The difference with the oblate case is now apparent. The last equation yields a value of Rtrue for any (R-p2p). Moreover, the first quantity only slightly exceeds the second for modest ratios. Reproducing the observations requires only that Rtrue lie between 0.4 and 0.5. That is, the structure can be rounder than in the oblate case.
We may recast the essential argument in non-mathematical terms. Even a razor-thin disk usually presents a rather high aspect ratio, when projected randomly onto the sky. A perfectly thin needle, on the other hand, looks just the same in projection, except when it is exactly poleon. Thus, a prolate object needs more intrinsic thickness for its projected image to have a sizable apparent thickness.
Which case is more reasonable: a highly flattened disk or a thick cigar? While not demanding the rounder, prolate shapes, the observations certainly favor them. The measured line widths in dense cores indicate that they are largely, though not totally, supported by the thermal pressure gradient, which acts isotropically. In addition, the dark cloud regions containing the cores often have a striated appearance (recall Figure 1.9). In a number of cases, there is clear alignment of the cores' long axes with these "fingers" of visual obscuration. Our geometric argument indicates that the larger structures, with their extreme axial ratios, are unlikely to be flattened. The same should then be true for their embedded cores. We will see in Chapter 12 that prolate clouds are also attractive theoretically as progenitors for binary stars.
Molecular lines are not the only tools for probing the cores' structure. The dust grains coexisting with the gas also emit continuum radiation as a result of their finite temperatures. Detection of this emission at millimeter wavelengths has yielded maps of high spatial resolution. Figure 3.16 shows the starless dense core L63, situated in a relatively isolated dark cloud about 160 pc away. The left panel is a map in the 1.3 cm line of NH3, while the right is the same object viewed in 1.3 mm continuum radiation. Although the second image covers a smaller spatial extent, it displays a similar elongation as the NH3 map. The continuum version also reveals more small-scale structure. Here, the angular resolution of the telescope is 12", corresponding to 0.01 pc at the estimated distance.
Both the continuum radiation and the molecular line are optically thin in this case, and thus yield conditions in the core's deep interior. As we have indicated, line emission can only be produced when the ambient density exceeds the critical value ncrit associated with that transition. The emission also tends to fall off significantly for n > ncrit, so that any given line samples a relatively narrow density range. Dust emission does not suffer from this limitation and requires only that the density and temperature along the line of sight be high enough for detection. The continuum maps are thus potentially useful for reconstructing the internal density profile, although detailed results still require accurate knowledge of the internal temperature distribution. The studies have indicated that starless cores like L63 have densities that rise steeply from the outside, but then reach a shallower plateau somewhat above 105 cm-3.
Since magnetic forces are expected to be significant in dense cores, it would be extremely valuable if field strengths could be ascertained directly by observation in these regions. Unfortunately, this is not yet the case. To date, measurements by Zeeman splitting are limited to a handful of dark clouds of larger size and lower density. Optical and near-infrared polarization maps using background field stars, which yield the direction but not the magnitude of B, are also confined to these sparser regions.
L63 Dense Core
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