THREE-DIMENSIONAL EXPLOSION GEOMETRY OF STRIPPED-ENVELOPE CORE-COLLAPSE SUPERNOVAE. I. SPECTROPOLARIMETRIC OBSERVATIONS^*

Core-collapse supernovae (SNe) are the explosions of massive stars at the end of their lives. Besides their importance as the end points of stellar evolution, SNe play an important role in the chemical and dynamical evolution of galaxies. However, the explosion mechanism of SNe has been unclear for a long time after the first concept proposed by Burbidge et al. (1957) and the first numerical simulation by Colgate & White (1966).

Modern detailed numerical simulations agree in that a successful explosion cannot be obtained in one-dimensional (1D) simulations (Rampp & Janka 2000; Liebendörfer et al. 2001; Thompson et al. 2003; Sumiyoshi et al. 2005). Recently, increasing attention is being paid to multi-dimensional effects, such as convection (e.g., Herant et al. 1994; Burrows et al. 1995; Janka & Mueller 1996) and Standing Accretion Shock Instability (e.g., Blondin et al. 2003; Scheck et al. 2004; Ohnishi et al. 2006; Foglizzo et al. 2007; Iwakami et al. 2008). In fact, some successful explosions have been obtained in two-dimensional (2D) simulations (Buras et al. 2006; Marek & Janka 2009; Suwa et al. 2010).

Thanks to the advancements in numerical simulations, some simulations have been done in a fully three-dimensional (3D) space (Nordhaus et al. 2010; Wongwathanarat et al. 2010; Takiwaki et al. 2012; Kuroda et al. 2012). Interestingly, Nordhaus et al. (2010) show that it is easier to obtain a successful explosion in 3D simulations than in 2D simulations (but see also Hanke et al. 2011).

Given these circumstances, it is important to obtain observational constraints on the multi-dimensional geometry of SN explosions. The most direct way is the observation of spatially resolved SNe. Isensee et al. (2010) and DeLaney et al. (2010) performed detailed observations of the Galactic SN remnant Cassiopeia A. Using a combination of imaging and spatially resolved spectroscopy, they reconstructed the 3D geometry of the SN. Similarly, direct imaging (Wang et al. 2002) and spatially resolved spectroscopy (Kjær et al. 2010) have been performed for SN 1987A in the Large Magellanic Cloud. However, the number of objects suitable for such observations is limited.

Observations of extragalactic SNe can also provide hints of explosion geometry, although at this distance SNe are point sources. For example, spectroscopic observations of SNe at >1 yr after the explosion provide information of explosion geometry. At such late epochs, SN ejecta are optically thin. Therefore, this method is sensitive to the geometry of the innermost ejecta. By analyzing line profiles of nebular emission, a bipolar geometry of SNe has been suggested (e.g., Mazzali et al. 2001, 2005; Maeda et al. 2002, 2008; Modjaz et al. 2008; Tanaka et al. 2009b; Taubenberger et al. 2009; Maurer et al. 2010, but see also Milisavljevic et al. 2010).

Polarization at early phases (≲50 days after the explosion) is another powerful method for studying the multi-dimensional geometry of extragalactic SNe (see Wang & Wheeler 2008 for a review). In contrast to late phase spectroscopy, polarization is sensitive to the geometry of the outer ejecta. In particular, spectropolarimetry can probe 3D geometry, as explained in the following section. In this paper, we study the multi-dimensional explosion geometry of SNe using spectropolarimetric observations.

Polarization in SNe is generated by electron scattering. Since the polarization degree of the scattered light reaches its maximum when the scattering angle is 90°, the expected polarization map for a spherical photosphere looks like (a) in Figure 1. However, SNe are observed as point sources, and thus all the polarization vectors cancel out, resulting in no polarization.⁹

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Absorption and re-emission by a bound–bound transition, which create a P-Cygni profile in the expanding ejecta, tend to cause depolarization (Jeffery 1989; Kasen et al. 2003). Thus, the polarization produced by the electron scattering is effectively concealed by the interaction with lines. If an ion is distributed spherically (b), polarization vectors still cancel out, and no polarization is expected at the wavelength of the line. Polarization would be detected at the wavelength of the absorption line only if the ion is distributed asymmetrically (c), because of the incomplete cancellation. Thus, we can identify an asymmetric ion distribution by detecting line polarization.

