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Astron. Astrophys. 344, 668-674 (1999)

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5. Discussion

5.1. Grain alignment in molecular clouds

Some general conclusions can be drawn from the above results. Firstly, the PTEAR mechanism has been shown to be less important than grain alignment in producing submillimetre polarization. The W3-IRS4 data indicate p(PTEAR)/p(magnetic) [FORMULA] 0.7, even for a very bright illuminating source. Another possible non-magnetic mechanism is scattering, which produces polarization in the optical/near-IR (e.g. Bastien 1996), but this is unlikely to be effective in the submillimetre, except in unusual sources where many of the grains are millimetre-sized.

Alignment mechanisms are made more effective if the grains are spun up to suprathermal speeds, but the mechanism using cosmic ray absorption is inconsistent with the Mon R2 data. Spin-up via radiative torque on asymmetrical grains is more promising, and the [FORMULA] Oph data suggest that radiation may even directly align grains. This would imply that it is not the magnetic field alone that determines the net grain orientation, and thus magnetic field maps deduced from the perpendiculars to polarization vectors (such as Fig. 1) could be misleading.

Other proposed grain alignment mechanisms include superparamagnetic and mechanical processes. Superparamagnetism occurs if small clusters of ferromagnetic or ferrimagnetic material exist in the grains (Draine 1996; Mathis 1986), but Roberge (1996) has found that unrealistically large temperature differences are required (e.g. [FORMULA] [FORMULA] 5 [FORMULA] [FORMULA]). In contrast, mechanical alignment is probable in our sources. Grains can be aligned by collisions with gas molecules, if the gas-grain drift speeds are supersonic (Gold 1951; Roberge et al. 1995; Lazarian 1996b). Mon R2, DR21 and W3 all contain extended outflow systems (Tafalla et al. 1997; Garden et al. 1991; Mitchell et al. 1991), where supersonic flow occurs within the clouds. In a related paper we discuss evidence for polarization correlated with gas-grain drift speeds (Greaves & Holland 1999).

We conclude that mechanical and radiative alignment mechanisms are the most consistent with the data, while superparamagnetism, scattering and PTEAR effects are not significant.

5.2. Wavelength dependence and grain populations

We have also used multi-wavelength observations to search for variations in polarization levels. These are unexpected in the submillimetre, because for optically thin sources in the Rayleigh-Jeans part of the blackbody spectrum, the flux density is proportional to [FORMULA] (where [FORMULA] is the frequency, and T and [FORMULA] are the grain temperature and opacity), and thus the flux density ratios are independent of wavelength. Then considering a mixture of two grain types, the net polarization can be expressed as


(Hildebrand 1988), where S and p are the flux densities and intrinsic polarization levels of the grain populations, numbered 1 and 2. (Note that this expression assumes that all the grains are aligned by the same magnetic field, so the p vectors can be added linearly.) Then if S1/S2 is constant in Eq. (1), [FORMULA] is constant, and measured values should be the same at all wavelengths.

However, in many sources the grains behave instead as greybodies, i.e. the opacity is frequency-dependent. In this case, the flux density will be proportional to [FORMULA], where [FORMULA] is the grain opacity index. Then if there are two grain populations with different [FORMULA] indices, S1/S2 is not constant, and [FORMULA] can be wavelength dependent. For the two population case, the difference in observed [FORMULA], for two wavelengths a and b, can be expressed as


where R is the flux density ratio S1/S2. Thus the observed change in polarization with wavelength, [FORMULA], depends on the two intrinsic p values and the two flux density ratios. Alternatively, this can be expressed as a dependence on the difference in beta indices, since [FORMULA] = [FORMULA].

The expression above shows that p could be wavelength dependent in any source with mixed grains. To determine whether such effects are common, we have searched for published JCMT polarimetry data on dust sources at wavelengths of 1100 and 800 [FORMULA]. Fig. 3 shows the distribution of the resulting fifteen values of p(1100)-p(800).

[FIGURE] Fig. 3. Distribution of p(1100)-p(800). The dashed portions of the histogram show sources where the polarization difference is less than its error. Data are from this work; Greaves & Holland (1998); Greaves et al. (1997, 1995, 1994); Holland et al. (1996); Minchin et al. (1995); Tamura et al. (1995); Vallée & Bastien (1995); Minchin & Murray (1994); Flett & Murray (1991).

