Distortion of Magnetic Fields in Barnard 335

Figure 1 shows the observed polarization vector map of B335 in the H band. B335 is visible near the center of the image as a dark region that obscures the stars behind it. The white circle marks the core boundary (R = 125'', Harvey et al. 2001). Inside the core radius, the bent structure of the polarization vectors can be seen, particularly in the northern part of the core. Outside of the core radius (i.e., the off-core region), polarization vectors generally flow from northwest to southeast. These polarization components can be regarded as off-core polarization, located along the same line of sight but unrelated to the B335 core.

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Considering the direction of galactic longitude (PA ∼ 30° in the equatorial coordinates at the position of B335), the off-core vectors deviate from the polarization direction of the galactic plane. As reported by Frerking et al. (1987), the B335 core is accompanied by diffuse envelope extending more than 20' in east–west direction (see Figure 1 of Frerking et al. 1987). The position angle of off-core vectors is roughly perpendicular to the elongation axis of the diffuse envelope, and may be associated with this structure.

The off-core vectors have ${P}_{H,\mathrm{off}}=1.43\pm 0.49$% and ${\theta }_{H,\mathrm{off}}=141\buildrel{\circ}\over{.} 9\pm 11\buildrel{\circ}\over{.} 8$ (Figures 2 and 3). The off-core polarizations are relatively well ordered (Figure 3) and have similar polarization degree (Figure 2). Following the same method described in our previous papers (Kandori et al. 2017a, hereafter Paper I), we fitted the off-core vectors onto the sky plane using the Stokes parameters Q/I and U/I. The distributions of Q/I and U/I values were modeled as $f(x,y)=A+{Bx}+{Cy}$, where x and y are the pixel coordinates, and A, B, and C are the parameters to be fitted. The estimated off-core vectors are shown in Figure 4. The regression vectors of off-core components were subtracted from the original vectors to isolate the polarization vectors associated with B335 (Figure 5). Comparing Figures 1 and 5, polarization vectors toward B335 are slightly rotated by the effect of ambient subtraction. After subtraction, polarization degrees of off-core vectors were successfully suppressed toward 0% (Figure 2), and approximately random distribution of polarization angles was realized (Figure 3).

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Some on-core polarization vectors increase in polarization degree after the subtraction of off-core vectors. This occurs for the on-core vectors with angles perpendicular to the off-core vectors, and is not an artificial effect. The increase of polarization degree for some on-core vectors is the result of the suppression of the depolarization effect by off-core vectors.

In Figure 5, the north part of off-core polarizations is not fully subtracted out and seems to be associated with the polarization pattern of the core region. As described above, these vectors may be associated with the diffuse envelope structure surrounding the B335 core.

In Figure 5, polarization vectors toward B335 are clearly oriented east–west, and a structure reflecting the bending of magnetic fields can be seen in the northern and (possibly) southern part of the core. The number of polarization vectors within the core radius of R ≤ 125'' and with P_H ≥ 1% is 24. This number is small, but sufficient to draft the magnetic field lines.

The most probable configuration of the magnetic field lines, estimated using a parabolic function and its rotation, is shown in Figure 6 (solid white lines). The field of view matches the diameter of B335 (250''), and the 24 polarization vectors having P_H ≥ 1% are shown in the figure. The threshold of P_H ≥ 1% was determined to avoid the systematic error caused by the subtraction of off-core polarizations. Note that most of the residual polarization vectors in the off-core region after the subtraction analysis are less than 1%, as shown in Figure 2. The coordinate origin of the parabolic function is fixed to the central position of the core/protostar system measured on the Atacama Large Millimeter Array (ALMA) dust emission map (R.A. = 19^h37^m0089, decl. = +7°34'100, J2000, Evans et al. 2015).

