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Continuum emission from H I I regions and dusty molecular clouds

Riccardo Cesaroni
INAF­Osservatorio Astrofisico di Arcetri, Firenze, Italy E-mail: cesa@arcetri.astro.it The classical description of the formation and evolution of ionized (HI I) regions associated with early-type stars is reviewed, with illustrative examples taking into account constant and power-law density profiles. The expression for the free-free continuum flux density from such HI I regions is then calculated and the main HI I region parameters are derived as a function of measurable quantities. The thermal emission from dust grains in molecular clouds is also briefly described and template spectral energy distributions in the simple case of homogeneous, isothermal, spherical clouds are discussed.

2nd MCCT-SKADS Training School Radio Astronomy: fundamentals and the new instruments August 26- September 4, 2008 Sigöijenza, Spain


Speaker.

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Riccardo Cesaroni

Figure 1: Continuum image (colour scale) at 20 cm with the Very Large Array interferometer of the Natio circle denotes a large HI I region which was resolved a map of the emission integrated under the J = 1 with the SEST telescope. Clearly, the molecular gas ionizing stars formed inside a molecular cloud.

of the high-mass star forming region S254/7 obtained nal Radio Astronomy Observatory. The yellow dashed out by the interferometer. The overlaied contours are 0 rotational transition of the 13 CO molecule obtained is enshrouding the ionized regions, as expected if the

1. Size of HI I regions
It is well known that early-type main-sequence stars are powerful emitters of Lyman continuum radiation. Such stars are born deeply embedded in the densest parts of molecular clouds, whose main constituent is molecular hydrogen (H2 ). Stellar photons with wavelengths < 912 å can first dissociate the H2 molecules and then ionize the H atoms, thus creating a so-called "HI I region" around the star. Figure 1 shows a number of these objects in the S254/7 star forming region, whose radio continuum emission has been imaged with the Very Large Array interferometer. The overlaied map of the 13 CO(1­0) line emission demonstrates that the HI I regions are still enshrouded by the dense molecular cloud where the O-B stars were born. While the shape and density of an HI I region is strongly dependent on the initial distribution of the circumstellar neutral gas, the size of it is determined in all cases by the balance between ionization and recombinations to the ground state of the H atom. In fact, recombinations to levels n > 1 are bound to produce photons with > 912 å, which are lost to the ionization process. This concept is expressed by the equation NLy = ne 2 2 dV (1.1)

where the integral extends over the whole volume of the HI I region and NLy is the number of ionizing photons emitted by the star per unit time, ne the electron density inside the HI I region, 2


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Riccardo Cesaroni

and ne 2 2 the number of recombinations per unit time and volume to levels n 2. The underlying assumptions are that the HI I region is fully ionized and all photons due to recombinations to the ground state, n = 1, are immediately re-absorbed by the H atoms ("on-the-spot" approximation). It is instructive to discuss the simple case of a spherical, isothermal HI I region and obtain the radius at which ionization equilibrium is attained, the so-called StrÆmgren radius, RS . In particular, in the following we consider a constant density HI I region and one with a power-law density profile. 1.1 Constant density HI I region In spherical symmetry, Eq. (1.1) can be rewritten as
R

NLy =

S

R

2 ne 2 4 R2 dR

(1.2)



with R distance from the star, R stellar radius, and 2 (cm3 s-1 ) 4.1 â 10-10 [Te (K)]-0.8 . Here Te is the electron temperature, which is quite constant (typically several 1000 K) inside the HI I region due to the nature of the balance between cooling and heating (see [6]). If the star lies inside a homogeneous molecular cloud with H2 volume density nH2 , one has also ne =constant. Taking into account that, in practice, RS R , one obtains RS 3 NLy 4 2 ne
1 3

2

(1.3)

with ne = 2nH2 . The factor 2 is due to the fact that each H2 molecule produces two electrons (and two protons), because molecular hydrogen is first dissociated into two H atoms (forming a layer of atomic hydrogen around the HI I region) and then each H atom is ionized into one electron and one proton. The condition of ionization equilibrium represented by this expression does not correspond to pressure equilibrium, because the external pressure due to the neutral cold H2 gas cannot balance the internal pressure of a 104 K bubble of ionized gas. One has, pHII = 2ne kTe pH2 = nH2 kTH
2

(1.4)

because the molecular gas temperature, TH2 , is typically a few 10 K and the electron density is twice the H2 density (see above). Due to this non-equilibrium situation, the HI I region undergoes expansion. Since the ionization time scale is much shorter than the dynamical time scale, during the expansion the ionization equilibrium condition expressed by Eq. (1.3) is satisfied at all times. 3 This implies that ne RS - 2 . One can calculate the StrÆmgren radius as a function of time (see [1]): 7CII RS (t ) = RS (0) 1 + t 4RS
4 7

(1.5)

where RS (0) is given by Eq. (1.3) and CII = (2kTe /mH )1/2 is the isothermal sound speed in the HI I region. Expansion will go on until pressure equilibrium is attained, when the final density satisfies the condition 2nf kTe = nH2 kTH2 . From this expression and the relation ne = 2nH2 (see above), one e obtains nf nf TH e = e = 2 10-3 . (1.6) ne 2nH2 4Te 3


