5 Discussion

   
5.1 Comparison to theoretical estimates of the B field

SO and OH results

Several basic formulations have been developed to estimate magnetic field strengths in clouds. All are based on meaningful physical footings, but approach the problem differently in the way assumptions are made about the equipartition of energies and source geometries. The various formulations require different ``observables'' ([ $N_{\rm H_2}$], [ $n_{\rm H_2}$, $\Delta V$], and [$\Delta V$, R]), some obtained by independent means. It is thus worthwhile to review all three theoretical methods and to compare the range of estimates they produce to the measured field limits.

In the absence of other large-scale supportive mechanisms (such as turbulence), the critical magnetic field needed to support a self-gravitating cloud against collapse is,

$${ B_{\rm crit} = 8 ~ 10^{-21}\;N_{\rm H_2}({\rm cm}^{-2})\phantom{xx}\mu {\rm G} ; }$$ (2)

this based on the derivation by Mouschovias & Spitzer (1976), where the expressions were tuned to the results of exact numerical models of initially uniform, spherical magnetic clouds, and using a mean molecular mass of $\mu = 2.33\,m_{\rm H}$.

A second means of estimating the magnetic field in a cloud is by its ``nonthermal'' velocity dispersion. There have been a number of mechanisms suggested to explain the nonthermal component, or that in excess of the thermal contribution to the linewidth, often observed in lines toward molecular clouds. Stellar winds or outflows, were among the first to be cited (Norman & Silk 1980; Larson 1981; Zuckerman & Evans 1974). Upon further consideration, however, such inputs are now believed to be relatively ineffective since the supersonic turbulence they produce should be quickly dissipated in shocks. Instead, long-wavelength and large-amplitude MHD waves within the cloud are thought to better fit the role, since these waves are long-lived so long as their propagation speed is below the local Alfven velocity. Thus, a means of estimating the upper limit to the B field is to compare the nonthermal contribution of a cloud's velocity dispersion, $\sigma_{\rm nt}$, to its Alfven velocity, $V_{\rm alf} = B/(4\pi\mu\,n_{\rm H_2})^{1/2}$, where $n_{\rm H_2}$ is the volume density. Following Myers & Goodman (1988), we adopt $\sigma_{\rm nt} = \sqrt{3}\cdot V_{\rm alf}$, leading to ( $\mu = 2.33\,m\rm _H$)

$${ B_{\rm alf} = 1.2\;n_{\rm H_2}^{0.5}(\hbox{cm$^{-3}$ })\;\sigma_{\rm NT}(\hbox{km$\,$ s$^{-1}$ })\phantom{xxx}\mu {\rm G}. }$$ (3)

The nonthermal velocity dispersion is taken to be the difference, in quadrature, between the observed velocity dispersion and the thermal component

$${\rm \sigma_{nt}^2 = \sigma_{obs}^2 + \sigma_{th}^2 = \frac{\Delta {\it V}_{obs}^2}{8\;ln2} - \frac{\it kT}{\mu_{obs}}, }$$ (4)

where $\Delta V_{\rm obs}$ is the FWHM linewidth of the observed molecule, $\mu_{\rm obs}$ is the molecule's mass, and T is the kinetic temperature of the gas. For the case of the SO emission lines, the thermal contributions to the FWHM linewidths are small ( $\Delta V_{\rm th} = 0.03~ T^{0.5}$) due to the relatively large mass of this molecule and the nonthermal component dominates in all sources; for the narrow CCS lines the correction is significant.

Another means of estimating the magnetic field in a spherical cloud with negligible thermal support is by assuming the conditions of both magnetic and virial equilibrium in the cloud. The resulting relation is,

$${ B_{\rm vir} = 15\,\alpha ^{-1} \Delta V^2 R^{-1}\; \mu {\rm G}, }$$ (5)

where R is the radius of the cloud in parsecs, $\Delta V$ the (non-thermal) FWHM of the line in km$\,$s^-1, and $\alpha$ a scaling factor (ranging somewhere between 1.1 and 1.3; Myers & Goodman 1988; McKee et al. 1993).

