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The Density Structure of Highly Compact H ii Regions
Jos'e Franco 1 , Stan Kurtz 1 , Peter Hofner 2 , Leonardo Testi 3 , Guillermo Garc'ia­Segura 1 and
Marco Martos 1
Received ; accepted
version : August 28, 2000
1 Instituto de Astronom'ia, Universidad Nacional Aut'onoma de M'exico, Apdo. Postal
70­264, 04510 M'exico D.F., Mexico
2 Department of Physics, University of Puerto Rico, P. O. Box 23343, Rio Piedras, PR
00931; and NAIC, Arecibo Observatory
3 Osservatorio Astrofisico de Arcetri, Largo Enrico Fermi 5, I­50125 Firenze, Italy

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ABSTRACT
We report the density structure of the ultracompact H ii (UC Hii) regions
G35.20\Gamma1.74, G9.62+0.19­E, and G75.78+0.34­H 2 O. The density profiles are
derived from radio continuum emission at wavelengths from 6 to 0.3 cm. In
the case of G35.20\Gamma1.74, a cometary UCH ii region with a core and a tail, the
spectrum of the core varies as S š / š 0:6 , implying that the density structure
is n e / r \Gamma2 . The emission from the tail has a flatter spectrum, indicating
that the density gradient is also negative, but shallower. For the case of
G9.62+0.19, which is an H ii region complex with several components, the
spectrum of the region designated component E is S š / š 0:95 , corresponding to
n e / r \Gamma2:5 . The steepest spectral index, S š / š 1:4 , is for the super UCH ii region
G75.78+0.34­H 2 O; its density stratification may be as steep as n e / r \Gamma4 . The
actual density gradient may be smaller, owing to an exponential (rather than
a power­law) density distribution or to the effects of finite spatial extent. The
contribution from dust emission and some of the possible implications of these
density distributions are briefly discussed.
Subject headings: ISM: clouds --- H ii regions -- radio continuum: ISM

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1. Introduction
Stellar groups form in the dense cores of molecular clouds. The structure of these
clouds has been unveiled with optical, infrared, and radio observations over the last three
decades, and a great effort has been made to understand the connections between cloud
properties and star­forming activity. In particular, knowledge of the gas density structure is
required to determine fundamental properties, such as the mass and stability of star­forming
cores. Extinction studies can be used to derive the density profiles of clouds but, given the
large opacities involved, they can only probe the outermost gas layers. Tracers occurring at
radio frequencies, on the other hand, can penetrate deeper into the cloud, and can reveal
the density stratification of dark clouds and cloud cores.
The information obtained from extinction and molecular­line studies shows that
molecular clouds are centrally condensed. For instance, moderate density (10 3 -- 10 5 cm \Gamma3 )
envelopes surrounding higher density (10 7 cm \Gamma3 ) cores suggest the existence of strong
density gradients within clouds (Hofner et al. 2000). The clouds can be approximated by
power­law density distributions of the form ae(r) / r \Gamma! , with exponents ranging from 1 to
about 3 (e. g. Young et al. 1982; Dickman & Clemens 1983; Arquilla & Goldsmith 1984;
Gregorio Hetem, Sanzovo & Lepine 1988), with 2 being the most common value (Arquilla
& Goldsmith 1985). The steepest gradients, with ! ¸ 3, are seen in star­forming clouds
and dark globules, whereas values between 1 and 1.5 seem to apply to rotating clouds
(Arquilla & Goldsmith 1985), and nearby small cloud groups (Myers, Linke & Benson 1983;
Cernicharo, Bachiller & Duvert 1985). The above molecular results provide the overall
density structure of clouds but the stratification of the actual star­forming cores is only
beginning to emerge. In particular, submillimeter imaging of dust emission in star­forming
cores suggests exponents between 1 and 2 (Hunter 1998; Chandler & Richer 2000; Hatchell
et al. 2000).

