A&A 377, 566-580 (2001)
DOI: 10.1051/0004-6361:20011137
G.-J. van Zadelhoff1 - E. F. van Dishoeck1 - W.-F. Thi1 - G. A. Blake2
1 - Leiden Observatory, PO Box 9513, 2300 RA
Leiden, The Netherlands
2 - Division of Geological and Planetary
Sciences, California Institute of Technology, MS 150-21, Pasadena, CA 91125, USA
Received 13 November 2000 / Accepted 9 August 2001
Abstract
Observations of submillimeter lines of CO, HCO+, HCN and
their isotopes from circumstellar disks around low mass pre-main
sequence stars are presented. CO lines up to ,
and HCO+and HCN lines up to
,
are detected from the disks around
LkCa 15 and TW Hya. These lines originate from levels with higher
excitation temperatures and critical densities than studied before.
Combined with interferometer data on lower excitation lines, the line
ratios can be used to constrain the physical structure of the disk.
The different line ratios and optical depths indicate that most of the
observed line emission arises from an intermediate disk layer with
high densities of
106-108 cm-3 and moderately warm
temperatures in the outer regions. The data are compared with three
different disk models from the literature using a full 2D Monte Carlo
radiative transfer code. The abundances of the molecules are
constrained from the more optically thin 13C species and indicate
depletions of
1-30 for LkCa 15 and very high depletions of
>100 for TW Hya with respect to dark cloud abundances. Evidence
for significant freeze-out (factors of 10 or larger) of CO and HCO+onto grain surfaces at temperatures below 22 K is found, but the
abundances of these molecules must also be low in the warmer upper
layer, most likely as a result of photodissociation. A warm upper
layer near the surface of a flaring disk heated by stellar and
interstellar radiation is an appropriate description of the
observations of TW Hya. LkCa 15 seems to be cooler at the surface,
perhaps due to dust settling. The density constraints are also well
fitted by the flared disk models.
Key words: stars: circumstellar matter - stars: pre-main sequence - stars: planetary systems: protoplanetary disks - accretion, accretion disks
Hubble Space Telescope (HST) observations of young low mass stars such as HH 30 and HK Tauri show edge on (silhouette) disks which indeed flare noticeably (Burrows et al. 1996). The radiation from the central star incident on the outer parts of the disk changes the temperature and chemistry in those regions, with the temperature change giving rise to a larger scale height and thereby flaring the disk. Recent models by Chiang & Goldreich (1997, 1999) and D'Alessio et al. (1997, 1998, 1999) include the irradiation of flared disks to derive self-consistent models with a warm outer layer. The models by Bell et al. (1997, 1999) take both the stellar radiation and re-processing of radiation in the disk into account. The latter models have an isothermal temperature in the vertical z-direction due to large flaring in the inner disk region, thereby shielding the the outer disk from stellar light. Comparison with the other models provides a good test case whether a high temperature upper layer is needed to satisfy the observational constraints. All three types of models are used in this work and will be discussed in more detail in Sect. 4.
An alternative method to derive the density and temperature structure in disks is through modeling of molecular line emission. Although the inferred solution from observations of a single line is not unique, data on a sufficiently large number of transitions of various molecules can be used to constrain the temperature and density independently. Moreover, careful analysis of the line profiles can provide positional information, since the center of a line probes a different radial part of the disk compared with the wings, unless the disk is nearly face-on. In addition, observations of various isotopomers can give information on different vertical regions of the disks due to their varying optical depths. To date, most data concern the lowest rotational J=1-0 and 2-1 transitions of 12CO and 13CO, which originate from levels at low energies (<20 K) and which have low critical densities (<5000 cm-3) (e.g., Dutrey et al. 1996). Data on molecules with larger dipole moments such as HCN and HCO+ have been limited to the 1.3 m band (Dutrey et al. 1997), except for the case of TW Hya (Kastner et al. 1997). In this paper, higher rotational lines in the 0.8 and 0.45 m atmospheric windows are presented, obtained with the James Clerk Maxwell Telescope (JCMT) and Caltech Submillimeter Observatory (CSO). These lines probe higher temperatures (up to 100 K) and higher densities (up to 107 cm-3) than do presently available spectra.