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Taking into account that ne R

S

-

3 2

, one finally gets nf e ne
-
2 3

RS f = RS

=

TH2 4Te

-

2 3

100

(1.7)

where RS f is the final StrÆmgren radius. This result shows that the final size of an HI I region is much larger than the initial one. 1.2 Power-law density profile In the real world, OB stars form in the densest parts of molecular clouds, where circumstellar material is free-falling onto the star. The free-fall velocity and conservation of mass are expressed by vff = 2GM R = 4 R2 mH2 nH2 vff M (1.8) (1.9)

from which, taking into account that ne = 2nH2 , one obtains the power-law density profile ne = In this case the solution of Eq. (1.2) is RS = R exp NLy 2 GM m 2 M 2
2 H
2

M2 8 2 GM m

2 H

R
2

-

3 2

.

(1.10)

(1.11)

which means that in spherical symmetry an HI I region is squelched (i.e. RS R ) if the mass accretion rate onto the star exceeds a critical value: M> NLy 2 GM m 2
2 H
2

.

(1.12)

2. Continuum emission from HI I regions
The main source of continuum emission from HI I regions at radio wavelengths ( > 1 cm) is bremsstrahlung (free-free) radiation. Here, we want to derive an expression for the continuum spectrum of such a region and show that important physical parameters of the HI I region can be obtained from this. In the simple case ne =constant, the solution of the radiative transfer equation along a line of sight crossing the HI I region is: I = B (Te ) 1 - e-ff (2.1) The free-free optical depth is equal to (see [4])
2

ff = 3.014 â 10-2 Te-1.5
8.235 â 10-2 Te-

-2

ln

4.955 â 10-
l.o.s.

+ 1.5 ln (Te )

l.o.s.

ne 2 dz

1.35 -2.1



ne 2 dz

4


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Figure 2: Template spectrum (black curve) of HI I region embedded in a dusty molecular cloud. The relevant physical parameters are given in the figure, where E M = 2RS ne 2 and RC is the radius of the cloud. Here, = 2 is adopted for the dust opacity. The blue and red curves denote the individual contributions respectively of the free-free and thermal dust emission.

where the integral is made along the line of sight (l.o.s.), Te is expressed in K, in GHz, ne in cm-3 , and z in pc. The quantity E M = l.o.s. ne 2 dz is called "emission measure". Although most of the following results hold in general, for the sake of simplicity we will refer to a spherical H 10-2 Te-
1.35 -2.1

II

region. In this case ff ( ) =

S

1-

S

2

with S 8.235 â

2RS ne 2 and the flux density of the HI I region is F = I d =

0
S

S

B (Te ) 1 - e- 2 S

ff ( )

2 d

(2.2) (2.3)

2 = S B (Te ) 1 +

e-S - 1 e-S + S

In practice, h kTe and the Rayleigh-Jeans approximation can be used for the black-body bright2 e ness: B (Te ) 2 kT2 . An example of free-free continuum spectrum from an HI I region is shown c by the blue curve in Fig. 2. It is useful to derive the approximate expression for Eq. (2.3) in the optically thin and thick limits. For S 1 one has
2 2 F S B (Te ) S Te 2

(2.4)

5


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Figure 3: Left panel: Flux density of expanding HI I regions as a function of the StrÆmgren radius, for the fiducial set of physical parameters listed in the top left. Different colours correspond to different stellar spectral types, as specified close to each curve. The thick part of each curve comprised between two solid dots represents the flux effectively emitted by the HI I region between the initial radius (given by Eq. 1.3 with ne = 2 nH2 ) and the final one (given by Eq. 1.7). For the sake of comparison the dashed lines denote the sensitivity at 22 GHz of the VLA interferometer in the most extended configuration, after 1 hour integration on source. Right panel: Flux density as a function of time for the same curves as in the left panel.

from which one can get an estimate of the HI I region angular radius S For S 1 2 2 2 F S B (Te ) S S Te-0.35 ne 2 RS -0.1 3 from which the electron density can be obtained: ne can be expressed as F Te-
0.35 F Te0.35 3 S d -2
0. 1

-1

F Te

. (2.5)

. Alternatively, the flux density (2.6)

RS 3 ne 2 d



-0.1

which makes it possible to derive the Lyman continuum luminosity of the star: NLy = 4 2 -0.45 0.1 . For convenience of the reader, we also express the electron density 32 3 RS ne 2 F d Te and Lyman continuum luminosity in terms of physical units commonly adopted in astronomy: ne (cm-3 ) 1.44 â 105 F (Jy)
0.1

(GHz) d
0.1

-1

- (kpc) S 3 (arcsec)

(2.7) (2.8)

NLy (s-1 ) 7.55 â 1046 F (Jy) d 2 (kpc) 2.1 Flux density of expanding HI I region

(GHz)