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Hot molecular cores are probably the most massive and dense condensations within
molecular clouds, and are thought to be the sites of massive star formation (e. g. Kurtz
et al. 2000 and references therein). Hence, the early stages of H ii region evolution occur
within the density structure of these cores, and the observed properties of the most compact
H ii regions may provide valuable information about cloud density stratification. For
instance, the evolution in uniform ambient densities has some well­known evolutionary
phases (e. g. , Kahn & Dyson 1965), Osterbrock 1989) but there are significant departures
from this simple model for non­uniform density distributions. For negative density
gradients, the expanding gas can greatly accelerate (sometimes producing internal shocks)
and, depending on the gradient, the ionization front can travel outward indefinitely (Franco,
Tenorio­Tagle & Bodenheimer 1989, 1990). Indeed, when the molecular cloud has a strong
negative density gradient, steeper than ae(r) / r \Gamma3=2 , the ionization front overruns it and
the entire cloud is photoionized. The spectral index of the radio­continuum emission from
the resulting H ii region, as long as the central regions are optically thick, depends on the
density gradient, and lies between \Gamma0.1 and 2 (Olnon 1975; Panagia & Felli 1975).
In this paper we make a first step toward unveiling the innermost density structure
of massive star­forming regions, and report the density profiles of the UCH ii regions
G35.20\Gamma1.74, G9.62+0.19­E, and G75.78+0.34­H 2 O. Their continuum spectral indices
indicate that their radio emission is produced by an optically thick plasma with a decreasing
density gradient. Their density structures can be approximated by a power­law n e (r) / r \Gamma! ,
where the exponents are larger than ! ' 2. Franco, Garcia­Barreto & de la Fuente (2000)
present a similar spectral index analysis of free­free emission from circumnuclear H ii regions
in barred galaxies, and find density gradients in the range 1:5 ! ! ! 2:5.
If no mechanism acts to maintain density inhomogeneities within the H ii region (of
radius R s and sound speed c II ), they will be smoothed out on the order of a sound­crossing

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time, R s =c II (Rodr'iguez­Gaspar, Tenorio­Tagle & Franco 1995). This is also true for the
initial density gradients, which are smoothed over time by the expansion. Thus, the density
gradients that we observe represent lower limits to the original distribution. Although
instabilities in the expansion process might lead to locally higher densities (e. g. fingers or
elephant trunks; see Garc'ia­Segura & Franco 1996) these are highly localized features and
are unlikely to significantly affect the density gradients we derive.
2. The Radio Emission of the H ii Regions
The UCH ii region G35.20\Gamma1.74, first imaged at high resolution by Wood & Churchwell
(1989), is a classic example of a cometary UCH ii region, 6 00 (0.09 pc) in size. The
ultracompact component coincides with the peak of a lower brightness and more extended,
2 0 (1.8 pc), cometary H ii region, whose symmetry axis is nearly perpendicular to that of
the ultracompact component (Watson, Hofner & Kurtz 2001). The emission measure is
greater than 10 8 pc cm \Gamma6 in the UC region (peak), rapidly drops to ¸ 10 7 pc cm \Gamma6 just
away from the UC component (tail), and then gradually falls to ¸ 10 5 pc cm \Gamma6 near the
edge of the extended region. Assuming that the ultracompact gas is embedded within
the extended gas, then the presence of both components (spanning nearly four orders of
magnitude in emission measure) already shows that the plasma is density­stratified on large
scales; here we consider only the 6 00 ultracompact core.
We used the 6 and 2 cm scaled­array maps of Wood & Churchwell (1989), the 3.6 map
of Kurtz, Churchwell & Wood (1994) and the 1.3 cm data of Hofner & Churchwell (1996)
to perform a spectral index analysis of the region. The highest resolution data (1.3 cm)
was sensitive to structures up to 7 00 in size, so little or no flux is missing from the UC
region for lack of short spacings. All four maps were convolved with a 1: 00 2 \Theta 0: 00 9 gaussian,
then adjusted to a common 0: 00 1 pixel size and aligned with the 2 cm map center. Flux