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The best-fit parameters are θ_mag = 90° ± 7° and C = 1.43 (±0.49) × 10⁻⁵ pixel⁻² (=7.06 × 10⁻⁵ arcsec⁻²) for the parabolic function $y=g+{{gCx}}^{2}$, where g specifies magnetic field lines, θ_mag denotes the position angle of the magnetic field direction (from north through east), and C determines the degree of curvature in the parabolic function. In the fitting procedure, the observational error for each star was taken into account in the calculations of ${\chi }^{2}={({\sum }_{i=1}^{n}({\theta }_{\mathrm{obs},{\text{}}{\rm{i}}}-{\theta }_{\mathrm{model}}({x}_{i},{y}_{i}))}^{2}/\delta {\theta }_{i}^{2}$, where n is the number of stars, x and y are the star coordinates, θ_obs and θ_model denote the polarization angle from observations and the model, respectively, and δθ_i is the observational error. The parabolic fitting seems reasonable, since the standard deviation of residual angles, θ_res = θ_obs − θ_fit, is smaller for the parabolic function ($\delta {\theta }_{\mathrm{res}}=21\buildrel{\circ}\over{.} 54\pm 0\buildrel{\circ}\over{.} 99$, Figure 7) than for the uniform field case ($\delta {\theta }_{\mathrm{uni}}=24\buildrel{\circ}\over{.} 94\pm 1\buildrel{\circ}\over{.} 00$). When $\delta {\theta }_{\mathrm{res}}$ and $\delta {\theta }_{\mathrm{uni}}$ are compared, the difference is statistically significant at the 3σ level, and the existence of the distorted field is most likely real.

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B335 is the first protostellar core to be associated with axisymmetrically distorted hourglass-shaped magnetic fields, and is the third dense core to be associated with such fields following the results of FeSt 1-457 (Paper I) and B68 (Kandori et al. 2020c ), which lack protostars. The obtained magnetic curvature value C = 7.06 × 10⁻⁵ arcsec⁻² = 6.4 × 10⁻⁹ au⁻² is similar to that obtained for FeSt 1-457 (C = 5.14 × 10⁻⁵ arcsec⁻² = 3.0 × 10⁻⁹ au⁻², Paper I) and B68 (C = 1.09 × 10⁻⁴ arcsec⁻² = 7.0 × 10⁻⁹ au⁻², Kandori et al. 2020c), suggesting the existence of similar mechanisms that create hourglass-shaped field distortions in dense cores/globules. The magnetic curvature of dense cores can be produced with the drag of magnetic fields by materials concentrating toward center, and the degree of curvature can be determined by the moving distance of mass that drags magnetic fields. The similar curvature values among the dense cores indicate that the spatial scale of core formation is similar in these cores. Core formation scale, i.e., initial contraction radius R₀, may be similar in dense cores/globules. Further observational studies to measure magnetic curvature of dense cores are needed.

The intrinsic dispersion, $\delta {\theta }_{\mathrm{int}}={(\delta {\theta }_{\mathrm{res}}^{2}-\delta {\theta }_{\mathrm{err}}^{2})}^{1/2}$, estimated using the parabolic fitting, is 2134 ± 100 (0.3725 radian), where δθ_err is the standard deviation of the observational error in polarization measurements. Note that the choice of a function form better than the function used here (i.e., parabolic form) can reduce the dispersion at residual angles (e.g., a mathematical model for hourglass fields by Ewertowski & Basu 2013; Myers et al. 2018).

If the magnetic field is assumed to be frozen in the medium, the intrinsic dispersion of the magnetic field direction, δθ_int, can be attributed to an Alfvén wave perturbed by turbulence. The strength of the plane-of-sky magnetic field (B_pos) can be estimated from the relation ${B}_{\mathrm{pos}}={C}_{\mathrm{corr}}{(4\pi \rho )}^{1/2}{\sigma }_{\mathrm{turb}}/\delta {\theta }_{\mathrm{int}}$, where ρ and σ_turb are the mean density of the core and turbulent velocity dispersion, respectively (Davis 1951; Chandrasekhar & Fermi 1953) and C_corr is a correction factor suggested by theoretical studies. In the original formulation, C_corr = 1 (Davis 1951; Chandrasekhar & Fermi 1953), whereas in this study, a value of C_corr = 0.5 (Ostriker et al. 2001, see also, Heitsch et al. 2001; Padoan et al. 2001; Heitsch 2005; Matsumoto et al. 2006) was adopted. Using the data for mean density (ρ = 2.30 (±1.74) × 10⁻¹⁹ g cm⁻³) calculated from Harvey et al. (2001), turbulent velocity dispersion (σ_turb = 0.085 km s⁻¹) from Zhou et al. (1990), and $\delta {\theta }_{\mathrm{int}}$ derived here, a relatively weak magnetic field was obtained as a lower limit of total field strength ($| B|$): B_pos = 19.4 ± 11.4 μG.

