A&A 455, 577-593 (2006)
DOI: 10.1051/0004-6361:20065311
M. Tafalla1 - J. Santiago-García1 - P. C. Myers2 - P. Caselli2,3 - C. M. Walmsley3 - A. Crapsi4
1 - Observatorio Astronómico Nacional, Alfonso XII 3, 28014 Madrid,
Spain
2 -
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge,
MA 02138, USA
3 -
Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125
Firenze, Italy
4 -
Leiden Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands
Received 29 March 2006 / Accepted 12 May 2006
Abstract
Context. Low mass starless cores present an inhomogeneous chemical composition. Species like CO and CS deplete at their dense interiors, while N2H+ and NH3 survive in the gas phase. As molecular line observations are used to determine the physical conditions and kinematics of the core gas, chemical inhomogeneities can introduce a serious bias.
Aims. We have carried out a molecular survey towards two starless cores, L1498 and L1517B. These cores have been selected for their relative isolation and close-to-round shape. They have been observed in a number of lines of 13 molecular species in order to determine a self-consistent set of abundance profiles.
Methods. In a previous paper we modeled the physical structure of L1498 and L1517B. Here we use this work together with a spherically-symmetric Monte Carlo radiative transfer code to determine the radial profile of abundance for each species in the survey. Our model aims to fit simultaneously the radial profile of integrated intensity and the emerging spectrum from the core center.
Results. L1498 and L1517B present similar abundance patterns, with most species suffering a significant drop toward the core center. This occurs for CO, CS, CH3OH, SO, C3H2, HC3N, C2S, HCN, H2CO, HCO+, and DCO+, which we fit with profiles having a sharp central hole. The size of this hole varies with molecule: DCO+, HCN, and HC3N have the smallest holes, while SO, C2S and CO have the largest holes. Only N2H+ and NH3 seem present in the gas phase at the core centers.
Conclusions. From the different behavior of molecules, we select SO, C2S, and CH3OH as the most sensitive tracers of molecular depletion. Comparing our abundance determinations with the predictions from current chemical models we find order of magnitude discrepancies. Finally, we show how the "contribution function'' can be used to study the formation of line profiles from the different regions of a core.
Key words: ISM: abundances - ISM: clouds - ISM: molecules - stars: formation - ISM: individual objects: L1498 - ISM: individual objects: L1517B
Recent observations and modelling of low-mass starless cores show that a number of molecular species deplete from the gas phase at the dense interior of these simplest star-forming regions (Kramer et al. 1999; Tafalla et al. 2002; Kuiper et al. 1996; Bergin et al. 2001; Bacmann et al. 2002; Pagani et al. 2005; Willacy et al. 1998; Caselli et al. 1999; Alves et al. 1999). The depletion of molecules in the pre-stellar material has important consequences for the study of the initial conditions of stellar birth. Molecular emission is routinely used to trace the physical properties and the kinematics of the star-forming gas, so the removal of certain species from the gas phase introduces a potential distortion in molecular line observations. Depletion, in addition, systematically increases with time and gas density. Thus, if properly understood, it can provide a reliable clock to time the contraction history of dense cores. Understanding the chemical composition of starless cores has therefore become necessary to reconstruct the earliest phases of low-mass star formation.
Previous studies of core chemical composition show that molecular depletion is a highly selective process. Molecules like CO and CS disappear rapidly from the gas phase, while species like N2H+ and NH3 survive much longer at high densities (see references above). As a result of this behavior, a core gradually develops a differentiated interior characterized by a center rich in depletion-resistant species surrounded by layers richer in depletion-sensitive molecules. Understanding this abundance pattern is critical to interpret molecular line observations of cores because different species will systematically trace different layers depending both on their response to depletion and their level excitation.
Chemical models already provide an indication of how the different molecular species will be distributed inside a core, but systematic molecular surveys are still needed to obtain a realistic picture of the core chemical composition. Such surveys should be carried out toward starless cores of simple geometry, so the abundances of the different species can be unambiguously derived from observations. In this paper, we present a survey toward two Taurus-Auriga cores, L1498 and L1517B. These two cores are reasonably isolated and present close to round shapes when observed in the millimeter continuum, so they constitute ideal targets for a systematic study. In Tafalla et al. (2004) (Paper I hereafter), we used millimeter continuum data together with C18O, CS, N2H+, and NH3 line observations of these cores to derive their radial profiles of density, temperature, turbulent linewidth, and line-of-sight velocity. Now we complement this study with additional observations of a number of molecules, and use the already derived physical models of the cores to determine in a self consistent manner the radial abundance distribution of 13 molecular species.
In the following sections, we present the details of our radiative transfer modeling of the observed lines and the set of abundance profiles derived from this analysis (Sect. 4). Using these abundances, we study the differences and similarities between the two cores, as well as their chemical relation with other cores of well determined abundances (Sects. 5.1-5.3). We also use our data to test core chemical models, in particular the recent work by Aikawa et al. (2005) (Sect. 5.4), and to study the different sensitivity of molecules to depletion (Sect. 5.5). We finish using our chemical determinations and the radiative transfer modeling to study how the different molecular lines originate at the core interior and therefore reflect (or miss) its internal structure when used to trace the core gas (Sect. 5.6). For this analysis we use as a tool the contribution function.
We observed L1498 and L1517B with the IRAM 30m telescope during several runs between 1999 October and 2002 November. We made maps of these cores in the lines shown in Table 1, always observing in frequency switching mode with several receivers simultaneously and a typical sampling of 20''. As a backend, we used a facility autocorrelator that provided a typical velocity resolution of 0.03-0.04 km s-1. During the observations, the telescope pointing was corrected making frequent cross scans of bright continuum sources, and the atmospheric attenuation was calibrated observing ambient and liquid nitrogen loads. The telescope intensity scale was converted into main beam brightness temperature using standard efficiencies. The FWHM of the telescope beam varied with frequency from 27.7'' at 90 GHz to 11'' at 230 GHz.
We observed L1498 and L1517B in HCO+(1-0), H13CO+(1-0),
and HCN(1-0) with the FCRAO 14m telescope
in 2001 April. We used the QUARRY array receiver in frequency switching
mode together with the facility correlator, which provided a
velocity resolution between 0.03 and 0.07 km s-1. Observations
of SiO masers were used to correct the telescope pointing, and
an ambient
load was used to calibrate the atmospheric attenuation. An efficiency
of 0.55 was used to convert the telescope units into mean beam
brightness temperature. The typical sampling of the maps was 30'',
and the FWHM of the telescope beam at 86-89 GHz was approximately 55''.
For the detailed line modeling presented here, accurate rest frequencies are required. We have searched for such frequencies using on-line compilations like the Cologne Database for Molecular Spectroscopy (CDMS, Müller et al. 2001) and the JPL Catalog (Pickett et al. 1998). In most cases, the CDMS and JPL frequencies are consistent with each other, so we have chosen the one quoted as having smaller uncertainty (where possible, we have referred to the original determination). For HCO+, however, the values in the two catalogs are inconsistent, probably due to the difficulty measuring frequencies in ions, and we have preferred the JPL value for its better agreement with our line data. For some transitions of CH3OH and SO, no accurate frequencies were found, and we have determined them by fitting our L1498 spectra assuming an LSR velocity of 7.80 km s-1 (as measured from the other lines). Fortunately, these few spectra present Gaussian lines, so the astronomical determination is likely to be accurate (about 20 kHz). A summary of the frequencies used in this work is presented in Table 1.
In Figures 1 and 2 we present the integrated intensity maps of all the lines observed toward L1498 and L1517B together with maps of the dust continuum emission and N2H+(1-0) already analyzed in Paper I. For each line, the integrated intensity map reflects the combined effect of excitation, optical depth, and molecular abundance, so its interpretation requires detailed radiative transfer modeling. Even without such an analysis we can appreciate the need for strong abundance variations by noting that most lines are not very optically thick and that their excitation increases with density toward the dust continuum peak. Thus, if the abundance of a molecule were spatially constant, its emission map would present a well-defined maximum at the dust peak. Although this occurs for N2H+ (whose abundance is close to constant, Paper I), it is not the case for the rest of the species.
In the larger and better resolved L1498 core, the maps of C3H2, H2CO, CH3OH, SO, HCO+ (plus isotopes), HCN (plus isotope), HC3N, and C2S all present distributions that differ from the centrally concentrated dust or N2H+. In a few species, like CH3OH, SO, and H2CO, the emission forms an almost-complete ring around the dust peak. In others, the emission presents discrete peaks, especially to the SW and NE of the dust peak, reminiscent of a broken ring. Such ring-like distributions are also seen in C18O and CS, and in Paper I it was shown that they reflect the presence of a central depletion hole (see also Kuiper et al. 1996; Willacy et al. 1998). The maps in Fig. 1 show now that depletion holes must be the rule for most species in L1498.