The observations are accompanied by a detailed analysis of the excitation and radiative transfer of the lines. In contrast with previous models (e.g. Gómez & D'Alessio 2000), our analysis uses statistical equilibrium (SE) rather than local thermodynamic equilibrium (LTE) since the surface layers of the disk may have densities below the critical density of various transitions. In addition, the two-dimensional (2D) code developed by Hogerheijde & van der Tak (2000) is used to calculate the full radiative transfer in the lines. The data can be used to test the disk models described above that are fit to the SEDs available for most T-Tauri and Herbig Ae stars. In addition to constraining the temperature and density, the observations and models also provide information on the depletion of different species.
The molecular abundances and excitation are studied by comparing
different isotopomers of CO, HCO+ and HCN for two sources: TW Hya
and LkCa 15. TW Hya is nearby (57 pc, Kastner et al. 1997) and has a
disk seen nearly face-on. LkCa 15 is located at the edge of the
Taurus cloud at 140 pc and has an inclination of
60
,
where 0
is face-on. Both sources show a wealth of molecular
lines and are well-suited for developing the analysis tools needed to
investigate disk structure.
The outline of this paper is as follows. In Sect. 2, we present the observational data. In Sect. 3, we perform a simple zeroth-order analysis of the observed line ratios to constrain the excitation parameters. The adopted disk models are introduced in Sect. 4.1, whereas the methods for calculating the level populations are explained in Sects. 4.2-4.5. Finally, the results of the analysis are given in Sect. 5 and summarized in Sect. 6.
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Figure 1:
Top: selected CO, HCO+ and HCN lines
toward LkCa 15. The profiles show a double-peaked structure
typical for a disk seen at an inclination of about
60![]() |
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Between September 1998 and December 2000, spectral line observations
were obtained for several pre-main sequence low mass stars surrounded by
circumstellar disks using the dual polarization B3 receiver at the
James Clerk Maxwell Telescope (JCMT)
in the 345 GHz (0.8 mm) band. The observations were obtained mostly
in single side band (SSB) mode using beam-switching with a typical
switch of 180
in azimuth. The spectra were recorded with the
Digital Autocorrelation Spectrometer (DAS) at a frequency resolution
of
0.15 MHz (
0.15 kms-1), and were converted to the
main-beam temperature scale using
.
See
http://www.jach.hawaii.edu/JACpublic/JCMT/ for details. The
calibration was checked regularly at each frequency setting against
standard spectra of bright sources obtained by the JCMT staff, and
were generally found to agree within 10%. Integration times ranged
from 30 min (ON+OFF) for 12CO 3-2 up to 120 min for
C18O 3-2, reaching rms noise levels on the
scale of
about 20 mK after adding the data from the two mixers together and
smoothing to 0.3 MHz resolution. A deep integration on the C18O 2-1 line was obtained with receiver A3 for LkCa 15, which has
.
These data are complemented by observations using the Caltech
Submillimeter Observatory (CSO) of the 12CO 6-5 line for the same sources. In addition, the 12CO 2-1 line
has been observed with the IRAM 30 m telescope
for LkCa 15. For the
12CO 6-5 line,
,
whereas for the IRAM
12CO 2-1 observations the raw data are divided by 0.55 (see http://www.iram.es/).
Interferometer maps of the lowest rotational transitions of several species toward LkCa 15 have been obtained by Qi (2000) using the Owens Valley Millimeter Array (OVRO). Some lines have also been imaged by Simon et al. (2000) and Duvert et al. (2000) with the IRAM Plateau de Bure interferometer. In addition, the Infrared Space Observatory (ISO) has detected the lowest rotational S(0) and S(1) lines of H2, which provide independent constraints on the temperature and mass of warm gas and which are discussed elsewhere (Thi et al. 2001). In this paper only the single dish results on CO, HCO+ and HCN for the sources LkCa 15 and TW Hya are presented.
Figure 1 shows some of the spectra observed toward
LkCa 15 and TW Hya. The double peaked profiles for LkCa 15 are
consistent with Keplerian rotation of the disk seen at an inclination
of
(Qi 2000; Duvert et al. 2000). Since TW Hya
is seen face-on, only narrow single-peaked lines are observed from
this source. For both stars, the 12CO lines disappear at
one beam offset from the source. Table 1 summarizes the measured line
parameters and beam sizes at the observed frequencies. The
upper-limits for LkCa 15 refer to a 2
rms noise level, with the
limit on the integrated line strengths obtained by using two separate
Gaussians each with a line-width of 1.3 kms-1, as found for
13CO 3-2. For TW Hya, the upper limits assume a Gaussian
with a width of 0.76 kms-1, similar to that observed for HCN and
H13CO+ 4-3. Note that our HCO+ 4-3 line toward TW Hya
is a factor of three weaker than that found by Kastner et al.
