Carbon monoxide observations of small dark globules: II. Stability analysis

New Astronomy 14 (2009) 451–460

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Carbon monoxide observations of small dark globules: II. Stability analysis H.G. Kim a,*, S.S. Hong b a b

Korea Astronomy and Space Science Institute, Taeduk Radio Astronomy Observatory, 61-1, Hwaam, Yuseong, Daejeon 305-348, Republic of Korea Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Republic of Korea

a r t i c l e

i n f o

Article history: Received 1 October 2008 Received in revised form 20 December 2008 Accepted 22 December 2008 Available online 31 December 2008 Communicated by G.F. Gilmore PACS: 98.38.Dq 33.20.t 47.50.Gj 94.05.Lk Keywords: ISM: clouds ISM: molecules Instabilities Turbulence

a b s t r a c t The stability of 12 small dark globules has been analyzed by using the full scalar virial theorem without magnetic field. We have applied the virial theorem to 18 sub-condensations identified from the column density maps of 12 globules. The sub-condensations are approximated by a uniform sphere of equivalent mass for simplicity. Based on the conventional simplified version of virial theorem, where the viral mass is compared with the LTE mass, we can only say that almost all the sub-condensations are approximately in a virial equilibrium. When we apply the full scalar virial theorem, where the sum of all the energy terms does not vanish and the time variation of the moment of inertia should be kept, one third of our sample cores are likely to collapse, one sixth of them are expected to expand, and the rest half of them are in a dense phase of an oscillatory equilibrium. The globules in the diffuse phase of the oscillatory equilibrium may not be detected by conventional means, because they are too rarefied to get CO molecules excited or to shield molecules from UV photons, or because they may not withstand the tidal disruption by neighboring clouds. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The small dark clouds have been known as possible birthplaces of stars (see the excellent review of Bergin and Tafalla (2007)), and one of the major objectives at studying dark clouds is to understand the star formation process. In order to understand the current dynamical status and future destinies of small dark clouds, Kim and Hong (2002; hereafter, Paper I) examined the internal structures of 18 sub-condensations of 12 small dark clouds. They found that the internal density structure of these clouds closely resembles that of truncated isothermal gas sphere, and the observed line widths varies with the cloud size in a power-law fashion similar to that for the other clouds or cores of GMC’s. These results suggest that the small dark clouds are generally dominated by turbulence, and can be described as quasi-equilibrium structure. With the observed internal density and velocity structures, however, it is difficult to judge the future destiny of a dark cloud. This motivated us to examine the cloud stability by balancing all the involved energies in detail. The virial theorem is a useful tool for describing an overall energy balance and checking the question * Corresponding author. Tel.: +82 42 865 3262; fax: +82 42 865 3272. E-mail addresses: [email protected] (H.G. Kim), [email protected] (S.S. Hong). 1384-1076/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.newast.2008.12.006

of stability. For the stability analysis the virial theorem without the moment of inertia term has been widely used for its simplicity (Larson, 1981; Leung et al., 1982; Solomon et al., 1987; Myers and Goodman, 1988; Langer et al., 1989; Simon et al., 2001; Shinnaga et al., 2004; Ward-Thompson et al., 2006). As discussed earlier by Spitzer (1978), the virial theorem involves balancing a number of energy terms including the gravitational energy, thermal and non-thermal kinetic energies, and also the terms from surface pressure, magnetic field, and cosmic rays. If sum of all the terms vanishes, then the cloud is said to be in the virial equilibrium. Observationally, it is difficult to know all of the terms exactly, because of the unknown internal structures and the irregular shape of interstellar clouds. Therefore, one normally takes a spherical geometry for the shape, and replaces the structure by uniform distributions of density (Ward-Thompson et al., 2006), or Gaussian density profiles (Stutzki and Güsten, 1990; Simon et al., 2001). To the dark clouds like the low mass dark globules we are studying, the full scalar virial theorem even with this level of simplifications has not been thoroughly applied because the available data has not been sufficient to calculate the moment of inertia term explicitly. Simon et al. (2001) checked the stability of the clumps of four molecular clouds from the Milky Way Galactic Ring Survey by simply comparing the virial and LTE masses. More

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detailed virial theorem was applied to find the stability of condensations in the Horsehead by Ward-Thompson et al. (2006). They estimated all the energy terms involved in the virial theorem. However, they did not take the time variation of the moment of inertia term into account. Dynamical nature of the dark globules still remains in question. For the 18 sub-condensations of Paper I, we now know masses, sizes, and internal velocity fields from the fully sampled maps of their CO emission. This body of information will enable us to use the full scalar virial theorem with the moment of inertia term in the stability analysis of the dark globules. In this paper we address ourselves to the following questions: (1) How much difference is there between LTE and virial masses for small dark globules? Mass is the most important information for the stability analysis. (2) What is the difference between the conventional and the full scalar virial theorem? (3) What fraction of the globules are in the virial equilibrium when the full scalar virial theorem was applied? This is an interesting question as regard to the star formation activity in the globules. Finally, (4) what are the dynamical destinies of the unstable globules? In Section 2 we briefly describe the data used for this analysis and present the resulting maps of the integrated intensity. In Sections 3

