The density and magnetic field of the dense cores of the molecular cloud L 1630

Chin. Astron. Astrophys. Vol.20, No. 4, pp. 437-444, 1996

A translation of Acta Astrophys. Sin. Vol. 16, No. 3, pp. 271-278. 1996 Copyright 63 1996 Elsevier Science Ltd

Pergamon

Printed in Great Britain. All rights reserved 02751062196 $32.00 + 0.00

PII: S@275-lfB62(%)00058-6

The density and magnetic field of the dense cores of the molecular cloud L 1630t ZHENG Department

HUANG

Xing-wu

of Astronomy,

Nanjkg

Yong-feng

University,

Nanjing

210008

Abstract

The density and magnetic field strength of the dense cores in the Orion B molecular cloud are derived from the observed radius and FWI-IM line width based on the model of a uniformly magnetic sphere. We obtain the average magnetic field strength of 1lOpG and the average density of 8 x 104/cm3 for the 39 cores, which agree closely with the observations. The method for deriving the density and magnetic field strength is applicable to the cores with R > 0.2~~. Key

words:

interstellar

medium-molecular

cloud-magnetic

field

1. INTRODUCTION

To study the physical process of star formation we must understand the physical conditions in the parent bodies-the molecular clouds that give birth to the stars. The parameters that describe the conditions are the temperature of the cloud, the number density, the density structure, the velocity field, the mass, the molecular abundances and the embedded magnetic field.

With

the aim of obtaining

reliably

these parameters,

the practical

astrophysicist

has

designed many clever and effective methods and after millions of measurements, obtained much valuable observational data. However, as research advances, the available methods begin to reveal certain limitations, and it remains a task for the practical astrophysicist to find new means of observation. The number density is probably the most important parameter. It is deduced from measured line ratios between two or more rotational transitions of some molecule, under some assumed model. For extended molecular clouds, the most effective spectral lines are optically thin lines of 13C0 and 12CO[‘-31. And the molecular lines of H&N, HCaN, H2C0, CS and NH3 have been widely used to determine the number density in the dense cores[4-6]. When using this method to estimate the number density, the following points arise: 1) The observations of the emissions of different transition must have the same filling factor. At present, different transitional lines are often obtained with different observing t Supported by National Natural Science Foundation kxeived

1995-06-19; revised vention 1995-12-20

431

ZHENG Xing-wu & HUANG Yang-feng

438

beamwidthss,

implying different filling factors.

2) We require the measured line intensity to

be proportional to the population at the appropriate level. If we use optically thick lines, the observed intensity is integrated intensity along the line of sight. Different transitions may integrate differently. Hence, to obtain a reliable estimate of the number density we require the observed line to be optically thin. 3) Wh en we find the number density by using observations of several transitions and solving the equation of radiative transfer, there is the basic assumption, Rdvldt >> AV, that is, the product of the cloud radius and the velocity gradient must be much greater than the line width. This requires the cloud to have a large velocity gradient, and this requirement is not satisfied in many cases. 4) To measure the number density in the dense cores, we require observations of lines of high dipole moments, such as NCO+, HCN, etc.. Another important parameter is the magnetic field. It plays an all-important role in the formation of young stars. At present, because of measuring difficulties, information on the magnetic field is scanty. The measurement is based on the Zeeman effect in the lines. For molecules with an electron orbital angular momentum or an electron spin, the Zeeman splitting is (Zg&h)B, g, ~0, h being the LandC g-factor, the Bohr magneton and the Planck constant. If we take (2gpo/h) t o h ave the value 1, the magnetic field J3 to be lOOpG, then for the OH molecular line (v = 1.665 GHz), the split is only 0.02 km/s, far smaller than the observed line width, AVI’l. At present, only the following atomic and molecular lines are used for magnetic field measurement in molecular clouds: HI, OH, CzS and HzO. These limitations make reliable information on the magnetic field difficult to obtain. Section 2 that follows is a simple presentation of the theoretical model. Section 3 sets out the basis for our selecting the L 1630 CS data for our calculation and analysis. Section 4 gives the basic results. The last Section analyses the results and outlines our ideas on future work.

