ANCHORING MAGNETIC FIELD IN TURBULENT MOLECULAR CLOUDS

The submillimeter polarimetry data we use here were obtained from two recently released archives (Dotson et al. 2009; Curran & Chrysostomou 2007) which respectively present data collected using the 350 μm Hertz polarimeter (Dowell et al. 1998) on the Caltech Submillimeter Observatory (CSO), and the 850μ SCU-POL polarimeter (Greaves et al. 2003) on the James Clerk Maxwell Telescope (JCMT). From these archives, we have selected those cores having at least five 3σ detections of polarization. For cores observed by both Hertz and SCU-POL, we find good agreement at most positions. Thus, for simplicity, we use only the Hertz data for such cores.

For each of the selected submillimeter cores, we chose a spherical "ICM region" centered at the core, and we studied the ICM magnetic field in this region using data from the optical polarimetry catalog of Heiles (2000). Since the accumulation length for a giant molecular cloud (GMC) is about 400 pc (Williams et al. 2000), and since many of our targets are dark cloud cores rather than GMC cores, our ICM regions should not be larger than 400 pc. On the other hand, to get enough stars for reliable statistics, the ICM regions should not be too small either. With these constraints in mind, we started by choosing ICM regions of 200 pc diameter. For cases where such regions contain fewer than five stars from the catalog, we expanded the diameter of the spherical ICM region to 400 pc. If this expanded region still contained fewer than five stars, then the corresponding core was eliminated from our study. Twenty five cores made our final list (Table 1), all within 2 kpc of the Sun.

Also listed in Table 1 are the means and interquartile ranges (IQRs) of the field directions inferred from the core and ICM polarimetry data. In evaluating the mean field directions, the smallest angular difference calculable between any two "vectors" is used (which is equivalent to the "equal weight Stokes mean" used by Li et al. 2006); for example, we use 20° and −10° instead of 20° and 170°. This is appropriate because polarization "vectors" are actually headless, in that they are ambiguous by 180°.

The IQR is defined as the difference between the 25th and 75th percentiles of a distribution. When there are extreme values in a distribution or when the distribution is skewed, as are the field distributions of some of the ICMs in our sample, the IQR is superior to the standard deviation as a measure of the spread. For a normal distribution, the IQR is about 1.34 times greater than the standard deviation.

Figure 1 shows submillimeter data from several cores in the Orion molecular cloud (OMC) region, together with the corresponding ICM results. The region is approximately 100 parsecs across (for an assumed distance of 450 pc). Within it, eight cores were mapped by Hertz at 350 μm, each with linear size of about 0.3 pc. It can be seen that even though the core separations exceed the core sizes by as much as a factor of 100, they are for the most part "magnetically connected," i.e., the cores' mean field directions are similar. Moreover, these directions are close to the mean field direction seen in the ICM for the OMC region (see Figure 1 and Table 1).

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The cores in the OMC region are not special cases. The mean field directions of the 25 cores in our sample (B_core) are plotted against those of the corresponding ICM regions (B_ICM) in Figure 2. The average IQR for the ICM regions (IQR_ICM) is about 52°, and this range is indicated in the figure using orange shading. Note that even if the ICM field directions were perfectly preserved during the processes of cloud and core formation, we should not expect |B_core − B_ICM| to be zero. This is because, given the very different scales of the cores and their corresponding ICM regions, one B_core can only sample a small portion of the magnetic field in the ICM regions. So only half of the B_core are expected to fall within 26° from the corresponding B_ICM because of the IQR_ICM. Figure 2 shows that almost 70% of the B_core are within 26° from B_ICM.

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Another indication of a significant correlation is that nearly 90% of the B_core/B_ICM pairs are more nearly parallel than perpendicular (i.e., nearly 90% have |B_core − B_ICM| < 45°). The probability for obtaining this correlation from two random distributions is less than 7 × 10⁻⁵. It might be argued that this calculation is flawed, since it ignores the fact that some of the cores in our sample belong to the same cloud, so their field directions might be correlated. However, note that this objection assumes that the field is sufficiently dynamically important to prevent field tangling by turbulence during the core formation process, which is the hypothesis we are currently testing.

It has been shown (e.g., Han & Zhang 2007; Reid & Silverstein 1990) that the sign of the line-of-sight component of the Galactic magnetic field is preserved in molecular clouds, using the correlation between the Zeeman splitting data of masers and rotation measures of pulsars. Our polarimetry data not only show that the components in the plane of the sky are preserved, but also put constraints on star and cloud formation theories, as described in the following section.

Ostriker et al. (2001) simulated molecular clouds with various Alfvénic Mach numbers (M_A ≡ v_t/v_A, where v_t and v_A are, respectively, turbulent and Alfvenic velocities), and the results (e.g., field orientations) were projected along lines of sight at various angles (θ) from the mean field direction. They divided the simulations into three categories: strong, moderate, and weak magnetic fields, which are corresponding to M_A ≈ 0.7, 2, and 7 for cases shown in their Figure 24. In that figure, distributions of cloud field orientations are plotted with respect to the most frequent directions. The observed data (B_core and B_ICM) are from cloud cores and ICM, while the simulated field orientations are mostly from the "bulk cloud volume," which are the regions between cores and ICM. To compare the observations with the simulations, we need to make assumptions, which are based on the fact that B_core and B_ICM are highly correlated.