Note that an asymmetric ion distribution is expected either from an asymmetric element distribution, which could be produced by a multi-dimensional explosion (e.g., Nagataki et al. 1997; Kifonidis et al. 2000, 2003; Maeda et al. 2003; Tominaga 2009; Fujimoto et al. 2011), or by an asymmetric ionization/temperature structure, which is caused by an asymmetric distribution of ⁵⁶Ni (Tanaka et al. 2007).

The properties of linear polarization are fully described by the Stokes parameters. Throughout this paper, we define the Stokes parameters as a fraction of the total flux: $Q \equiv \hat{Q}/I$ and $U \equiv \hat{U}/I$, where $\hat{Q}$ and $\hat{U}$ can be expressed as $\hat{Q} = I_{0} - I_{90}$ and $\hat{U} = I_{45} - I_{135}$, respectively. Here I_θ is the intensity measured through the ideal polarization filter with an angle θ. From the Stokes parameters Q and U, the polarization position angle θ is obtained by

$$\begin{equation} 2\theta = {\rm atan}(U/Q). \end{equation} \tag{ 1 }$$

The position angle is measured from north to east in the sky.

Figure 2 shows the expected line polarization in a 2D bipolar model (top) and a 3D clumpy model (bottom). The polarization spectra are calculated by the Monte Carlo method for arbitrary line optical depth distributions, under assumptions similar to those in Kasen et al. (2003) and Hole et al. (2010). The details of the simulations will be given in a forthcoming paper. The left panels show the distribution of line optical depth. In the 2D model, the opacity is enhanced by a factor of 10 in the polar region, while in the 3D model, the opacity is similarly enhanced in the randomly distributed clumps.

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In both cases, the polarization changes at the wavelength of the line (middle panels), which is caused by the asymmetric distribution of the line optical depth. However, the behavior of the polarization angle is different. In the 2D case, the polarization angle is constant (some fluctuation is caused by the Monte Carlo noise), while in the 3D case, it largely changes across the line. Since the SN ejecta expand homologously and the velocity can be used as a radial coordinate, a change in the polarization angle means a radially dependent ion distribution.

This effect is more clearly seen in the Q – U plane. The right panels in Figure 2 show the same simulated polarization spectrum in the Q – U plane. Different colors represent different Doppler velocities, according to the color bar above the plots. The polarization for the 2D model shows a straight line in the Q – U plane, while that for the 3D model shows a "loop," which corresponds to a change in the polarization angle.

Therefore, a change in the polarization angle or the loop in the Q – U plane is indicative of a 3D ion distribution. In order to fully utilize the power of spectropolarimetry, we should obtain high-quality data to discriminate such a feature from noise. Loops in the Q – U plane have been commonly observed in Type Ia SNe (Wang et al. 2003a; Kasen et al. 2003; Chornock & Filippenko 2008; Patat et al. 2009) and in some core-collapse SNe (e.g., Maund et al. 2007b, 2007c, 2009).

In this paper, we show our spectropolarimetric observations of two Type Ib/c SNe with the Subaru Telescope (Sections 2 and 3). The first target is the Type Ib SN 2009jf (discovered in NGC 7479 by Li et al. 2009 on UT 2009 September 27.33 and classified by Kasliwal et al. 2009; Sahu et al. 2009). See Sahu et al. (2011) and Valenti et al. (2011) for more details. The other target is the Type Ic SN 2009mi (discovered in IC 2151 by Monard et al. 2009 on UT 2009 December 12.91 and classified by Kinugasa et al. 2009).

We then summarize the spectropolarimetric data of stripped-envelope SNe published so far (Section 4). We argue that a non-axisymmetric, 3D geometry is common in stripped-envelope SNe. We find a relation between line polarization and the depth of absorptions: i.e., a stronger line tends to show a higher line polarization. Even after correcting for the effect of the absorption depth, there remains a dispersion in the polarization degree among different objects. The implications of this dispersion are discussed. Finally, we give conclusions in Section 5.