We compared the obtained plane-of-sky magnetic field strength with previous studies (${134}_{-39}^{+46}$ μG, Wolf et al. 2003; 12–40 μG, Bertrang et al. 2014). Note that these previous studies used no theoretical correction factor in the Davis–Chandrasekhar–Fermi calculation. If we apply C_corr = 0.5 to their data, the values of the magnetic field strength are reduced to ${67}_{-20}^{+23}$ μG (Wolf et al. 2003) and 6–20 μG (Bertrang et al. 2014), respectively. The value by Bertrang et al. (2014) is roughly consistent with our result, and the value by Wolf et al. (2003) is larger than our result. Note that the submm polarimetry by Wolf et al. (2003) covered only the central region of the core, and the obtained magnetic field strength is not the mean value for the entire core.

For the 3D magnetic field modeling, we followed the procedure described in our previous papers (Kandori et al. 2017b, hereafter Paper II, see also Kandori et al. 2020a, hereafter Paper VI). The 3D version of the simple parabolic function employed in Paper I, $z(r,\varphi ,g)=g+{{gCr}}^{2}$ in the cylindrical coordinate $(r,z,\varphi )$, was used for modeling the core magnetic fields, where g specifies the magnetic field line, C is the curvature of lines, and φ is the azimuth angle (measured in the plane perpendicular to r). In this function, the shape of magnetic field lines is axially symmetric around the r axis. Thus, the function $z(r,\varphi ,g)$ has no dependence on the parameter φ.

The 3D model was virtually observed after rotating in the line-of-sight (γ_mag) and the plane-of-sky (θ_mag) directions. The analysis was performed in the same manner described in Section 3.1 of Paper VI (see also Sections 2 and 3.1 of Paper II). The resulting polarization vector maps for the 3D parabolic model for various viewing angles ($90^\circ -{\gamma }_{\mathrm{mag}}$) are shown in Figure S1 of Paper VI. The density distribution of the model core was calculated based on the Bonnor–Ebert sphere model (Ebert 1955; Bonnor 1956) with a solution parameter of ξ_max = 12.5 (Harvey et al. 2001). Since we obtained a linear polarization–extinction relationship for B335 (see Section 3.5), it is reasonable to assume that the polarization arisen in a cell in the model core is proportional to the density in the cell. The model map can be compared with observations for each parameter set (see Figure S1 of Paper VI). Since the polarization distributions in the model core differ from one another depending on the viewing angle toward the line of sight, χ² fitting of these distributions with the observational data can be used to restrict the line-of-sight magnetic inclination angle and 3D magnetic curvature. Note that the resolution of the final model 3D map is enough for comparison with observations. The nearest neighbor distance of stars in observations is 44 pixels on the average, whereas the sampling grid width of the model is 9 pixels (≈4'').

Figure 8 summarizes the distribution of ${\chi }_{\theta }^{2}=({\sum }_{i=1}^{n}({\theta }_{\mathrm{obs},i}\,-{{\theta }_{\mathrm{model}}({x}_{i},{y}_{i}))}^{2}/\delta {\theta }_{i}^{2}$, where n is the number of stars, x and y show the coordinates of stars, θ_obs and θ_model denote the polarization angle from observations and the model, and δθ_i is the observational error), calculated by using the model and observed polarization angles. The optimal magnetic curvature parameter, C, was determined at each inclination angle γ_mag to obtain ${\chi }_{\theta }^{2}$. In Figure 8, the minimization point is γ_mag = 45°. It is clear that the high-γ_mag region (especially γ_mag ≳ 80°) is unlikely. Note that the large γ_mag model approaches the pole-on magnetic field geometry and shows a radial polarization pattern that does not match observations.

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To check consistency of the χ² analysis based on polarization angle, we made χ² calculations using both polarization angles and degrees. We determined the best model parameters for each γ_mag by minimizing the difference in polarization angles, and we calculated ${\chi }_{P}^{2}={({\sum }_{i=1}^{n}({P}_{\mathrm{obs},i}-{P}_{\mathrm{model}}({x}_{i},{y}_{i}))}^{2}/\delta {P}_{i}^{2}$, where P_obs and P_model show the polarization degree from observations and the model, and $\delta {P}_{i}$ is the observational error in polarization degrees (Figure 9). In the procedure, the relationship between the model core column density and polarization degree was scaled to be consistent with observations. In Figure 9, the minimization point is γ_mag = 55°. It is clear that the low-γ_mag region (especially γ_mag ≲ 30°) is unlikely. Note that the small γ_mag model approaches the edge-on magnetic field geometry and shows a hourglass-like field structure with weak depolarization pattern.

