 for large grains of a≫ a_ align and adequately approximates the numerical results from <cit.> as well as results from inverse modeling of starlight polarization (; ).
    
    
    
     §.§ Polarization curves
    
    
    Figure <ref> (upper panel) shows the polarization curve given by the cloud at 1 pc with time for  S_ max=10^7^-3 (upper panel) and S_ max=10^8^-3 (lower panel) as a result of RAT alignment and RATD. At t≲1 days, dust grains are aligned by the diffuse interstellar radiation, so the maximum polarization at λ_ max is around 0.55. After that, SNe radiation dominates and makes smaller grains to be aligned. As a result, the UV polarization is increased rapidly, and the peak wavelength is shifted toward the blue. The degree of polarization at long wavelengths (λ > 0.5) is slightly increased. After t∼ 10 days, grain disruption by RATD begins (see Figure <ref>), reducing the abundance of large grains. Therefore, the degree of optical-NIR polarization decreases substantially, which results in a narrower the polarization profile compared to the original polarization curve. 
    
    Figure <ref> (lower panel) shows the similar results, but with S_ max=10^8^ -3. The same trend as the upper panel is observed, but the significant decrease of optical-NIR polarization is only seen at a later time of t ≳ 14 days because grains of stronger material require more time to disrupt (see Figure <ref>). One note that the polarization curves at 10 days and 14 days are very similar, revealing that grain alignment is saturated. 
     
    
    
    Figure <ref> shows the polarization curve calculated at t=20 days for the different cloud distances, assuming S_ max=10^7^-3 (upper panel) and S_ max=10^8^-3 (lower panel). The optical-NIR polarization decreases significantly with decreasing the cloud distance. This arises from the fact that grains closer to the source experience stronger disruption due to the dependence of the radiation flux as 1/d^2 (see Figure <ref>). At the same time, the peak wavelength λ_ max is smaller for grains at smaller distances due to more efficient alignment of grains by RATs. 
    
    Above we have assumed that only silicate grains are aligned with the magnetic field.  Nevertheless, the peak wavelength of polarization would be not much different when carbonaceous grains are assumed to be aligned because it mostly depends on the alignment function.
    
    
    
     §.§ Time-variation of dust polarization and peak wavelength
    
    
    Figure <ref> shows the temporal variation of P(λ,t)/P(λ,0) from FUV to R bands for the different cloud distances. During the initial stage, the ratio P(λ,t)/P(λ,0) is constant, however, this stage is rather short, between 1-5 days corresponding to the alignment timescale t_ align (see Figure <ref>; also ). After that, the polarization degree increases gradually and this rising period continues until t∼ 5-30 days until the grain disruption begins (i.e., at t=t_ disr) for d=0.5-3 pc. The polarization degree then declines rapidly when grain disruption by RATD begins to occur. The polarization degree achieves a saturated level when RATD ceases, which occurs after t∼ 20 days for d=0.5 pc, 40 days for d=1 pc, respectively. In summary, due to RAT alignment and RATD, the polarization degree increases from t_ align to t_ disr, and it decreases rapidly at t> t_ disr.
    
    
    
    Figure <ref> (upper panel) presents the variation of the peak wavelength with time for dust clouds at different locations and S_ max=10^7^-3. Initially, λ_ max∼ 0.55 is produced by grains aligned by the interstellar radiation field (). In the presence of strong SNe radiation, smaller grains can be aligned, resulting in a rapid decrease of λ_ max. Dust grains at farther distances receive lower radiation energy density such that λ_ max start to decrease later and stops at higher λ_ max. For example, dust at 0.5 pc give λ_ max≤ 0.5 after ∼ 1 day and stop at 0.1, while dust at 3 pc needs ∼ 8 days to begin and give the minimum λ_ max∼ 0.15.
    
    Figure <ref> (lower panel) shows the comparison of λ_ max during 60 days with (solid line) and without (dashed line) dust disruption. As shown, λ_ max in the two cases are similar in the early times and become different when grain disruption begins. Their difference is larger at a shorter distance because grain disruption is stronger.
    
    The dashed line shows that alignment grain size saturates earlier than disruption grain size, such that λ_ max remains unchanged just after 10 day (for grain at 0.5 pc) and 20 day (for grain at 2 pc). The slower disruption rate at the distant cloud will makes λ_ max to be constant longer.
    