For the more compact L1517B core, the pattern of line emission is similar to that in L1498, although less distinct for some species because of angular resolution. In this core, most molecules present a single emission peak offset to the west from the dust/N2H+ peak, being the most extreme example that of SO. This pattern is again similar to that found in C18O and CS (Paper I), and for the same reasons as in L1498, it requires a central depletion hole.
A simple inspection of the maps in Figs. 1 and 2 shows that different molecular species have depletion holes of different sizes. Using again the larger L1498 core as a reference, we note that CH3OH presents a rather prominent hole, while DCO+ has a central depression barely resolved by our observations. This variety of hole sizes most likely arises from the different sensitivity to depletion of the different species and from the complex chemical changes resulting from the freeze out of CO, the main source of carbon in the gas phase. The freeze out of CO decreases the abundance of the different C-bearing species by different degree, while it enhances the abundance of deuterium-bearing species (Dalgarno & Lepp 1984). In Sect. 5.2 we will present a quantitative comparison of hole sizes using the results of a radiative transfer analysis.
Given the large effect of depletion in the maps of Figs. 1 and 2, it seems clear that in most species the shape of the emission reflects more the chemical composition of a core than its true physical structure. As chemical inhomogeneities seem the rule in starless cores, this high sensitivity to chemistry of the maps should be carefully considered when deriving physical properties from observations of line emission. The risk of overlooking it is graphically illustrated by the maps of L1498 in species like CH3OH, SO, or C3H2. These maps show highly fragmented distributions with multiple peaks along an approximately elliptical shell, and a naive interpretation of the emission peaks as distinct physical structures would lead to a picture of a highly clumped core. This is of course in contradiction with the the distribution of the most reliable tracers (1.2 mm dust continuum, N2H+, and NH3), which shows that the core is smooth and centrally concentrated. Any correspondence between map peaks and core substructure, therefore, requires careful consideration core chemistry and a self consistency check using multiple species. The traditional warning against the use of optically thick tracers to infer physical properties of cores should now be expanded to avoid using depletion-sensitive species for the same purpose. Comparing in Figs. 1 and 2 the similar maps of thick and thin tracers like HCO+ and H13CO+ (also HCN and H13CN) with the distinct distributions of N2H+ and the dust continuum, we see that the danger of using depletion-sensitive molecules can sometimes exceed that of using thick or even self-absorbed tracers.
To improve the qualitative abundance analysis of the previous section we need to model the emission of the observed species. Modeling this emission requires first determining the physical structure of each core and then following the transfer of radiation. For the first step, we make a physical model of the core that describes its density, temperature, and gas motions, and we do so assuming spherical symmetry because of the close-to-round shape of the continuum maps. Once the core is modeled, it becomes like a laboratory of known physical conditions where the line intensities can be inverted into abundance estimates. For this second step, we use a non-LTE Monte Carlo code that solves numerically the transfer of the line emission. The details of these two steps are described in the next section.
The physical models of L1498 and L1517B were derived
in Paper I from the simultaneous fit of the dust continuum,
C18O/C17O, CS/C34S, N2H+, and NH3 emissions.
These data constrain the core density, temperature, and
kinematics, and here we use the same parameterization for
consistency. As mentioned in Paper I, we search for
the simplest expressions consistent with the data.
We select the core center from the continuum emission,
and fit the radial profile of this emission assuming
a density profile of the form
To solve the radiative transfer inside each core we also
assume spherical symmetry.
In Paper I we saw that the maps of C18O and CS are not
circularly symmetric, in contrast with the maps of more reliable
tracers like the mm continuum, N2H+, and NH3. Figures 1
and 2 show now a similar situation for the species of our survey:
in L1498 the emission is systematically brighter towards
the SE (some species present a secondary maximum toward the
NW), and in L1517B the western half of the core is brighter than
the eastern half. These distributions suggest that the
abundance of most species is not spherically symmetric,
despite the symmetric gas distribution inferred
for the two cores. As studied in Paper I, the deviations
seem correlated with the velocity of the gas, and this can be understood
as the result of differential depletion caused by a
non spherical contraction of the cores. Modeling these
asymmetric distributions requires a 2D or even 3D radiative transfer
code, which exceeds the scope of this paper. In the following
discussion we assume spherical symmetry and fit for each species
a circular average of the emission. In this way, our abundance
estimates represent azimuthal averages over the core and therefore
correspond to the result of a mean contraction.
As in Paper I, we solve the radiative transfer inside each core with the non-LTE Monte Carlo code of Bernes (1979), which we have modified to include additional molecules (see Appendix A for a summary of the molecular parameters used in this work). For each molecular species, we run the code together with the core physical model to produce a set of expected intensity distributions for different abundance profiles. These intensity predictions are convolved with the appropriate Gaussian beam and compared with the observed radial profile of integrated intensity and the central emerging spectrum for as many transitions of the species as we have observed (usually two); the best fit model is the one that fits all these constraints simultaneously. As the only free parameter in this process is the abundance profile, observations of one transition are in principle enough to constrain the solution. Fitting at least two transitions simultaneously, as we do, checks the self-consistency of the process.
As when fitting the core physical parameters, we aim for the simplest abundance profiles consistent with the data. For each species, we start with a constant abundance model that fits the integrated intensity at a fiducial outer radius. This radius is chosen as 75'' for L1498 and 55'' for L1517B and represents a compromise between the need of a large enough radius (to detect central molecular depletion) and the need of a bright enough signal (to make a reliable fit). As shown by the dashed lines in Figs. 3 and 4, the constant abundance models systematically overestimate the intensity toward the core center.
To improve the fit, we decrease the abundance toward the core center
by introducing a step-function at .
The abundance inside
is taken to be
negligible (10-4 of the outer abundance), and the value of
is used as a free parameter to improve the quality of the fit.
This is the same approach used in Paper I to derive abundance
profiles for C18O and CS, and for most species,
it provides a reasonable fit.
In a few cases, a slightly different parameterization
is needed, and these exceptions are detailed below.
Parameterizing the abundance profiles with a step function
is clearly a simplification of the expected
depletion behavior of molecules under realistic core
conditions (e.g., Aikawa et al. 2005; Bergin & Langer 1997).
However, given the fast abundance drops predicted by these models
and the finite spatial resolution of our observations,
this parameterization is almost as good a description, and
it has the advantage of allowing an easy comparison between
molecules by comparing the outer abundance and
values of
their fits. In
Sect. 5 we will use this approach to study different aspects
of the behavior of molecules under high density conditions.
In the following paragraphs, we present further details on
the fitting procedure of each individual species. Readers not interested
in such a detailed view can simply inspect Figs. 3 and 4 and
move to Sect. 5.
CH
OH. Two transitions were observed
for each of the A and E forms
of this molecule, so we have fitted the two forms independently and
determined
E/A ratios of 1 for both L1498 and L1517B. As Figs. 3 and 4 show,
the constant CH3OH abundance models fail to fit the central intensity
by about a factor of 2, while the models with
a central hole provide reasonable fits to all data simultaneously.
SO. SO is the only non C-bearing molecule observed in
this survey (N2H+ and NH3 were studied in Paper I),
so its behavior provides information complementary to that of
all other species.
Three SO transitions were observed toward L1498 and L1517B,
but the SO(-21) line
was not detected in either core. Our fits therefore only take into account
the emission from SO(32-21) and SO(43-32),
as the 12-21 non detection is trivially fitted by all models.
Again, the constant abundance models fail to
fit the observed SO intensity toward the core centers by
a wide margin. Models with a central
hole, on the other hand, reproduce the intensities both at large
and small radius, and show that this species is one of the most sensitive
indicators of molecular depletion (Sect. 5.5).
HC
N.
In L1498, the model with constant HC3N abundance
does not reproduce the observed combination of compact emission
and a central hole. Such a pattern requires that the HC3N abundance
increases inwards at intermediate radii, and that it has a sharp
depletion hole near the core center. We fit this
behavior by introducing an inward jump in the abundance by a
factor of 10 at
cm (85'' at 140 pc)
together with the step central hole
used for the other species. This type of profile
fits well both the J=4-3 and 10-9 data (Fig. 3), together with
the central 12-11 spectrum (not shown). It also agrees with
the prediction of the chemical model by Ruffle et al. (1997), who
find a late-time HC3N peak caused by CO depletion.
In L1517B the situation is
less clear due to the smaller size of this core. A constant
abundance model barely fits the data and is inconsistent
with the lopsided emission seen in the 4-3
map. To improve the fit, we have introduced a
central hole with a small radius (
)
and a factor of 2 outer abundance drop at
cm
(60''). This small inner hole, at the limit of our resolution, is
consistent with the smaller-than-average hole found in L1498. As
will be further discussed in Sect. 5.2, HC3N seems to survive
in the gas phase to higher densities than other species.