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densities were measured near the cometary arc (peak) and in the lower density tail of the
region. Integration boxes about one synthesized beam area in size; i. e., about 16 millipc
or 3200 AU on a side were used, one centered on the peak position, the other in the tail. A
least­squares fit to the four flux density points yields a spectral index of ff = 0:6 \Sigma 0:1 for
the peak position and ff = 0:2 \Sigma 0:1 for the tail position (see Fig. 1a).
The G9.62+0.19 complex of H ii regions has been extensively studied at radio
wavelengths (e. g. Garay et al. 1993; Cesaroni et al. 1994; Hofner et al. 1994, 1996; Testi
et al. 2000) and also in the near infrared (Testi et al. 1998). It contains several massive
star formation sites, possibly at various stages of development. One of these, component
E, lies along a ridge of dense molecular gas reported by Hofner et al. (1996). They noted
that the radio continuum flux density distribution of component E could be explained by
either a power­law density gradient of n e (r) / r \Gamma2:7 , or a homogeneous H ii region with
dust emission contributing to the 2.7 mm flux density. The recent measurement at 3.6 cm
by Testi et al. (2000) shows that the homogeneous model does not work for this source.
Hofner et al. also report dense (n H 2
? 10 7 cm \Gamma3 ), warm (TKin = 108 K), molecular gas
coincident with the ionized region, with a spatial extent of about 28 millipc or 6000 AU.
The most homogeneous data set available for this source is that of Testi et al. (2000),
at 3.6, 2, 1.3, and 0.7 cm. In addition, we use the 2.7 mm flux density from Hofner et al.
(1996). The source is marginally resolved by the highest resolution data of Testi et al. (at
1.3 cm) with a resolution of 0: 00 22; this corresponds to a linear size of ¸1500­2000 AU or
7­10 millipc. A least­squares fit to the five flux density values gives a spectral index of
ff = 0:95 \Sigma 0:06 (see Fig. 1b).
The G75.78+0.34 UC H ii region (also known as ON­2) was mapped at 6 cm by Wood
& Churchwell (1989). Subsequent water maser observations by Hofner & Churchwell (1996)
detected a group of water masers located 1: 00 5 (0.034 pc linear projected distance) from

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the cometary arc of the H ii region. Radio continuum emission coincident with the clump
was detected by Carral et al. (1997) at 2 and 0.7 cm and by Kurtz et al. (2001) at 3.6
and 1.3 cm. Carral et al. discuss the possibilities of pure free­free emission and free­free
emission plus a contribution from dust emission at 0.7 cm. In the former case, measured
parameters suggest an ionized region about 0.01 pc in size, with an electron density of
about 5 \Theta 10 4 cm \Gamma3 . At the highest angular resolution ( !
¸ 0: 00 5 or 2300 AU for the 1.3 cm
data) the source is resolved into two components of roughly equal brightness; we use the
sum of the individual flux densities in our least­squares fit. Using the flux densities at these
four wavelengths, we calculate a spectral index of 1.4\Sigma0.1 (see Fig. 1c).
3. Results and Conclusions
The radio continuum spectral index of ionized regions with a decreasing density
gradient has been analyzed by Olnon (1975) and Panagia & Felli (1975); the application to
collimated outflows is presented by Reynolds (1986). The main result of these studies is
that the radio spectral index, ff, for unresolved sources with power­law forms n e / r \Gamma! , and
other functional dependences, is in the range \Gamma0:1 Ÿ ff Ÿ +2. For a pure power­law, which
has a singularity at the center, the plasma in the central regions is optically thick at all
frequencies and the spectral index is given by the simple formula ff = (2! \Gamma 3:1)=(! \Gamma 0:5);
see Olnon (1975). For a flattened power­law, which has a finite, constant­density core (or
any other density distribution that is bounded at r = 0), the optical depth toward the
nebular center depends on frequency, and the expression for the spectral index has a more
complicated functional form. At frequencies below a critical value, š c , where the central
core becomes optically thick, the spectral index for both the flattened and pure power­laws
tends to the same value and the expression above applies.
For simplicity, and given that the density structure of nearby clouds is approximated