and 4 we test the small dark clouds with the simple scalar virial theorem. In Section 5, the full scalar virial theorem is introduced, and applied our sample dark clouds. The last section concludes the paper. 2. Data All the data were taken from Paper I. A brief description for the observations is as follows. Two radio telescopes were used in the mapping observations of the 12 dark clouds listed in Table 1. For the smaller clouds, the Taeduk Radio Astronomy Observatory (TRAO)’s 14 m telescope was used for its small beam size, and for the larger ones, the large beam of the University of Nagoya’s 4 m telescope was used. The half power beam width (HPBW) of the TRAO 14 m radio telescope is about 5000 at the rest frequency 110.2014 GHz of the 13 COðJ ¼ 1 ! 0Þ transition. All the spectra were collected in the mapping mode, in which the telescope was pointed periodically to the center and reference positions. The sampling rate for mapping was 10 . The HPBW of the Nagoya 4 m antenna is about 2.70 at the rest frequency of the 13 COðJ ¼ 1 ! 0Þ transition. Spectra were taken in frequency switching mode with

(a) B130

(b) B133

0.5

0

-20

0

-20

20

0

-20

20

0

Δ RA [arcmin]

Δ RA [arcmin]

(c) B134

(d) B34N

-20

0.5

0.7

0

0.7

20

Δ DEC [arcmin]

20

0

0.4

Δ DEC [arcmin]

1.2 1.6

Δ DEC [arcmin]

0.8

0.8

Δ DEC [arcmin]

20

0.4

20

-20

-20

20

0

Δ RA [arcmin]

-20

20

0

-20

Δ RA [arcmin]

Fig. 1. 13 COðJ ¼ 1 ! 0Þ integrated intensity contours for (a) B130, (b) B133, (c) B134, and (d) B34N. The dashed lines in all the frames mark the boundary of the area observed in 13 COðJ ¼ 1 ! 0Þ. B130 is composed of three distinct sub-condensations, which are labeled as B130-a, -b and -c in sequence from upper region. The lowest contour levels are 0.5, 0.4, 0.5 and 0.4 K km s1, and the contour intervals are 0.3, 0.4, 0.5 and 0.3 K km s1, respectively. 13 COðJ ¼ 1 ! 0Þ integrated intensity contours for (e) B361, (f) B6, (g) L134, and (h) L1523. The lowest contour levels are 4.0, 1.0, 1.0 and 0.5 K km s1, and the contour intervals are 2.0, 0.4, 0.6 and 0.4 K km s1, respectively. 13 COðJ ¼ 1 ! 0Þ integrated intensity contours for (i) L392, (j) L400, (k) L460, and (l) SCHO93. The lowest contour levels are 0.5, 1.0, 1.4 and 0.4 K km s1, and the contour intervals are 0.3, 0.4, 0.3 and 0.3 K km s1, respectively.

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(e) B361

(f) B6

0

-20

1.0 1.4

4.0

0

8

Δ DEC [arcmin]

20

1.

Δ DEC [arcmin]

20

1.0

-20

20

0

-20

20

Δ RA [arcmin]

0

-20

Δ RA [arcmin]

(g) L134

(h) L1523

20

1.6

20

1.3

5 0.

Δ DEC [arcmin]

1.0 4.0

-20

0

0.9

0

1.7

Δ DEC [arcmin]

2.2 2.8 3.4

-20

20

0

-20

20

Δ RA [arcmin]

0

-20

Δ RA [arcmin] Fig. 1 (continued)

a spatial sampling rate of 20 . The resulting maps of the 13 COðJ ¼ 1 ! 0Þ integrated intensity are shown in Fig. 1a–l for the 12 dark globules. The dashed line demarcates the area that was actually observed in the 13 COðJ ¼ 1 ! 0Þ emission. The line width data were also taken from Paper I. The line width DV DL ðpÞ averaged over the same projected distance p from the center is plotted in Fig. 2 against the projected distance. In some condensations the line width remains almost constant over the entire cloud surface, while in many condensations the line width shows a decreasing trend from center to some value pcrt , and then stays more or less constant beyond it. This tendency may possibly be due to systematic motion or may reflect real variation of turbulence velocity dispersion in the sub-condensations. In this study, we simply assumed that the excessive line broadening toward the center is entirely due to the systematic motion of expansion or collapse. In Fig. 2, the horizontal dashed line represents the turbulent line width which is an average of the line width over the outskirts. Assuming that the total line width toward center DVðp ¼ 0Þ is related with the line width due to the systematic and turbulent line widths by

DV 2tot ðp ¼ 0Þ ffi DV 2sys þ DV 2tur :

ð1Þ

_ and DV tur are listed The results of DV sys at the globule center ð¼ 2RÞ in Table 2.