2. THEORETICAL

MODEL

In their series of studies on the cores of molecular clouds, Myers and GoodmanI have discussed in detail the model of uniform magnetized cloud. Here, we give the relations between the required number density and magnetic field on one hand and the radius of the cloud and the observed line width on the other. The motion of the cloud, its internal heat energy, its magnetic field, the action of external force and the interaction between the cloud and its surrounding matter are linked together by the virial equilibrium relationI131, 1 D21 -=2T+3II+m+W+& 2 Dt2

j(r.B)B.da-/(P+E)r.ds * *

)

(1)

where I is the generalized rotational inertia, and T, II, m, W are the well-known kinetic energy, internal energy of random motion, magnetic energy and gravitational energy. We assume the cloud to an isolated, self-gravitating sphere, of uniform density n, uniform magnetic field B and radius R. That is, there is no energy interchange at the cloud boundary. Under equilibrium, equation (1) can be simplified into a relation between the internal, mag-

Molecular Cloud Cores

netic,

and gravitational

energy

densities,

439

KT, KNT, UC:

L is Boltzmann’s constant, T the gas temperature, G the gravitational constant, M and m, the cloud mass and the mean molecular weight (2.33amu). For a cloud of temperature T, the thermal velocity dispersion is UT = dm, and for a magnetic field B, the Alfvkn velocity is VA = B/SW, and we can define a nonthermal velocity dispersion by ONT = B,fZJG = V_/fi. Then, equation (2) can be re-written as

where

(3) From this we derive the expressions for the number density and magnetic field for an isolated, uniform magnetized sphere in terms of the thermal and nonthermal velocity dispersions:

(4) n= We assume the observed line width by the nonthermal dispersion:

15 1 -4 aGR2m is partly AV2 -=-

&+a& caused

>

.

by the thermal

dispersion

and partly

KT

8ln2

/.A +u’T

(6)

’

where p is the mass of the molecule

of the observed line. For a cloud of a given temperature, we can calculate bT and bNx, then from (4) and (5) we can calculate the magnetic field and the number density. The discussion below will show that the cloud temperature has little effect on the estimates of the magnetic field and number density. Hence, we can take a typical

value, T = 10K.

3. SELECTION Observations

by different

antennas

differ in regard

OF DATA to sensitivity,

and spatial

and frequency

resolutions. Large systematic differences may therefore exist between data from different antennas. If we mix them together in our analysis, we may fail to detect any statistical trends. In this paper we opt to use the observed data of CS (J = 2 -+ 1) of the molecular cloud L 1630113~141. It is a survey of the L 1630 region by the 7m telescope of AT&T Bell Laboratory. The total area scanned is 3.6 sq deg, the distance of the source being 4OOpc, the scanning interval is 1’ or 0.12 pc, and the resolution is 1.8’ or 0.21 pc. 39 dense cores were identified in this cloud above the 3a level. Table 1 lists, for each core, its effective radius (defined as the geometric mean of its long and short axes, for its actual shape is usually very complicated) and the FWHM width of its molecular line. The data in Table 1 show that the observed line widths are far greater than the thermal as due to ‘the magnetic field. In hi width (0.05 km/s), so we can regard the broadening

440

treatment

ZHENG Xing-wu & HUANG Yong-feng

of the same data,

Ladall

did not consider

the effect of magnetic

field in the

virial equilibrium. The reason for our choosing this particular data set is two-fold. First, the angular radii of the 39 dense cores and the CS line widths were all observed by the same telescope, and such homogeneous data can be expected to give more reliable results. Second, as we included magnetic field in the equilibrium consideration, we can compare our results with those obtained without including magnetic field.

4.

CALCULATION

For the calculation, we use practical and nonthermal velocity dispersions, density

of hydrogen

AND

RESULTS

units. In these units, the expressions the magnetic field, the cloud radius

for the thermal and the number

are U$ = 3.55 x 10e3T flj%~T= 0.18(A~)~

B

,

(km/s)2

- 1.87 x 10e4T 2c+,

=,,.,$(I+

-

4 n = 4.83 x 103-

1

(7) ,

(km/s)2

(8)

2 _ I 112 >

erg + u&

R2 (

PC

I

1

mm3

7

.

