I I

Velocity Width AVpy^jM (km s_1)

Figure 3.13 Distribution of dense cores as a function of their NH3 velocity width AVfwhm . The top histogram shows those cores whose nearest star lies outside their boundaries, again as seen in NH3. The bottom histogram is for cores with embedded stars.

In particular, examination of several dozen dense cores reveals mean axial ratios of about 0.6, similar to the L1489 case shown here. This ratio, as well as the orientation of the long axis, does not vary greatly from line to line for a given core. The question now is which three-dimensional structures can yield such shapes when projected onto the sky.

As a first approximation, we assume that all cores have a single intrinsic shape, which we take to be spheroidal. Figure 3.14 shows how both oblate (flattened) and prolate (elongated) spheroids can appear to have identical axial ratios when projected on the sky. However, if we further assume that the spheroids are randomly oriented, then the oblate configurations are less likely.

The argument here is a bit technical but worth some effort. Refer to Figure 3.14, and define true and apparent (i. e., projected) axial ratios for the oblate spheroid as Rtrue = b/a and Rapp = b'/a, respectively. Note that Rtrue < Rapp < 1 for any angle i between the line of sight and the spheroid's axis of rotation. Figure 3.15 shows more explicitly the relation between b' and the true axes. It also defines a length ya, which is related to b' by b' = ya sin i. As an exercise in geometry, the reader may verify that a2 cot2 i + b2 , from which we deduce We thus find that

sin2 i

1 Rtrue oblate

Consider now the average observed ratio for a number of randomly oriented spheroids, all with identical Rtrue. Denoting by { ) the average over solid angle and recalling that observer's view prolate observer's view side view side view Figure 3.14 Projected views of oblate and prolate spheroids. Both objects generally appear as ellipses in the plane of the sky. Figure 3.15 Geometric relations within the cross section of an oblate spheroid.

(sin2 i) = 2/3, we see that no Rtrue, however small, will reproduce a given {R"ipp) unless {R-Ipp) > 1/3. The observations to date marginally satisfy this latter criterion and demand that Rtrue be about 0.2, i. e., that the putative oblate spheroids be highly flattened. Turning to the prolate case, the analogous relation to equation (3.26) is

where the axis labels are defined in Figure 3.14. Since Rtrue is still b/a but Rapp is now b/a', we find

cos21

true app

prolate

Recognizing that (cos2 i) = 1/3, the averaged form of equation (3.29) may be written