3. Scalar virial theorem The line widths observed from molecular clouds are in general much broader than the thermal line widths expected from the kinetic temperatures. This non-thermal line width is thought to be caused by some sort of turbulent motions. Although the turbulent motion can cause the turbulent fragmentation (Ballesteros-Paredes, 2006) in some circumstances, the turbulence is generally considered as a main agent for supporting the clouds against the external pressure and the self-gravity. We will therefore assume that all the kinetic energy helps to support the cloud against collapse for simplicity. In this case the scalar virial theorem states 2

1 d I ¼ W þ 3P þ 2K  2 dt2

Z

P~ r  d~ S;

ð2Þ

surf

where P; I; W; P, and K mean the pressure, moment of inertia, gravitational energy, random kinetic energy of the turbulence motion, and the kinetic energy associated with the mass motion inside the cloud, respectively. The magnetic energy in dark cloud core is in general small compared with either gravitational or turbulent energies by a few factors (Troland and Crutcher, 2008). Hence in this study we ignore magnetic fields for simplicity. For a uniform spherical cloud of mass M and radius R, we can express the moment of inertia I and the gravitational energy W as

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(i) L392

(j) L400

0.5

1.7

0.5

-20

1.0 2.2

0

0 Δ RA [arcmin]

-20

20

(k) L460

0 Δ RA [arcmin]

-20

(l) SCHO93

20

1.4

2.02.3 1.4

1.7 1.4

-20

1.0 1.6 1.3

1.4

2.3 2.0

0.4 0.7

Δ DEC [arcmin]

20

Δ DEC [arcmin]

1.4

-20

20

0

1.0 1.8

2.0

0

Δ DEC [arcmin]

0.8

20

1.1 1.4

Δ DEC [arcmin]

20

0

-20

20

0 Δ RA [arcmin]

-20

20

0 Δ RA [arcmin]

-20

Fig. 1 (continued)

Table 1 Selected 12 dark globules. ID No.

Name

aB1950.0a

dB1950.0a

V LSR (km s1)

Optical size (arcmin2)

Telescope used

1 2 3 4 5 6 7 8 9 10 11 12

B6 SCHO93 L1523 B34N L134 L392 L400 L460 B130 B133 B134 B361

03 05 05 05 15 17 17 17 18 19 19 21

55 59 21 32 43 21 31 40 00 32 38 18 04 42 00 08 24 26 08 15 00 03 46 07 05 37 12 06 58 07 06 19 22 47 10 28

2.0 6.9 7.0 4.2 2.6 6.8 10.7 11.5 10.5 11.8 11.3 2.0

15  10 10  15 12  15 10  20 25  20 10  15 10  10 20  10 15  20 10  15 6 5 15  15

TRAO 14 m Nagoya 4 m Nagoya 4 m TRAO 14 m TRAO 14 m Nagoya 4 m Nagoya 4 m Nagoya 4 m Nagoya 4 m Nagoya 4 m Nagoya 4 m TRAO 14 m

a

I¼

52 01 03 40 51 50 52 55 59 03 04 10

38 00 00 09 30 33 48 28 06 31 07 40

Units of right ascension are hour, minute, and second, and units of declination are degree, arcminute, and arcsecond.

3 MR2 5

ð3Þ

and

P¼ 2

W¼

effective pressure P inside the cloud can be given in terms of the turbulence as

3 GM ; 5 R

ð4Þ

respectively. In interstellar clouds the turbulence dominates the cloud energy over the internal heat and the external pressure. The

1 qhV 2tur i; 3

ð5Þ

hV 2tur i1=2

where is the rms dispersion of the turbulence motion. Then the internal energy becomes

P

Z

PdV ¼

1 MhV 2tur i: 3

ð6Þ

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2

1.5

<ΔV(p)> [km/s]

<ΔV(p)> [km/s]

2

1.5

(a) B130 - a σr

1 .5 0

2

4 6 p [arcmin]

8

0

<ΔV(p)> [km/s]

<ΔV(p)> [km/s]

4 6 p [arcmin]

1.5

(b) B130 - b σr

1 .5

8

10

(h) B34N - c σr

1 .5 0

0 0

2

4 6 p [arcmin]

8

0

10

2

4 6 p [arcmin]

8

10

8

1.5

<ΔV(p)> [km/s]

2 <ΔV(p)> [km/s]

2

2

1.5

(c) B130 - c

σr

1 .5

6

(i) B361 σr

4 2 0

0 0

2

4 6 p [arcmin]

8

0

10

3

2

4 6 p [arcmin]

8

10

2 <ΔV(p)> [km/s]

<ΔV(p)> [km/s]

.5

10

2

(d) B133

2 1

0

5 10 p [arcmin]

(j) B6 - a

1.5

σr

0

σr

1 .5 0

15

0

3

2

4 6 p [arcmin]

8

10

2 <ΔV(p)> [km/s]

<ΔV(p)> [km/s]

1

0

0

(e) B134

2

1.5

σr

1 0 0

5 10 p [arcmin]

σr

(k) B6 - b

1 .5 0

15

0

2

2

4 6 p [arcmin]

8

10

3

1.5

<ΔV(p)> [km/s]

<ΔV(p)> [km/s]

(g) B34N - b σr

(f) B34N - a σr

1 .5

(l) L134 σr

2 1 0

0 0

2

4 6 p [arcmin]

8

10

0

2

4 6 p [arcmin]

8

10

Fig. 2. The line width, after being corrected for the broadenings due to rms noise and optical saturation, is plotted as a function of projected distance p for (a) B130-a, (b) B130-b, (c) B130-c, (d) B133, (e) B134, (f) B34N-a, (g) B34N-b, (h) B34N-c, (i) B361, (j) B6-a, (k) B6-b, and (l) L134; for (m) L1523, (n) L392, (o) L400, (p) L460-a, (q) L460-b, and (r) SCHO93. Note that the center pixel has a single value, and denoted as a filled circle. The error bar at the center represents an average of the errors at all the non-central positions.