In Table 2, we summarize line polarization measurements obtained so far for stripped-envelope SNe. Here the line polarization is defined as a relative change from the continuum level, and thus, not affected by the ISP. We show six stripped-envelope SNe with high-quality data, namely, Type Ib SNe 2005bf (Maund et al. 2007b; Tanaka et al. 2009a), 2008D (Maund et al. 2009), 2009jf (this paper), Type Ic SNe 2002ap (Kawabata et al. 2002; Leonard et al. 2002; Wang et al. 2003b), 2007gr (Tanaka et al. 2008), and 2009mi (this paper). As also noted by Wang & Wheeler (2008), it is clear that all SNe show non-zero polarization, which means that stripped-envelope SNe generally have asymmetric explosion geometry.

For SNe 2005bf and 2008D, Maund et al. (2007b, 2009) found a loop in the Q – U diagram at strong lines, which is indicative of a 3D geometry (Kasen et al. 2003; Maund et al. 2007b, 2007c; see also Figure 2). Our new data for SNe 2009jf and 2009mi also show a loop in the Q – U diagram. We have checked the literature about SN 2002ap (Kawabata et al. 2002; Leonard et al. 2002; Wang et al. 2003b). Although they do not explicitly mention the loop, the data in Kawabata et al. (2002) show a loop in the Ca ii line. Thus, loops are quite common in stripped-envelope SNe; five of six stripped-envelope SNe show the loop. This implies that a non-axisymmetric, 3D geometry is common in stripped-envelope SNe.

Figure 6 shows the observed line polarization as a function of epoch from maximum brightness. There is a large variety in the polarization degree. It is clear that the Ca ii line (blue) tends to be more polarized than the other lines. There is no clear trend in the time evolution; the line polarization could increase or decrease with time.

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Figure 7 shows the same polarization data as a function of the fractional depth (FD) of the absorption. The FD is defined as

$$\begin{equation} {\rm FD} = \frac{F_{\rm cont} - F_{\rm abs}}{F_{\rm cont}}, \end{equation} \tag{ 3 }$$

where F_cont and F_abs are the fluxes at the continuum and absorption minimum, respectively.

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Figure 7 shows that stronger lines tend to show a higher polarization degree. In fact, there are no observational points at the top left of the plot, i.e., FD <0.3 and P_obs > 1%. This seems natural because the absorption, unless it is quite strong, cannot cause a significant incomplete cancellation of the polarization (Figure 1).

We can analyze this relation in a simple, analytic way. We assume a spherical photosphere projecting an area S on the sky. Within this photospheric disk, the flux per unit area I and the polarization degree are assumed to be uniform. The total flux from the photospheric disk is F_cont = IS. When light is emitted from a point on the photospheric disk on the sky, it can undergo line absorption with an absorption fraction x_abs ≡ 1 − exp (− τ). At the wavelength of this line, the flux is F_abs = (1 − x_abs)IS.

If the distribution of the line opacity is uniform, no polarization is expected at the line, i.e., P_abs = 0 (Section 1.2). To introduce asymmetry in the line opacity, we assume that the absorption fraction is enhanced by a factor of f in a region ΔS (e.g., the shaded region in Figure 1(c)). Then, the total flux at the wavelength of the line is

$$\begin{eqnarray} F_{\rm abs} &=& (1-x_{\rm abs}) I (S-\Delta S) + (1-fx_{\rm abs}) I \Delta S \nonumber \\ &=& [(1-x_{\rm abs}) - (f-1)x_{\rm abs} \Delta S/S] {\it IS}, \end{eqnarray} \tag{ 4 }$$

and the FD (Equation (3)) can be written by using Equation (4) as

$$\begin{equation} {\rm FD} = x_{\rm abs} + (f-1)x_{\rm abs} \Delta S/S. \end{equation} \tag{ 5 }$$

The asymmetry introduced above results in an incomplete cancellation of the polarization. If this region has a constant polarization direction, the amount of non-canceled flux is equivalent to the excess absorption in the region, i.e.,

$$\begin{equation} P_{\rm abs}F_{\rm abs} = (f-1)x_{\rm abs} I \Delta S. \end{equation} \tag{ 6 }$$

Then, the polarization degree can be expressed as a function of the FD:

$$\begin{equation} P_{\rm abs} = \frac{(f-1)x_{\rm abs} \Delta S/S}{1 - {\rm FD}}. \end{equation} \tag{ 7 }$$