(10)

In the derivation of these expressions we took the mean molecular mass to be m = 2.33 amu, the mass of the CS molecule to be 44.1 amu, and a typical temperature of 10 K for the dense cores[r51. r , . ,

I.,.,.,.-(

a.0

02

0.4

Od

0.1

Radius(pc) Fig. 1 Correlation

between size and number density of the dense cores of L 1630. The curves are

model calculations

for three values of the magnetic

field, 50, 150 and 300 rG

441

Molecular Cloud Cores

Table Core

R

FWHM

1

Physical Parameters

B

KNT/'%

of Dense Cores in L1630 core

R

FWHM

B

n

KNT/UG

1

0.29

0.73

32

7.4e3

0.72

21

(PC) ('ds) 0.54 1.90

2

0.06

0.83

195

2.11~5

0.77

22

0.71

1.38

43

3.6e4

0.90

3

0.10

0.96

152

9.6e4

0.82

23

0.50

0.88

26

3.3e3

0.79

4

0.06

0.71

147

1.7e5

0.71

24

0.24

0.97

65

1.7e4

0.82

5

0.10

0.65

76

5.3e4

0.67

25

0.22

0.82

52

1.5e4

0.77

6

0.24

1.21

98

2.5e4

0.88

26

0.14

0.93

103

4.7e4

0.81

7

0.15

1.90

374 -1.5e5

0.94

27

0.32

1.34

89

1.9e4

0.90

8

0.16

1.07

117

4.5e4

0.85

28

0.16

1.42

199

7.5e4

0.91

9

0.24

1.79

208

5.le4

0.94

29

0.25

1.54

149

3.6e4

0.92

10

0.42

1.85

127

1.8e4

0.94

30

0.48

2.34

176

2.le4

0.96

11

0.20

1.16

108

3.3e4

0.87

31

0.06

0.99

269

2.8e5

0.83

12

0.10

1.40

310

1.9e5

0.90

32

0.09

1.51

399

2.6e5

0.92

13

0.71

1.82

73

6.0~~3

0.94

33

0.57

2.30

143

1.5.d

0.96

14

0.10

1.20

231

1.4e5

0.87

34

0.18

1.67

242

8.Oe4

0.93

(~4

b/4

(43

(:-'I

bG) (a-') 104 l.le4

0.94

15

0.10

1.80

505

3.0e5

0.94

35

0.10

1.84

527

3.le5

0.94

16

0.32

0.86

39

7.9e3

0.78

36

0.31

1.76

156

3.0e4

0.94

17

0.40

1.19

59

8.7e3

0.87

37

0.10

0.93

144

9.le4

0.81

18

0.56

2.19

132

1.4e4

0.96

38

0.14

1.62

294

1.2~~5

0.93

39

0.28

1.03

62

1.4e4

0.84

19

0.25

1.49

140

3.3e4

0.91

20

0.16

2.11

431

1.6e5

0.95

The physical parameters of all the 39 dense cores are given in Table 1. Columns 4 and 5 give the calculated values of the number density of hydrogen and the magnetic field. From the data shown we get an average core radius of 0.26 pc, an average line width of 1.39 km/s, an average number density of 8.0 x lo4 cmM3, and an average magnetic filed of 110 PG. Fig. 1 shows the relation between the number density and the core radius. According to equation (lo), the theoretical relation is n - R-*. This is graphed for’ three values of the magnetic field, 50, 150 and 300pG. Fig. 2 shows the relation between the number density and the magnetic field. According to equations (9) and (lo), the theoretical relation is (B/B,-,) = (r~/no)‘.~, the s&ix 0 referring to the original interstellar gas. The full curves are for an original number density of 1 molecule/cm3,

and magnetic

fields of 0.5 and 1 PG.

I+om (3) we find the ratio of the nonthermal gravitational energy UC = 3GMmn/5R to be

magnetic

energy

KNY = 3mna&.

to the

(11) This ratio is given in Column 6 of Table 1. It measures the fraction of the energy of gravitational contraction that is balanced by the magnetic energy in the virial equilibrium. For the 39 cores of L 1630, this ratio averages 87%. This means that magnetic energy is dominant in the equilibrium of such cores. Fig. 3 shows the relation between KNT/UG and

442

ZHENG Xing-wu & HUANG Yong-feng

the core radius.

The full curves

correspond

to model

calculation

for magnetic

fields of 50,

150 and 300j~G. 12

LO

0)

2

Y

2

OS

0.4

a2

0.0 T

0.0

Fig. 2 Correlation between the magnetic field and the number density in dense cores

5. 5.1

Magnetic

Correlation between KNTIUG and the size of dense cores

DISCUSSION

Filed

In the L 1630 region, we used the model

the magnetic

Fig. 3

I

of uniform

field in 39 dense cores from their observed

magnetized

sphere

and determined

core radii and spectral

line widths.