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3 <ΔV(p)> [km/s]

<ΔV(p)> [km/s]

2 (m) L1523

1.5 σr

1 .5 0 2

4 6 p [arcmin]

8

1

10

0

2

2

4 6 p [arcmin]

8

10

3 σr

1.5

(n) L392

<ΔV(p)> [km/s]

<ΔV(p)> [km/s]

σr

0 0

1 .5 0

(q) L460 - b

2

σr

1 0

0

5 10 p [arcmin]

15

0

2

4 6 p [arcmin]

8

10

2

3

<ΔV(p)> [km/s]

4 <ΔV(p)> [km/s]

(p) L460 - a

2

1.5

(o) L400 σr

2 1 0

(r) SCHO93 σr

1 .5 0

0

2

4 6 p [arcmin]

8

10

0

2

4 6 p [arcmin]

8

10

Fig. 2 (continued)

Table 2 Derived parameters for 18 sub-condensations. Name

M LTE ðM Þ

Angular size (arcmin)

Radius (pc)

DV tot ðp ¼ 0Þ ðkm s1 Þ

DV sys ðkm s1 Þ

DV tur ðkm s1 Þ

B130-a B130-b B130-c B133 B134 B34N-a B34N-b B34N-c B361 B6-a B6-b L134 L1523 L392 L400 L460-a L460-b SCHO93

15 12 18 90 20 8 10 7 440 50 14 96 17 112 288 35 31 12

3.98 ± 0.48 5.12 ± 0.33 3.56 ± 0.33 4.90 ± 0.39 4.03 ± 0.47 4.20 ± 0.22 3.98 ± 0.34 3.65 ± 0.46 6.95 ± 1.34 5.99 ± 0.72 5.63 ± 0.72 7.10 ± 0.98 5.72 ± 0.24 8.38 ± 0.08 7.60 ± 0.24 9.67 ± 0.18 9.15 ± 0.36 7.77 ± 0.29

0.40 ± 0.05 0.52 ± 0.03 0.36 ± 0.03 0.57 ± 0.05 0.47 ± 0.05 0.24 ± 0.01 0.23 ± 0.02 0.21 ± 0.03 1.21 ± 0.23 0.35 ± 0.04 0.33 ± 0.04 0.23 ± 0.03 0.23 ± 0.01 0.73 ± 0.01 0.99 ± 0.03 0.56 ± 0.01 0.53 ± 0.02 0.32 ± 0.01

1.06 ± 0.14 1.10 ± 0.11 1.16 ± 0.18 1.23 ± 0.17 1.61 ± 0.23 0.84 ± 0.07 0.87 ± 0.12 0.76 ± 0.10 2.41 ± 0.80 1.16 ± 0.18 1.08 ± 0.09 1.40 ± 0.23 0.75 ± 0.10 1.04 ± 0.08 1.70 ± 0.37 0.89 ± 0.16 1.05 ± 0.16 0.69 ± 0.07

0.80 ± 0.14 0.90 ± 0.10 0.82 ± 0.17 0.67 ± 0.15 1.23 ± 0.11 0.70 ± 0.13 0.64 ± 0.16 0.57 ± 0.17 0.00 ± 0.25 0.92 ± 0.16 0.80 ± 0.15 0.85 ± 0.11 0.44 ± 0.13 0.50 ± 0.10 0.96 ± 0.16 0.13 ± 0.10 0.00 ± 0.09 0.28 ± 0.08

0.69 ± 0.10 0.63 ± 0.06 0.82 ± 0.16 1.03 ± 0.17 1.04 ± 0.04 0.47 ± 0.13 0.59 ± 0.12 0.50 ± 0.13 2.43 ± 0.41 0.71 ± 0.13 0.73 ± 0.21 1.11 ± 0.03 0.61 ± 0.07 0.91 ± 0.11 1.40 ± 0.11 0.88 ± 0.02 1.06 ± 0.04 0.63 ± 0.03

The third term on the right hand side of Eq. (2) is the kinetic energy associated with the mass motion inside the cloud, and can be expressed as

K¼

1 2

Z

qðrÞVðrÞ2 d3~r:

ð7Þ

Although some of our sample dark clouds show rotation (e.g. B361, L134), the rotational kinetic energy in the low mass cloud cores is in general very small about 104 to 0.07 (Caselli et al., 2002), compared with the gravitational energy, indicating the contribution of rotation to the overall stability of the cloud is insignificant. There-

fore, in this study we assumed systematic motion of expansion or contraction dominates the kinetic energy. For a cloud of uniform density, the velocity of expansion or contraction at distance r from the cloud center is often assumed to behave as a self-similar man_ ner, and can be described as VðrÞ ¼ r_ ¼ Rðr=RÞ (Weber, 1976; Tatematsu and Fujimoto, 1976; Tilley and Pudritz, 2003; Shen and Lou, 2005). We take this type of homologous motion as a first approximation for the internal dynamics of the equivalent uniform sphere. In this case the total kinetic energy associated with the expansion or R contraction of the cloud as a whole, ð1=2Þq 4pr2 drR_ 2 ðr=RÞ2 , simply 2 _ becomes ð3=10ÞMR .

