Three-dimensional Explosion Geometry of Stripped-envelope Core-collapse Supernovae. II. Modeling of Polarization

We perform 3D radiation transfer simulations to study the properties of the line polarization. For this purpose, we use a simple, 3D Monte Carlo radiation transfer code. The code takes into account the electron scattering and the line scattering. We treat only a single line at a single epoch rather than modeling the time evolution of full spectra since we aim to obtain the connection between explosion geometry and the properties of line polarization (see Hole et al. 2010 for a similar strategy). The code computes the polarization spectrum of the line for an arbitrary 3D distribution of the line optical depth. More details of the code are given in the Appendix.

We use 100 × 100 × 100 linearly distributed Cartesian meshes. The velocity is used as a spatial coordinate thanks to the homologous expansion (r ∝ v). The maximum velocity is 25,000 km s⁻¹, and thus the resolution is 500 km s⁻¹, giving the resolution of λ/Δλ = c/Δv = 600, which is comparable to a typical spectral resolution of low-resolution spectropolarimetric observations.

We start simulations by generating unpolarized photon packets from the spherical inner boundary ($v={v}_{\mathrm{in}}$). The electron-scattering optical depth from the inner boundary to infinity is set to ${\tau }_{\mathrm{in}}$. In this paper, we adopt ${\tau }_{\mathrm{in}}=3$ as in Kasen et al. (2003) and Hole et al. (2010). Note that the photosphere ($v={v}_{\mathrm{ph}}$) is defined as the position where the electron-scattering optical depth is unity, and thus the inner boundary of the computation is located inside the photosphere.

The photon packets are then tracked by taking into account the electron scattering and the line scattering. For the electron scattering, we use a power-law electron density profile, ${n}_{e}\propto {r}^{-n}$. The electron density is assumed to be spherically symmetric. For the power-law index, we use n = 7, which describes the line-forming region of hydrodynamic models of stripped-envelope SNe (Iwamoto et al. 2000; Mazzali et al. 2000). Although the very outermost ejecta have a steeper slope (n ∼ 10, Matzner & McKee 1999), we use a single power-law profile since the outermost ejecta do not have a strong contribution to absorption lines. Note that the polarization pattern is not affected by the slope if the slope is steep enough (n ≳ 5, Kasen et al. 2003). We assume the photospheric velocity (${v}_{\mathrm{ph}}$) and the time after the explosion (t), which give the photospheric radius ${r}_{\mathrm{ph}}={v}_{\mathrm{ph}}t$. Then, with the condition that the electron-scattering optical depth is unity at the photosphere, the normalization of the electron density is obtained. We adopt ${v}_{\mathrm{ph}}={\rm{8000}}$ km s⁻¹ and t = 20 days as typical values for stripped-envelope SNe around the maximum light.

For the line scattering, we use the Sobolev approximation (Castor 1970), which is a sound approximation in the SN ejecta with a large velocity gradient. For the Sobolev line optical depth, we assume a power-law radial profile above the photosphere, ${\tau }_{\mathrm{line}}={\tau }_{\mathrm{ph}}{(r/{r}_{\mathrm{ph}})}^{-n}$. Here ${\tau }_{\mathrm{ph}}$ is the Sobolev optical depth at the photosphere. For simplicity, we use the same power-law index with the electron density (n = 7).

In addition to the spherical component of the line optical depth, we assume an enhancement by a factor of ${f}_{\tau }$ in some regions, such as a torus or clumps. Note that this is different from the treatment by Hole et al. (2010), where the line opacity is set to zero outside the clumps. Such a treatment seems more suitable for Type Ia SNe (as they applied it to), where a strong line is formed dominantly in a certain layer; for example, Si lines are produced mostly in the Si-rich layers. On the other hand, for strong lines in stripped-envelope core-collapse SNe, such as those of Ca and Fe, both pre-SN and newly synthesized elements contribute to the absorption. Therefore, we assume an enhancement in addition to the spherical component. In the models presented in this paper, we adopt ${f}_{\tau }=10.0$. The implication of this choice is discussed in Section 4. The parameters for the models are summarized in Table 1.