(1997). We adopt our values in the analysis. The HCN 4-3 integrated
intensity is comparable to that found by Kastner et al. (1997) within
10%. There is a hint of a 12CO 6-5 line toward TW Hya, but
this is treated as an upper limit.
line | ![]() ![]() |
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Beam | Telescope | Date |
K km s-1 | K | km s-1 |
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|||
LkCa 15 | ||||||
CO 6-5 | 0.53 | 0.29/0.28 | 2.0 | 14.5 | CSO | Jun. 00 |
CO 3-2 | 1.39 | 0.60/0.56 | 3.3 | 13.8 | JCMT | Nov. 99 |
CO 3-2 | 1.17 | 0.37/0.39 | 2.2 | 25.7 | CSO | Feb. 98 |
CO 2-1 | 1.82 | 0.74/0.76 | 2.9 | 10.5 | IRAM | Dec. 98 |
13CO 3-2 | 0.39 | 0.13/0.15 | 3.4 | 14.4 | JCMT | Sept. 98 |
13CO 1-0e | 7.43 |
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OVRO | |||
C18O 3-2 | <0.14b | <0.05c | 14.5 | JCMT | Nov. 99 | |
C18O 2-1 | <0.20b | <0.07c | 22.2 | JCMT | Nov. 98 | |
C18O 1-0e | 0.70 |
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OVRO | |||
HCO+ 4-3 | 0.26 | 0.14/0.14 | 3.3 | 13.4 | JCMT | Sept. 98 |
HCO+ 1-0e | 4.19 | 4.5![]() |
OVRO | |||
H13CO+ 4-3 | <0.13b | <0.05c | 13.7 | JCMT | Jan. 00 | |
H13CO+ 1-0e | 0.07 |
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OVRO | |||
HCN 4-3 | 0.25 | 0.09/0.08 | 3.3 | 13.5 | JCMT | Sept. 98 |
HCN 1 2-01e | 3.04 |
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OVRO | |||
H13CN 3-2e | 1.49 |
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OVRO | |||
H13CN 1 2-01e | 1.20 |
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OVRO | |||
TW Hya | ||||||
CO 6-5 | <3.22 | <1.19 | 2.46 | 14.5 | CSO | Jun. 00 |
CO 4-3d | 5.0 | 11.0 | JCMT | |||
CO 3-2 | 1.98 | 2.94 | 0.63 | 13.8 | JCMT | Nov. 99 |
CO 3-2 | 1.00 | 0.77 | 1.23 | 25.7 | CSO | Jun. 00 |
CO 2-1d | 1.02 | 20.0 | JCMT | |||
13CO 3-2 | 0.24 | 0.29 | 0.78 | 14.4 | JCMT | Feb. 99 |
13CO 2-1d | 0.14 | 20.0 | JCMT | |||
HCO+ 4-3 | 0.49 | 0.72 | 0.63 | 13.4 | JCMT | Nov. 99 |
H13CO+4-3 | 0.07 | 0.08 | 0.76 | 13.7 | JCMT | Dec. 99 |
HCN 4-3 | 0.49 | 0.60 | 0.76 | 13.5 | JCMT | Dec. 00 |
H13CN 4-3 | <0.04f | <0.05c | 0.76 | 13.5 | JCMT | Dec. 00 |
HCN 3-2d | 0.45 | 20.0 | JCMT |
a Width of best single Gaussian fit to total profile.
b Calculated assuming the line is double peaked consisting of two separate Gaussians, each with a width of 1.3 kms-1. c Listed value is 2 ![]() d Values from Kastner et al. (1997). e Values from Qi (2000). f Calculated assuming a line width of 0.76 kms-1. |
Although the observed line intensities are a complex function of the physical structure of the disk and the line/continuum optical depth, useful insights can be obtained from a simple one dimensional analysis of the line ratios. For constant temperature and density models such as presented by Jansen et al. (1994) and Jansen (1995), the data provide constraints on both parameters. To compare data obtained with different beams the intensities were scaled to the same beam (see Sect. 4.4.2).
