Energetics of the molecular cloud W40

Chin.Astron.Astrophys. 2 (1981) 413-417 dct.dstrophys.Sin. 1 (1981) 234-242

ENERGETICS XU

Lan-ping,

OF

THE

XING

Pergamon

MOLECULAR

jun,

CLOUD

VIU Yue-fang

Beijing University, XIE Shu-ding, Observatory, Academia Sinica. Received

1980 September

Press. Printed in Great Britain 0275-1062/81/040413-05$07.50/O

W40

Department ZHAO

Jing-zhi

of Geophysics Beijing

1

ABSTRACT In this paper we study the main processes of energy exchange in the molecular lines and infrared data, we estimate the basic physical parametersofthe cloud and calculate its cooling and heating rates by gas and by dust. Based on the results of calculation, we discuss the energy constraints among the dust, the gas and the embedded infrared source.

1. INTRODUCTION Following the development of observing techniques in the millimetre and infrared ranges, a series of HI1 region - infrared sources - molecular cloud complexes have been discovered in It is now generally recognized that the association of the three and their the Milky Way. close physical connections are an important sign that stars have recently been formed or are in the course of being formed. A study of the various processes of energy absorption, release and transfer by the main components (gas and dust) of the molecular cloud in such a complex will give a glimpse of the sources of energy and the physical conditions therein. It will also provide energy constraints on the dynamical processes that may eventually reveal the evolution of the cloud. W40 (G28.8 + 3.5) is one of the complexes with the above features. Its exsistence was first detected in the radio wave range (X-cm neutral hydrogen line) by Westerhout in 1958 111. At the end of the '60s , Reifenstein et al. /2/ observed the H109a recombination line in its HI1 region at 5 GHz and, combining the free-free continuum in the vicinity, they In 1974, Olthof /3/ measured the far derived a set of physical parameters of the region. infrared flux of its infrared source using balloon. In 1978, Zeilik and Lada I41 gave an intensity map of the infrared source at 2.2~, a spectral brightness distribution in the range 2-ZOu, an intensity map in the laCO line. Using available data, we have determined the basic physical parameters in the molecular cloud of W40 (kinetic temperature, density, etc.) and, under certain simplifying physical assumptions, calculated the gas cooling rate, the gas heating rate, the dust tolling rate Following an analysis, we shall discuss the energy exchange and and the dust heating rate. transfer between the various components of the cloud and the question of the original energy source. 2. PHYSICAL

PARAMETERS

OF THE MOLECULAR

CLOUD

1. Geometrical Assumptions In order to evaluate the physical parameters of the molecular cloud, we followed Evans et al. /5/ and made the following assumption concerning the geometry We assume that the area a. contained in an equal-intensity of the cloud. contour of Fig. 1 to be the projected area of a spherical legion, which would then have a radius of

, a surface area of S. = 4a. and a volume of V. = 0.75a?". Successive c&tours tken enclose shells of volkme AV': = vi - vi_l and a'i1 physic:1 properties are assumed to be the same within each shell.' The distance of the W40 molecular cloud is taken to be 0.7 kpc, after the calculations Shaver and Goss /6/.

P. = 0.56a:"

of

2. Physical Paraters 1) Kinetic Temperature T Because the velocity gradient of the cloud collapse is sufficiently high, the Sobo ev approximation holds; also, the "CO spectral line is optically thick and thermalized, 'I = m, TK = Tex, hence TK is given directly by the antenna temperature TJ of '*CO: 1 ehdKTbb

(1) -

1 I ’

Energetics of W40

414

= 2.7

K,

the calculated temperature in

the var ous shells is shown in Column With T5P

3 of

Table 1. 2) Volume Density of Hydrogen 7ZHp. According th Zeilik and Lada 141, at the "CO peak location, the brightness temperature is Tb = 9.3 K, AV = 2.2 km/s, and the column density of '=cu is rtlsz= 5.6 x fO~~lcln2. Applying the relation between the column densities of "CO and HI, derived by Dickman from dark dust cloud,

I

I

16'2Sn.14' 28"50‘

13

I

28"26‘ 28°C 32’

RA(i950) Fig.

Intensity map of W40 after !4!.

(2) on,1= 5.0 x lO%,J to the molecular cloud in W40 gives a column density of Hn of 2.8 x 10za/cm2. Equating z to the diameter of the sphere for TA = 15Kf4.2 x 10'aem), we have = 6.7 x 103/cm3. %a Comparing the values of nHzfrom (2) for other clouds with those directly found from fomaldehydemeasurements, we note that the values using the "CO data are probably underestimates.