Table 1.  Summary of the Models

Model	${\tau }_{\mathrm{ph}}$ a	${\alpha }_{\mathrm{cl}}$ b	${f}_{\mathrm{cl}}$ c
2D-bipolar-30degd	10.0	⋯	0.13
2D-torus-20dege	10.0	⋯	0.35
3D-a0.5-f0.3	3.0, 10.0, 30.0, 100.0	0.5	0.3
3D-a0.25-f0.3	3.0, 10.0, 30.0, 100.0	0.25	0.3
3D-a0.125-f0.3	3.0, 10.0, 30.0, 100.0	0.125	0.3
3D-a0.5-f0.06	30.0	0.5	0.06
3D-a0.5-f0.2	30.0	0.5	0.2
3D-a0.5-f0.5	10.0	0.5	0.4
3D-a0.5-f0.7	10.0	0.5	0.7

Notes.

^aSobolev line optical depth at the photosphere. ^bSize parameter of the clumps for 3D models (${\alpha }_{\mathrm{cl}}={v}_{\mathrm{cl}}/{v}_{\mathrm{ph}}$). ^cCovering factor of the clumps. ^d2D model with the two polar blobs with the half opening angle of 30°. ^e2D model with an equatorial torus with the half opening angle of 20°.

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We study the explosion geometry of stripped-envelope SNe by comparing the results of our simulations with observations. Figure 1 shows an example of spectropolarimetric data of stripped-envelope SNe (Type Ib SN 2009jf, Tanaka et al. 2012). In this paper, we define Stokes parameters as a fraction of the total flux: $Q\equiv \hat{Q}/I$ and $U\equiv \hat{U}/I$, where $\hat{Q}$ and $\hat{U}$ are polarized fluxes, or $\hat{Q}={I}_{0}-{I}_{90}$ and $\hat{U}={I}_{45}-{I}_{135}$, respectively (${I}_{\psi }$ is the intensity measured through the ideal polarization filter with an angle ψ). From Stokes parameters Q and U, the position angle of the polarization, θ, is obtained by $2\theta =\mathrm{atan}(U/Q)$.

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The properties of line polarization in stripped-envelope SNe can be summarized as follows:

• 1.  
  Nonzero line polarization is common, and the polarization feature shows an inverted P-Cygni profile that peaks at the flux absorption minimum (e.g., Kawabata et al. 2002; Leonard et al. 2002; Wang et al. 2003a; Maund et al. 2007a, 2007b, 2009; Tanaka et al. 2008, 2009a, 2012; Mauerhan et al. 2015, 2017; Stevance et al. 2016).
• 2.  
  When the polarization data across the line (middle panel of Figure 1) are plotted in the Stokes Q – U diagram (right panel), the observed data commonly show a loop (e.g., Maund et al. 2007a, 2007b, 2009; Tanaka et al. 2012; Mauerhan et al. 2015, 2017; Stevance et al. 2016).⁸
• 3.  
  The degree of line polarization, that is, the maximum polarization level at the absorption lines, is generally a few percent and tends to be higher for stronger lines (Tanaka et al. 2012).

The degree of polarization depends on the strength of the absorption. The absorption strength is mainly determined by the global properties of SNe, such as ejecta mass, temperature, and element abundances, and not directly by the explosion geometry. Therefore, it is important to compare features with a similar absorption strength to discuss the explosion geometry. Here we define a fractional depth (FD) of absorption at the absorption minimum, $\mathrm{FD}=({f}_{\mathrm{cont}}-{f}_{\mathrm{abs}})/{f}_{\mathrm{cont}}$, where ${f}_{\mathrm{abs}}$ and ${f}_{\mathrm{cont}}$ are the flux at the absorption minimum and at the continuum near the absorption line, respectively. Tanaka et al. (2012) showed that, in a simple configuration, the observed polarization (${P}_{\mathrm{obs}}$) can be approximately described as ${P}_{\mathrm{obs}}\simeq {P}_{\mathrm{cor}}[\mathrm{FD}/(1-\mathrm{FD})]$, where a corrected polarization ${P}_{\mathrm{cor}}$ is defined as the polarization level if FD = 0.5.