Consider first the observed ratios of 12CO and its
isotopomers. The 12CO 3-2/13CO 3-2 ratios of 3.3 and 7.6 for
LkCa 15 and TW Hya, respectively, indicate that the CO lines are
optically thick, assuming a normal isotope ratio of
in the solar neighborhood. On the other
hand, the 13CO emission has an optical depth of only a
few, since the C18O 3-2 emission is not detected. The ratio of
peak 3-2 temperatures of >3 (using the 2
limit for C18O) and the observed beam-corrected 1-0 ratio of 5.0 are only
marginally smaller than the isotope ratio [13C]/[18O] of 8.3 (Wilson & Rood 1994).
The temperature can be determined from the 13CO 3-2/1-0 ratio
of
,
which gives temperatures of
20-40 K in
LkCa 15. Care should be taken with the interpretation of this result since
the emission of the two lines most likely comes from different regions
of the disk due to the difference in optical depth of the two lines
(see Sect. 5.1). The beam-corrected ratio for 13CO 3-2/2-1 of 0.9 for TW Hya indicates that the bulk of the gas in this
source is colder than 25 K for densities >105 cm-3. The
12CO 6-5 line probes higher temperatures since its upper level
lies at an energy of 116 K. The observed ratio
for LkCa 15 also
suggests the presence of gas with a
temperature
in the range 20-40 K while the upper limit
of CO 6-5/3-2 <1.0 for TW Hya gives T<150 K (cf.
Fig. 4 of Jansen et al. 1996). The 4-3 (JCMT)/3-2 (CSO) ratio of
0.91
+0.45-0.31 suggests temperatures
40 K whereas
the ratio of both JCMT lines indicates somewhat higher temperatures. The
different optical depths of the 3-2 and 6-5 lines imply that they
probe different vertical layers of the disk. Such vertical structure
may affect these conclusions, though not by large factors (see Sect. 5.1).
probed by molecules with a large dipole moment
such as HCO+ and HCN. The measured HCO+/H13CO+ 1-0
ratio of 6.2 for LkCa 15, and the 4-3 ratio of 6.7 for TW Hya and
>1.9 for LkCa 15 indicate that the main isotopomeric lines are again
optically thick. No limits on HC18O+ exist, but H13CO+may be close to optically thin (see Sect. 5.1). The limit on
the H13CO+ 4-3 line toward LkCa 15 together with the
1-0 line detected with OVRO gives a 4-3/1-0 ratio of
less than <2.4 and
constrains the density to <107 cm-3 in the HCO+ emitting
region. The optically thick HCO+ 4-3/1-0 ratio of 0.75
+0.38-0.25 suggests n > 106 cm-3 at T=20-30 K.
The HCN 12-01 to H13CN 12-01 ratio of 1.4
indicates that both lines are severely optically thick. The HCN 4-3
line has an even higher critical density than that of HCO+ 4-3.
The observed HCN 4-3/1-0 ratio of 1.0
+0.5-0.3 indicates
densities
cm-3 in the LkCa 15 disk.
For the TW Hya disk, the HCN/H13CN 4-3 ratio has a lower limit of 12.3, indicating that the HCN lines are optically thin or nearly optically thin. The HCN 4-3/3-2 ratio of 0.6 +0.3-0.2constrains its density to lie in the 106 to 108 cm-3 range, and, as the lines are (nearly) optically thin, this should refer to the regions in the disk where HCN is most abundant.
In summary, the simple analysis indicates that the main isotope lines are optically thick, but that the lines of 13C isotopomers of CO and HCO+ have at most moderate optical depths. The bulk of the gas is cold, but the presence of a warm layer is suggested from the CO 4-3 line for TW Hya. The inferred densities in the region where the lines originate are high, at least 106 cm-3, but not sufficient to thermalize all transitions, especially those from high dipole moment species.