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The last term in Eq. (2) is originated from the PdV work - the work done by the cloud against the external pressure. For a spherical cloud under a uniform external pressure P ext , the surface integral becomes simply

Z

P~ r  d~ S ¼ 4pR3 P ext :

ð8Þ

surf

Substituting all these terms into Eq. (2), we rewrite the scalar virial theorem in the following form: 2 3 € ¼  3 GM þ MhV 2 i  4pR3 Pext : MRR tur 5 5 R

ð9Þ

_ Multiplying R=R to both sides of Eq. (9) and integrating with respect to time, we may have

3 1 _ 2 3 GM2 4p  MR ¼ þ MhV 2tur i ln R  P ext R3 þ E; 5 2 5 R 3

ð10Þ

where the integral constant E is to be determined from the current _ and the turvalues of the cloud radius, the systematic velocity jRj, bulent velocity hV 2tur i1=2 . We will call this form of virial balance as the full scalar virial theorem. 4. Stability tests with the simple virial theorem _ term in the left hand side of the full It is difficult to know the jRj scalar virial theorem. For a stationary cloud, however, one can simply ignore it, and Eq. (10) takes a simpler version of the virial theorem

4pR3 Pext ¼ MhV 2tur i 

3GM2 ; 5R

ð11Þ

in the usual stability test. For a typical example of the dark globules, where R ’ 0:5pc; M ’ 50M ;hV 2tur i1=2 ’ 0:5kms1 , and Pext ’ 5  1013 dynes cm2, the surface pressure term amounts to 2:3  1043 ergs, while the internal energy and the gravitational energy become 2:6  1044 ergs and 2:6  1044 ergs, respectively. Therefore, one may further ignore the surface pressure term, and utilizes in the stability analysis the simplest version of the scalar virial theorem:

hV 2tur i ’

3GM : 5R

ð12Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Leung et al. (1982) compared DV FWHM with MLTE =R for their sample of dark clouds. For comparison purpose we do the same type of stability analysis for our 18 sub-condensations. In our case, the observed line width toward the cloud center is used instead of DV FWHM , because they are related to each other by a simple multiplication factor. The results are shown in Fig. 3, where two solid lines represent ±30% deviations from the virial equilibrium. Because some of the globules are not in the condition of strict iso-turbulence, we attached error bars to take into account for the variation of line width over the projected distance from the cloud center. If all the quantities are known exactly, only the sub-condensations lying on the dashed line are said to be in a virial equilibrium: those in the upper side of dashed line would expand, while ones in lower side would collapse. But, considering the uncertainty in the physical quantities, the clouds lying within the ±30% lines can be said to be in an approximate virial equilibrium. According to this stability criterion, most of the sub-condensations are approximately in a virial equilibrium, B130 and B134 are to expand, and L134 is likely to collapse. Another stability test based on the simple scalar virial theorem is to use the virial mass instead of using R and DV FWHM . The virial mass is the mass that makes the equality in Eq. (12) hold true with given R and DV FWHM :



 5R ½DV FWHM 2 8 ln 2  G   2 R DV FWHM ’ 210 M  : pc km s1

Mvir ’

ð15Þ

One may then check whether a cloud is in the virial equilibrium or not by simply comparing the LTE mass with the virial one. For the 18 sub-condensations the virial masses were calculated from their turbulence dispersions and sizes, and compared with the LTE masses. Table 3 lists the masses and the parameters used for the virial mass estimation. Fig. 4 compares the two masses in logarithmic scale. To the filled circles we attached arrows which represent the virial masses one would have if the central line width is fully taken for the turbulence velocity.

3

Since we can observe only one component of the velocity dispersion along the line of sight, the turbulent velocity hV 2tur i1=2 is related to the observed full line width at half maximum DV FWHM approximately as

hV 2tur i ¼ 3hV 2z;tur i ¼ 3



DV FWHM 2

2

  1 3 DV 2FWHM : ¼ 2 ln 2 8 ln 2

2 ð13Þ

Here hV 2z;tur i is the component along the line of sight, and the pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ln 2 factor comes from the conversion of standard deviation of the Gaussian profile. Using the relation in Eq. (13), we can rewrite the simplest virial theorem in terms of observable quantities only,

   1=2 8 ln 2 3G M  3 5 R  1=2 M pc ’ 0:069 km s1 : M R

B361

L400

B134 L134 B133

1

L460-b B130-c B130-b B130-a B6-b

L392 B6-a

L460-a B34N-b B34N-a B34N-c SCHO93

DV FWHM ’

L1523

ð14Þ

When the mass, radius and turbulent line width are known from the observations, Eq. (14) is often used for testing if a cloud is in an approximate virial equilibrium or not. The line width of the 13 COðJ ¼ 1 ! 0Þ transition is widely used for this purpose. The 13 CO line width reflects velocity field across the entire cloud in a line of sight, because it is optically thin.

0 0

10

20

30

Fig. 3. The mean observed velocity widths of the 13CO line are plotted as a function of ðMass½M =Radius½pcÞ1=2 . The error bars represent the velocity range between broadest and narrowest ones.