In Cols. 9-11 of Table 1, we provide the critical magnetic field estimates, determined using the three previously discussed methods, for the twelve SO/OH sources observed in this study using the ``observables'' from Cols. 2-5 ($T_{\rm K}$, d(size), $n_{\rm H_2}$, and $N_{\rm H_2}$). The critical field estimates $B_{\rm crit}$, $B_{\rm alf}$, and $B_{\rm vir}$ for any given source vary by as much as a factor of a few with G10.62 being the extreme case with a $B_{\rm crit}$ and $B_{\rm vir}$ that differ by a factor of seven. While the $B_{\rm crit}$ formulation generally gives the largest critical field results (Table 1, Col. 9) of the three formulations, it is the most straightforward requiring the fewest assumptions on equipartition and source geometry and requiring only an estimate of the source column density ( $N_{\rm H_2}$), the most reliable of our observed parameters. We therefore adopt the $B_{\rm crit}$ values for comparison to our observational $B_{\scriptsize {\parallel}}$ field limits and for the discussion to follow.

In all but one (Orion-KL) of the eight SO sources for which we were able to calculate the $B_{\rm crit}$ field values (Table 1, Col. 9) the observationally determined 3$\sigma$ $B_{\scriptsize {\parallel}}$ field limits (three times the value of Col. 8 in Table 1) fall well below their corresponding $B_{\rm crit}$values. Even if our SgrB2(N) data does indicate a $B_{\scriptsize {\parallel}}$ field of 1.2 mG, which we very much question (Sect. 4.2), it is still well below the $B_{\rm crit}$of $\sim$8 mG of estimated for this region. Moreover, while we conservatively use 3$\sigma$ values as the upper limit to the $B_{\scriptsize {\parallel}}$ field in these regions, we are, in fact, confident that the V-spectra show no magnetic field signatures down even at the 1$\sigma$ level. If we were to go by the less stringent 1$\sigma$ criteria, then in no instance would the observed $B_{\scriptsize {\parallel}}$ magnetic field limits exceed the critical B field determinations for the sources, whether it is that of $B_{\rm crit}$, $B_{\rm alf}$ or $B_{\rm vir}$.

The 3$\sigma$ observed limits for the H$\,$ II regions G10.62 and DR21(OH) ( $B_{\parallel} = 3.6$ and 6.0 mG, respectively), determined from the OH $^2{\rm\Pi}_{\rm 3/2}\;\; J=7/2$ observations, also fall far short of their $B_{\rm crit}$ values, implying supercriticality in these clouds. Furthermore, the observations indicate no fields comparable to that of W3(OH), found in our earlier study. The SO observations of G5.89 and G10.62 provide even tighter constraints on the fields in H$\,$ II regions, with 3$\sigma$ $B_{\scriptsize {\parallel}}$ limits averaging 1.6 mG. Given the results of this survey, the W3(OH) region - earlier reported to have a line-of-sight B field of $\sim$3 mG in its dense thermal gas (Güsten et al. 1994) - appears to be the exception, rather than the norm, in terms of its magnetic field.

The outflow sources NGC2071A and VLA1623 are the few of its kind to have been observed in Zeeman studies. Their observations also provide among the lowest field limits ( $B_{\scriptsize {\parallel}}$(3$\sigma$) = 0.3 and 0.75 mG) in our study. Unfortunately, we have little physical information on the NGC2071A and VLA1623 regions - either from our observations or the literature - and, therefore, no means of determining any of the critical B field values ( $B_{\rm crit}$, $B_{\rm alf}$ or $B_{\rm vir}$) by any of the methods. While the SO lines measured toward the other eight sources consistently agree with those of other high density tracers, implying that this transition traces hugh volume and column densities, according to chemical/shock models (Neufeld & Dalgarno 1989; Mitchell 1984; Harquist et al. 1980) and some observations (Martin-Pintado et al. 1992), compounds such as SO and SiO may also be strongly enhanced in post shock environments. This is likely the situation toward VLA1623 and NGC2071A where the SO emision is enhanced along the edge of the outflow. The critical field determinations of these regions and their comparison to our measured field values is thus deferred until the source parameters are better known.