Unless the enhanced opacity dominates the absorption, FD ≃ x_abs (Equation (5)). Thus,

$$\begin{equation} P_{\rm abs} \simeq (f-1)\Delta S/S \frac{{\rm FD}}{1 - {\rm FD}} = P_{\rm cor} \frac{{\rm FD}}{1 - {\rm FD}}, \end{equation} \tag{ 8 }$$

where we define a corrected polarization degree P_cor ≡ (f − 1)ΔS/S. This corrected polarization degree is equivalent to the polarization with FD = 0.5. It can be interpreted as a rough indicator of the product of the amount of the enhancement (f − 1) and the fractional area of the enhanced region ΔS/S under the assumption that the enhanced region only has one polarization direction. The gray dashed line in Figure 7 shows the line polarization expected from Equation (8) with P_cor = 2.0%.

Note that the analytic equations here involve significant simplifications, i.e., a constant intensity in the photospheric disk and a constant direction of polarization in the enhanced region. But if we use ΔS as an intensity-weighted area, in which polarization is not canceled, the equation holds even for general cases with a non-constant intensity and many opacity-enhanced regions.

From Equation (8), it is expected that stronger lines with a larger FD tend to have higher polarization, which is actually found in Figure 7. It should be emphasized that the variety of the observed line polarization is largely generated by a variety of the absorption depth. The absorption depth is primarily determined by the density, chemical abundance, or ionization in the ejecta, which can be varied even in the 1D explosion (reflecting the mass and kinetic energy of the ejecta, and the temperature in the ejecta).

Figure 8 shows the observed polarization (left) and polarization corrected for the absorption depth (right) for each stripped-envelope SN. It is obvious that after correcting for the absorption depth, the Ca ii line is not always more polarized than the other lines. This demonstrates the importance of the correction. To compare the polarization of different lines or different objects correctly, one must correct for this effect, which is not directly related to the geometry. This fact has been overlooked in previous analyses. It would be interesting to study this effect also on the line polarization of Type Ia SNe, which has been suggested to be correlated to the declining rate of the light curve (Wang et al. 2007).

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It is clear that there is a remaining dispersion in the line polarization among different objects even after correcting for the absorption depth (right panel in Figure 8). We first examine if the corrected polarization is related with the epoch. The left panel of Figure 9 shows the corrected line polarization as a function of epoch. Although there is a weak trend of the polarization decreasing with time, it is not clear if all the variety in the corrected polarization can be explained by the effect of the epoch.

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The remaining dispersion could also be understood by the multi-dimensional properties of each SN. It is thus worth studying the relationship between the line polarization and the [O i] line profile in the nebular spectra, which is another geometric probe of the SN ejecta (Mazzali et al. 2001, 2005; Maeda et al. 2002, 2008; Modjaz et al. 2008; Tanaka et al. 2009b; Taubenberger et al. 2009; Maurer et al. 2010).

The right panel of Figure 9 shows the relation between the corrected line polarization and the [O i] line profile. SNe 2002ap and 2007gr show a single-peak profile (Leonard et al. 2002; Foley et al. 2003; Mazzali et al. 2007, 2010; Hunter et al. 2009) while SNe 2005bf and 2008D show a double-peak profile (Maeda et al. 2007; Modjaz et al. 2009; Tanaka et al. 2009b). The [O i] profile of SN 2009jf is complex; it shows both single- and double-peak components (Sahu et al. 2011; Valenti et al. 2011). Following the classification by Maeda et al. (2008), we classify it as "transition." With only five objects, it is difficult to draw a firm conclusion. It is therefore important to increase spectropolarimetric samples to further study this relation.

The dispersion may also reflect a difference between Type Ib and Ic. In fact, three Type Ic SNe in our sample tend to have a lower polarization degree than three Type Ib SNe (left panel of Figure 8). By increasing the number of samples, we will be able to study the effect of the existence of the He envelope on the explosion geometry.

One natural scenario is that the dispersion is caused by random line-of-sight effects. As discussed in Section 4.1, stripped-envelope SNe generally have a 3D geometry. If we assume a clumpy structure as in the left bottom panel of Figure 2, a variety of polarization is naturally expected for different lines of sight. Since the effect of the line of sight is very sensitive to the size and the number of clumps, it can be used as an observational probe to study a typical 3D geometry of SNe. This will be further studied in a forthcoming paper.