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Table 3 Comparison of virial to LTE masses. Source

Radius (pc)

DV FWHM ðkm s1 Þ

DV tur ðkm s1 Þ

MLTE ðM Þ

M vir ðM Þ

B130-a B130-b B130-c B133 B134 B34N-a B34N-b B34N-c B361 B6-a B6-b L134 L1523 L392 L400 L460-a L460-b SCHO93

0.40 0.52 0.36 0.57 0.47 0.24 0.23 0.21 1.21 0.35 0.33 0.23 0.23 0.73 0.99 0.56 0.53 0.32

1.06 1.10 1.16 1.23 1.61 0.84 0.87 0.76 2.41 1.16 1.08 1.40 0.75 1.04 1.70 0.89 1.05 0.69

0.69 0.63 0.82 1.03 1.04 0.47 0.59 0.50 2.43 0.71 0.73 1.11 0.61 0.91 1.40 0.88 1.06 0.63

15 12 18 90 20 8 10 7 440 50 13 96 17 112 288 35 31 12

40 43 51 127 107 11 17 11 1505 37 37 59 18 127 410 91 126 26

B361

3 L400

L460-b

2

B134

B133

L392

L460-a B130-b

L134

B130-c B130-a B6-b

B6-a

SCHO93

Q

B34N-b L1523 B34N-c

1

B34N-a

-P P -Q

0 0

1

2

3

Fig. 4. Virial masses of the 18 sub-condensations are plotted against their LTE masses. The dashed line represents the location of a strict virial equilibrium. The tips of the arrows indicate the virial masses one would obtain if the line widths at globule centers are entirely due to the turbulence motion.

we assume the conversion factor could be different from its true value by ±50%. Except for the uncertainty in the conversion factor, all the other sources tend to make the LTE mass an underestimate. Therefore, in extreme case, the LTE mass can detect only 36% ½¼ ð1  0:2Þ  ð1  0:5Þ  ð1  0:1Þ of the true mass. In the case of the other extreme, the LTE mass becomes 1:08 ½¼ ð1  0:2Þ  ð1 þ 0:5Þ  ð1  0:1Þ times the true mass. In conclusion the true mass M is expected to be in the range ð0:9 2:8ÞM LTE . This error of systematic nature is marked, in the lower right corner of Fig. 4, by a horizontal bar with two tick marks. Uncertainties in the size and the turbulence dispersion directly affect the virial mass. From Table 2, we may notice the relative errors of the size and the turbulence dispersion are 10% and 20%, respectively. Therefore, the total uncertainty in the virial mass is ±50%, and the true virial mass is expected to be in the range (0.5–1.5) times its current value. This error of randomness is marked, in the lower right corner of Fig. 4, by a vertical bar with two tick marks. The locus of strict virial equilibrium could have drawn the dashed line in Fig. 4, if there are no errors in the estimates of M vir and M LTE . The upper dotted-line is obtained by moving the ~ . Because the methdashed line by vector ~ P, and the lower by Q ods of estimating M vir and MLTE introduce inevitable errors in the mass, even the clouds under strict virial equilibrium will show up within the band marked by the two dotted-lines. Therefore, the stability test based on the simple version of the virial theorem can only say that the sub-condensations of the dark globules are not far from the virial equilibrium. Because some of the sub-condensations seem to undergo motions of systematic expansion or contraction, the R_ term should be kept in the scalar virial theorem (Eq. (10)). As was first pointed out by Hong et al., 1991, the R_ term introduces interesting alternatives to the stability analysis. In the previous tests the systematic motion was also treated as the turbulence. It is unnatural for any globule to be in a static equilibrium; they are likely to be in an oscillatory equilibrium. In this sense the external medium plays an important role in the globule dynamics. Therefore, one should include both the possibility of systematic motion and the role of the external pressure. This forces us to use the full version of the scalar virial theorem in the stability analysis. 5. Stability test with the full virial theorem For given conditions of hV 2tur i and mass M, there is a maximum value of external pressure that the cloud can withstand. Under this critical condition the cloud has the size R0

#2  " 4 GM M km s1 3 ’ 3:439  10 ½pc R0 ¼ 5 hV 2tur i M hV 2tur i1=2

ð16Þ

and the external pressure Except for two objects B6-a and L134, the virial masses of the sub-condensations are all larger than their LTE masses. This does not necessarily mean that 16 out of the 18 sub-condensations are currently expanding, because there are various sources of uncertainty in the mass estimation. The LTE analysis underestimates mass by 20% in the core part of clouds and by up to 50% near boundary (Arquilla and Goldsmith, 1985), and the LTE mass would be a lower limit in most case of molecular clouds (Andersen et al., 2004; Shinnaga et al., 2004). There is also undetected mass in the outskirt of each sub-condensation, because of the sensitivity limit in detecting the 13 COðJ ¼ 1 ! 0Þ emission. The photo-dissociation by the UV radiation near the cloud boundary also makes the LTE mass an underestimate. The undetected mass is assumed to be about 10% of the detected LTE mass. The relative abundance of 13CO to H2 is also uncertain;