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by power­laws, we assume that the observed UCH ii regions can also be approximated by
a (flattened) power­law. The density gradients are derived in the regime š ! š c , so the
difference between pure and flattened power­laws is not significant. As discussed below, the
data from G35.20 indicate that the value of the exponent is not constant over the whole
photoionized region, but decreases with distance from the core. This variation is expected
because the density structure of a molecular core must merge at some point with the more
tenuous inter­core medium of the parental cloud, which can be taken as a more­or­less
uniform density medium. Taken thus, the gradient would be steepest at the central parts
of the UCH ii region and would drop to lower values near the inter­core medium. That a
power­law is the most likely density distribution can be seen from Fig. 2, where we plot
theoretical spectra for the free­free emission from a uniform sphere, a gaussian distribution,
and a flattened power­law (r \Gamma2 ), for the peak position of G35.20. The power­law clearly
provides the best agreement with the data.
For density distributions with exponents above ! = 1:5, the ionization front overruns
the density gradient, but it becomes trapped when the exponent is less than 1.5 (Franco et
al. 1989). Thus, once the ionization front of the UCH ii region enters into the steep central
gradient, it can only be stopped at the external inter­core region, where the value of the
exponent becomes small. Here we are interested in optically thick regions with exponents
above ! = 1:5, for which the spectral index is above ff = \Gamma0:1 (the value for the optically
thin case; see Mezger & Henderson 1967). For instance, using the expression for ff given
above, the case with ! = 2 (similar to a stellar wind with constant mass loss and velocity)
corresponds to a spectral index ff = +0:6, and it increases to ff = +1:16 at ! = 3 (Olnon
1975; Panagia & Felli 1975).
The G35.20 region has spectral indices of ff = +0:6 \Sigma 0:1 (peak) and ff = +0:2 \Sigma 0:1
(tail), appropriate for power­law exponents of ! = 2 and ! = 1:7, respectively. The average

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electron density in the peak region is about 10 5 and drops to about 10 4 cm \Gamma3 in the tail. For
the G9.62­E region we find a spectral index of ff = 0:95 \Sigma 0:06, corresponding to a density
gradient proportional to r \Gamma2:5 . The 1.3 cm data of Testi et al. (2000) suggest an electron
density of about 6 \Theta 10 5 cm \Gamma3 , which could rise to above 10 6 cm\Gamma3 in a uniform core region.
The largest spectral index, ff = 1:4 \Sigma 0:1, is for G75.78­H 2 O. This spectral index would
imply an extremely large, probably unrealistic, density gradient of r \Gamma4 . Possible alternatives
may be: (i) the density stratification in this case might be better approximated by a
gaussian (or sech 2 ) function with warm dust contributing to the 0.7 cm flux density, or (ii)
the photoionized region may be truncated before the tail of the density gradient is reached.
In the first case, for a gaussian distribution, the radio emission for š ! š c is proportional
to š 2 [constant \Gamma ln š] (see Olnon 1975), which results in values of ff closer to 2. In the
second case, of a truncated distribution, there would be little or no contribution from the
outer, lower density (and hence optically thin) layers. This would bias the spectral index
toward the black body value of the inner, denser layers, resulting in an over­estimation
of the density gradient (see Simon et al. 1983). Further observations of this source are
needed to understand its internal structure. In this Letter, it is sufficient to note that the
continuum emission from G75.78­H 2 O shows clear evidence for density stratification, with
an equivalent power­law index larger than ! = 1:5.
For G35.20, we note that the integrated flux density (S 6cm = 1.93 Jy; Wood &
Churchwell 1989) is much greater than typical values for stellar winds from massive stars
(on the order of a few mJy; Bieging, Abbott & Churchwell 1989). Reasonable assumptions
for a massive star stellar wind (cosmic abundances, singly­ionized He, V1 = 10 3 km s \Gamma1 ,
Twind =17,000 K; Bieging et al. ) yield a mass­loss rate of order 10 \Gamma2 M fi yr \Gamma1 , which is
several orders of magnitude higher than those found in ``classical'' stellar winds. Thus,
although the indicated density gradient is very nearly the r \Gamma2 expected for a constant
velocity wind, the high flux density implies that we are detecting an ambient density