C
H
.
In L1498, C3H2 presents the same combination of
compact emission and central hole seen in HC3N.
A constant abundance model that reproduces the emission
at intermediate
radii fails to fit the observations both at large and small
radii (Fig. 3). As with HC3N, we introduce a factor of
10 inward jump in the abundance at
cm
together with a central hole. This model
fits well the emission, including the
slightly self-absorbed 212-101spectrum towards the core center (Fig. 3).
We note that although there are no theoretical
predictions for a
late-time enhancement of C3H2 (Ruffle et al. 1997, do not
present results for this species), the chemistry
of C3H2 and HC3N seem related (e.g.,
Cox et al. 1989). It is therefore possible that a single
process explains the observed behavior of the C3H2 and
HC3N in L1498.
Unfortunately, only one C3H2 transition was observed
toward L1517B. Lacking the brighter 212-101transition, we have fitted the C3H2 abundance in this core
with only a central hole and no abundance jump
at intermediate radii. Observations of additional lines
of this molecule are needed to clarify the behavior of
C3H2 in L1517B.
C
S. Only the
-56 transition was observed toward
both cores and, as Figs. 3 and 4 show, its radial
distribution is inconsistent with a constant abundance profile.
A model with a central abundance hole fits the emission
both at large and small radii in both cores and shows that, like SO,
C2S is highly sensitive to depletion. Our model for
L1498 also fits nicely the emission of the
-10 (22 GHz),
43-32 (45 GHz), and 87-76 (94 GHz) lines observed
by Wolkovitch et al. (1997), who with Kuiper et al. (1996), first
found central depletion for this molecule.
A detailed comparison with these published
data is presented in Appendix C.
HCN and H
CN. The HCN(1-0) spectra show evidence for
self absorption towards both L1498 and L1517B (see also Sohn et al. 2004).
The HCN emission is therefore dominated by
the core outer layers and it cannot be used to derive
an abundance profile inside the core. We thus base our abundance
determination on the thinner H13CN(1-0) emission, which
clearly shows a need for a central abundance drop in both cores.
As in Paper I, we assume
a 12C/13C ratio of 60, and we model the main isotope emission by
scaling the H13CN abundance profile by that factor. This produces
a fit of the HCN(3-2) emission in both cores but
underestimates
slightly the depth of the 1-0 self absorptions. To improve
the fit, we introduce in the constant density envelope that surrounds each
core (
cm) an abundance enhancement
of a factor of 4 in L1498 and a factor of 2 in L1517B (the
enhancement has no effect on the H13CN emission or even on the 3-2 line).
It should be noted that
cm
corresponds to radii larger than 190'', which are outside
the region where our dust-based determination of the density is
reliable. For that reason, the outer abundance enhancements
should be considered a fitting convenience that corrects the
simple core+envelope model that we have used in the Monte
Carlo radiative transfer, and not necessarily an indication of an abundance
change in the outer core. It is in fact remarkable that such a simple
parameterization can simultaneously fit both the radial profiles
and the central spectra of lines with such different optical
depths as H13CN(1-0), HCN(1-0), and HCN(3-2) (Figs. 3 and 4).
H
CO. Our two H2CO lines,
212-111 and 211-110, show evidence for self-absorption
and, unfortunately, no thin isotopologue of this species was observed in
the survey. Despite this, constant abundance models can be easily ruled
out from the shape of the radial profiles,
and a central abundance drop is needed to fit simultaneously
the inner and outer emission. To properly
model the self absorption, we need again an abundance
increase in the envelope (r > 190'', and factors 15 and 5 for L1498 and
L1517B, respectively), to which the same caveats mentioned for HCN apply.
Young et al. (2004) have observed two additional
lines towards L1498 (111-110 and 312-211)
and have also concluded
that a central abundance drop is needed to fit the data.
As shown in Appendix C, our best fit model also fits the
H2CO(312-211) data from Young et al. (2004),
while it predicts a 111-110self-absorption that is weaker than observed.
This 111-110 absorption originates at the core outer
envelope (Appendix C), so the failure in the fit
results again from our simplified parameterization of the
core outer layers.
HCO+, H
CO+. Both the J=1-0 and 3-2
transitions of HCO+ are deeply self absorbed, so our abundance
determination relies on the thinner H13CO+(1-0) line
and assumes again a 12C/13C ratio of 60.
As in all other species, a central drop in the HCO+abundance is needed to fit the shape of the radial profiles.
To fit the HCO+ self absorption, we again
need to include an outer envelope (r> 190'')
abundance enhancement of a factor of 4 in the L1498 model
(still the predicted 1-0 line is
brighter than observed mostly due to poor modelling
of the red component) while
no abundance enhancement is needed for L1517B.
DCO+. A central abundance hole is also required to fit the DCO+ data, but its radius is smaller than the HCO+ hole radius. This difference most likely results from a central increase in the deuterium fractionation caused by the CO depletion (e.g., Caselli et al. 1999), which partly compensates the DCO+ freeze out at the inner core. The effect is of course incomplete because the DCO+ abundance end ups falling at the highest densities. Still, the presence of DCO+ inside the CO and HCO+ depletion holes suggests that a small amount of CO and HCO+ survives at high densities. Comparing the DCO+ and HCO+ abundances in the outer core, we derive deuteration ratios of 0.02 and 0.03 in the outer layers of L1498 and L1517B, respectively. Each of these values is a factor of 2 lower than the deuteration ratios measured toward the inner core by Crapsi et al. (2005) using N2H+ and N2D+, in good agreement with the expectation of a degree of deuteration that increases toward the core center.
Although the abundance enhancements at large radii required to fit the self absorptions in HCN, H2CO, and HCO+need confirmation using a model of the large-scale structure of the cores, we note that such enhancements are not totally unexpected. Absorption line studies by Lucas & Liszt (1996) and Liszt & Lucas (2001) reveal that species like H2CO, HCO+, and HCN present large column densities in diffuse clouds (but not N2H+, for example). It is therefore possible that the lower density gas surrounding L1498 and L1517B has a chemical composition similar to the diffuse clouds studied by Liszt & Lucas. This would naturally explain the large amounts of low-excitation gas needed to produce the observed self absorptions.
For species like NH3, H2CO, and C3H2, our observations and modeling only determine the abundance of the ortho or para form of the molecule, and we need to assume an ortho-to-para ratio (OPR) to convert our estimate into a total molecular abundance. This OPR depends on the formation history of the molecule, and is therefore somewhat uncertain. Here we assume that the three species are formed by gas-phase reactions, because grain production will require a mechanism to release the products to the gas phase, a difficult task given the strong freeze out observed. (This of course does not imply that NH3, H2CO, and C3H2 do not form on dust grains, but that the gas-phase chemistry is separate from the dust-grain production, as seen in the models of, e.g., Aikawa et al. 2005.) If this is the case, the OPR should be close to the high-temperature limit, because the energy difference between ortho and para forms (2.4 K for C3H2, 15.2 K for H2CO, and 23.4 K for NH3) is significantly lower than the energy released during molecule formation (e.g., Dickens & Irvine 1999). Indeed, OPRs close to 3 (high temperature limit) have been found for H2CO in starless cores, including L1498, by Dickens & Irvine (1999). Takakuwa et al. (2001) measure a C3H2 OPR of 2.4 towards TMC1, also close to the high temperature limit. We thus assume OPRs of 3 for H2CO and C3H2 and 1 for NH3.
In addition to OPRs, we need to assume isotopic ratios to convert rare isotopologue abundances into main species values. Along this paper and in Paper I we have used the standard isotopic ratios of 12C/13C = 60, 32S/34S = 22, and 18O/17O = 3.65, to which we now add the terrestrial 16O/18O ratio of 500. Using these values, we estimate the final main isotopologue abundances presented in Table 2. In the following discussion we will refer only to the main molecular forms, although it should be remembered that most abundances are based on the rare (and optically thin) isotopologues.
Finally, we stress the dependence of our abundance
estimates on the assumed dust parameters. As mentioned in
Paper I, the values of the dust temperature and emissivity
are still poorly known (e.g., Bianchi et al. 2003; Kramer et al. 2003), and
their uncertainty propagates
into the determination of the gas density profile and
total core column density. These determinations, in turn, affect
the abundance estimates through their combined effect on the
level excitation and the H2 column density. For most species,
the abundance estimate uses
optically thin lines from low-energy levels, so the effect of
the density on the excitation is smaller than its effect
on the column density. Thus, to first order, we can
correct for dust parameter changes with the following simple equation:
L1498 and L1517B present similarities and differences in their
physical properties. They have almost the same kinetic temperature,
level of turbulence, and total gas column density, while they
differ by a factor of 2 in central density and half
maximum density radius (Paper I). Analogously,
the two cores present similarities and differences in their chemical
composition. To examine them here, we compare the X0 and values derived for the different species.