Of these fifteen data points, eight are statistically significant (sources within the solid histogram bars). Of these, seven have p(1100) [FORMULA] p(800), and one source has p(800) [FORMULA] p(1100). Therefore in the majority of cases p increases with wavelength, while in most of the remaining sources the p-differences are small or not significant above the error. It could be argued that p(800) and p(1100) differ because of resolution effects, as the beam sizes ranged from 14" to 19". However, there is no general trend of p(1100)-p(800) with source distance, as would be expected if complex field structure were better resolved in closer sources.

We now compare the observed [FORMULA] values for 1100 and 800 [FORMULA] with the values that can be derived using Eq. (2). The range of [FORMULA] values used was 0 to 2, corresponding roughly to the observed range in interstellar clouds. Fig. 4 shows the model results for two grain populations, in a plane of p1-p2 versus [FORMULA]-[FORMULA]. The observed mean p(1100)-p(800) is 0.48 [FORMULA] 0.33% (standard error of the fifteen observations), and this value is reproduced in the area within the curved lines. The required grain characteristics are p1-p2 [FORMULA] 1% and/or [FORMULA]-[FORMULA] [FORMULA] 0.3. An additional effect could arise if one of the grain populations is very cold, so that the emission is not in the Rayleigh-Jeans tail and the fluxes are proportional to [FORMULA](T), not T. However, even for 10 K grains the required p1-p2 differences are large ([FORMULA] 7% for p(1100)-p(800) = 0.48%), so this effect is likely to be less significant.

[FIGURE] Fig. 4. Plot of the p1-p2, [FORMULA]-[FORMULA] plane showing the region corresponding to the observed mean p(1100)-p(800) value. For two grain populations with equal flux densities at 800 [FORMULA], this region lies between the solid curved lines, and both bounds are moved upwards for other flux ratios. An upper boundary is indicated by the dashed line at 6.5% polarization, which is the maximum p(800) seen in observations (Minchin & Murray 1994).

Fig. 4 shows that the grain population with the larger intrinsic polarization, p1, must have a smaller opacity index ([FORMULA]). This is in good agreement with previous studies: elongated grains can have a low value of [FORMULA] (Beckwith & Sargeant 1991), and also be more efficiently aligned (Roberge 1996). However, it should be noted that [FORMULA] indices may also reflect grain size and composition (Beckwith & Sargeant 1991; Mannings & Emerson 1994).

The published JCMT data also show some wavelength dependence in the polarization directions. Of the fifteen sources in Fig. 3, thirteen have position angles determined at both wavelengths, and of these seven have a [FORMULA](800)-[FORMULA](1100) difference that is significant at the 1[FORMULA] level or more. For all thirteen sources, the mean [FORMULA](800)-[FORMULA](1100) is [FORMULA] (standard error).

The grain mixture hypothesis is also relevant to variations in [FORMULA] (although some of the larger differences are best explained by unresolved field structure, e.g. Greaves & Holland 1998). At any particular wavelength the polarization is biased towards the grain population with the highest fraction of the polarized flux. Then if the grain populations are not cospatial, and the field is not uniform along the line of sight, different [FORMULA] can result at different wavelengths. Since grain properties are expected to vary with position in a cloud (e.g. Goodman 1995), it is likely that grain populations will indeed be non-cospatial. Alternatively, the grains could be mixed but one population might be aligned radiatively (see Sect. 4.3), and thus [FORMULA] (and [FORMULA]) changes would occur at the wavelength where these grains dominate the flux. This hypothesis would suggest larger changes with more illumination, but a counter-example occurs in DR21, where the south point has more marked p,[FORMULA] variations with wavelength than the centre position. Further modelling is needed to determine the importance of this effect.

Finally, to determine if the grain mixture hypothesis is really applicable to molecular clouds, we have searched the literature for [FORMULA] indices of the sources in Fig. 3. Values were found for twelve of the sources (not the three positions in DR21), and the average [FORMULA] is 1.3 with a 1[FORMULA] range of [FORMULA] 0.4. This is reasonably consistent with our requirement for mixed [FORMULA] values, as the overall [FORMULA] value should then be intermediate between 0 and 2.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999