• 1.  
  B_ICM is the mean field orientation in the initial condition of cloud formation, and is close to the most probable field direction in a cloud.
• 2.  
  B_core is a good representative of field directions in the bulk volume of a cloud.

With these assumptions, the offset of B_core from B_ICM is comparable to cloud field orientations with respect to the most frequent directions. A histogram of B_core − B_ICM and two field distributions (M_A ≈ 0.7 and 2; both with θ = 45°) from Figure 24 of Ostriker et al. (2001) are shown in Figure 3.

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The strong field case fits well to the central 80% (−40° to 40°) of the B_core − B_ICM histogram. The more significant "tails" of the observational data could be because some clouds have M_A > 0.7 and/or θ < 45°. Some of the disagreement is no doubt due to factors that are not included in the simulations and due to the oversimplified assumptions. For example, the observed B_ICM has a significant IQR, while the B field in the initial condition of the simulations is uniform. Also, the observed B_core values may be significantly altered by stellar feedback (e.g., H ii regions and outflows, which are not simulated) and possibly no longer correlated with the fields in the bulk cloud volumes.

On the other hand, even though some random factors are not included in these simulations, the moderate field case (M_A ≈ 2) cannot produce a distribution as peaked as the B_core − B_ICM histogram, no matter how θ is adjusted (see Figure 24 of Ostriker et al. 2001 for other θs). The kurtoses of B_core − B_ICM and the strong and moderate field distributions shown in Figure 3 are 3.7, 3.2, and 2.8, respectively. Falceta-Goncalves et al. (2008) presented similar simulations, and their super-Alfvénic (M_A ≈ 2) case shows a random angle distribution (Figure 3).

The lowest polarization fraction from a cloud core usually happens at or close to the total flux peak (see Figure 4 for an example from DR 21). This phenomenon is called a "polarization hole." Low grain alignment efficiencies in high-density environments have been proposed to explain polarization holes. Some (e.g., Padoan et al. 2001) suggested that grains cannot be aligned for A_v > 3 mag. If this were true, submillimeter polarimetry definitely could not probe the cloud cores in our sample, whose typical A_v is above 10³ mag.⁷ However, in the following, we argue that a low grain alignment efficiency is not the only possible reason for polarization holes, and that a cutoff at A_v = 3 mag is not supported by observations.

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The Chandrasekhar–Fermi method is widely used to estimate magnetic field strength (B) in molecular clouds using cloud density (ρ), turbulent velocity dispersion (σ), and field deviation (α) from a mean direction:

$$\begin{eqnarray} &&\frac{B(G)}{\sqrt {\frac{{4\pi }}{3}\rho ({\rm g}\;{\rm cm}^{ - {\rm 3}})} } = \frac{\sigma ({\rm cm}\;{\rm s}^{ - {\rm 1}})}{\alpha ({\rm rad})}. \end{eqnarray} \tag{ 1 }$$

Here we use this relation slightly differently. Given a fixed ratio of B to $\sqrt \rho$, α is proportional to σ, which roughly increases with column density (e.g., Pillai et al. 2006; Burkhart et al. 2009)⁸ Thus, α can also increase with flux. An increase of α within a telescope beam will decrease the polarization fraction as shown in the following. Assume that the degree of polarization is a constant P₀ for α = 0 (i.e., uniform grain alignment efficiency and field orientation). For simplicity, we also assume that field orientations in a telescope beam are evenly distributed from θ − α to θ + α, with a mean field direction θ and angle deviation 0 < α < π/2. The mean Stokes parameters of this group of vectors are

$$\begin{eqnarray} &&q = \frac{{P_0 }}{{2\alpha }}\int_{ - \alpha }^\alpha {\cos 2(\theta + \phi)d\phi } {\rm,}\;{\rm and}\;u = \frac{{P_0 }}{{2\alpha }}\int_{ - \alpha }^\alpha {\sin 2(\theta + \phi)d\phi },\nonumber\\ \end{eqnarray} \tag{ 2 }$$

and the mean polarization fraction can be shown as

$$\begin{eqnarray} &&\bar P = \sqrt {q^2 + u^2 } = P_0 \frac{{\sin 2\alpha }}{{2\alpha }} = P_0 \frac{\sin\, 2\sigma /c} {2\sigma / c}, \end{eqnarray} \tag{ 3 }$$

