«ABSTRACT The shapes of isolated Bok globules and embedded dense cores of molecular clouds are analyzed using a nonparametric kernel method, using the ...»
The Shapes of Dense Cores and Bok Globules
Barbara S. Ryden 1
Department of Astronomy, The Ohio State University,
174 W. 18th Ave., Columbus, OH 43210
The shapes of isolated Bok globules and embedded dense cores of molecular clouds
are analyzed using a nonparametric kernel method, using the alternate hypotheses
that they are randomly oriented prolate objects or that they are randomly oriented oblate objects. In all cases, the prolate hypothesis gives a better ﬁt to the data. If Bok globules are oblate spheroids, they must be very ﬂattened; the average axis ratio is γ ≈ 0.3, and no globules can have γ 0.7. If Bok globules are prolate, their ∼ intrinsic ﬂattening is not as great, with a mean axis ratio γ ≈ 0.5. For most data samples of dense cores embedded within molecular clouds, the randomly-oriented oblate hypothesis can be rejected at the 99% one-sided conﬁdence level. If the dense cores are prolate, their mean axis ratio is in the range γ = 0.4 → 0.5. Analysis of the data of Nozawa et al. (1991) reveals that dense cores are signiﬁcantly diﬀerent in shape from the clouds in which they are embedded. The shapes of dense cores are consistent with their being moderately ﬂattened prolate spheroids; clouds have ﬂatter apparent shapes, and are statistically inconsistent with a population of axisymmetric objects viewed at random angles.
Subject headings: ISM: clouds – ISM: globules – ISM: structure
1. Introduction The molecular gas in our galaxy shows structure on a wide range of scales. The largest structures detected in the molecular gas are giant molecular clouds (GMCs), of which the largest have diameters of ∼ 100 pc and masses of more than 106 M. High resolution studies of molecular clouds, however, reveal that they have internal structure on all scales, and are typically clumpy or ﬁlamentary. Direct imaging of CO in nearby clouds reveals structure on all scales down to lengths of ∼ 0.01 pc and masses of ∼ 0.01 M (Falgarone, Puget, & P´rault 1992; Langer et al. 1995).
e Studies of the time variability of absorption lines indicates the presence of structure in the dense 1 National Science Foundation Young Investigator; email@example.com –2– gas on scales down to lengths of ∼ 5 × 10−5 pc and masses of ∼ 5 × 10−9 M (Marscher, Moore & Bania 1993; Moore & Marscher 1995).
In this hierarchy of sizes, however, not all scales are of equal interest to astronomers. Stars form by gravitational collapse of dense regions within molecular clouds; much interest is therefore focused on the scale corresponding to the mass of protostars. Surveys of nearby clouds (within 500 parsecs of the earth) have focused attention on dense cores with typical diameters of ∼ 0.1 pc, masses of ∼ 30 M, and densities of ∼ 2 × 104 cm−3 (Myers, Linke, & Benson 1983; Benson & Myers 1989). In addition to dense cores embedded within GMCs, our galaxy also contains isolated dense clouds known as Bok globules (Bok & Reilly 1947). Neither dense cores nor Bok globules are spherical, as a general rule. The projected axis ratio, q ≡ b/a, of small Bok globules has an average value q ≈ 0.6 (Clemens & Barvainis 1988, hereafter CB; Bourke, Hyland, & Robinson 1995; hereafter BHR). Embedded dense cores are similarly ﬂattened, with q ≈ 0.5 − 0.6 (Myers et al. 1991; Tatematsu et al. 1993).
The projected axis ratios of globules and cores are of interest because they place constraints on the intrinsic shapes of these objects. The intrinsic shapes of isolated globules and embedded cores are determined by a variety of physical processes. Globules and cores are sculpted by the self-gravity of the gas which they contain and by thermal pressure, turbulent pressure, and magnetic pressure. Since star formation occurs in these dense regions, their physical properties are further modiﬁed by outﬂows and winds from the protostars which may be embedded within them.
Recent optical surveys of Bok globules (CB; BHR; Hartley et al. 1986) and millimeter surveys of dense cores (Benson & Myers 1989; Loren 1989; Lada, Bally, & Stark 1991; Nozawa et al. 1991) provide data sets of apparent axis ratios. As a cautionary note, however, it should be pointed out that a core or globule is not a solid, well-deﬁned object with easily measurable axis ratios. It is merely one scale in a hierarchy of structure. The shape of a dense core as deﬁned by the emission of one molecular line, moreover, will not be precisely the same as its shape deﬁned by the emission of a diﬀerent molecule. Readers are advised to regard the shapes measured in surveys with a certain amount of skepticism.