In this work, the line emission from three recently published disk models is calculated and compared with observations. Although each of these models has limitations, they are representative of the range of temperatures and densities that may occur in disk models, even if not specifically developed for the large radii probed in this work. The three disk treatments analyzed in detail are:
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Figure 2:
Comparison of the density in cm-3(left three figures)
and the temperature in K (right three figures) for the models
of D'Alessio et al. (1999) (top), Chiang & Goldreich (1997)
(middle) and Bell (1999) (bottom). All models have a gas+dust mass of 0.024 ![]() |
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The adopted models are not tailor-made for the two sources
studied here. For example, they do not fit in detail the observed
SEDs: the Bell (1999) model is too cold on the outside to reproduce
the mid-infrared emission. Chiang et al. (2001) have presented
models for LkCa 15 and TW Hya which fit the observed SEDs. However,
both models have significant settling of the dust. The gas may still
flare out to higher vertical distances, but the SED does not provide
observational constraints. For this reason, we adopted the original
Chiang & Goldreich (1997) model which has no settling of dust so that
the temperature is defined over the entire disk. Other parameters
entering the models are the disk accretion rate and the luminosity and
effective temperature of the star. For the accretion rate, which
enters the D'Alessio et al. models, a value of 10
yr-1 and
was chosen, which is higher than the
observed values of 10-9 and
yr-1 for LkCa 15 and TW Hya, respectively (Hartman et al. 1998,
Muzerolle et al. 2000). However, the observed molecular lines probe
the outer region of the disk whereas the accretional heating due to
the high values of
and the accretion rate will only affect
the inner few AU. The models refer to a 0.5
star with
K. For comparison, LkCa 15 has a mass
of roughly 1
and an effective temperature of 4400 K (Siess
et al. 1999), while TW Hya has a mass of 0.7
and
K (Muzerolle et al. 2000).
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Figure 3: Comparison of the fraction of mass in a given temperature and density interval for the models of D'Alessio et al. (1999) (top figures), Chiang & Goldreich (1997) (middle figures) and Bell (1999) (lower figures). The distributions in the disks are plotted in Fig. 2. |
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The radiative transfer in the molecular lines from disks is calculated in two steps. First, the abundances of the molecules in the disk are estimated using a ray-tracing method in the vertical direction through the disk and adopting statistical equilibrium calculations. The ratio of the different modeled lines constrains the range of depletions. This calculation does not take into account the inclination of the source and assumes that the ratio of different lines is less sensitive to inclination than the integrated intensity of a single line.
Once the depletions are constrained, a full 2D radiative transfer
calculation is performed using the accelerated Monte Carlo (AMC) code
of Hogerheijde & van der Tak (2000), whose results are compared to
the observations. The motivation for this elaborate scheme is given in
Sect. 4.3 and is driven by the huge computational time
involved in the latter calculation. The AMC code has been compared
with other radiative line transfer codes in a workshop in Leiden in
1999, where the populations of the levels and convergence have been
tested for a set of one-dimensional problems. The comparison of the
various codes is described in van Zadelhoff et al. (in preparation).
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Figure 4:
The relative populations of levels of the CO
molecule (J=3 and J=6) and the HCO+ molecule (J=1 and
J=4) for three different calculations (LTE (dotted), SE with no
stimulated emission or absorption (dashed) and SE with full 2D
radiative transfer (solid)). The levels are calculated for the
D'Alessio et al. (1999) model and plotted as a function of height Zat a radius of 175 AU. The assumed abundances are 10-4 for CO and
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The level populations xu and xl can be calculated in various ways. The simplest approximation is that of Local Thermodynamic Equilibrium (LTE), which is valid for all levels that are collisionally excited in a gas with densities higher than the critical density for that level. The latter is given by
![]() |
(1) |
The populations of the levels are calculated by solving the equation:
Test calculations have been performed for three cases. The first is
LTE, where the populations are given by the Boltzmann equation. In
this assumption the populations are dominated by collisions and
therefore depend only on the local temperature. The second is
Statistical Equilibrium without stimulated radiative effects
(SE[
]), in which the populations of the levels
are no longer assumed to be dominated by collisions and are calculated
explicitly. In this case, Eq. (2) is solved under the
assumption that
for all radiative
transitions. The third method is the full Statistical Equilibrium (SE)
solution using a Monte Carlo code (Hogerheijde & van der Tak 2000) to calculate
the mean intensity
for each radiative transition
iteratively, taking the line emission and absorption throughout the 2D disk into account.
In Fig. 4, the relative populations for the levels
J=3 and 6 of CO and J=1 and 4 of HCO+ are plotted for the
LTE, SE(
= 0) and SE for the D'Alessio et al.
(1999) model. This model is chosen because it has a smooth temperature
gradient but does show a temperature inversion in the z-direction.