    
    
     §.§ Effect of size-dependence tensile strength
    
    Till now, we calculated the grain disruption size, R_ V and λ_ max by assuming that the maximum tensile strength is constant for all grain sizes. A more realistic situation is that small grains would have compact structures and thus have a higher S_ max. To see how the size-dependent tensile strength affects our results, we now assume S_ max=10^ 7^-3 for grains larger than ∼ 0.05 and S_ max=10^9^-3 for grains smaller than 0.05.
    
    Figure <ref> (top panel) shows the grain disruption size for constant S_ max (dashed lines) and changing S_ max (solid lines) for the different cloud distances. In the latter case, the grain disruption size is decreased to a final value of 0.05 if the radiation energy density is not strong enough to destroy further. 
    
    Same as the top panel, but Figure <ref> (middle panel) shows R_ V versus time for constant (dashed lines) and changing S_ max (solid lines). The R_ V values in two cases decrease rapidly with time, but the latter case has higher terminal values of R_V∼ 1.45 due to higher disruption sizes (see top panel).
    
    Figure <ref> (bottom panel) shows the time dependence of λ_ max during 60 days for constant (dashed lines) and changing S_ max (solid lines). For the latter case, the terminal peak wavelength is slightly larger than the former case, e.g.,  λ_ max=0.13 at 1 pc and λ_ max=0.15 at 2 pc. From the middle and top panels, it appears that the effect of changing S_ max has a more important effect on R_V than on λ_ max.
    
    
    
    § EFFECT OF RATD ON THE SNE IA LIGHT CURVE
    
    In Section <ref>, we show the disruption of large grains into smaller ones by RATD increases the dust extinction in the NUV band and decreases in the optical and NIR band with time. As a result, the observed radiation spectrum of SNe Ia would be different from the case where dust properties are constant, i.e., in the absence of RATD. In this section, we use the new extinction value calculated from Section <ref> to predict how the SNe Ia light curve changes with time in the presence of RATD.
    
    Let us first derive a simplified intrinsic light curve of the SNe Ia. The explosion of SNe Ia releases a huge amount of energy and eject all material into space in the short time. Its radius will increase dramatically, and the temperature will drop during the free expanding phase. For our simple model, we assume the total energy E∼ 10^51  is the constant during this phase, and the ejecta of mass M_ ej=1.4M_⊙ will move with the constant velocity: v = √(2E/M_ ej)∼ 10^4E_ 51^1/2(M_⊙/M_ ej)^1/2 km^-1 with E_51=E/10^51. The total bolometric luminosity of SNe Ia is given by:
    
        L_ bol =  4 π R^2σ T_ SN^4,
    
    where R=vt is the radius of SNe Ia after t time and T_ SN is its effective temperature. Using L_ bol in Equation (<ref>), the effective temperature of SNe Ia decreases with time as
    
        T_ SN≃ 10^4(L_ bol/10^9L_⊙)^1/4(10^4 km^-1/v)^1/2(t/1  days)^1/2.
    
     
    The intensity of SNe radiation after propagating through a dusty cloud at wavelength λ is governed by
    
        I_λ(t) = I_λ^0e^-τ(λ,t),
    
    where 
    
        I_λ^0=2 hc^2/λ^51/exp(hc/λ kT_ SN)-1,
     
    is the intrinsic specific radiation intensity, and the optical depth τ(λ,t)=A(λ,t)/1.086. Here, the infrared thermal emission from dust grains is disregarded because we are interested only in the UV-NIR spectrum of SNe Ia. With A(λ,t) obtained in Section <ref>, one can calculate the observed radiation intensity via Equation (<ref>).[Here, to illustrate the effect of time-varying dust extinction, we disregard the rising phase of the SNe light curve. With A(λ,t) available from Section <ref>, one can predict the observed intensity for an accurate intrinsic intensity without difficulty.]
    
    Figure <ref> shows the UV-NIR spectrum of a SN Ia observed at the different times (dashed and dot-dashed lines) compared with the intrinsic spectrum (black solid line). We assume the visual extinction estimated for SN 1986G of A_ V=2.03 mag to derive the total gas column density N_ H = 3.14 × 10^21^-2. The black dashed line represents the observed SNe spectrum with constant dust extinction of A_ V(t=0)=2.03 (no disruption), and the color dot-dashed line shows the results when RATD is taken into account. Without RATD, the observed SNe spectrum appears constant with time but will vary in the presence of RATD. Indeed, due to RATD, large grains are disrupted in smaller grains, increasing the extinction at the NUV band, such that SNe Ia become redder. In contrast, the decrease of the extinction in the optical and NIR band due to grain disruption makes SNe Ia brighter than with time. The radiation intensity at long wavelengths (λ∼ 3-5) is not affected by dust disruption because of optically thin regime (see Section <ref>).
    