TABLE 1. 5

(+)A%

,4j(CO}A.yj (to>

31 25 20 I5 10

34.5 28.5 23.5 18.5 13.4

1.08

4.70 28.9k2.17 138 526

5.79x10-4 2.61x10-’ 2.39x10-’ 1.15x10-‘ 3.13x104

cw 47.7 33.1 21.0 14.2 10.2

1.92x10-4 8.03x10-' 6.8X10" 0.54x102.75x10-~

3) Infrared parameters To calculate the dust cooling rate we need the optical thickness in the far infrared 'FIR. Using Evans' empirical relation 1o-‘%,3t rFIR = (3) = 0.056 for the central region of the cloud. we find T FIR Using Olthof's observations /3/, we find that the infrared luminosity of the embedded infrared source in the range 20 - 20011,LZO-ZOOM = 3.3 X 104Llp 3. PROCESSES OF ENERGY EXCHANGE IN THE MOLE~U~R CLOUD In a molecular cloud, the gas temperature and density are both low, transfer of energy through collision between gas and dust has a low efficiency, so that the relaxation time for such collisions is very long and the gas and dust each wfll keep their own temperature over a long time. Hence, when we discuss the energetics of the cloud, we can regard the gas and dust as two separate systems. We now proceed to consider these separately. 1. Energetic6 of the Gas We believe that inelastic collisions between gas molecules and dust particles and collapse of the cloud are probably the main mechanisms for compensating the cooling by the gas molecular line emission and keeping the gas at the observed temperature. 1) Gas Cooling Rate The cloud temperature being between a little over ten degrees K and a few tens of degrees K, the thermal motion cannot cause any electronic or vibrational transitions of the hydrogen molecules it can give rise only to the rotational transitions of polar molecules (mainly CO). Therefore, the cooling of the hydrogen gas is through the collision between hydrogen and CO molecules in which energy is passed to the CO molecules which is then radiated away in the CO rotational lines. The cooling rate by the CO line

415

Energetics of W40

emission can be found from Evans' formula Ai

iiii 2 X 10Fn X T:

(erg/cm3* S),

(4)

which is valid in regions where nCOnH >> 10' /cm". The molecular cloud of W40 is divfded into shells according to the equal-intensity contours of Fig. 1. The calculated contribution of each shell to the cooling rate is shown in Column 5 of TABLE 1. The total cooling rate is c Ai(CO)AVi= 0.46Lo 2) Gas ieating Rate The gas loses energy through CO emission. In order that the gas is not cooled, we must have an equal amount of heat input. One of the possible heating mechanisms is the heating from cloud collapse. During a collapse, gravitational potential energy overcomes thermal pressure and does work on the gas, part of the potential energy is transformed into the thermal motion of the molecules. The rate of energy input through Evans f5/ has given an collapse is usually described in terms of the free-fall time T ff expression . For rff, which we have verified to be reasonable under the cloud conditions

Here,

dV J.7rl_E =p (>dc T dr c is the volume of the cloud (= (4/3!7R', R being the cloud radius),

V

?!I!= 4&'_!@=3ys dt R ” dr tts being the speed of collapse at the cloud surface, PT is the thermal pressure ( = NH,kTX = (Z/3) E, B being the internal energy per unit volume of the cloud. Hence,

=2 with ~~~ = R/2vc.

E = -$ r..

rl, = R/&,,

For a unifo:m density, we have

1-o;=4 aR’Gp = -4 ~~R’GNH,~H,, 3 2 3 and after substitution we have rf,= 3.66 X lO”N~~(s), ,J

To assess the effectiveness of this mechanism, we list in TABLE 2 the heating rate and the cooling rate in each of the shells. From this we see that the rate of heating through collapse is smaller than the gas cooling rate for all the shells, so that some other heating mechanism must be considered. TARLF 2. NO. .4c0(10-+rg/s)

I

1 82

3 26

2 46

4 12.7

5 4.8

One other possible mechanism is through inelastic collisions between dust and gas. The embedded infrared sources W40IR heats up the dust in the molecular cloud, making the dust temperature higher than the gas temperature. Inelastic collisions between the dust particles and the gas molecules then transfer part of the energy of the dust to the gas. The rate of energy transfer can be described in terms of the relaxation time t . Suppose each collision gives the hydrogen molecule an average energy increment of (3/2)f;(Td- TK), Td being the dust temperature, and in unit volume and unit time there are N v o Hz H2 bvd collisions (7-J is the mean thermal velocity of the hydrogen molecules, Nd is the number HZ density of dust particles, oJ is the cross section of the dust particles). Then

u

. +(T,

-

TR) =

;

NnI * K(Tai -

t, Taking the standard values Nfld = 2 X li)-a'Nn,(cmz) and un,=

we have

t,= 4.5 x 10’6N~~T~‘~(s)