The top panel of Fig. 5 shows the ratio between the outer molecular abundances derived in L1498 and L1517B for each molecule in our survey (values for C18O, CS, N2H+, and NH3 are from Paper I). As can be seen, there is good agreement between the X0 parameters estimated for the two cores, with most differences being smaller than a factor of 2. The main outliers in the plot are C2S (4 times more abundant in L1498 than in L1517B) and C18O (3 times more abundant in L1517B), and although their abundance differences could be real, it should be noted that their X0 determination is specially prone to error: C2S is depleted up to such a large radius in L1498 that X0 is determined using very limited information of the outer core, and the X0 value for C18O can be contaminated by the extended emission from the ambient cloud. Given these and other uncertainties, it is rather remarkable that most X0 values agree within less than a factor of two, and we take this fact as an indication that the two cores share very similar molecular compositions in their outer (undepleted) layers.
The bottom panel of Fig. 5 presents a
comparison between the radius of the central abundance holes
in L1498 and L1517B. In addition to some scatter, the figure reveals
a trend for
to be larger in L1498 by a factor of about 1.5.
This trend agrees with the impression from the maps of Figs. 1 and 2
that L1498 has a larger depletion hole than L1517B for most
or all molecular species. Unfortunately,
the scatter and the uncertainties in the analysis
make it impossible to determine
whether the
ratio varies with molecule.
The similar outer abundances of L1498 and L1517B
suggest that both cores have contracted from ambient gas
with similar initial compositions. It is unclear however
whether the different
values result from the
cores being at different evolutionary stages or from
them having followed different contraction paths.
L1517B is more centrally peaked, denser at the center, and more circularly
symmetric than L1498, so it would seem to be at a more
advanced evolutionary stage. If this is the case,
and both cores are following the same contraction path,
the smaller
of L1517B would imply that the depletion hole
in a core shrinks as it contrasts; this could result from
the contraction time scale being shorter than
the freeze out time scale during the late stages of contraction.
Alternatively, L1517B may have
contracted faster than L1498, so its central gas has had less
time to freeze out. Further studies of core composition using a
larger sample of systems are needed to clarify this important
issue.
The radiative transfer results of Table 2 suggest that different molecules deplete at different radii. In addition to the extreme behavior of N2H+ and NH3, which do not present abundance holes, several molecules have holes that are systematically smaller or larger than average in both L1498 and L1517B. Identifying these molecules can help better understand the details of depletion. It can also provide a set of tracers to selectively sample cores at different depths.
To compensate for the systematically larger
holes in L1498, we normalize the hole radii. We do this
by dividing the radius of each species by the average hole radius
in the core, which is calculated from the mean radius of all
species with small dispersion (i.e.,
we exclude CO and CS, see below, in addition to
N2H+ and NH3). This normalized hole radius is
presented in Fig. 6 for all species in our survey.
As the figure illustrates, the hole radii span a range of
about a factor of 2. The CO and CS estimates
present large scatter, which could result from
a real difference between the cores or from artifacts like
contamination by the extended cloud in the case of CO and
the fact that L1498 the CS analysis is based on the optically
thin C34S emission while the analysis in L1517B used the thicker CS line
(as C34S was too weak). All other species, on the other hand,
present a reasonably low scatter between their L1498 and
L1517B results. This low scatter allows distinguishing between
species with relatively large and small depletion holes. In the first
group we find SO, C2S, and CH3OH, which in both cores have
larger-than-average abundance holes. In the second group we find
HC3N, DCO+, and HCN, whose central holes are more than
50% smaller than the holes in the first group. Such a relatively
large difference in
suggests a corresponding difference in the
depletion behavior of these species. It also shows that
HC3N, DCO+, and HCN survive in
the gas phase up to higher densities than most other species,
and this makes them interesting
line tracers of the middle layers of dense cores.
Recent infall searches in starless cores (Lee et al. 2004; Sohn et al. 2004),
do use this property of DCO+ and
HCN to penetrate deeper than using traditional tracers
like CS or HCO+ (e.g., Lee et al. 2001). Our L1498 and L1517B
analysis, however, shows that HCN and DCO+
are no substitute for N2H+ or NH3, as they
end up depleting at densities of about 105 cm-3.
TMC1 is the core whose chemical composition has been studied with
most detail,
and its abundances are commonly used as standards for comparison
with other low-mass star-forming regions. Current estimates of the TMC1
abundances, however, do not consider the effect of
molecular depletion, so a comparison with our L1498 and L1517B
estimates can only be approximate. Until a more
realistic analysis of TMC1 is available, the most complete study
is that of Pratap et al. (1997), who have presented FCRAO observations
and analysis of more than a dozen molecular species,
8 of them common to our L1498/L1517B survey. Pratap et al. (1997) normalize
their abundances to HCO+, and provide values for three
positions along the TMC1 filament: the cyanopolyyne peak, the ammonia peak, and
the SO peak. To compare with these values, we select the L1498 and L1517B
abundances outside the depletion hole and
normalize them to HCO+. The resulting values agree
well with those of TMC1, and in no case the difference exceeds
one order of magnitude. The best agreement occurs
for the cyanopolyyne peak, where the average ratio between the L1498
and TMC1 abundances for the 8 species is .
This position also
provides the best match to the L1517B abundances, although the ratio
is significantly larger (
)
because
our estimate of the HCO+ in L1517B is half of that in L1498.
Given the very different methods
and the uncertainties in the two analysis, the agreement with
the TMC1 estimates seems rather good. The
fact that the best match to the abundance in the
undepleted outer parts of L1498 and L1517B occurs for
the cyanopolyyne peak is also in good
agreement with the interpretation that this TMC1 position is the most
chemically young of the filament (Hirahara et al. 1992; Suzuki et al. 1992).
It also suggests that the abundances of TMC1 are not anomalously high
(Howe et al. 1996),
but representative of the population of dense cores in the Taurus cloud.
A different set of abundance determinations in starless cores has been recently provided by the Texas group using a technique similar to ours: density estimates using continuum (SCUBA) data followed by Monte Carlo modeling of the line emission (Lee et al. 2003; Evans et al. 2001). As a result of this work, Lee et al. (2003) have estimated the abundance of HCO+, H13CO+, and DCO+ in L1512, L1544, and L1689 by fitting step functions with non zero value at the core center. Comparing the outer abundances estimated by these authors with those in L1498 and L1517B, we find reasonable (factor of 2) agreement for HCO+ in L1512 and L1544, the two Taurus cores, but almost one order of magnitude difference for H13CO+ and DCO+ in the same objects. This result is somewhat surprising, especially for H13CO+, as our abundance values agree with a standard isotopic 12C/13C ratio of 60, while the Lee et al. (2003) values imply a ratio one order of magnitude lower. The origin of this discrepancy is not clear, although it most likely results from a combination of a higher dust emissivity, lower gas temperature, and smaller radius coverage by Lee et al. (2003). A similar (factor of 4) discrepancy occurs between our H2CO abundance determination in L1498 and that by Young et al. (2004) in the same core, despite our model fitting the H2CO(312-211) emission from these authors (Appendix C). This discrepancy most likely arises from a different choice of dust emissivity, which as discussed by Shirley et al. (2005), makes our L1498 models differ from that of the Texas group by a factor of 3 in H2 column density. Such a systematic discrepancy between models highlights the urgent need for an accurate determination of the dust emissivity at millimeter and submillimeter wavelengths.
Better agreement between our abundance determinations and those in other
starless cores occur for N2H+. Caselli et al. (2002a) have estimated
an N2H+ abundance of
for a sample of
25 starless cores assuming virial equilibrium, and this result is in
excellent agreement with our determination. Also, Keto et al. (2004)
have carried out a detailed analysis of the N2H+ emission
in 3 starless cores, one of them being L1517B. They derive for this object
a central density within a factor of 2 of ours, and an
N2H+ abundance that differs from ours by less than 10%.
Even if the larger abundance differences between cores are real, it seems that in most cases they are well within the factor of 3-4 range. This of course does not imply that starless cores form an homogeneous family. The gradient across TMC1 reveals differences in abundance that are unlikely to be explained simply as the result of differential depletion, and a number of cores are known to have significantly lower abundance of certain late-time species like NH3 and N2H+ (Suzuki et al. 1992). L1521E, for example, has negligible depletion of C-bearing species and NH3 and N2H+ abundances one order of magnitude lower than L1498 and L1517B (Hirota et al. 2002; Tafalla & Santiago 2004; Suzuki et al. 1992). Our molecular survey of the L1521E core (in preparation) seems to suggest, however, that most of chemical differences between cores can be explained as a result of differential depletion plus time evolution of the N-bearing species, and that the undepleted abundances in the outer layers do not present extreme (more than a factor few) core to core variations. A systematic survey of a larger sample of cores is still needed to confirm this result.