    
    
    To see more detail about time variation of SNe colors, in Figure <ref>, we show the light intensity at FUV (0.15), MUV (0.25), NUV (0.3), U, B, V and R bands for the different dust cloud locations, assuming S_ max=10^7^-3. In the absence of RATD, the radiation intensity of SNe Ia at the different bands only decreases with time due to the decrease of T_ SN. However, in the presence of RATD, the radiation intensity from optical-NIR wavelengths exhibits an abrupt increase (i.e., SNe become re-brighter), whereas it has a drop at FUV-NUV bands (solid lines) due to the effect of grain disruption. The "rebrighter" time in optical-NIR is longer for more distant clouds. For instance, the re-brighter time t_rebrighter∼ t_ disr∼ 10 days for d=0.5 pc and t_re-brighter∼ 30 days for d=3 pc. 
    
     
    Figure <ref> presents the same as Figure <ref> but with grains with S_ max= 10^8^-3. One can see that the large disruption grain size of the stronger grains gives a smaller effect on the dust extinction in UV-NIR band than the light grain. As a result, the 'rebrighter' and 'redimmer' feature due to the reduction of the extinction in the visible band and increase in the FUV-NUV band smaller and nearly no effect if the cloud is farther than 2 pc.
    
    Note that to highlight the effect of RATD on the observed SNe Ia light curve, we have modeled the intrinsic SNe Ia light curve as a fireball with a constant expansion velocity. However, the radioactivity of synthesized Ni^56 and Co^56 () can provide more radiative energy, such that the peak luminosity of SNe Ia and the effective temperature would be different from our simplified model. Nevertheless, we expect the effect of RATD on the SNe light curve is similar.
    
    
    
    § DISCUSSION
    
    
     §.§ Comparison of R_ V and λ_ max from our models with SNe Ia data
    
    
    Using RATD theory, we have computed the extinction curves A(λ,t) by dust grains which have the grain size distribution being modified by RATD at the different times and dust cloud distances in Section <ref>. Our obtained results show that, due to RATD, R_ V rapidly decreases with time, from an original value of R_ V=3.1 to small values of R_ V∼ 1.5 after a timescale t_ disr∼ 50-60 days if the dust cloud is located within d∼ 4 pc from the supernova (see Figure <ref>). The final value R_V achieved when RATD ceases is smaller for the dust cloud closer to the supernova, but it increases with the tensile strength of grains (see Figures <ref> ). Comparing our results with the observational data of SNe Ia (see Table <ref>), one can see that the RATD can successfully reproduce the unusual low values of R_ V observed toward many SNe Ia (see Table <ref>; ; ).  
    
    We also modeled the polarization of SNe Ia light by grains aligned with the magnetic fields using RAT mechanism in Section <ref>. Our results show that the peak wavelength λ_ max decreases from the standard ISM value ∼ 0.55 to λ_ max<0.2 due to RAT alignment after a timescale of t_ align∼ 5-20 days for clouds at distance within 4 pc (see Figure <ref>, upper panel). These results can reproduce the low values of λ_ max observed toward SNe Ia (see Table <ref>; ; ). We note that the characteristic timescale for achieving terminal λ_ max is much shorter than that for terminal R_V because t_ align<t_ disr.
    
    From lower panel of Figure <ref>, one can see that the main reason for the decrease of λ_ max from 0.5 to 0.2 is due to the alignment of small grains by RAT mechanism as suggested by <cit.>. Therefore, to obtain a higher λ_ max, one can increase the critical limit for suprathermal rotation, i.e., T_ gas.
    
    
    
    Figure <ref> shows the peak wavelength of the dust polarization for the different gas temperatures and the dust cloud at 3 pc from the source. Here the grain disruption is not considered in order to focus on the effect of grain alignment. The peak wavelength increases with increasing the gas temperatures because higher temperatures result in larger ω_T such that stronger radiation energy is required spin-up grains to suprathermal rotation and align grains efficiently. For example, a dust cloud at 3 pc will give λ_ max∼ 0.3 if T_ gas = 10^5, two times larger than the case T_ gas =10^3. 
    