where c is $B /\sqrt {4\pi \rho /3}$. Zeeman measurements (e.g., Vallée 1997, Crutcher 1999) show that c can be fitted to within the errors by a constant. Though this correlation is approximate because only one component of B is measured (and possibly some other reasons; see Basu 2000 and Myers & Goodman 1988), it should still be good enough to reveal a trend of $\bar P$ versus σ, i.e., a polarization hole. Overlapped on the $\bar P$–flux plot of DR 21 in Figure 4 is a $\bar P$–σ plot from the same sky positions. We note that the σ data in Figure 4 are based on HCN (4-3) lines (Kirby 2009), which could be optically thicker than the 350 μm dust emission, from which the polarimetry data are derived. So the correlation may change when optically thinner lines are used for σ, and the trend may be enhanced. Kirby (2009) estimated the sky component of the mean field in this region as 2.5 mG, and Jakob et al. (2007) estimated the n(H₂) as 10⁶ cm⁻³. Based on these two parameters, Equation (3) is also plotted in Figure 4, assuming a mean molecular mass of 2.3 and P₀ = 3%, which is estimated by the mean of the polarizations from the lowest 10% of the flux with 3σ polarization detections from DR 21 Main (Dotson et al. 2009).

An interesting fact is that submillimeter polarization holes show at very different scales: ∼10 pc (Li et al. 2006), 0.1–1 pc (this work), and below 0.1 pc (e.g., Girart et al. 2006). The average A_vs from these scales are different in order of magnitude, so switching off grain alignment above one particular A_v cannot explain polarization holes from all these scales. Even when one ignores the effect of velocity dispersion discussed above and charges polarization holes completely to grain alignment efficiency, A_v = 3 mag is still too low for a cutoff on alignment efficiency to explain many observations. In our sample, the mean polarization is 2.55% for the 955 3σ detections from the Hertz archive. With the maximum possible polarization around 10% (Hildebrand & Dragovan 1995) and the typical A_v for our samples over 10³ mag, 2.55% requires grains to be aligned to as deep as A_v = 255/2 mag, assuming α = 0 and that the polarized mass is evenly distributed to the near and far sides of a core. So, even A_v = 125 mag is a conservative estimate of the cutoff point. Interferometer polarimetry also needs to be explained. Take NGC 1333 IRAS 4A for an example, the 800 μm flux from the central 33 arcsec² region is about 6 Jy according to the Submillimeter Array (SMA) observation (Girart et al. 2006) and is approximately 11 Jy from the JCMT (Sandell et al. 1991; scaled down from 12.9 Jy over 39 arcsec²). The 5 Jy missing flux filtered out by the SMA is corresponding to a fore/background column density N(H₂) around 7 × 10²³ cm⁻² (Girart et al. 2006), which is about A_v = 700 mag (Harjunpää et al. 2004). Therefore, the grains are aligned deeper than A_v = 350 mag.

Our goal is to distinguish clouds between globally super- and sub-Alfvénic by comparing the field orientations from the two ends of the density spectrum within a bulk cloud volume (Section 3.2). For this purpose, knowing that grains are aligned to at least hundreds of magnitude in A_v is good enough, because even for a GMC, the typical A_v of the bulk volume is in the order of 10 mag (McKee & Ostriker 2007).

Any single polarization measurement gives the mean field direction along the line of sight, weighted by the density and the alignment efficiency of dust particles. In principle, the optical polarimetry data will only reliably give the field orientation in each ICM region if we first correct these data for the foreground contribution to the polarization.

In this work, we assume that the foreground effects are not dominant, and make no correction for foreground polarization. Our reasons for ignoring foreground effects are two-fold. First, accurate foreground subtraction requires one to identify a significant number of "foreground stars" lying very close in the sky to each "background star" whose polarization is attempting to correct. Starting from the limited number of stars in the Heiles catalog, this is often not possible. For example, in our earlier work (Li et al. 2006), we searched for foreground stars for each background star, and we imposed the following restrictions on foreground correction: (1) there should be more than three foreground stars within 15 of each background star; and (2) they should lie approximately one accumulation length (400 pc) closer to us in comparison with the background star. With these restrictions, only 10 out of 77 nearby regions studied were found to have five or more background stars that could be corrected for foreground polarization.

Our second reason for ignoring foreground subtraction is that during the course of carrying out the work reported in Li et al. (2006), we noticed that foreground subtraction typically tends to reduce the polarization fraction while not introducing significant changes to the polarization direction. This effect agrees with Figure 1 of Andreasyan & Makarov (1989), where the degree of optical polarization for stars located within 2 kpc tends to increase with the distance, even though the polarization direction stays nearly constant, for most sight lines.

Most importantly, our conclusion that core fields are correlated with ICM fields does not depend on the assumption that foreground polarization is negligible. To see this, assume that indeed our B_ICM values are primarily measuring the foreground polarization. In this case, Figure 2 implies that the B_core values are correlated with ICM fields over distances larger than the accumulation length. It is hard to imagine how this could be possible unless the B_core values were also correlated with the ICM fields within an accumulation length surrounding the cloud.

In other words, foreground effects merely add noise to Figure 2. If one assumes that the foreground effects must be important, then this noise must be large. In this case, the underlying correlation between the B_core values and the ICM fields within an accumulation length surrounding the cloud must be even stronger, so that one can obtain the observed correlation of Figure 2 despite the large noise level.