The size of the clumps is of interest in studying the origin of the 3D structure in the SNe. We show the first attempt to quantify the size of the clumps by comparing the results of the modeling and the observed polarization degrees, that is, the maximum polarization level at the absorption line. We calculate the polarization spectra with different sizes of clumps by keeping the covering factor ${f}_{\mathrm{cl}}=0.3$ and other parameters the same. The middle and bottom panels in Figure 3 show the results for the 3D models with ${\alpha }_{\mathrm{cl}}=0.25$ and 0.125, respectively.

As shown in the figures, for a given covering factor, models with smaller clumps show a lower polarization. In such models, the photospheric disk is hidden by many small clumps, and polarization vectors tend to be canceled out (Figure 4). This behavior was also pointed out by Hole et al. (2010) in the context of Type Ia SNe. Since stripped-envelope SNe generally show nonzero line polarization, the typical size of the clumps should not be too small.

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Since the polarization degree depends not only on the geometry but also on the absorption strength, it is important to compare models and observations for similar absorption strengths. Therefore, in Figure 5, we compare models and observations in the plane of the polarization degree and the fractional absorption depth. The black points with error bars are observational data of the Ca ii (filled) and Fe ii (open) lines for six Type Ib and Ic SNe analyzed in Tanaka et al. (2012). The small dots show the polarization degree of the models for 100 lines of sight. In each panel, we show four sets of the models with the same size and distribution of the clumps but with a different line optical depth at the photosphere (${\tau }_{\mathrm{ph}}=3.0$, 10.0, 30.0, and 100.0 from left to right).

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When the clump is as small as ${\alpha }_{\mathrm{cl}}=0.125$ (bottom panels of Figures 3 and 5), the polarization degree cannot be >0.5% for any line of sight. For the larger sizes of the clumps, a higher polarization can be obtained. When the size of the clumps is ${v}_{\mathrm{cl}}=0.25$ (middle panels), the polarization degrees of these models are still short of some of the observed polarization. When the size of the clumps is relatively large, ${\alpha }_{\mathrm{cl}}=0.5$ (top panels), the polarization degree can be as high as >1% for the FD of 0.5.

Ideally the polarization properties of the models should be compared with the statistical distribution of the observed polarization. Although the number of objects with good data is still small, Figure 6 shows a cumulative distribution of polarization properties of six Type Ib and Ic SNe. To define one characteristic polarization for each object, we take the average of the corrected polarization (${P}_{\mathrm{cor}}$) for the Ca ii and Fe ii lines. Color lines show the cumulative distribution of the modeled polarization for 100 lines of sight. We choose models with ${\tau }_{\mathrm{ph}}=30.0$, which approximately give FD ∼ 0.5 (Figure 5).

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The comparison in the cumulative distribution clearly shows that the model with too-small clumps (${\alpha }_{\mathrm{cl}}=0.125$) is not consistent with the observations. The p value for a Kolmogorov–Smirnov (KS) test is ${p}_{\mathrm{KS}}=0.0016$. Since the number of objects is so small, we cannot distinguish the model with the clump size of ${\alpha }_{\mathrm{cl}}=0.25$ (${p}_{\mathrm{KS}}=0.54$) and ${\alpha }_{\mathrm{cl}}=0.5$ (${p}_{\mathrm{KS}}=0.94$). Nevertheless, the model with ${\alpha }_{\mathrm{cl}}=0.25$ is already short of explaining the polarization level of >1%, and this clump size seems too close to the lower limit to explain the observations. Here it is noted that our models adopt an enhancement factor of ${f}_{\tau }=10.0$. For a higher enhancement factor, the polarization degree is not largely affected because models with ${f}_{\tau }=10.0$ already give an optically thick absorption in the clumps near the photosphere. On the other hand, for a smaller enhancement factor, the polarization degree decreases for a given FD. In such cases, even larger clumps are required to reproduce a high polarization degree. Therefore, we conclude that a typical size of the 3D clumps should be ≳25% of the photospheric radius to reproduce the observed polarization degrees.