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The distributions of ${\chi }_{\theta }^{2}$ and ${\chi }_{P}^{2}$ show relatively large scatter, especially for small or large γ_mag. This can be due to the complicated patterns of hourglass-like fields projected onto the sky plane, and the relatively small number of data points (N = 24) of the B335 background stars.

The 1σ error estimated at the minimum ${\chi }_{\theta }^{2}$ and ${\chi }_{P}^{2}$ points are 125 and 47, respectively. These values are similar to the difference between χ² minimization points based on ${\chi }_{\theta }^{2}$ and ${\chi }_{P}^{2}$ (10°). We conclude that the most likely value for γ_mag is 50° with uncertainty of 10°. The magnetic curvature obtained at γ_mag = 50° is C = 1.73 × 10⁻⁴ arcsec⁻². Note that the χ² values in Figures 8 and 9 are relatively large. This seems to come from the existence of polarization angle scattering mainly caused by Alfvén waves, which cannot be included in the observational error term in the calculation of χ².

Figure 10 shows the best-fit 3D parabolic model with the observed polarization vectors. The direction of polarization vectors in the model generally agrees with observations at the same level compared with the results of 2D fitting, although there is some deviations in between model and observations particularly in the north part of the core. The standard deviation of the angular difference in plane-of-sky polarization angles between the 3D model and observations is 1512, which is less than the fitting result in 2D, as well as in the uniform field case of δθ_uni = 2494.

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Figure 11 is the same observational data as Figure 10, but with the background image processed using the line integral convolution (LIC) technique (Cabral & Leedom 1993). We used the publicly available IDL code developed by Diego Falceta-Gonçalves. The direction of the LIC "texture" is parallel to the direction of magnetic fields, and the background image is based on the polarization degree of model core.

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Using the magnetic inclination angle θ_inc of 50° ± 10° obtained as described in the previous section, we determined that the mean magnetic field strength of B335 is ${B}_{\mathrm{pos}}/\cos ({\gamma }_{\mathrm{mag}})=19.4/\cos (50^\circ )=30.2\pm 17.7$ μG. The ability of the magnetic field to support the core against gravity was investigated using the parameter $\lambda ={(M/{\rm{\Phi }})}_{\mathrm{obs}}/{(M/{\rm{\Phi }})}_{\mathrm{critical}}$, which represents the ratio of the observed mass-to-magnetic flux ratio to a critical value, ${(2\pi {{\rm{G}}}^{1/2})}^{-1}$, suggested by theory (Mestel & Spitzer 1956; Nakano & Nakamura 1978). We found that λ = 3.24 ± 1.52 (magnetically supercritical) and that the magnetic critical mass of the core is 1.13 ± 0.69 M_⊙, which is lower than the observed core mass of M_core = 3.67 M_⊙.

The critical mass of B335, evaluated using both magnetic and thermal/turbulent support against collapse, M_cr ≃ M_mag + M_BE (Mouschovias & Spitzer 1976; Tomisaka et al. 1988; McKee 1989), is 1.13 + 2.24 = 3.37 ± 0.94 M_⊙, where 2.24 ± 0.64 M_⊙ is the Bonnor–Ebert mass calculated using the kinematic temperature of the core of 13 K, turbulent velocity dispersion of 0.085 km s⁻¹ (Zhou et al. 1990), and external pressure P_ext of 1.35(±0.77) × 10⁵ K cm⁻³ calculated from Harvey et al. (2001). Though the obtained M_cr is slightly less than the core mass M_core, we concluded that the stability of B335 is near the critical state, with M_cr ∼ M_core.

Thus, the protostellar core B335 appears to have started its contraction from a condition near equilibrium between gravity and magnetic and thermal/turbulent support. We speculate that, in general, the (spontaneous) low-mass star formation in globules is initiated from the state close to the critical condition. This is consistent with the observations of subsonic infall motion in dense cores (Lee et al. 1999, 2001). If the formation of stars were to start from a condition far out of equilibrium, the dense core would collapse suddenly with strong acceleration, and supersonic infall motion would be widely observed toward dense cores (see, e.g., Aikawa et al. 2005). The infalling gas motion in B335 was successfully modeled (e.g., Zhou et al. 1993; Choi et al. 1995; Evans et al. 2005, 2015) using the spherical inside-out collapse model (Shu 1977), in which the collapse starts from the end of quasi-static evolution. This is consistent with our conclusion that B335 is in a nearly equilibrium state.