    
    
     §.§ Test RATD and RAT alignment mechanisms with time-variation of SNe Ia extinction and polarization
    
    
    Our modeling results of extinction by RATD reveals that the extinction and R_V remains constant until the grain disruption occurs at t=t_ disr spanning 5-30 days depending on the distance (right panel of Figure <ref>) and grain tensile strength S_ max (Figure <ref>). Beyond t_ disr, the optical-NIR extinction and R_V decrease rapidly and achieve saturated levels after 50-60 days for the cloud distance d<4 pc. Therefore, by observing SNe Ia at early times and infer time-dependence R_V, one can test the RATD mechanism.
    
    On the other hand, our modeling of dust polarization due to grains aligned by the RAT mechanism show that the dust polarization starts to increase after a timescale corresponding to the alignment timescale, t_ align, which spans between 3-10 days (see Figure <ref> and <ref>). When the RATD begins at t∼ t_ disr> t_ align, the polarization starts to decrease rapidly over time, and its time-variation is similar to that of the extinction. Therefore, the best strategy to test RAT alignment mechanism is to observe the SNe Ia polarization at earliest times t<5-15 days if the dust cloud is located within 4 pc from the source. This is important for interpreting the photometric and polarimetric data of SNe Ia with high dust reddening.
    
    
    
     §.§ Effect of RATD on SNe Ia colors and light curves
    
    
    Dust extinction is considered the most important systematic uncertainty for SNe Ia cosmology (). Understanding better the intrinsic light curves will provide the better study the physical properties and the evolution of SNe Ia () and allows a precise measurement of cosmological constant and more precise constraint on dark energy. 
    
    Usually, to infer the intrinsic light curve of SNe Ia, one adopts a typical model of constant dust extinction and fits the model to observational data. However, here we show that the properties of dust within 4 pc from SNe Ia are modified by supernova radiation via RATD, thus, and their extinction curves are fundamentally different from the typical dust model if there exists some dust clouds within 4 pc from the source. Therefore, to achieve an high accuracy of the intrinsic SNe Ia light curve, we suggest to conduct observations at the earliest times as possible, ideally within the first ten days after the first light when grain disruption has not occurred yet.
    
    The situation becomes more complicated for deriving intrinsic SNe Ia colors because the intrinsic colors of SNe Ia are expected to vary with time (see ). Moreover, as shown in Figure <ref>, the color excess E(B-V) by dust extinction first increases with time and then it declines slowly. This trend may already be seen toward many SNe Ia (see ). 
    
    
    
     §.§ Constraining the distance and properties of dusty clouds toward SNe Ia with observations
    
    Understanding the distance and physical properties of dust clouds to SNe Ia is essential for inferring the intrinsic SN light curve as shown previously (; ) as well as in this paper.
    
    In Section <ref>, we show that SNe Ia become re-brighter at optical-NIR bands accompanied by a sudden dimmer at UV bands when grain disruption begins. The rebrighter time is essentially the same as the disruption time t_ disr which is a function of the cloud distance to the source (). Therefore, one can constrain the cloud distance by observing the SNe Ia colors and identify the re-brighter time. On the other hand, the polarization at optical-NIR bands decreases substantially due to RATD. As a result, a simultaneous decrease in the polarization together with re-brighter would be a smoking gun for RATD and for inferring the cloud distance.
    
    We note that the cloud distance of some SNe Ia were constrained in <cit.> using the time-variation of color excess E(B-V) caused by scattering of photons by dust. Yet, their analysis assumed a standard grain size distribution from the Milky way which cannot reproduce the unusual low values of R_V and λ_ max. 
    
    Physical and chemical properties of the dusty cloud near SNe Ia are also poorly known. Usually, one derives the gas column density of the dust cloud using the standard relationship N_ H/A_V≈ 5.8× 10^21^-2 mag^-1/R_V (). Yet, the optical extinction decreases with time from the original value if the cloud is located within 4 pc from the source (see Figure <ref>). Therefore, the estimate of N_ H using the observed A_V would yield a lower value than the actual column density.
    