To obtain possible constraints on the number or the covering factor of the clumps in the ejecta, we vary the covering factors of clumps, keeping their size at ${\alpha }_{\mathrm{cl}}=0.5$. Figure 7 shows the model input (left) and cumulative distributions of the resultant polarization (right). For the models, we choose the line strength at the photosphere to have FD ∼ 0.5, that is, ${\tau }_{\mathrm{ph}}=30$ for the models with ${f}_{\mathrm{cl}}=0.06$, 0.2, and 0.3, and ${\tau }_{\mathrm{ph}}=10$ for the models with ${f}_{\mathrm{cl}}=0.4$ and 0.7. The observed distribution is the same as in Figure 6.

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For the model with a smaller covering factor (${f}_{\mathrm{cl}}=0.06$), the probability of having a high polarization is also low. Then, by increasing the covering factor of the clumps, a higher polarization can be more frequently observed (${f}_{\mathrm{cl}}=0.2\mbox{--}0.5$). However, if the covering factor of the clumps is too large (${f}_{\mathrm{cl}}=0.7$), the distribution of the resultant polarization shifts toward a lower value again since the system restores the symmetry again.

Since the observational samples are small, it is difficult to draw a firm conclusion on the covering factor of the clumps. However, the models with ${f}_{\mathrm{cl}}=0.06$ and ${f}_{\mathrm{cl}}=0.7$ are already at the edge of the distribution. By taking into account the fact that models with ${f}_{\tau }=10.0$ tend to give an upper limit of the polarization level (see Section 4.1), it seems that the current data do not support models with too-small covering factors (${f}_{\mathrm{cl}}\lesssim 0.05$) and too-large covering factors (${f}_{\mathrm{cl}}\gtrsim 0.8$).

We set up the 3D Cartesian spatial mesh with the 100 × 100 × 100 meshes. The velocity is used as a spatial coordinate because the SN ejecta expand homologously (r ∝ v). The outer velocity of the grid is ${v}_{\max }={\rm{25,000}}$ km s⁻¹, and thus the resolution is Δv = 500 km s⁻¹. This spatial resolution gives the wavelength resolution of λ/Δλ = c/Δv = 600, which is sufficient to make a comparison with the observed data.

Since the code computes only one (arbitrary) line, the wavelength range used in the computation is very small. If the rest wavelength of the line is λ₀, we compute the spectrum only at the wavelength range between ${\lambda }_{0}(1-{v}_{\max }/c)$ and ${\lambda }_{0}(1+{v}_{\max }/c)$. Within this wavelength range, the energy spectrum is assumed to be constant ($\lambda {F}_{\lambda }=\mathrm{const}$).

Our code assumes a sharply defined inner boundary and solves radiation transfer above the boundary by tracking the photon packets in the expanding ejecta. Every photon packet has assigned energy, wavelength, and Stokes parameters. In particular, each photon packet in the simulation has a constant energy, irrespective of the wavelength of the packet. Because of this treatment, no photons are lost during the simulation, which results in the accurate energy conservation (Lucy 1999; Kasen et al. 2006; Kromer & Sim 2009).

The position of the inner boundary is determined so that the electron-scattering optical depth from the inner boundary to infinity is ${\tau }_{\mathrm{in}}$. In the simulations used in the main text of the paper, we always adopt ${\tau }_{\mathrm{in}}=3$ (Table 1) as in Kasen et al. (2003) and Hole et al. (2010). The radiation from the inner boundary is assumed to be thermalized and thus to be unpolarized:

$$\begin{eqnarray}&&{\boldsymbol{I}}=\left(\begin{array}{c}I\\ Q\\ U\end{array}\right)=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right).\end{eqnarray} \tag{ 1 }$$

The direction of the photon is determined by $\mu =\sqrt{z}$ (Mazzali & Lucy 1993; hereafter we use z to denote a random number, 0 < z ≤ 1), where μ is the cosine of the angle between the radial and photon directions. The azimuthal angle around the radial direction ψ is uniformly distributed: $\psi =2\pi z$.