P0 ¼

 4 3 1 5 1 hV 2 i4 4 4p tur G3 M 2 7

’ 3:310  10

" #8   2 hV 2tur i1=2 M ½dynes cm  ; km s1 M 2

ð17Þ

respectively. If R is less than R0 , the cloud would undergo a collapsing motion, and if Pext is larger than P0 , it would do the same. Taking R0 and P 0 as normalization units, we define dimensionless radius r and dimensionless external pressure pext as

r  R=R0 ;

ð18Þ

pext  Pext =P0 :

ð19Þ

With the dimensionless variables, the full version of the scalar virial theorem takes the following form

459

H.G. Kim, S.S. Hong / New Astronomy 14 (2009) 451–460

3 3 1 1 M R_ 2 ¼ MhV 2tur i þ MhV 2tur i ln r  MhV 2tur ipext r 3 þ E: 10 4 r 12

ð20Þ

Dividing both sides of Eq. (20) by ð3=4ÞMhV 2tur i, we have

1 4R_ 2 1 4 1 ¼ þ ln r  pext r 3 þ ; 10 hV 2tur i r 3 9

ð21Þ

where  is now dimensionless. We introduce a function Wðr; pext Þ defined by

1 r

Wðr; pext Þ ¼  

4 1 ln r þ pext r3 : 3 9

ð22Þ

The function is dimensionless effective potential energy of the system, and the energy conservation (Eq. (21)) can be expressed as

1 DV sys  ¼ Wðr; pext Þ þ 10 hV 2tur i1=2

!2 ð23Þ

:

_ and  is a dimensionless enHere we have used DV sys instead of 2R, ergy of the system. The DV sys is the systematic motion. The function Wðr; pext Þ is illustrated in Fig. 5 for three cases of external pressure. The ordinate represents the logarithm of the total energy or the effective potential energy, and the abscissa the logarithm of the normalized radius. The uppermost curve is for the case with pressure pext ¼ 1, the middle pext ¼ 0:2 and the lowest pext ¼ 0. Let us suppose an iso-turbulent globule of mass M with given hV 2tur i is embedded in a medium of external pressure P ext . Eqs. (16) and (17) fix the normalization parameters R0 and P0 , and we may then select a curve corresponding to a proper value of pext . As a first example, let us suppose that a globule of a radius r is in a medium of an external pressure pext ¼ 0:2, and is currently at rest (DV sys ¼ 0 in Eq. (23)). Then the total energy of the globule is simply equal to Wðr; pext Þ, and it comes on position A in Fig. 5. Although this globule is currently at rest, it will oscillate within the potential well of smaller width. As a second example, let us suppose another globule has the same radius, mass, and turbulence dispersion as the first one, but undergoes a systematic motion with a non-zero DV sys . Then the total energy consists of Wðr; pext Þ and the kinetic energy equal to the second term on the right hand side of Eq. (23).

This globule is marked B in Fig. 5. The energy difference between B and A is due to the kinetic energy associated with the systematic motion. The globule will oscillate within the wider potential well having configurations of radii rD and rR . When the globule takes a smaller configuration ðr ¼ rD Þ, it is at the dense phase of the oscillatory equilibrium; when it takes a larger size ðr R Þ, it is at the rarefied phase of the oscillatory equilibrium. As a third example, a globule under the same pext has the same total energy as B but is found at radius rU . This globule is currently at rest, but a slight perturbation will cause it to collapse gravitationally. Under a given pext a globule having total energy B may take one of the three stationary configurations, whose radii are rU ; rD and rR , respectively. The configuration with r U is gravitationally unstable, while the ones being with r being between r D and r R are oscillatory stable. Under this condition of given B and pext , the globule cannot take the size in the range from rU to rD , nor the size larger than r R . One may ask what would happen if the total energy is larger than Wmax as marked by C in Fig. 5. If the globule is currently expanding, it will continue to expand until its radius reaches r max and then contract forever. If the globule is currently contracting, it will continue to do so. Please note that the extremum values of potential energy, Wmax and Wmin , depend on the external pressure pext . When pext < 1, there are well defined extrema for the potential function to take. As long as Wmax < B < Wmin under given pext , a globule can be either gravitationally unstable or oscillatory stable. If B > Wmax , the globule is gravitationally unstable. Only if the total energy is between Wmax and Wmin , it will have a dynamically stable configuration. A note should be made on the role of external pressure; the external pressure plays a very important role, no matter how small it is. Without the external pressure, a globule may have only two stationary configurations, both of them are not dynamically stable. One with the larger size r L would expand, while the other with size very close to rU would contract gravitationally. This means that a globule may not have a stable configuration; it either expands or contracts, depending on its current radius. On the contrary, for a normalized external pressure larger than 1, a globule may have

0.6 0.7 pext = 1.0 0.5

pext = 1.0

L392 L1523

pext = 0.2

SCHO93 L400

0.6

B

B2

εB

A

log energy

log energy

B1

0.2

B6-a

C Ψmax

B34N-a B133B34N-c B34N-b

B130-a L460-a

L134

B6-b B130-c

0.5

B130-b

0.1 B361

Ψmin

L460-b

pext = 0.0 B134

0.4 0.5 -0 .5

rU

rL rD

r

rR rmax

0 .5

-0.4

-0.2

0.0 log [ r ]

0.2

0.4

0.6

log [ r ] Fig. 5. A schematic illustration of function Wðr; pext Þ. Logarithm of Wðr; pext ) is plotted as a function of log ½r for pext ¼ 0, 0.2 and 1.0. In case of pext ¼ 0:2, the filled circles at B1 and B2 represent the configuration of static globule. See the text for details.