    
    
     §.§ On the origin of anomalous K-λ_ max and K-R_V relationships of SNe Ia
    
    
    The wavelength-dependence polarization of distant stars in the Milky Way is usually fitted by the Serkowski law ():
    
        P_SerK(λ) = P_ maxexp[-K ln^2(λ_ max/λ)],
    
    where P_ max is the maximum degree of polarization, λ_ max is the wavelength at the maximum polarization (hereafter peak wavelength), and K is a parameter given by  (; ):
    
        K = 0.01 ± 0.05 + (1.66 ± 0.09)λ_ max.
    
    
    Nevertheless, <cit.> and <cit.> showed that the observational data of SNe Ia do not follow the standard relationship (Eq. <ref>). Specifically, the K value for a given small λ_ max is much larger than predicted Equation (<ref>). This puzzle once again reveals the abnormal properties of dust in the local environments of SNe Ia.
    
    Our modeling results in Section <ref> reveal that the polarization curves tend to become narrower, i.e., the parameter K tends to increase, whereas the value of λ_ max decreases in the presence of grain disruption by RATD (see Figures <ref>, <ref> and <ref>). To see whether the RATD mechanism can reproduce the observed anomalous relationship of K-λ_ max, we fit the Serkowski law (Equation <ref>) to the polarization curve calculated by Equation (<ref>) to obtain K and λ_ max (see Appendix <ref> for more details) and compare with the observational data. 
    
    
    
    
    In Figure <ref> (upper panel), we show the peak wavelength λ_ max and the parameter K for several values of the grain disruption size from a_ disr=0.2 to 0.05. For each a_ disr, the alignment grain size is varied from a_ align=0.05 to 0.002 to account for the effect of enhanced alignment by SNe light (see Appendix <ref> for more details). Observational data from SNe Ia and some stars in our galaxy  (see Table <ref> and <ref>) are shown for comparison. We see that, for a given a_ disr, K decrease rapidly with decreasing λ_ max due to the decrease of a_ align. Moreover, for a given a_ align, λ_ max tends to decrease with decreasing a_ disr. In particular, we can see that the high K values of SNe Ia can be reproduced by our models with RATD at the different a_ disr. For example, SN 2006X can be explained with the maximum grain size a_ disr∼  0.2 and alignment grain size a_ align∼ 0.03, or SN 2010ev can be explained with a_ disr= 0.2 and a_ align≈ 0.04. 
    
    Lastly, our results in Section <ref> and <ref> show that under strong radiation field, both extinction curve and polarization curves are modified from the standard curves in the ISM. This suggests that there is some correlation between R_V and K. In Figure <ref> (lower panel) we show the values of R_V and K calculated during 100 days with the cloud at 1 pc for several maximum tensile strengths and compare with observation data (see Table <ref> and <ref>). The increase of K with decreasing R_V as a result of RAT alignment and RATD appears to be consistent with the observational data. 
    
    
    
    § CONCLUSIONS
    
    In this paper, we have studied the effects of rotational disruption and alignment of dust grains by radiative torques on the extinction, polarization, and light curves of SNe Ia. The main findings of our study are summarized as follows:
    
    
      1. Bases on the RATD mechanism, grains can be disrupted when radiative torques spin up it up to critical speed during 200 days after the supernova explosion. The grain disruption size decreases with time at the fixed distance and increases with increasing the cloud distance to the source. The higher maximum tensile strength makes grain is more difficult to disrupt and the gas density is independent to the dust disruption size in the very strong radiation field as SNe Ia.
    
    
      2. Using the grain disruption size from RATD, we model the extinction of SNe light by dust absorption. We find that the conversion of large grains into smaller ones by RATD results in a rapid decrease in the optical-NIR extinction but increase in the UV extinction. This results in the rapid decrease in the the total-to-selective visual extinction ratio R_V over a timescale of t_ disr∼ 50 days. This can successfully explain the unusually low values observed toward SNe Ia.
    
    
      3. Using RAT theory, we study the alignment of grains by SNe radiation. We find that the intense SNe radiation can align small grains, and the alignment size decreases with time. We model the polarization of SNe light by aligned grains. We find that the polarization degree increases first due to enhanced alignment by SNe radiation and then drops rapidly when grain disruption begins. We find that the peak wavelength decreases rapidly from the original standard value to very small values of λ_ max<0.4 within less than t_ align< 10 days.
    
    
      4. We find that due to RATD and RAT alignment, the K-λ_ max cannot be described by the standard relationship in the Galaxy, but it can qualitatively explain the K-λ_ max reported for SNe Ia. 
     