The chemical evolution of cores as they contract under
gravity has been studied by different authors. Bergin & Langer (1997)
and Charnley (1997) were first to model the differential
depletion of C and N-bearing species as a result of
their freeze out onto the dust grains together with
a lower binding energy of N2.
More complete chemical networks coupled to
realistic contraction physics have been used to
improve on this earlier work
(Aikawa et al. 2003; Shematovich et al. 2003; Li et al. 2002; Aikawa et al. 2001; Lee et al. 2003). Very recently,
(Aikawa et al. 2005, AHRC05 hereafter)
have presented the most up-to-date chemical
model of starless core contraction. These authors
have followed the evolution of
two dense cores initially having Bonnor-Ebert density profiles,
one subject to a small over-density factor (their
case, where
is the gravity-to-pressure ratio), and the
other highly over dense (
).
As intuitively expected, the core with the small over-density factor
loses equilibrium and contracts slowly (timescale
1 Myr), while the highly over dense core contracts in one tenth of
the time. Due to these very different time scales,
the two cores develop significantly different chemical
compositions by the time their central densities reach values like those
of L1498 and L1517B. The
core has lost
most molecular species at the center, while the
core still retains a significant fraction of depletion-sensitive
molecules in the gas phase. From this dichotomy, AHRC05 argue that the
model approximates the evolution of quiescent, heavily depleted
cores like L1498 and L1517B, while the
model
simulates dense, but not depleted cores like L1521E.
Our molecular survey of L1498 and L1517B can be used to test the
predictions of the different chemical models of core contraction.
We choose the AHRC05 model because it is the most complete
model available and because it presents predictions of both
the core velocity field and its chemical composition.
To test this model, we have run a series
of Monte Carlo radiative transfer calculations
using the observed density profiles of L1498
and L1517B together with
the abundance and velocity profiles predicted by the
AHRC05 models
at the time when the core central density
reaches
cm-3 (note that
AHRC05 quote densities of H nuclei while we use densities of H2molecules). At this time, the density profiles of the AHRC05
models are close to those of L1498 and L1517B, although
there are still factor-of-2 differences between the density profiles
that should be kept in mind when comparing with our observations.
Concerning the velocity field, we find that the
model predicts for NH3 lines much broader than observed (0.38 km s-1
versus 0.2 km s-1, Paper I), and for species with central depletion
like C18O and CH3OH, the model predicts spectra with two peaks.
These peaks correspond to the front
and the back sides of the core, and are separated by 0.4 km s-1 because
each side is moving toward the center with a velocity of about 0.2 km s-1(Fig. 1e in AHRC05). Such broad, double-peaked profiles are not observed in
L1498 or L1517B, and they rule out a fast contraction model for
these cores. The
model, on the other hand, predicts narrower
spectra, similar to those observed. In the absence of turbulence, this model
predicts the correct NH3 linewidth toward the center
for both L1498 and L1517B (0.2 km s-1), but for species with central
depletion, it predicts lines that have two peaks separated
by 0.15 km s-1, and this is not observed.
Although these peaks can be blended into a single feature by adding a small
amount of turbulence, the lines are already broader than
observed and the extra turbulence further degrades the fit. For the
narrow-lined
CH3OH emission, for example, the AHRC05 model without turbulence
predicts lines that are broader than measured by 20% in the case of
L1517B and by 70% for L1498. This disagreement
may be exaggerated by the imperfect match between the
density profiles of our cores and the AHRC05 models, but also
illustrates the fact that in the AHRC05
model, densities
like those of L1498 and L1517B are reached at relatively late
stages, when the core contraction has started to accelerate. For
this reason, the model predicts that the L1498/L1517B phase
will only last 10% of the core time life, or just 105 yr, which is
too short for a typical starless core (e.g., Lee & Myers 1999).
The relatively fast contraction of the model is therefore not supported
by the observations. Adding a magnetic field can help to slow down the
contraction, but this may introduce additional problems with the chemistry.
A long period of slow contraction, for example 1 Myr,
would imply CO depletion at densities as low as 104 cm-3(Léger 1983), which is lower than observed. Accretion of new
(undepleted) material from the surrounding molecular cloud may
help mitigate this difficulty. An additional problem with
chemical models of magnetized clouds is that they predict
N2H+ to trace the quiescent and thermally supported core nucleus
in contracting starless
clouds such as L1544 (Shematovich et al. 2003), in disagreement with
observations (Caselli et al. 2002b).
![]() |
Figure 7:
Correction factors needed to make the abundances
predicted by
the Aikawa et al. (2005)
![]() |
To test the AHRC05 abundance predictions, we now fix the velocity field of
each core to the best fit value determined in Paper I, and we use the Monte
Carlo code to estimate predicted radial profiles of integrated intensity for
all species observed in L1498 and L1517B and
predicted by model
(
cm-3).
If the AHRC05 abundance profile does not fit
the observations at the fiducial outer radius defined in Sect. 4.2 (75'' for L1498 and 55'' for L1517B),
we estimate the global factor by which
the model abundance needs to be multiplied to match the data
at that radius. This factor
measures the deviation of model from the observations,
and is shown in Fig. 7 for both the L1498 and L1517B models.
As expected from the good agreement
between the L1498 and L1517B abundances (Sect. 5.1), the
correction factors derived from the two cores agree within a
factor of two. This shows that the corrections are almost
independent of the exact structure of the core.
The correction factors in Fig. 7 cover more than three orders of magnitude, and this indicates that some species are seriously over or under predicted by the model. Considering all the uncertainties of our fitting procedure, we take any correction smaller than a factor of two as a reasonable match between model and data. This occurs for C3H2 (although the model misses the observed outer abundance drop), HCO+(although L1498 requires a factor of 3 correction), and CO. Correction factors between 2 and one order of magnitude are considered "clear deviations'' between model and data, and include SO, DCO+, C2S, CS, and N2H+. Interestingly enough, three of these species are S-bearing, and all of them are under predicted by a factor of a few (5-7.5 for CS, and 2-4 for SO and C2S). This suggests that the model overestimates the depletion of S in its initial atomic conditions by a similar factor. Finally, correction factors larger than one order of magnitude indicate a "serious deviation'' between model and data, and occur for CH3OH, H2CO, HCN, and NH3. Given its large value, the deviation of H2CO is likely real, although thin isotopologue observations are needed to better quantify the error. On the other hand, the large under production of CH3OH is not surprising given the lack of a known gas-phase production mechanism for this molecule (Luca et al. 2002). As for HCN, it is possible that the addition of photochemistry to the model may help correct the overproduction of this molecule.
The need for order-of-magnitude corrections does not necessarily imply that the AHRC05 model fails to reproduce all aspects of the chemical structure in cores like L1498 and L1517B. Most abundance profiles, when scaled appropriately, reproduce the shape of the radial distribution of intensity, and this suggests that the model captures at least part of the process of molecular depletion as the core contracts. To study how well the AHRC05 model predicts the radial variation in the abundance of the different species, we compare the shape of the observed intensity profiles with the shape predicted by the model after correcting the abundances by the appropriate global factor. We do this by defining a "concentration factor'' as the ratio between the intensity at the core center and the intensity at a fiducial outer radius (75'' for L1498 and 55'' for L1517B, see Sect. 4.2). Such a factor measures how centrally peaked the emission is, and can be easily estimated both in the model predictions and in the data. The ratio between the model and data concentration factors will equal one in a perfect match, will be larger than one if the model under-predicts the central abundance drop (so its emission is more centrally peaked), and will be less than 1 the other way around.