The adopted CO and HCO+ abundances are 10-4 and
respectively, and the turbulent line width is assumed to be
0.2 kms-1. Figure 4 shows that the
differences between the three methods are small in the midplane but
that they become significant in the lower density upper layers. For
the J=1 level of HCO+, the influence of the Cosmic Microwave
Background 2.7 K radiation is apparent since its relative population
continues to rise toward the outside in the SE calculation compared to
the SE(
)
calculation.
Even though only the full 2D SE calculation describes the populations
accurately, its calculation is an enormous computational task due to
the large column densities and steep gradients in density and
temperature coupled with a narrow velocity profile. In these cases,
the convergence criteria of a numerical code become very important.
The SE(
)
calculation provides better agreement
with the SE calculation compared to LTE, especially at larger
distances from the star where most of the observed radiation
originates. Therefore the SE(
)
method is adopted
in the ray-tracing calculations.
The abundances and depletion of various molecules are taken into account in two different ways:
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Figure 5:
The two ways in which the depletion of molecules
compared to the interstellar value is taken into account in the
computations. On the left a constant depletion ![]() ![]() |
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![]() |
(3) |
The equation to be solved thus becomes
![]() |
(4) |
The optical depth
is the sum of the attenuation by dust
and gas along the step-length and is equal to
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(5) |
![]() ![]() |
(6) |
![]() ![]() |
(7) |
![]() |
(8) |
The continuum mass absorption coefficient
is taken
from Ossenkopf & Henning (1994) and extended to wavelengths
longer than 1.3 mm as
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(9) |
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= | ![]() |
(10) |
![]() |
= | ![]() |
(11) |
The models are calculated using the ray-tracing Eq. (4) which are weighted according to its emitting surface area
The radius of the disk is taken to be 200 AU in all models, or 400 AU
diameter. The size of the LkCa 15 disk (d=140 pc) suggested by the
OVRO 13CO maps of Qi (2000) is slightly larger (420 AU530 AU). For TW Hya, no millimeter interferometer observations are
available, but mid-infrared and VLA 7 m images suggest a disk
size of
100 AU (Wilner et al. 2000). Scattered light images observed
with the Hubble Space Telescope suggest an outer radius of at
least 200 AU, however (Weinberger et al. 1999; Krist et al. 2000). We
therefore adopt a similar disk size in AU as for LkCa 15, but with
d=57 pc.
For the calculation of the populations in SE using full radiative
transfer, the Monte Carlo code developed by Hogerheijde & van der Tak
(2000) is used. In this code, Eq. (2) is solved in an
iterative fashion, where all photons start at the outer boundary with
an intensity given by the 2.728 K Cosmic Background radiation. In
this calculation, the inferred abundances from the
SE(
)
method are adopted. The calculated
populations at each position in the disk are used to compute the
complete line profiles of selected molecules using a program which
calculates the sky brightness distribution. The profiles are
calculated by constructing a plane through the origin of the disk
perpendicular to the line of sight, with a spatial resolution small
enough to sample the physical and velocity distributions. Both the
spatial resolution and the velocity resolution can be specified. A
ray-tracing calculation is performed through this plane from
to
,
keeping track of the intensity in the different velocity
bins.
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Figure 6:
The ![]() ![]() |
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The models are calculated initially using standard dark cloud abundances
of CO of
,
HCO+ of
,
and HCN of
relative to H2. For the isotope ratios the
following values are used throughout:
and
.
These models are referred
to as
.
Subsequently, the values of
and
are varied (see Sect. 4.3).
The line
intensities have been calculated assuming a micro-turbulence of 0.2 kms-1. For TW Hya, this results in calculated line widths
of
0.8 kms-1, in good agreement with observations.
Significant insight into the observational results can be obtained by
investigating the regions of the disk where the different lines become
optically thick. At each radius the effective emission region for
each line is calculated using the SE(
)
method by
integrating from the top layer down until
in line + continuum
is reached. Although the
level is chosen arbitrarily and
radiation from deeper in the disk may still escape, it provides a
useful measure of the volume of the emitting region for each molecular
transition. This calculation is performed only for a face-on disk for
simplicity and is thus only applicable for the TW Hya case. It does,
however, give an indication of the parts of the disk from which the
molecular emission arises in more general cases. For the model by
D'Alessio et al. (1999), a contour-plot of the
surfaces of
the observed CO and HCO+ lines is given in Fig. 6,
where the former are overplotted on the temperature distribution and
the latter on the density distribution. The line emission is
dominated by densities and temperatures above the
contour,
which can then be compared to the values derived from the constant
temperature and density models given in Sect. 3.