    
      5. We suggest a possible way to constrain the cloud distance by performing extreme early time observations. The time of the abrupt decrease in optical-NIR extinction and polarization would put a strong constraint on the cloud distance. An increase in the polarization degree and decrease in the peak wavelength is useful to test the RAT mechanism.
    
    
      6. We predict an abrupt increase in the SNe brightness at optical-NIR bands but a decrease in UV extinction due to RATD. This can be tested with photometric and polarimetric observations at early times.
    
    
      7. Our results suggest that to achieve a precise measurement of cosmological distances as well as precise constraint on dark energy, one needs to account for the time-variation of dust extinction if there exist dust clouds within several parsecs from the supernova.
    
    
    T.H. acknowledges the support from the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (2017R1D1A1B03035359).
    
    
    
    
    § GRAIN SIZE DISTRIBUTION MODIFIED BY RATD
    
    In Section <ref>, we mentioned that there are three possible ways to calculate the grain size redistribution function dn/da. We have shown results from model 1 where the slope is changed but the normalization constant C is constant. We now study the results for model 2 where the slope α is kept constant but the constant C changes accordingly.
    
    With the case keeping an order is α=-3.5 (model 2), the new constant with the new maximum size is:
    
        C_ new = C (a_ max^0.5 - a_ min^0.5/a_ disr^0.5 - a_ min^0.5).
    
    
    Figure <ref> shows the value of α (model 1) and C (model 2) for silicate grains with time if the cloud stays at 1 pc for the different values of S_ max. As shown, the slope increases (i.e., becomes steeper) from the original value of α=-3.5 to α∼ -3.62 when RATD begins. This implies that RATD transports dust mass from large sizes to smaller sizes. The right panel shows the value of the constant normalization C which increases with time due to RATD and reaches the terminal value when RATD ceases. In general, the constant varies from 10^-25.11^2.5 to 4× 10^-25^2.5 with the cloud in 1 pc.
    
    
    
    Figure <ref> (left panel) shows the extinction curve after 45 days with the cloud at 0.5 pc and 1 pc. One can see that model 1 and 2 give similar extinction, except at optical-NIR wavelengths.
    
    Figure <ref> (right panel) shows the variation of R_ V with time for the two models. Model 2 gives slightly lower extinction values than model 1 after grain disruption.
    
    
    
    § OBSERVATIONAL DATA OF SNE IA AND STARS
    
    
    Table <ref> shows the physical parameters for 13 SNe Ia available in the literature. 
     
    
    Table <ref> shows the physical parameters for 15 normal stars with low R V but normal peak wavelength taken from <cit.>.
    
    
    
    § RELATIONSHIP BETWEEN K-Λ_ MAX AND R_V-Λ_ MAX
    
    
     §.§ Fitting polarization curves with Serkowski law
    
    
    We fit the theoretical polarization curves with the Serkowski law (Eq. <ref>) to derive the best-fit parameters K and λ_ max, which are summarized in Table <ref>. 
    Figure <ref> shows the best fit Serkowski law for the theoretical polarization curves in Figures <ref> and <ref> for the different times (left panel) and different cloud distances (right panel). One see that due to grain disruption and grain alignment by RATs begins, the K values increases significantly accompanied with the decrease of λ_ max. In other word, there is a simultaneous variation in K and λ_ max with time but they follow the opposite trend.
    
    
    
     §.§ K-λ_ max and R_V-λ_ max
    
    To understand in more detail how K and λ_ max change with grain disruption and alignment by RATs, we model the dust polarization by varying a_ disr from 0.2 to 0.03 and a_ align from 0.05 to 0.002 and derive the best-fit K and λ_ max parameters as in the previous section.
    
    In Figure <ref> (left panel) we show the resulting K-λ_ max values. Each point represents a couple (K, λ_ max) predicted by a given value of (a_ disr,a_ align). Several values of K at a given λ_ max represents the different values of P_ max when a_ align decreases, and the dashed line is the best fit to these results. 
    
    
    
    In Figure <ref> (right panel), we plot R_ V against λ_ max evaluated at t= 60 days for S_ max=10^7^-3 with different gas temperatures. The black solid line shows the tentative relationship reported by <cit.>. As shown, both data of SNe Ia and our modeling results do not exhibit correlation between R_V and λ_ max, in agreement with the finding in <cit.>.