The emitted photon packets experience electron scattering and line scattering, which are treated in a way similar to that of Mazzali & Lucy (1993). For the electron scattering, we assume a power-law density structure with the power-law index n. We also have the photospheric velocity (${v}_{\mathrm{ph}}$) and the epoch from the explosion (${t}_{{\rm{d}}}$) as input parameters. The photospheric radius (${r}_{\mathrm{ph}}={v}_{\mathrm{ph}}{t}_{{\rm{d}}}$) is defined to be the radius where the optical depth for the electron scattering is unity. By setting ${v}_{\mathrm{ph}}$ and ${t}_{{\rm{d}}}$, the normalization of the electron density is determined.

For the line scattering, we use the Sobolev approximation (Castor 1970), and we assume a power-law optical depth profile with the same index n. The parameter for the line scattering is ${\tau }_{\mathrm{ph}}$, the optical depth at the photosphere. In addition, we assume enhancement of the optical depth by a factor of ${f}_{\tau }$ in some regions. The parameters used in the simulations are summarized in Table 1.

A photon packet propagating in one computational grid can have three possible events: (1) escape from the grid, (2) electron scattering, and (3) line scattering. The event that actually occurs is judged by calculating the length to the three events. It is simple to compute the length to the next grid ${l}_{\mathrm{grid}}$ for the given position and the direction vector of the photon packet. The direction to the electron scattering event is computed by the randomly selected event optical depth ${\tau }_{R}=-\mathrm{ln}(z)$. When the optical depth reaches this value, a scattering event occurs. Thus, the distance to the electron scattering ${l}_{\mathrm{elec}}$ can be computed by ${\tau }_{R}={n}_{e}({\boldsymbol{r}})\sigma {l}_{\mathrm{elec}}$. When ${l}_{\mathrm{elec}}$ is shorter than ${l}_{\mathrm{grid}}$, the electron scattering occurs if there is no contribution of line scattering.

Since the line scattering is treated as a resonance, the distance to the line scattering event is ${l}_{\mathrm{line}}={{ct}}_{{\rm{d}}}({\lambda }_{0}-\lambda ^{\prime} )/{\lambda }_{0}$, where $\lambda ^{\prime}$ is the comoving wavelength of the photon packet. If ${l}_{\mathrm{line}}$ is shortest among the three lengths, the line scattering is taken into account. The line scattering event actually occurs when the sum of the line-scattering optical depth (${\tau }_{\mathrm{line}}({\boldsymbol{r}})$) and the electron-scattering optical depth in ${s}_{\mathrm{line}}$ (${\tau }_{e}={n}_{e}({\boldsymbol{r}})\sigma {l}_{\mathrm{line}}$) exceeds ${\tau }_{R}$. If this sum does not reach ${\tau }_{R}$, then the electron scattering opacity is evaluated and added again, and the fate of the packet is electron scattering or escape from the grid. For an illustration of this process, see Figure 1 of Mazzali & Lucy (1993).

When the scattering event occurs, the next direction vector of the photon packets is determined. For electron scattering, this scattering angle depends on the polarization, which is discussed in the next section. For line scattering, the direction is determined by the isotropic probability function in the comoving frame.

The energy and the wavelength of the packet are changed by the scattering event. For the energy, by the energy conservation in the rest frame,

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{out}}={\epsilon }_{\mathrm{in}}\displaystyle \frac{1-{\mu }_{\mathrm{in}}v/c}{1-{\mu }_{\mathrm{out}}v/c},\end{eqnarray} \tag{ 2 }$$

where ${\epsilon }_{\mathrm{in}}$ and ${\epsilon }_{\mathrm{out}}$ are the rest-frame energy of the incoming and outgoing packets, respectively. Here, ${\mu }_{\mathrm{in}}$ and ${\mu }_{\mathrm{out}}$ are the cosines of the angles between the radial direction and the incoming/outgoing propagating directions, respectively. Similarly, the change in wavelength is given by

$$\begin{eqnarray}&&{\lambda }_{\mathrm{out}}={\lambda }_{\mathrm{in}}\displaystyle \frac{1-{\mu }_{\mathrm{out}}v/c}{1-{\mu }_{\mathrm{in}}v/c},\end{eqnarray} \tag{ 3 }$$

where ${\lambda }_{\mathrm{in}}$ and ${\lambda }_{\mathrm{out}}$ are the rest-frame wavelengths of the incoming and outgoing packets, respectively.