It is seen that, for standard abundances, the 12CO lines become
optically thick in the upper, warm layer of the disk where T>40 K.
On the other hand, the 13CO lines probe into the colder regions
around 20-30 K. Thus, the 12CO excitation temperature should be
higher than that of 13CO, which must be taken into account in the
analysis of isotopomeric line ratios. Similarly, the higher frequency
3-2 and 6-5 lines generally have higher optical depths than the 1-0 lines, and thus probe better the warmer upper layer. Even C18O is
not fully optically thin, but has
.
The low temperature of 20-30 K probed by 13CO is consistent
with the simple analysis of the data in Sect. 3.
For the standard HCO+ abundance, the 1-0 to 4-3 lines are optically
thick in the outer layers, whereas the H13CO+ lines are close
to optically thin throughout the disk. Thus, the HCO+ lines probe
densities of order
106-107 cm-3, below the critical density
of the 4-3 transition. For H13CO+, the populations will be
closer to LTE because its emission arises primarily from regions with
densities of
107-108 cm-3. If the HCO+ abundance is
depleted by a constant factor
,
the HCO+ lines
become optically thin in the outer regions of the disk and now trace regions
with densities of
107-108 cm-3. The HCN 1-0 to 4-3 lines
show a similar behavior to HCO+. The densities of
106-108 cm-3 derived from the observed HCO+ and HCN lines in Sect. 3 are
consistent with this analysis for modest depletions of both species.
D'Alessio et al. (1999) | Chiang & Goldreich (1997) | Bell (1999) | |||||
LkCa 15 | TW Hya | LkCa 15 | TW Hya | LkCa 15 | TW Hya | ||
CO | ![]() |
[3, 15] | >30 | [3, 30] | >100 | [1, 500] | [10, 1000] |
![]() |
[3, 30] | >1 | [1, 15] | >1 | >1 | >1 | |
HCO+ | ![]() |
[3, 80] | >80 | [10, 100] | >100 | [1, 80] | [2, 1000] |
![]() |
>1 | >1 | >10 | >1 | [1, 100] | >1 | |
HCN | ![]() |
[1, 400] | [4, 600] | [10, 200] | [10, 800] | [1, 500] | [4, 500] |
![]() |
>1 | >1 | >1 | >1 | >1 | >1 |
In this section, the relative line intensities obtained in the
SE(
)
method are used
to constrain the abundances of the molecules and the level of
depletion. Since lines of different isotopomers arise from different
regions, their line ratios will depend on the local depletion values.
By calculating models for a range of depletions, the abundances
can be derived by varying both the overall depletion
and the jump depletion
as described in Sect. 4.3.
In the comparison of the line ratios, the difference in beam dilution
for the two lines must be taken into account (Sect. 4.4.2).
![]() |
Figure 7:
The CO/13CO 3-2 (top) and 13CO/C18O 3-2
(bottom) line intensity ratios as functions of the jump depletion
![]() ![]() |
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Isotope ratios are more sensitive to both
and
as compared
to ratios of different species. For instance the 12CO emission
remains optically thick up to large values of
and therefore does
not probe the region below 22 K, whereas 13CO becomes sensitive
to
for modest values of
.
In Fig. 7 the
CO/13CO 3-2 and 13CO/C18O 3-2 line intensity ratios are
plotted as functions of
and
.
The observed values for
LkCa 15 are plotted with dashed lines, and indicate a range
when both plots are combined. The ratio for TW Hya (dot-dashed
line) indicates a larger depletion with
and
.
Combining similar plots for all species and lines, the resulting
values of
and
are shown in Table 2 for the
three models of interest for both sources. The inferred ranges for the
disk models are large and it is difficult to give accurate values for
molecules for which few lines have been measured. The abundances of
all molecules in the TW Hya disk seem to be depleted by a large
factor. In general
is taken as a best fit for both
sources. In cases where no constraints are available,
has
been assumed.