CCS results

In those dark cloud cores observed by their CCS transition, the thermal linewidth components correspond to the kinetic temperatures derived from the NH₃ excitation studies (Fiebig 1990), suggesting that thermal support plays a major role in the stability of these cores. In such case, the relations for $B_{\rm crit}$ and $B_{\rm vir}$ discussed previously (Eqs. (2) and (5), respectively) do not apply. We therefore take a somewhat different approach from above in estimating and comparing to the critical field limits of these clouds.

In Table 2 we characterize the dense cores by their virial expressions for the gravitational energy ${{\cal W} = (3/5)aGM^{2}/R}$, the kinetic energy ${\rm {\cal T} = (3/2){\it M}\sigma^{2}_{H_{2}}}$, and the magnetic energy ${{\cal M} = (1/3)bB^{2}R^{3}}$ (e.g. Crutcher 1999, with a = 1.2 and b = 0.3 following Mckee et al. 1993). The physical parameters of the clouds are derived from NH₃(Fiebig 1990) and C₃H₂ studies (Cox et al. 1989; Cox, priv. comm.). The Jeans, $M_{\rm J}$, is presented in Col. 6 and the thermal ( $\sigma_{\rm (H_2)}$) and nonthermal linewidths ( $\sigma_{\rm nt}$) in Cols. 7 and 8, respectively.

Within the uncertainties, the mass of the cores compare to their Jeans mass, the ratio of the kinetic to gravitational energy $2{\cal T/W}$ is $\sim$1, suggesting that the cores are evolving in approximate equilibrium. If we assume the non-thermal velocity dispersion, $\sigma_{\rm nt}$ (Col. 8), provides some measure of the turbulent field component, $B_{\rm alf}$, then $2{\cal T/M}_{\rm alf} \sim\ 42 - 92$ (Col. 12). Our observational limits to the large scale field strength, $B_{\parallel}$, as sensitive as they are, do not allow a critical assessment of any large scale field on the energetics of the core since $2{\cal T/M}_{\rm obs} \sim 1$ (Col. 13) (assuming $B^{2} = 3^{\star}B_{\rm par}$ - a statistical approach that may not apply to the small number of cores observed), i.e., $B_{\parallel}$ exceeds $B_{\rm alf}$ by a factor of $\sim$6.

The large scale magnetic field

The $B_{\scriptsize {\parallel}}$ upper limits found in this study, most being well below the critical B field values determined, tend to support the assertion that the magnetic fields in these dense clouds are supercritical (not capable of supporting the cloud). However, as with all Zeeman studies we give the cautionary notice that the sample is limited and there are a number of factors involved with Zeeman measurements that can cause the underestimation of the actual B field strength or avoid detection entirely, such as, (1) the alignment of an ordered magnetic field component out of the line of sight of the observer, and (2) substantial tangling and sign reversals of the field within the beam or region sampled by the beam. Nevertheless, we do not believe that in this study these effects significantly alter our results or conclusions.

Because Zeeman line measurements measure the component of the B field parallel to the line of sight $B_{\parallel} = B \cdot \cos (\theta)$, one can only infer the total field strength statistically and there is the risk of missing an ordered field, even a strong one, if oriented sufficiently close to within the plane of the sky. The total field (B), or total field limit, is expected to be twice as large on average for a Zeeman derived sample of randomly oriented fields. It is, however, highly unlikely in a sufficiently large sample that all the fields observed are fortuitously aligined such that they avoid detection entirely. Adopting the statistical methods of Crutcher et al. (1993), we find that the probability of not detecting any field above the 3$\sigma$ levels obtained, assuming total field values ( $B_{\rm tot}$) at the expected $B_{\rm crit}$ levels, is only 0.04%.