As previously discussed, an HI I region is in general not in pressure equilibrium and undergoes expansion. Consequently, its flux density will change with time. The exact expression of F as a function of t can be obtained by substituting Eq. (1.5) into Eq. (2.3). This is illustrated in Fig. 3 where F is shown both as a function of RS and t , for a given set of fiducial parameters of the HI I region and molecular gas. It is instructive, though, to discuss the solution in the optically thick and thin limits. At a given frequency, the free-free opacity is bound to decrease with time, because we have seen that 1 3 ff RS ne 2 and ne RS - 2 , which implies ff RS - 2 . Therefore, initially the HI I region is likely 6


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optically thick and its flux is F RS 2 , while later on it will become thin with F RS 3 ne 2 NLy . This means that after undergoing a rapid increase, the HI I region flux will soon saturate at a value depending only on NLy , i.e. on the spectral type of the star.

3. Dust emission in molecular clouds
Dust grains are a fundamental component of the interstellar medium, despite their mass density being only 1/100 of that of the gas. The goal of this contribution is not to review the origin and physical properties of dust (for which the reader can refer, e.g., to [2]), but to describe the characteristics of the continuum emission from a typical molecular, dusty cloud and show how knowledge of the corresponding spectral energy distribution (SED) can provide us with an estimate of the mean physical parameters of the cloud itself. The dust emissivity beside depending on the density and temperature of the dust grains is also a function of their composition and shape. While a detailed description of the dust optical properties goes beyond our purposes, one can naÎvely describe the dust opacity from the radio to the optical as a decreasing function of wavelength, mostly due to pure absorption in the (sub)millimeter regime (where most grains are larger than the relevant wavelength) and to both absorption and scattering in the IR and optical. Both observations and theory (e.g. [3]) indicate that a suitable approximation of the dust absorption coefficient in the (sub)millimeter is , with ranging approximately from 0 to 2 depending on the mean grain size. As already done for the free-free emission from HI I regions, it is instructive to consider a toy model consisting of a homogeneous, isothermal, spherical cloud and calculate the SED of the dust continuum emission. The flux density is given by Eq. (2.3) where Te and ff must be replaced respectively by the dust temperature, Td , and opacity, d . This is often referred to as "grey body" emission as it may be naÎvely described as a black-body spectrum becoming optically thin at long wavelengths. An example is shown by the red curve in Fig. 2. For a practical example, we consider a cloud with radius R = 0.5 pc, Td =30 K, gas density = 3 â 10-19 g cm-3 , a mass gas-to-dust ratio of 100, (cm-1 ) = 0.005 (g cm-3 ) ( /230.6GHz) , and = 2. For these fiducial values, d =1 at = 90 µ m and the Rayleigh-Jeans approximation 2 d (B (Td ) 2 kT2 ) holds for 500 µ m. For 90 µ m (i.e. in the optically thick regime) c F 2 B (Td ) with angular source size and is known, e.g., thin limit), one (3.1)

radius of the cloud. This shows that the measured flux depends only on the apparent on the dust temperature, so that one can obtain an estimate of the latter if the former from a map of the source. On the other hand, for 90 µ m (i.e. in the optically has 2 F 2 B (Td ) d (3.2) 3 which in the Rayleigh-Jeans regime ( 500 µ m) takes the form F Td M d
-2



+2

(3.3)

with M mass of the cloud. One can easily obtain from the ratio between two values of F measured at two different wavelengths (provided 500 µ m), while Td can be estimated in the optically thick limit, as explained above. Then, Eq. (3.3) can be used to calculate the cloud mass. 7


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While this idealized example is very useful for illustrative purposes, it must be kept in mind that real molecular clouds are much more complex, which makes the derivation of their physical parameters a challenging task. In particular, temperature gradients make questionable the assumption of constant temperature. Inhomogeneities, density gradients, and complex cloud geometries all contribute to cause significant deviations of the SED from the ideal grey-body approximation. This is especially relevant at short wavelengths, where clumpiness may cause strong variations of the measured flux depending also on the orientation of the cloud with respect to the observer. Complex numerical models have been developed to elaborate more realistical SEDs (see [5]).

References
[1] J.E. Dyson, D.A. Williams, The physics of the interstellar medium, The Graduate Series In Astronomy, Bristol 1997 [2] J.S. Mathis, Interstellar dust and extinction, ARA&A 28 (1990) 37 [3] V. Ossenkopf, Th. Henning, Dust opacities for protostellar cores, A&A 291 (1994) 943 [4] P.G. Mezger, A.P. Henderson, Galactic H II Regions. I. Observations of Their Continuum Radiation at the Frequency 5 GHz, ApJ 147 (1967) 471 [5] T.P. Robitaille, B.A. Whitney, R. Indebetouw, K. Wood, Interpreting Spectral Energy Distributions from Young Stellar Objects. II. Fitting Observed SEDs Using a Large Grid of Precomputed Models, A p JS 1 6 9 ( 2 0 0 7 ) 3 2 8 [6] L. Spitzer, Physical Processes in the Interstellar Medium, Wiley, New York 1978

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