We evaluated the stability parameter ${\lambda }_{\mathrm{cr}}\equiv {M}_{\mathrm{core}}/{M}_{\mathrm{cr}}\,={M}_{\mathrm{core}}/({M}_{\mathrm{mag}}+{M}_{\mathrm{BE}})$, and found that λ_cr is 1.09 for B335, which can be compared to λ_cr of 0.94 for FeSt 1-457 (Papers I, II, VI) and 0.91 for B68 (Kandori et al. 2020c). Two starless cores and a protostellar core all show λ_cr values close to unity, again supporting our conclusion that the initial condition of star formation in isolated globules may be in the condition very close to the critical state. If this is true, a slight decrease in turbulence (Nakano 1998) and/or ambipolar diffusion can initiate the onset of star formation. However, the formation mechanism of nearly critical dense cores/globules remains unknown and is an open problem.

The relative importance of magnetic fields in supporting the core against gravity was investigated using the ratio of the thermal and turbulent energy to the magnetic energy, $\beta \equiv 3{C}_{{\rm{s}}}^{2}/{V}_{{\rm{A}}}^{2}$ and ${\beta }_{\mathrm{turb}}\equiv {\sigma }_{\mathrm{turb},3{\rm{D}}}^{2}/{V}_{{\rm{A}}}^{2}=3{\sigma }_{\mathrm{turb},1{\rm{D}}}^{2}/{V}_{{\rm{A}}}^{2}$, where ${C}_{{\rm{s}}}$, σ_turb, and V_A denote the isothermal sound speed at 13 K, turbulent velocity dispersion, and Alfvén velocity, respectively. These ratios were found to be β ≈ 4.39 and β_turb ≈ 0.69, respectively. Thus, B335 is dominated by thermal support with a smaller contribution from static magnetic fields. Turbulence seems dissipated and makes a contribution comparable with or smaller than that of magnetic fields. These values were previously derived for FeSt 1-457 (β ≈ 1.27 and β_turb ≈ 0.12 with the application of the core's inclination angle; Paper I; Paper II) and B68 (β ≈ 3.15 and β_turb ≈ 0.70, Kandori et al. 2020c). Comparison of the results for all three dense cores indicates that they are mainly thermally supported with comparably smaller contributions from magnetic fields. Though the support from turbulence seems to be minor, this does not necessarily mean that turbulence is unimportant. Considering the nearly critical state of dense cores, even slight further dissipation of turbulence can initiate core contraction (e.g., Nakano 1998).

Figure 12 shows the relationship among the core's elongation axis (θ_elon ∼ 0°), outflow (θ_out ∼ 90°), and magnetic field (θ_mag = 90°) superimposed on the map obtained by submm dust emission polarimetry (Wolf et al. 2003). Like FeSt 1-457 and B68, the orientation of the elongation of the optical obscuration of B335, as well as in the H¹³CO⁺ (J = 1 − 0) line map (Kurono et al. 2013), is clearly perpendicular to the magnetic axis. This geometrical relationship is consistent with the picture of mass accretion along magnetic field lines suggested by theories (e.g., Galli & Shu 1993a, 1993b). This geometry is also consistent with the magneto-hydrostatic configuration (e.g., Tomisaka et al. 1988) as the starting condition for core contraction. The outflow axis is toward east–west (Hirano et al. 1988) on the plane of sky and is nearly perpendicular to the line of sight, with an inclination angle of 87° (Stutz et al. 2008). Though the outflow axis and magnetic axis are aligned perfectly on the plane of sky, the magnetic inclination angle of γ_mag = 50° ± 10° means that a gap probably exists between the two axes in the line of sight.

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As described in the previous paper (Kandori et al. 2018a, hereafter Paper III, see also Paper VI), the relationship between dust dichroic polarization (P) and extinction (A) in molecular clouds and their cores is important for (1) interpreting the obtained angle of interstellar polarizations, which is closely related to the direction of magnetic fields pervading the observed region, and (2) investigating dust alignment mechanisms in the dense environment. The former point is essential to ensure that our polarization observations can trace the dust alignment and magnetic fields deep inside B335. The latter point is important for the comparison of observations with dust alignment theories.

The observed polarizations toward B335 are a superposition of the polarization from the core and the ambient medium surrounding the core. The polarizations arising from the ambient off-core medium must be subtracted from the observed polarizations in order to extract the polarization associated with the core (Section 3.1). As described in Section 3.3, distorted magnetic fields surrounding the core can produce depolarization, particularly at the equatorial plane of the core. Furthermore, the line-of-sight inclination angle in magnetic axis weakens the observed plane-of-sky polarizations. These effects must be corrected in order to determine the true relationship between polarization and extinction in dense cores.