7’~) 3K ~- TK J- mH,

in agreement with the expression given by Goldreich and

Energetics of W40

416

Rwan 171. For the central part of the cloud of W40, we have t = 1.15 x 10L2 s , which is of the same order of magnitudeas 'I found above. Thus, it sgems that, in the central part of the cloud, heating by dust is aJX' ther important mechanism alongside heating through collapse. Considering that the value of ?z determined from "CO data may be too low, we should raise it suitably and the correspon8 4ng values of r and t, will be lessened, that is, the

ff

two heating mechanisms will be more important. On the other hand, in peripheral regions with lower nHa, other heating mechanisms may be important and should be considered. On one hand the dust component of the molecular cloud takes 2. Energetics of the Dust energy from the embedded.infraredsource, on the other, it loses to the gas component and through its own radiation. The balance between the two keeps the dust temperature constant. 1) Dust Heating Rate Observations on the infrared source W4OIR show that it emits a vast quantity (3.3 x 1O“Ls) in the far infrared range 2 - 2OOu. This is the main source of heating the dust. We have scrutinized the sky surrounding W40 and found that all other objects are at some distance away, hence their effect can be neglected in this discussion. The expression for the dust temperature is derived from thermal equilibrium considerations. Let the mean absorption coefficient of the dust in the far infrared range be a, its emissive power is then 6.=.=&$---. 51 Since, at the temperature concerned, the thermal emission is concentrated in the far infrared, the total emissive power is E s aT2, a being the Stefan constant. The emissive power per dust particle is em = a,+*a&': = mT:. P where K is the particle absorption coefficient. On the other hand, since the temperature is kept constant, the energy absorbed by each shell must be equal to the energy emitted by it. In the case of a source-free shell, the total luminosity entering the shell must then be equal to the total luminosity L of the central infrared source. Consider the shell between r7:and ri + dr?:. Here, the incident intensity is1 = L/47rrzand the total absorption power of the dust and the absorption per particle are, respectively, L pabs c&f=x---= 42~: nNdfdri = rlNddVi , Pgg = 4nr:tii 43cr: l

When the dust temperature is not extremely low, thermal emission by the dust far exceeds the amount given to the gas, i.e., Pm % pabs, hence the dust temperature in the shell is

Tdi = y;(:,"'

= 0.87(3"

where L is in units of L, 'andP. in PC. TABLE 1 Column 6 lists the galculated dust temperature in the various shells of W40. For the two shells at the centre, the dust temperature is greater than the gas temperature, hence energy flows from the dust to the gas, at the rate of 1.63 x lo-*' erg/s for shell 2. Because we did not include the emission in the long wave range ( > 200~) of the embedded source, nor the heating by the ambient radiation field, especially the heating of the outer shells of the cloud, and also because we have neglected the effects of the interstellar absorption and of stars in the vicinity, the above dust temperature estimates are lower bounds. 2) Dust cooling Rate There are two causes of dust cooling. One is the transfer of energy to the gas through inelastic collision. From the above discussion on the energetics of the gas, we see that the total energy passed on to the gas is always less than that corresponding to the gas cooling rate (0.46Ls). The other is the energy release through thermal emission. In the shell between ri and ri + dri, the number of dust particles is Nd4.rrri2dri , and the thermal emission power is em Ci = PemNd4ar: dri

-

MT: Na4ir;

dri

= aT:<$;

where S. is the surface area of the shell and T. is the mean optical thickness in the wavethe central portion of W40, we have length 'range concerned. As mentioned above, f;";r = 0.056. For the peripheral regions, the "CO column density is lower, hence so is ri 'FIR but for a rough estimate, we shall still take 0.56. We then have the total dust cooling

Energetis

rate F

417

of W40

9.4 x 1O’Lo.

ci=

Since thermal emission increases as the fourth power of temperature, we see that, at the temperature of the cloud, the thermal radiation far exceeds the energy transferred to the gas. Hence in the calculation of the dust cooling rate, we can neglect the gas altogether and consider only the balance between thermal emission and the heat input. 4. CONCLUDING

REMARKS

From the above calculations and analysis, we conclude 1. The total energy given to the gas from heating by dust and from collapse of the cloud This is less than the amount carried away by the CO molecular emission (0.46L,) is 0.34L,. Hence, heating by dust and by cloud collapse are probably two main heating mechanisms, but in regions of higher temperature and lower density, there may be some other important heating mechanism operating. 2. Since the total gas cooling rate is 0.46L, while the total emission by the dust is 9.4 x 10SLB the main path of energy transfer in the cloud considered as a whole, is through the dust. 3. This being so, the infrared source that provided the dust with energy must be identified as the main energy source of the cloud. 4. Because the dust in the molecular cloud is optically thin in the far infrared range, it absorbs only 28 percent of the total energy of the infrared source (9.4 x 10' / 3.3 Y. 10' = 0.28).

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