Another concern is if the field has complex structure or numerous sign reversals within the beam, leading to a diluted average of its actual value or a nondetection. Beam averaging is especially a consideration in HI Zeeman studies where the beamsizes are typically very large and the diffuse gas that is being probed is extended, often filling the entire beam. In this study, however, the CS and NH₃ clump sizes (Table 1, Col. 3), which we take as roughly representative of the SO clump sizes we observe, are far smaller than the antenna beam FWHM of 63''. We thus avoid averaging over large spatial areas (few pcs), but only on the scales of the H$\,$ II regions, dense cloud cores, or accretion disks themselves (all sub-parsec except for Sgr B2(N)).

Unfortunately, very little is known regarding the typical coherence scale of magnetic fields on the level of dense molecular clouds and their dense sub-cores. Submillimeter and far infrared polarization studies, which infer the field direction by the alignment of grains, are one of the few means of detailing the fields on the scales of protostars (Holland et al. 1995) and dense condensations. Typical polarization studies have beamsizes of the order 15'' (the JCMT beam at 800 $\mu$m), comparable to those of the clumps sampled by this study. Such studies often infer hour-glass shaped fields centered on the condensation, consistent with theoretical treatments and the idea of an originally large scale ordered field that is tied to and that is being concentrated along with the collapsing cloud/subclump. Non-zero polarization values toward the centralmost or ``pinch'' areas imply coherence even in the smallest observable regions. Recent polarization studies suggest a few cases of sharp field reversals, speculated to occur at the boundary of the dusty stellar disk. We are, however, insensitive to any changes that may occur at or within the dusty stellar disk since the gas densities there are far in excess of that being sampled here.

We concede that the SgrB, the active starforming region in the Galactic center, is perhaps the one case where we may be failing to detect the field due to sign reversals in the beam. In a H I Zeeman line study performed with the VLA (effective resolution of 10''-20''), Crutcher et al. (1996a) detected a line-of-sight field of $B_{\parallel} = -0.5$ mG that varies by about a factor of $\sim$50% across the 1'-diamter complex. Our failure to detect a field toward the SgrB2 region ( $B_{\parallel} < 1.8$ mG) is not particularly surprising given that this is the one instance where the emission from the dense gas almost certainly fills our entire 63''beam. The extremely large linewidths observed from some molecules toward the SgrB region ($\sim$60 km$\,$s^-1) reflect tremendous inputs of mechanical energy into the gas by means of multiple outflows, expanding shells, etc, and thus great potential for modification of the field.

5.2 Summary

Much effort has been expended over the last decade to measure the magnetic flux to mass (or column density) ratio in dark clouds. The relatively diffuse clouds (<10³ cm^-3), or cloud envelopes traced by OH (1665 and 1667 MHz) and $\lambda$21 cmH I line emission, appear to be critical or at least approximately so in terms of support by their B fields against collapse (Roberts et al. 1995; Troland et al. 1995; Crutcher et al. 1993; Goodman et al. 1989). The matter is potentially different toward the moderate density clumps ( $n_{\rm H_2} < \,\sim\!10^{4}$ cm^-3), where in the handful of sources sampled thus far, the majority of them (Ori-KL and S106, Crutcher et al. 1996b; B1, Crutcher et al. 1993; $\rho$ Oph, Troland et al. 1996; SgrB2(N), Crutcher et al. 1996a) have line of sight fields, $B_{\parallel}$, that on the average fall a factor of $\sim$3 or so below the critical limit. With the stated uncertainties in mind, we maintain that the observed $B_{\scriptsize {\parallel}}$ field limits determined in this study add weight to that implied by the earlier handful of data points, that the magnetic fields in dense clouds are supercritical or not capable of supporting those clouds against collapse.

There is a great need for more Zeeman observations, coupling to high gas densities, to assess the role the fields play in the late stages of the stellar formation process. Few radicals are known to be suitable and while Crutcher et al. (1996b, 1999) explored the use of CN, we explored the potential of SO and CCS. Despite using the most sensitive telescope available, a dedicated receiver, and spending many hundreds of hours of observation/integration time, we failed to detect any field via the SO and CCS transitions. It is difficult to forsee any technological advancements in the near future that will allow us to approach the critical sensitivity regieme with SO and CCS Zeeman line splitting observations. The more promising approach appears to be follow-up observations of CN, as is being done by Crutcher et al.