Figure 13(a) shows the observed P_H versus $H-{K}_{s}$ relationship with no correction. The stars with R ≤ 125'' and P_H/δP_H ≥ 6 were used. We did not employ the threshold of P_H ≥ 1% as used in the parabolic fit analysis in Section 3.2. Though several stars with P_H ≥ 1% were included in the plot, this does not change the important conclusions. Note that the strongest polarization vector in Figure 1 was dropped in Figure 13(a) due to the large uncertainty in P_H. In the figure, the polarization generally increases with increasing extinction up to H − K_s ∼ 1.2 mag. The slope of the relationship is 3.30 ± 0.14% mag⁻¹. The observed polarization is the superposition of the polarization from the core and ambient medium. Stars located outside of the core radius, R > 125'', were used to estimate the off-core polarization vectors. After subtraction, a relatively linear P_H versus $H\mbox{--}{K}_{s}$ relationship was obtained, as shown in Figure 13(b). The slope of the relationship is 4.67 ± 0.17% mag⁻¹, which is close to the average interstellar polarization slope (Jones 1989). Note that, since the ambient vectors are roughly north–south and the B335 vectors are roughly east–west, the slope of the P–A relationship after subtraction is greater than the original value.

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B335 is associated with inclined distorted magnetic fields, as discussed in Sections 3.1–3.3, which provides depolarization effects inside the core. Based on the known 3D magnetic field structure, a depolarization correction factor was estimated, as shown in Figure 14. This figure was created simply by dividing the polarization degree map of the distorted field model with edge-on geometry (γ_mag = 0°) by the inclined distorted field model (γ_mag = 50°). The correction factor map can thus simultaneously correct for both the depolarization and line-of-sight inclination angles. In Figure 14, the factors distributed around the equatorial plane (north–south direction) are less than unity (indicating depolarization). This is due to the crossing of the polarization vectors at the front and back sides of the core along the line-of-sight (see the explanatory illustration in Figure 7 of Kataoka et al. 2012).

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Figure 13(c) shows the depolarization and inclination corrected P–A relationship obtained by dividing the Figure 13(b) relationship by the correction factor map (Figure 14). The slope of the relationship is 12.73 ± 0.43% mag⁻¹. The final corrected slope is larger than those obtained for FeSt 1-457 (6.60 ± 0.41% mag⁻¹, Paper III and Paper IV) and B68 (4.57 ± 0.11% mag⁻¹, Kandori et al. 2020c), and it is comparable to the statistically estimated upper limit of interstellar polarization efficiency (Jones 1989, ${P}_{H}/{E}_{H-{K}_{s}}\sim 14$).

The correlation coefficients obtained in these analysis are 0.80 (ambient polarization subtraction, Figure 13(b)) and 0.87 (ambient polarization, depolarization, and inclination correction, Figure 13(c)). These are comparable to the value of 0.85 for the original relationship. For B335, our analysis did not greatly improve the tightness of the P–A correlation, but it changed the slope value.

Comparison of Figures 13(a) and (c) shows that the corrections produced a dramatic change, particularly in slope. The final result is a relatively linear relationship between polarization and extinction in the range up to A_V ∼ 15 mag. This linear relationship verifies that our NIR polarimetric observations trace the overall polarization (magnetic field) structure of B335.

Figure 15 shows the P_H/A_V versus A_V diagram. Reflecting the relatively linear slope in Figures 13(c), distributions of data points seems flat especially for A_V ≳ 5 mag. To reject too high P_H/A_V data for very low A_V, the data points with ${P}_{H}/{A}_{V}\gt 4$% mag⁻¹ were not plotted. The dotted line shows the fitting of the data using the power law ${P}_{H}/{A}_{V}\propto {A}_{V}^{\alpha }$, resulted in the α index of −0.22 ± 0.22. The relatively shallow α index indicates that the polarization (and magnetic field) structure toward B335 is traced in our observations. The dotted–dashed line shows an observational upper limit by Jones (1989). The relation was calculated based on the equation ${P}_{K,\max }=\tanh {\tau }_{{\rm{p}}}$, where ${\tau }_{{\rm{p}}}=(1-\eta ){\tau }_{K}/(1+\eta )$, and the parameter η is set to 0.875 (Jones 1989). τ_K denotes the optical depth in the K band. The polarization efficiency of B335 is quite high, as high as the interstellar upper limit value.

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