![]() |
Figure 8: Comparison between the concentration ratios predicted by the Aikawa et al. (2005) model and the observations of L1498 (red squares) and L1517B (blue triangles). A value equal to 1 indicates that the model predicts emission with the same central concentration as observed, a value larger than 1 indicates that the model is more concentrated than the data (small central hole), and a value less than 1 indicates a model prediction flatter than the data (too large a hole). There is an overall (factor of 2) agreement between model and data, although the L1517B data seems systematically lower than 1. The model therefore overestimates the central hole seen in L1517B. |
Figure 8 presents the ratio of concentration factors (model over data) in both L1498 and L1517B for all available species. For L1498, the AHRC05 model predicts on average the correct size of the central hole, and the average model-to-data ratio is approximately 1. For L1517B, the model over predicts the size of the central hole, as would be expected due to its smaller dimension (Sect. 5.1), and the mean ratio of concentration factors is about 0.75. This systematic over prediction for L1517B reflects a global problem modeling the core, so in the following discussion we concentrate on the L1498 results. As the figure shows, SO and HCN present the largest ratio of all, indicative that the model under predicts their central abundance hole by the largest factor. This occurs in SO because this species has a relatively large hole, while the AHRC05 model predicts an average value. For HCN, the data show a relatively small central hole, but the AHRC05 model predicts negligible depletion at the time when the core has the central density of L1498 and L1517B. (Note that the model correctly predicts a well defined, relatively smaller HCN hole at later times.) DCO+, on the other hand, is the species with smallest ratio in the figure, indicating that the AHRC05 model predicts a hole larger than observed. This is most likely due to an under prediction of the central deuterium enhancement, as the model predicts the correct depletion hole for the main isotopologue HCO+. Finally, the radial behavior of the two species without central freeze out, N2H+ and NH3, is well predicted by the model after dividing the N2H+ abundance by a factor of 3.5 and the NH3abundance by a factor of 20. This general over prediction of N-bearing species probably results from an underestimate of the binding energy to grains (see below), while the improved behavior of NH3 arises from the new treatment of the N2H+ dissociative recombination thanks to the work by Geppert et al. (2004).
A final problem affecting most current chemical models (including AHRC05, but also Bergin & Langer 1997) is that they owe most of their success in explaining the different depletion behavior of C-bearing and N-bearing species to the assumption that CO and N2 have significantly different binding energies to grains. Recent laboratory measurements by Öberg et al. (2005) and Bisschop et al. (2006), however, show that the two binding energies differ by less than 10%, and this may seem insufficient to account for the different depletion behavior of CO and N2H+(note however that Aikawa et al. 2001 find differential depletion even when using similar binding energies for CO and N2). Flower et al. (2005) have proposed an alternative explanation in terms of a lower sticking coefficient for N2 or N. While the former alternative seems ruled out by laboratory measurements (Bisschop et al. 2006), the latter is still a viable solution, although no laboratory measurements exist yet to confirm or refute the idea. This remaining uncertainty in our understanding of the process behind the differentiation of C-bearing and N-bearing species - the most visible feature of core chemistry - illustrates how it is still premature to use current chemical models to assign contraction ages to cores. A new generation of models is still needed to fulfill the promise of providing core studies with a reliable chemical clock.
The simplest use of depletion as a qualitative indicator of core age is the classification of cores as chemically evolved if they show evidence for molecular depletion and as chemically young if they do not. Most dense cores, in fact, suffer from severe depletion (e.g., Tafalla et al. 2002; Bacmann et al. 2002), while only a minority seem unaffected by it (Hirota et al. 2002; Tafalla & Santiago 2004). This suggests that young cores are rare, either because they are absent from clouds or because an observational bias limits our ability to recognize them in surveys. In either case, it is of interest identifying the most sensitive tracer of molecular depletion to use it for systematic searches of young cores. Choosing such a tracer requires some consideration. Sensitivity to depletion depends not only on the relative size of the central hole, but on the response of the emission to dense gas and on possible optical depth effects. In this section, we use our radiative transfer analysis of L1498 and L1517B to assess the effect of depletion on the emission of different molecules under realistic core conditions, in order to find the species most sensitive to central depletion.
Strictly speaking, the sensitivity to depletion depends not on the molecule but on its transitions. These may greatly differ in critical density and optical depth, especially in molecules with complex level structure. However, as we will see below, the results tend to agree within a factor of two. To measure quantitatively the sensitivity to depletion of a molecular transition, we compare the intensity toward the core center predicted by our best-fit Monte Carlo model with the prediction from a model of constant abundance equal to the outer value in the best fit (solid and dashed lines in Figs. 3 and 4). The ratio of the constant abundance result over the depletion result measures how much brighter a core with constant abundance would appear compared with the core suffering real depletion.
Figure 9 presents the depletion sensitivity ratio for all
transitions observed toward L1498 (squares) and L1517B (triangles).
For most molecules, the L1498 and L1517B values agree within a
factor of two even when multiple transitions
have been observed. As the figure shows, a sensitivity
factor of 2 (dashed line) is common for most species, indicating that
molecular depletion typically halves the expected
intensity towards the core center. Several species, however,
consistently present sensitivity ratios larger than 2. The
SO lines reach the highest values of all, and have a mean factor
of 4. This behavior is somewhat expected from
the systematically larger depletion radius of this molecule
(Sect. 5.2) and its moderately large dipole moment
(1.55 D), which makes it sensitive to the presence or
absence of SO in the inner core. Not surprisingly, the
highest SO ratios correspond to the -32 transition
(138 GHz), which has an Einstein A coefficient larger than
the other observed SO line,
at 99.3 GHz. Second in
sensitivity to depletion is C2S, with small scatter and also
a mean value
of 4. This molecule again combines a larger-than-average
depletion radius with lines of relatively large Einstein A.
In third position, and
with a sensitivity ratio of about 3, lies CH3OH, which also
presents a depletion radius slightly larger than average,
sizable A coefficients, and intensities similar to those of the
SO lines. Interestingly, the molecule with largest depletion
radius, C18O, has a sensitivity factor marginally better than
2, which results from a combination of a low dipole
moment and moderate optical depth. The thinner C17O (not
shown) is significantly more sensitive, although still
suffers from contamination by the extended cloud.
SO, C2S, and CH3OH seem therefore the top three choices
for any systematic search of young, undepleted
cores.
Our L1498 and L1517B models reproduce the observed molecular emission, so they are expected to provide good approximations to the core internal excitation and radiation transfer. We can therefore use them to investigate how the line emission of different molecules is produced inside the core, and how the emission propagates and is distorted by optical depth effects as it travels towards the outside. Understanding these processes is critical when interpreting observations of similar systems for which a full radiative transfer analysis is not available, and can only be done by examining the internal properties of well-calibrated models.
![]() |
Figure 10: Normalized contribution functions of representative species in L1517B (the observer is at large radius, i.e., to the far right of each panel). The red solid and dotted lines are the contribution functions, while the black dashed lines indicate the normalized density profile shown for reference (the normalized square of the density profile is also shown for CH3OH). In the best fit models ( right column), the solid line corresponds to our simple abundance step function, while the dotted lines are the contribution functions expected for the abundance profiles predicted by the Aikawa et al. (2005) models (scaled appropriately to fit the L1517B data). In each panel, the vertical dashed lines indicates the radius at which half of the total emission is reached. Note how in the moderately thick C18O(2-1) and extremely thick HCO+(1-0) lines most of the emission arises from the front outer layer of the core. |
The general problem of line formation in a core is complex
because of the non linear nature of the radiative
transfer equation. In this section, we concentrate on
the question of how the different layers of a core contribute
to the emerging intensity, and on how different molecules can
sample (or miss) the internal structure of the core. To
quantify the discussion, we make use of the "contribution function''
(CF) commonly used in the study of stellar atmospheres
(e.g., Gray 2005; Magain 1986). This function is derived from
the formal solution of the radiative transfer equation. According to
this solution, the emerging intensity from the core at a given
angle and frequency (without the background contribution) is
The emergent core intensity is the line-of-sight integral of
,
so the CF is true to its name in the sense that it measures the contribution
of a given core element to the observed line intensity. Strictly speaking, the CF
depends on wavelength because different regions of the core may contribute
to different parts of the line profile. However, because of the small velocity
gradients in L1498 and L1517B (Paper I), and for the sake of simplicity,
here we will only deal with the frequency-integrated CF, which measures the
contribution of a line-of-sight element to the integrated line profile.
Also for simplicity, we will only study the central line of sight of the core,
although the generalization of the CF for non-zero impact parameters is
straightforward.
With these assumptions, it can be easily shown that the CF of an optically thin
line (where
)
in an isothermal core
has two simple limits. If LTE applies ("high density''),
the CF is proportional to the density n(r), so the emergent intensity is
proportional to the column density. If
(where
Aul is the Einstein A coefficient and Cul is the collision
coefficient, "low density'' case), the CF is proportional to n(r)2,
as each emerging photon is the result of a collision (which has a
probability proportional to n(r)2). In this case, the emergent
intensity is proportional to the neutral analog of the
"emission measure''. Monte Carlo tests using different species and transitions
under realistic core conditions show that
the CF for a thin line usually lies somewhere between the above two limits.
To illustrate the variety of CFs found during our radiative transfer
modeling, we
present in Fig. 10 a series of normalized CFs for different species in
L1517B (plots for L1498 are similar). Each panel represents a cut
along the central impact parameter of the core, has the density
peak at r=0, and assumes that the observer is located to the far right
of the plot (at
); a normalized density profile in dashed lines
indicates the LTE optically thin limit of the CF. Constant abundance
models are presented in the left column and best-fit abundance models
appear on the right; we first discuss the constant abundance case.