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gradient rather than a wind generated at the atmosphere of a star. That both phenomena
give rise to an r \Gamma2 density profile (and spectral index ff = +0:6) should not confuse the fact
that they are, essentially, different phenomena. Less certain, is whether we can distinguish
a remnant density gradient in an expanding H ii region from some other form of ionized
outflow, i. e., , a collimated disk wind (Reynolds 1986; Hollenbach et al. 1994; Yorke &
Welz 1996).
A contribution from dust emission may be present in G9.62­E. Its spectrum (Fig. 1b)
can be fit by free­free emission with the density profile described above and becoming
optically thin around 7 mm. In this case, the higher flux density at 3 mm would arise from
thermal emission from warm dust, with a much steeper spectral index. This dust emission
would not change our basic conclusions because the spectral index from 3.6 to 1.3 cm
--- where dust emission is unlikely to contribute --- is sufficient to establish the density
gradient.
The data discussed here indicate that G35.20, G9.62­E, and G75.78­H 2 O have density
stratifications that can be approximated by power­laws with exponents larger than ! = 1:5.
The expansion of these objects will modify the original density structure and, as time
passes, the gradient will become more gradual. We are currently performing numerical
models of H ii region expansion that include the gravitational field of the star­forming core.
Gravity allows for stratification during the expansion. If an H ii region reaches pressure
equilibrium while still inside the cloud, the original stratification is maintained, otherwise,
the gradient decreases with time. Thus, except for the case of G75.78­H 2 O, the derived
exponents represent lower limits to the initial density gradients of the parental cloud cores.
This, in turn, suggests that some star­forming cores may have density gradients steeper
than the isothermal, self­gravitating distribution of r \Gamma2 . Here we have presented three
examples that illustrate the values that are likely to be found; a more complete study

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of a larger sample of UCH ii regions should be made. Particularly worthwhile would be
studies of H ii regions which are embedded within hot molecular cores, as these objects may
provide the best indication of the natal density gradient. Such an observational program
must be carried out in the millimeter regime, primarily because of the high densities (and
hence large optical depths) that are expected. The Extended VLA and ALMA will be
necessary instruments for such a study. To characterize (and subtract) the emission from
warm dust will require high resolution far infrared observations, thus FIRST might also
make significant contributions.
Acknowledgements
It is a big pleasure to thank Tony Garc'ia­Barreto, Niruj Mohan, and Robert Joynt
for stimulating discussions and comments during the development of this project. JF,
GGS, and MM acknowledge partial support by DGAPA­UNAM grant IN130698, and an
R&D CRAY Research grant. SK acknowledges partial support by DGAPA­UNAM grant
IN117799 and CONACyT, Mexico.

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Fig. 1.--- (Fig. 1a) The flux density distribution of G35.20\Gamma1.74. Flux densities are plotted
for 4.885, 8.415, 14.965, and 22.232 GHz. The solid lines are least­squares fits, giving a
spectral index at the peak position of ff = 0:6 \Sigma 0:1 and at the tail position of ff = 0:2 \Sigma 0:1.
The lower spectral index in the tail region may be indicative of a transition from a steeper
core gradient to a uniform inter­clump medium.
Fig. 2.--- (Fig. 1b) The flux density distribution for G9.62+0.19­E. Flux densities are plotted
for 8.415, 14.965, 22.232, 43.34, and 110 GHz. The solid line is a least­squares fit, yielding
a spectral index of ff = 0:95 \Sigma 0:06.
Fig. 3.--- (Fig. 1c) The flux density distribution for G75.78+0.34­H 2 O. The upper limit is
for 4.885 GHz; the remaining points are at 8.45, 14.96, 22.23 and 43.34 GHz. The least
squares fit indicates a spectral index of 1.4\Sigma0.1.
Fig. 4.--- (Fig. 2) Theoretical spectra for G35.20\Gamma1.74. The data points are for the peak
position, as shown in Fig. 1a. The three curves correspond to the expected flux densities
for a uniform sphere of ionized gas, for a gaussian distribution, and for a flattened power
law (i.e., a gradient with a uniform core). In accordance with the spectral index of the peak
position, we adopt a density gradient of r \Gamma2 . All three curves have been scaled to match
the flux density at 1.3 cm, with the assumption that this point corresponds to the turn­over
frequency.

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