Previous studies (David & Verschueren 1987; Myers et al. 1991; Fleck 1992) have placed constraints on the permitted intrinsic shapes of dense cores. For instance, the mean apparent axis ratio q ≈ 0.5 − 0.6 for cores is fully consistent with a population of prolate objects, but only marginally consistent with an oblate population (Myers et al. 1991; Fleck 1992).
More sophisticated analytic tools for examining sets of axis ratios have now been developed by investigators studying the intrinsic axis ratios of elliptical galaxies and other stellar systems.
For instance, starting with a nonparametric estimate for the distribution f (q) of apparent axis ratios, it is possible to ﬁnd an estimate for the distribution N (γ) of intrinsic axis ratios, given the assumption that the systems considered are either oblate or prolate spheroids and are randomly oriented with respect to the observer (Tremblay & Merritt 1995; Ryden 1996).
Similarly, one can test the hypothesis that cores or globules are randomly oriented oblate objects. The available kinematic information, however, indicates that dense cores are not rotationally supported oblate spheroids (Goodman et al. 1993). Large scale maps also show that many dense cores are aligned with larger ﬁlamentary structures, indicating that a prolate geometry is more likely in such cases (Myers et al. 1991). Thus, it should not be surprising when the randomly oriented oblate hypothesis turns out to be emphatically rejected for samples of dense cores.
In section 2, I give a brief outline of the mathematical techniques used to ﬁnd the distribution of intrinsic axis ratios, given either the prolate or oblate hypothesis – a fuller description, for those who desire it, is given in Ryden (1996). In section 3, I apply these techniques to Bok globules, and in section 4, I apply them to dense cores embedded within molecular clouds. Generally speaking, for each data set considered, the prolate hypothesis gives a better ﬁt than does the oblate hypothesis, given the assumption that the globules and cores are randomly oriented. In section 5, I discuss the implications of this ﬁnding for the origin and evolution of dense cores and globules.
If the globules or cores in the sample are all randomly oriented oblate spheroids, then the ˆ estimated distribution NO (γ) for the intrinsic axis ratio γ is given by the relation
If the globules or cores are assumed to be randomly oriented prolate spheroids, then the estimated ˆ distribution NP (γ) for the intrinsic axis ratio is
The conﬁdence intervals derived from bootstrap resampling represent only the error resulting ˆˆ ˆ from ﬁnite sample size. Additional errors in the estimates f, NO, and NP will be present as a result of errors in the measured values of q. If a typical measurement error σq is smaller than the smoothing length h, then the eﬀects of measurement error can be ignored. The errors in the axis ratios of dense cores observed at millimeter wavelengths are relatively small, if the eﬀects of the beamwidth are correctly subtracted (see section 4 below). However, the errors in the published axis ratios for surveys of Bok globules (CB; BHR) are signiﬁcantly larger than h. A further complication is added by the fact that errors in q are generally not Gaussian, so they cannot simply be added in quadrature to the kernel width h. The presence of non-Gaussian errors in ˆ the measurement of the axis ratio of nearly circular objects can signiﬁcantly aﬀect the shape of f
ˆ, in turn, can aﬀect the conclusions which are drawn about thewhen q ∼ 1. This distortion of f intrinsic shapes of the observed objects. A cautionary tale is related by Franx & de Zeeuw (1992), who deduced the intrinsic ellipticity of disk galaxies from their observed axis ratios. Assuming Gaussian errors in q, their best model had = 0.06; however, the introduction of non-Gaussian errors led to a best ﬁtting model with = 0. In the following section, it will also be seen that the ˆ introduction of non-Gaussian rounding errors distorts the distribution f (q) for Bok globules.