As Fig. 10 shows, the constant abundance model for N2H+(1-0) (and that of NH3, not presented) has a CF that closely follows the density profile, with a a slight shift to positive radius due to optical depth effects (see below). This good behavior of the CF shows that the N2H+(1-0) emission responds linearly to density and therefore traces faithfully the core structure. To quantify this property, we compare the fraction of integrated CF that arises from the "inner core'' (as defined within the half maximum density radius) to the fraction of gas column density contained in the same region (0.62 for L1517B). For both the N2H+ and NH3 constant abundance models, we find that the inner core contributes to the emission a percentage indistinguishable from the ideal 62% value. In contrast, the CF of CH3OH(30-20) in a constant abundance model (Fig. 10, second left panel) follows closely the n(r)2 curve, and is therefore significantly biassed toward the high density gas. For this line, the inner core contributes 85% to the emergent emission, which is almost 40% more than the column density fraction.
The CF for the C18O constant abundance model shows signs of non negligible optical depth in the form of a forward shift of the peak. This shift indicates that most of the observed C18O(2-1) emission arises from the front part of the core, and that the emission from the back is heavily attenuated as it travels through the core. In fact, only 15% of the emerging radiation originates from the backside (r<0), and the inner core contributes with only 38% of the emission (a 40% decrease from the ideal value). C18O, therefore, is a biassed tracer of the core emission despite its LTE excitation, although most of these problems can be minimized using C17O transitions (but see below for models with depletion). As a final example of a constant abundance model, we consider the heavily self absorbed HCO+(1-0) line, although similar results are obtained for CS(2-1). The emission from this line is so dominated by the outer core that half of it originates from radii larger than 90'', and the inner core only contributes 17% of the emergent intensity (23% in the case of CS(2-1)). These numbers illustrate the difficulty of modelling self absorbed lines like HCO+(1-0), CS(2-1), or HCN(1-0): the region that dominates the line emission is poorly constrained by the continuum observations that are the basis of our density profile, and its structure is simply guessed using an extrapolation of the central core parameters. These very thick lines are therefore excellent tracers of the outer velocity fields (like infall), but they become poor tools for deriving any line of sight structure, even of the infall velocity, unless there is a very accurate description of the outer core and the surrounding cloud.
The presence of depletion introduces a
substantial distortion in the CF. This is illustrated by the right-hand
panels of Fig. 10, which are based on the best-fit abundance profiles
of Sect. 4.
The N2H+ CF is of course the same as in the constant abundance
case because no abundance gradient was found for this molecule. The
CH3OH, C18O, and HCO+ CFs, on the other hand, present sharp
drops near the core center that correspond to
the abundance holes found for these species. Because of these
drops, the CF in the inner core vanishes or becomes so small that a
negligible part of the emergent emission arises from the dense gas:
5% in HCO+(1-0) and less than 1% in both CH3OH(30-20)
and C18O(2-1). The spectra towards the core center for these
species is therefore overwhelmed by gas outside the core, mostly from
an innermost ring if the emission is not very thick (the case of CH3OH
and C18O) or from the front part of the core in the thicker HCO+(1-0)
line. Although it can be argued that the abundance drops in our models
are artificially steep because of the use of a step function,
theoretical considerations of core chemistry still predict
very sharp abundance decreases. To test the effect of smoother, but
realistic abundance gradients, we have produced a set of CFs using the
model of Aikawa et al. (2005) corrected by the
appropriate scaling factors to match the observed
radial profile. The new CFs (shown as dotted
lines in the right hand panels of Fig. 10) again present very steep holes toward
the core centers and predict inner-core contributions to the emerging
intensity of 15% (CH3OH) and less (C18O and HCO+). The inner
core contribution to the emission for these and other depleted
species is therefore a minor fraction of the total emergent intensity.
The above small sample of CFs illustrates some of the basic phenomenology
of line formation in cores like L1517B. As the plots show, even in the
simple case of constant abundance, the emission is not generated at
a "
surface'', but arises from a broad range of radii that
in the simplest case (thin LTE) has a weight that
mimics the core density distribution, and that
in a thick line is strongly biassed toward the front part of the core.
When realistic abundance profiles are considered,
if the species is sensitive to depletion,
all hope of tracing the inner core is lost no matter how much
spatial resolution is achieved. Only depletion-resistant species like
N2H+ and NH3 trace the inner gas, at least to densities of
a few 105 cm-3. Among these tracers, N2H+(1-0)
seems specially reliable because of a combination of constant abundance,
reasonably low optical depth (helped by the hyperfine structure), and
well-behaved excitation. Ignoring the NLTE ratios between the J=1-0
hyperfine components (a 10% effect most likely due to a population
redistribution), this line consistently produces a CF very close in
shape to the density profile. This is somewhat surprising given the
significant radial gradient in the excitation temperature
of this line (more than a factor of 2 change between center and edge
in L1517B), and it results from a population of the J=1 level that follows
closely the density law (the population of higher N2H+ levels
falls faster than the density law, and the CFs of higher transitions
is closer to n(r)2). This fortunate behavior of the N2H+
molecule explains the striking similarity between the mm continuum and
N2H+(1-0) maps in many starless cores (e.g.,
Tafalla et al. 2002; Crapsi et al. 2005; Bacmann et al. 2002; Caselli et al. 1999, this work),
and makes the 1-0 line the ideal choice
for studying core interiors as long as its optical depth remains low.
We have presented observations of 13 molecular species (plus a number of isotopologues) toward the Taurus-Auriga starless cores L1498 and L1517B. Combining Monte Carlo radiative transfer modeling with a physical description of the cores, we have derived a self-consistent set of molecular abundance profiles. From the analysis of these profiles, we have reached the following main conclusions.
Acknowledgements
We thank the staffs of the IRAM 30m and FCRAO telescopes for support during the observations, Claudia Comito for providing us with the HC3N(4-3) data, Yuri Aikawa for useful comments on chemical modelling, Kaisa Young and Jeong-Eun Lee for information on their H2CO data of L1498, and the referee, Sébastien Maret, for a thorough and useful review. MT acknowledges partial support from grant AYA2003-7584, and PC and CMW acknowledge support from the MIUR grant "Dust particles as factor of galactic evolution''. This research has made use of NASA's Astrophysics Data System Bibliographic Services and the SIMBAD database, operated at CDS, Strasbourg, France.
CS.
Dicarbon monosulfide is a linear radical with two unpaired
electrons (electron spin
), and therefore fine structure
in its spectrum. Its rotational levels are characterized by
N and J, the quantum numbers of rotational angular momentum and
total angular momentum, respectively (e.g., Wolkovitch et al. 1997).
We model its radiative transfer using the
C2S molecular parameters from
the JPL catalog (Pickett et al. 1998) together with the C2S-H2collision rates from Wolkovitch et al. (1997)
(downward rates for 10 K together with upward rates
derived from detailed balance). We include
all energy states up to approximately 60 K, which results
in a total of 37 levels and 79 transitions.
CHOH.
Methanol is a slightly asymmetric top with a hindered internal
rotation (Townes & Schawlow 1955).
It consists of two symmetry species denoted A and E (e.g.,
Lees 1973) whose rotation energy levels are characterized by
their total angular momentum J and its component along the
near symmetry axis k (an additional + or - is required
for the A species if
). For our calculations, we have
used the energy and line strength determinations in
the JPL catalog (Pickett et al. 1998) together with the recent CH3OH-para H2
collision
rates of Pottage et al. (2004) (downward rates for 10 K together with upward rates
derived from detailed balance). For both species, we include
all energy states up to approximately 70 K, which results
in a total of 31 levels and 92 transitions for E-CH3OH,
and 27 levels and 66 transitions for A-CH3OH.
c-CH
.
Cyclopropenylidene, a ring molecule, is an oblate asymmetric top
with ortho and para species (Vrtilek et al. 1987). Its rotational
levels are characterized by the total angular momentum J and by
K-1 and K+1, its two projections on the symmetry axis
for the prolate and oblate symmetric top limits (Townes & Schawlow 1955).
For our calculations, we have used the C3H2
parameters in the JPL catalog
(Pickett et al. 1998) together with the C3H2-He collision rates
for 10 K from
Avery & Green (1989) (multiplied by 21/2 to simulate collisions with
H2, as recommended by the authors). Only transitions of
ortho C3H2 were observed, so our modeling is restricted to
that species. This modeling includes all states with available
collision rates (up to Eu=41 K): 16 levels and 32 transitions.
HCO.
Formaldehyde is a slightly asymmetric prolate rotor with ortho and
para species, and its rotation levels are characterized (like those
of C3H2) by J, K-1, and K+1. For our calculations,
we have used the H2CO parameters from the on-line
Cologne Database for Molecular Spectroscopy (CDMS, Müller et al. 2001),
together with the 10 K H2CO-He collision rates of Green (1991).