Scattering events change the polarization properties of the photon packets. For electron scattering, the phase matrix can be written as follows (Chandrasekhar 1960):

$$\begin{eqnarray}{\boldsymbol{R}}({\rm{\Theta }})=\displaystyle \frac{3}{4}\left(\begin{array}{ccc}{\cos }^{2}{\rm{\Theta }}+1 & {\cos }^{2}{\rm{\Theta }}-1 & 0\\ {\cos }^{2}{\rm{\Theta }}-1 & {\cos }^{2}{\rm{\Theta }}+1 & 0\\ 0 & 0 & 2\cos {\rm{\Theta }}\end{array}\right),\end{eqnarray} \tag{ 4 }$$

where Θ is the scattering angle on the plane of the scattering. This matrix should be operated in the scattering frame. In general, the rotation matrix for the Stokes parameters is written as follows (Chandrasekhar 1960):

$$\begin{eqnarray}{\boldsymbol{L}}(\phi )=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos 2\phi & \sin 2\phi \\ 0 & -\sin 2\phi & \cos 2\phi \end{array}\right).\end{eqnarray} \tag{ 5 }$$

By using these matrices, the effect on the Stokes parameters is given by

$$\begin{eqnarray}&&{{\boldsymbol{I}}}_{\mathrm{out}}={\boldsymbol{L}}(\pi -{i}_{2})R({\rm{\Theta }}){\boldsymbol{L}}(-{i}_{1}){{\boldsymbol{I}}}_{\mathrm{in}}.\end{eqnarray} \tag{ 6 }$$

Here, ${{\boldsymbol{I}}}_{\mathrm{in}}$ and ${{\boldsymbol{I}}}_{\mathrm{out}}$ are the Stokes parameters in the rest frame before and after the scattering, respectively. The angles i₁ and i₂ are the angles on the spherical triangle defined as in Chandrasekhar (1960, see Figure 1 of Code & Whitney 1995).

Equation (6) means that the angle dependence of the intensity of the scattered light depends on the polarization properties of the incident radiation. From Equation (6), the probability distribution function (pdf) of the total intensity is

$$\begin{eqnarray}{\rm{p}}.{\rm{d}}.{\rm{f}} & = & \displaystyle \frac{1}{2}({\cos }^{2}{\rm{\Theta }}+1)+\displaystyle \frac{1}{2}({\cos }^{2}{\rm{\Theta }}-1)\\ & & \times (\cos 2{i}_{1}{Q}_{\mathrm{in}}/{I}_{\mathrm{in}}-\sin 2{i}_{1}{U}_{\mathrm{in}}/{I}_{\mathrm{in}}).\end{eqnarray} \tag{ 7 }$$

By using this function with the rejection method as outlined in Code & Whitney (1995), we determine the scattering angle of the electron scattering.

We assume that the line scattering works as a depolarizer: the emission is turned into the unpolarized state by the line scattering, as assumed in previous studies (see Höflich et al. 1996; Kasen et al. 2006; Hole et al. 2010).

For the computation of polarization for the electron scattering, the code was tested with the analytic formulae of Brown & McLean (1977) for optically thin cases, and with the numerical results from Code & Whitney (1995) for optically thick cases. For both cases, we got an excellent agreement. For the application to an SN, that is, expanding ejecta with a steep density slope, we checked our results with those of Kasen et al. (2003). We confirmed that our code gives results consistent with the radial profile of polarization for several power-law indexes (n) and inner boundaries (${\tau }_{\mathrm{in}}$).