3. Isolated Bok Globules
Bok globules are not, in general, spherical. Bok & Reilly (1947) deﬁned globules as “approximately circular or oval dark objects of small size”, as contrasted to the “wind-blown wisps of dark nebulosity” which can also be found in the interstellar medium of our galaxy. CB constructed a catalog of small Bok globules with declination δ −36◦. They searched Palomar Observatory Sky Survey (POSS) plates for isolated, opaque globules with diameters less than
10. The axis ratio of the globules was explicitly not a selection criterion. Their complete catalog contains 248 objects, with a mean angular size of 4. The apparent axis ratio of each globule was determined by approximating its shape as an ellipse, and then measuring the minor and major axes of the ﬁtted ellipse. The axis ratios measured for the complete sample ranged from q = 1.00 to q = 0.14, with a mean q = 0.59 and standard deviation σq = 0.23.
To begin the analysis, I na¨ ıvely take the values for the axis length a and b published by CB, ˆ ˆ and use the resulting values of q = b/a to ﬁnd the estimated distribution f (q). The function f determined in this way is shown as the solid line in the upper panel of Figure 1. In this Figure (and in every subsequent Figure in this paper), the dashed lines indicate the 80% conﬁdence band and the dotted lines show the 98% conﬁdence band, as determined by bootstrap resampling. The ˆ estimated distribution NO of intrinsic axis ratios, given the randomly oriented oblate hypothesis, ˆ is shown in the middle panel of Figure 1. The estimated distribution NP of axis ratios, given the randomly oriented prolate hypothesis, is shown in the bottom panel. The most striking aspect of ˆ the estimate f (q) found in this manner is that it is bimodal. In addition to a broad maximum around q ∼ 0.55, there is a second peak at q = 1. Does this bimodality indicate that there are –6– two populations of globules, one ﬂattened and one nearly spherical? No. The peak at q = 1 is an artifact, the result of rounding errors in q. CB, in measuring the axis lengths of globules on the POSS plates, rounded a and b to the nearest millimeter, corresponding to 1.12 in angular scale.
Many of the globules in the survey are quite small; 42 out of 248 have a ≤ 2 mm. Consequently, rounding of the axis lengths can have a large eﬀect on the measured value of q. For instance, a globule whose true size on the POSS plate is 1.6 mm × 2.4 mm will be tabulated as being
2.0 mm × 2.0 mm. Its axis ratio will be erroneously computed as q = 1.00 instead of its true value of q = 0.67. A major eﬀect of rounding errors is that small ﬂattened globules will be incorrectly classiﬁed as being circular.
The eﬀect of the non-Gaussian rounding errors can be approximately compensated for. To the values of a and b tabulated by CB for each globule, I add an error term ∆ drawn uniformly from the interval −0. 56 ∆ 0. 56. I then compute the new values of q after the error terms are ˆ added on, and compute f (q). After repeating this process 800 times, with a diﬀerent seed for the ˆ random number generator each time, I take the average value of the 800 estimates f (q) as the new best estimate of the underlying distribution f (q), taking into account the errors introduced by rounding the axis lengths a and b. The best estimate found in this way is given as the solid line in the top panel of Figure 2. The 80% conﬁdence bands (dashed lines) and the 98% conﬁdence bands (dotted lines) include both the errors due to ﬁnite sample size and the errors in q due to rounding.
Note that when the rounding errors are compensated for, the peak at q = 1 disappears. The ˆ distribution f now has a single maximum at q = 0.6. The mean value of q is q = 0.57 and ˆ the standard deviation is σq = 0.22. The derived values of NO, given the oblate hypothesis, and ˆ NP, given the prolate hypothesis, are shown in the middle panel and bottom panel of Figure 2.
The oblate hypothesis cannot quite be rejected at the 99% one-sided conﬁdence level, but can be rejected at lower conﬁdence levels. If the globules are all oblate spheroids, then they must be ˆ quite ﬂattened. Few or no globules, if they are oblate, can have γ 0.7, and the peak in NO is at ∼ γ ≈ 0.25. The observed axis ratios, when corrected for rounding errors, are fully consistent with the hypothesis that globules are all prolate objects with random orientations. The distribution ˆ NP (bottom panel of Figure 2) is everywhere positive, yielding a mean axis ratio γ = 0.48 and standard deviation σγ = 0.20. The prolate hypothesis yields a broader range of intrinsic axis ratios than does the oblate hypothesis, with prolate globules ranging in shape from ﬁlamentary structures with γ ∼ 0.1 to nearly spherical globules with γ ∼ 1.
The catalog of CB in the northern sky is complemented by that of BHR in the southern sky.