These collision rates were multiplied by 2.2 to convert them
into H2CO-H2 rates, as recommended by Green (1991).
We only observed lines of ortho H2CO,
so only this species has been modeled. For this, we have
taken into account all energy states with
energy below 115 K, resulting in a calculation with 14 levels
and 19 transitions.
HCN.
Cyanoacetylene is a linear molecule. Its rotational
levels have hyperfine (hf) structure due to the nuclear
spin of the nitrogen atom, so they are characterized
by the quantum numbers J and F. For our radiative
transfer calculation we have used the molecular
parameters from Lafferty & Lovas (1978) and the collision
rates with He of Green & Chapman (1978) (multiplied by
21/2 to simulate collisions with
H2, as recommended by the authors). The calculation
treats the hf structure in a simplified manner
by solving the equations of statistical equilibrium
for the pure rotational levels after
dividing by 3 the effect of trapping,
to simulate the decrease of optical depth
due to the three-fold hf splitting
(we find this to be a minor effect).
The resulting population of
each rotational level is then divided between the
hf sub levels assuming LTE to compute the emergent
spectrum (see Tafalla et al. 2002
for a discussion of a similar approach with N2H+). All
24 levels up to an energy of 130 K are considered
(23 transitions).
HCO+ and isotopologues. The formyl cation HCO+ and its isotopologues H13CO+ and DCO+ are linear ions. We have modeled their radiative transfer using the parameters in the JPL catalog together with the 10 K HCO+-H2 collision rates of Flower (1999). Our calculation considers a set of 7 levels (6 transitions), up to an equivalent energy of 120 K. To take into account the small additional broadening due to hyperfine structure in H13CO+, we have artificially broadened this line by an extra 0.133 km s-1(see Schmid-Burgk et al. 2004).
HCN and H
CN.
Hydrogen cyanide is a linear molecule with hyperfine
structure due to the nuclear spin of the nitrogen atom.
We have modeled its radiative transfer using the molecular
parameters in the CDMS together with the HCN-He collision
rates of Monteiro & Stutzki (1986) (for J up to 4) and the HCN-He rates
of Green & Thaddeus (1974) (remaining levels
up to J=7). A factor of 2 was used to convert the He collision
rates into H2 rates. The hyperfine structure of this species
(and that of its isotopologue) has been considered explicitly
in the calculations, although overlap effects were ignored
and the collision rates between hf sub levels with J>4were estimated from the Green & Thaddeus (1974) rates
assuming independence of the F number (Guilloteau & Baudry 1981). A total
of 22 levels and 39 transitions were included, corresponding to
all levels with energy lower than 119 K (up to J=7).
SO. Sulfur monoxide is a diatomic molecule with two unpaired electrons and therefore a fine structure similar to that of C2S. We have modeled its radiative transfer using the molecular parameters in the CDMS together with the SO-He collision rates from Lique et al. (2005). Following the authors's recommendations, a factor of 1.38 was used to convert the SO-He collision rates into SO-H2 rates. All 24 levels up to an energy of 85 K and their corresponding 72 transitions were included in the model.
Convergence tests. When solving the radiative transfer, we divide the core in 200 equally spaced shells. We normally use 2000 photon packages to simulate the radiation, and allow 40 iterations for the solution to converge (no radiation reference field was used, see Bernes 1979 for details on the code). To check the convergence of the solution, several tests were carried out. Independence on the initial level population was tested by comparing the results from runs that started with Boltzmann equilibrium at 10 K (kinetic temperature) and at 2.7 K (cosmic background); no differences in the final solution were found. Independence on the internal Monte Carlo parameters was tested by repeating the best fit solutions using 5000 photon packages and 300 iterations. Again, no appreciable differences were found in the predicted intensities.
The line of sight toward the L1498 core is complex.
In addition to the narrow-line component that corresponds to the
core studied in this paper, spectra of abundant species like
12CO and 13CO show extra emission toward the red extending up
to about
km s-1, which is 1.7 km s-1 higher than
the core velocity (Fig. B.1). This additional component has a complex spatial
distribution (see also Kuiper et al. 1996).
At low velocities (
km s-1), it lies toward
the south west of the core (as also found by Lemme et al. 1995), while at high
velocities (
km s-1), it appears toward the NW. This
distribution around the L1498 core suggests some
relation with the dense gas, although the velocity difference implies
that the red gas is not bound to the core. From the C18O(1-0) spectrum
in Fig. B.1 and assuming an excitation temperature of 5-10 K together
with a "standard'' C18O abundance of
Frerking et al. (1982),
we deduce an approximate H2 column density for this component
of
cm-2 (less than 10% of the core column density).
Although the red component does not affect the narrow optically thin
lines used in our abundance determinations, it can distort
the shape of very thick spectra like HCN(1-0), shown in
Fig. B.1. These spectra were also modelled in Sect. 4 by assuming
standard isotopic ratios, and they sometimes required anomalous abundances
in the outer core (Sect. 4.2). As stressed before, these optically
thick lines were only fitted to prove self consistency, and not to
determine the conditions of the outer core layers. To model
these lines, however, some assumptions had to be made about the red
component, and a main one was its location along the line of sight with
respect to the L1498 core. A foreground position was preferred based on
hints from several optically thick spectra, like the HCN(-01) line
shown in Fig. B.1. This spectrum shows a deep self absorption that removes
completely the red part of the line and replaces it by a low-level plateau that
continues smoothly into the red-component range (from
8 to 9 km s-1). Such a feature is best understood if the red component
contributes to the absorption of the core emission and therefore arises
from gas located in front of the core. To model such a component, we have
red-shifted
the front outer core layers of L1498 ("envelope'' in Paper I) by 0.35 km s-1and broadened them to
km s-1(note that because of a typo, Table 3 in Paper I gives the wrong
envelope velocity, which should be 8.15 km s-1); such a simple
parameterization reproduces approximately the shape of most optically
thick lines.
![]() |
Figure C.1: Comparison of L1498 data taken from the literature (solid squares) with the predictions of the best fit model of Sect. 4.2 (red lines). The top three panels show that the model fits well the C2S radial profiles generated from the data of Wolkovitch et al. (1997). The bottom two panels show that the model fits well the H2CO(312-211) line of Young et al. (2004), but fails to fit the H2CO(111-110) line (from the same authors), which appears in absorption against the cosmic background. To illustrate the sensitivity of the 111-110 absorption to the core low density gas, a model with a backside envelope is shown in blue dashed lines (see text). |
We have compared our L1498 model predictions with
data for this core available in the literature (no
similar data were found for L1517B). Wolkovitch et al. (1997)
have presented observations of L1498 in three C2S transitions,
-01 (22 GHz),
-32 (45 GHz), and
-76 (94 GHz), made
with the telescopes of NASA's Deep Space Network and the FCRAO,
and having spatial resolutions of about 50''. From the linewidths
and peak intensities reported by these authors in their
Table 2, we have reconstructed the line
integrated intensities for all positions available, and with them,
we have generated the radial profile of emission for each transition.
When we compare these profiles with the prediction from the best abundance
model derived in Sect. 4.2, we find the excellent match illustrated
by the three top panels in Fig. C.1. This match shows that our model
predictions are fully consistent with the observations of
Wolkovitch et al. (1997).
Other L1498 data of interest for our study are those of Young et al. (2004), who have observed L1498 in H2CO(111-110) (6 cm) with the Arecibo telescope and in H2CO(312-211) (1.3 mm) with the CSO (spatial resolutions of 60'' and 32''). To reconstruct the integrated intensities of these lines, we have used the values reported by the authors in their Table 2 (1.3 mm line) and data kindly provided by Kaisa Young and Jeong-Eun Lee (6 cm line). With them, we have generated a set of radial profiles, and in Fig. C.1 we compare them with the predictions from the best fit model of Sect. 4.2. As the figure shows, the model reproduces well the 312-211 emission, which is sensitive to the high density gas inside the core. The model, however, fails to reproduce the observed intensity of the 111-110 transition, although it agrees with the data in producing an absorption line against the cosmic background. The poor fit occurs because the absorption depends sensitively on the low density gas outside the core, and this gas is not well parameterized by our extrapolation of the inner core model (Sect. 4.2). To illustrate this sensitivity to the low density gas, we have added to the L1498 model an envelope in the back that has the same properties as the front envelope used to simulate the red shifted component (Appendix B). This back envelope doubles the 111-110 absorption, and brings the model prediction closer to the observations (blue dashed lines in Fig. C.1). Such strong sensitivity to the low density gas shows that the central core is close to invisible in the 111-110 transition (note the flat radial profile), and that improving the fit requires a re-parameterization of the envelope, not the core. Given the few observational constrains of the low density gas, such a parameterization lies outside the scope of this work.