Fig. 6. Logarithm of energy Wðr; pext Þ is plotted against the logarithm of normalized radius. The filled dots represent the current position in the (log[r], ) plane of the 18 sub-condensations. Wðr; pext Þ as a function of log ½r for 5 different set of the normalized external pressures is plotted together as a reference. The open circle represents the position of the effective potential energy only (log ½r; Wðr; pext Þ).

460

H.G. Kim, S.S. Hong / New Astronomy 14 (2009) 451–460

only one stationary configuration regardless of the total energy it has. Even in this case the globule would eventually contract. Having outlined the contents of the full virial theorem in the energy versus size plane, we now want to see where the 18 sub-condensation would come in the same plane. Their energy and size status is marked in Fig. 6, where the function Wðr; pext Þ is plotted for 5 different values of the normalized external pressure. For the external pressure Pext , we took the value from Myers and Goodman (1988)

Pext ¼ 5  1013 dynes cm2 :

ð24Þ

In Fig. 6 we marked the two possible energy status of each globule by filled and open circles. The filled circle represents the total energy including the kinetic energy associated with the systematic motion; while the open circle is for the effective potential energy only. If the broadening of the line profile at the cloud center is originated from the systematic motion, the total energy of the globules B6-a, L1523, L392, B34N-a,b,c are higher than their local maxima of potential energy. These 6 sub-condensations will eventually collapse, regardless of the direction of its current motion. One third of the 18 sub-condensations are found to be in this situation. Those located on the right hand side of the local maxima are expected to undergo an oscillatory motion. Some of them (e.g. B134) are expected to expand. When their sizes become larger than the tidal radii, they will be dissipated into general interstellar medium. About one half of the 18 sub-condensations seem to be in the oscillatory equilibrium. It should be pointed out that the globules are found near the dense phase of the oscillatory equilibrium. In the diffuse phase the globules are likely to be disrupted by the tidal interaction of neighboring clouds. Furthermore, in the diffuse phase, the density becomes low to the degree that the CO molecules are hard to be excited. Therefore, the globules in the diffuse phase of the oscillatory equilibrium, even if they exist, can not be detected by using the 13 COðJ ¼ 1 ! 0Þ emission lines. Although the energy status of L134 is located on the left hand side of the energy maxima, we cannot judge its future dynamics confidently, because of the line blending at the central direction.

103 —104 cm3 (Troland and Crutcher, 2008). If magnetic field contributes significantly to cloud support, then additional classes of oscillation may exists (Hennebelle, 2003; Galli, 2005), and this may help to stabilize the cloud against collapse. Further analysis with magnetic field would be necessary as a future work. Another notice should be done on the damping of the oscillation. The global large scale oscillation assumed in our analysis may suffer a significant damping, and will be dissipated away faster than the sound crossing or free-fall times of the globules. This would slow down the oscillation and thereby help collapsing. The recent numerical simulations (Broderick et al., 2007; Broderick et al., 2008; Keto et al., 2006) show that the damping rate of nonradial oscillations of embedded globules is only a weak function of the density of the external medium. According to their analysis, the energy dissipation would be more significant in case the density contrast between the core and the exterior bounding medium is small or the oscillations are radial. In our analysis, however, all the sub-condensations are well isolated and the density contrasts are large, and therefore damping effect was not included in this study. From our analysis we may draw following conclusions: When a globule is approximated by uniform sphere of equivalent mass embedded in an external medium, it may possibly be in an oscillatory equilibrium. About one half of our sample globules show possible oscillatory motions which may prevent the globules from collapsing. This fact suggests that globules showing large-amplitude oscillations may be common and are stable objects. The globules in the diffuse phase of the equilibrium may not be detected, because they are too rarefied to get CO molecules excited, or because they may not withstand the tidal disruption by neighboring clouds. Acknowledgements We wish to thank the referee, Dr. D. Ward-Thompson for a helpful report and comments. H.G.K. thank Dr. J.S. Kim for his careful reading of the manuscript. References

6. Conclusion By using the scalar virial theorem, we have performed the stability analysis for the dark globules. Major results of the analysis are summarized in the followings: The test based on the simplified versions of the virial theorem can only tell us that the 18 sub-condensations of the dark globules are all in an approximate virial equilibrium. We have pointed out an importance of the external pressure and the systematic motion for some of the globules, and kept the term for the second derivative of the moment of inertia in our stability analysis. Out of the 18 sub-condensations, six are expected to collapse, three to expand, and the remaining nine are in oscillatory equilibrium. The nine globules are all observed currently in the dense phase of the oscillatory equilibrium. We note that we restricted our attention to the simplest case of non-magnetized, non-rotating oscillation of pressure-bounded spheres supported by turbulent and kinetic energies only. In general, the magnetic energy in dark cloud core is small compared with either gravitational or turbulent energies by a few factors, and we disregard it for simplicity. Even in this case, the clouds can be in an oscillatory equilibrium, and this can be a possible explanation of the long lifetime of starless cores. However, the magnetic energy turned to be non-negligible at densities of order

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