The field varies from 50pG to 500pG, with a mean of 110pG. For comparison, that the field from measurements of the Zeeman splitting of the HI line and the line at 1665 and 1667 MHz, was about 80 pG lr6~‘fi. This measured value is close to derived from our model. The difference is probably due to the different scales

we recall main OH the value involved:

HI and OH reflect the properties of the more extended diffuse gas, while CS, those of the smaller scale, denser cores of the cloud. From the 30% difference we can estimate an internal turbulent velocity of 0.5 km/s inside the cores. This is possibly caused by internal cascading turbulence, or by cloud-cloud collision or stellar windsl’s~‘gl. In Fig. 2 the two full curves correspond to fields of 0.5n112 and 1.0n1f2 pG, thus showing that, for an original interstellar density of 1 molecule/cm3, the background magnetic field is in the range 0.5-1.0 PG. This value is very close to previous estimates of the background filedl171. The number distribution of the field intensities is shown by the histogram of Fig. 4 and is as follows: of the 39 dense cores, 31% are in the range 50-100 /JG, 23% in loo-150 pG, 18% in 150-200 pG, and 28% above 200 PG. The distribution looks like a power law. Future

Molecular Cloud Cores

work may show whether

it also applies

to other molecular

443

clouds.

Magnaic Field (pG)

Fig. 4

Distribution of the field strength for dense cores

5.2

Number Density Our calculated values of the number density for the 39 cores fall in the range 3.3x 1023.1 x lo5 cmv3. The mean is 8~10~crn-~. Fig. 1 shows that, for cores with R > 0.2pc, that the there is a good agreement with the theoretical curve. In Section 1 we mentioned method of deducing the number density from observations at two transition frequencies has certain limitations.. The above conclusion means we now have a supplementary method for deriving the number density, namely, for clouds with R > 0.2pc, we can get a reasonable estimate from the observed line width and cloud radius, under the uniform model. This method, however, may not be applicable to cores with R < 0.2~~. We found the effect of the temperature on the calculated number density is small, and the main effect comes from the cloud radius. This suggests that the shape of the core may be another cause for the discrepancy.

Also, a non-uniform

distribution

of the core may be another

factor,

since, for

example, a cloud with a core-halo structure will have different equilibrium conditions from a uniform magnetized sphere. The total mass of the molecular cloud Orion B from CO observations was 8.3 x lo4 lv1~I~~1. This value refers to the total mass of the cloud which covers a,region of 19 sq deg. The dense cores observed by CS account for only 20% of the total surface distribution of the CO cloud, hence, proportionally, the total mass of the CS dense cores is 1.7x lo4 M,, and this ia in good agreement with our calculated value of 1.3x lo4 M,. The NH3(1,1) and (2,2) observations of the Orion B region I24 gave a number density of 2x lo4 cmv3, and this is very close to our average calculated value. 5.3

Temperature In our model, the core temperature is an adjustable parameter, but it is not a sensitive parameter for the calculation of the number density and the magnetic field. For example, for a core with R = 0.25~~ and AV = 1.5 km/s, the values of the number density and magnetic field calculated assuming T = 20 K or T = 10 K differ from each other by only 8%

ZHENG King-wu & HUANG Yong-feng

444

and 5%. Also, the mean kinetic and (2,2) observations, temperature

temperature

was less than

of the cores of this region,

15 K1231. Hence, it seems adequate

found

by NHs(l,l)

for us to assume

a

of 10K in our model.

6. CONCLUSION Applying a model of uniform, magnetized sphere to the 39 dense cores in the molecular cloud region L 1630, we have estimated from their observed radius and line width, their number density and magnetic field. Comparing our estimates with the actually measured values, we found our method to be effective for cores with radii R > 0.2~~. On conservative estimates, our calculated values should not differ from the observed values by more than a factor of 2. The cause for the discrepancy is complicated and varied, mainly the shape of the source region, the density structure and the mechanisms of line broadening. ACKNOWLEDGMENT We thank Dr Myers for the many throughout this work and for his valued opinions.

discussions

we had with him

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