Model | ![]() |
![]() |
CO 6-5 | CO 3-2 | 13CO 3-2 | C18O 3-2 | HCN 4-3 | H13CN 4-3 | HCO+ 4-3 | H13CO+ 4-3 |
Aa | 1 | 1 | 0.78 | 1.15 | 0.39 | 0.18 | 0.39 | 0.12 | 0.54 | 0.15 |
Bb | 1 | 1 | 0.076 | 0.21 | 0.16 | 0.11 | 0.15 | 0.092 | 0.17 | 0.10 |
Cc | 1 | 1 | 0.59 | 0.80 | 0.42 | 0.28 | 0.42 | 0.20 | 0.48 | 0.24 |
A | 5 | 10 | 0.61 | 0.93 | 0.24 | 0.074 | 0.25 | 0.030 | 0.37 | 0.043 |
B | 5 | 10 | 0.046 | 0.16 | 0.089 | 0.036 | 0.096 | 0.011 | 0.11 | 0.018 |
C | 5 | 10 | 0.36 | 0.61 | 0.23 | 0.086 | 0.26 | 0.049 | 0.33 | 0.062 |
A | 10 | 10 | 0.44 | 0.68 | 0.16 | 0.044 | 0.17 | 0.018 | 0.26 | 0.026 |
B | 10 | 10 | 0.041 | 0.15 | 0.074 | 0.022 | 0.081 | 0.006 | 0.095 | 0.009 |
C | 10 | 10 | 0.25 | 0.46 | 0.16 | 0.057 | 0.19 | 0.031 | 0.24 | 0.041 |
LkCa 15d | 0.53 | 1.39 | 0.39 | <0.14 | 0.25 | ![]() |
0.26 | <0.13 | ||
Model | ![]() |
![]() |
CO 6-5 | CO 3-2 | 13CO 3-2 | HCN 4-3 | HCN 3-2 | H13CN 4-3 | HCO+ 4-3 | H13CO+ 4-3 |
A | 100 | 10 | 0.88 | 1.84 | 0.29 | 0.36 | 0.22 | 0.024 | 0.52 | 0.038 |
B | 100 | 10 | 0.12 | 1.07 | 0.14 | 0.18 | 0.13 | 0.004 | 0.26 | 7.0E-3 |
C | 100 | 10 | 0.60 | 1.46 | 0.36 | 0.46 | 0.24 | 0.044 | 0.58 | 0.065 |
A | 200 | 10 | 0.68 | 1.81 | 0.23 | 0.29 | 0.17 | 0.013 | 0.42 | 0.021 |
B | 200 | 10 | 0.069 | 0.95 | 0.083 | 0.11 | 0.08 | 0.002 | 0.18 | 3.4E-3 |
C | 200 | 10 | 0.57 | 1.73 | 0.32 | 0.43 | 0.21 | 0.025 | 0.55 | 0.040 |
B | 10 | 10 | 0.38 | 1.41 | 0.15 | 0.54 | 0.48 | 0.045 | 0.62 | 0.06 |
TW Hyad | <3.2 | 1.98 | 0.24 | 0.49 | 0.45 | <0.04 | 0.49 | 0.07 |
a D'Alessio et al. (1999) model. b Bell (1999) model. c Chiang & Goldreich (1997) model. d The observed values have an estimated uncertainty of 20%; all values refer to the original beam size of the observations (see Table 1). |
Model | ![]() |
![]() |
13CO 1-0 | C18O 1-0 | HCN 1-0 | H13CN 1-0 | HCO+ 1-0 | H13CO+ 1-0 |
Aa | 1 | 1 | 6.18 | 1.90 | 4.47 | 0.85 | 4.84 | 0.20 |
Bb | 1 | 1 | 4.04 | 1.57 | 2.58 | 0.76 | 2.68 | 0.17 |
Cc | 1 | 1 | 8.80 | 2.88 | 5.72 | 0.96 | 5.94 | 0.25 |
A | 5 | 10 | 3.60 | 0.53 | 2.70 | 0.11 | 3.11 | 3.3E-2 |
B | 5 | 10 | 2.04 | 0.32 | 1.39 | 5.5E-2 | 1.56 | 1.8E-2 |
C | 5 | 10 | 3.56 | 0.62 | 2.52 | 0.14 | 3.06 | 4.0E-2 |
A | 10 | 10 | 2.31 | 0.31 | 1.75 | 6.0E-2 | 2.11 | 1.8E-2 |
B | 10 | 10 | 1.55 | 0.18 | 1.11 | 2.8E-2 | 1.31 | 9.3E-3 |
C | 10 | 10 | 2.40 | 0.38 | 1.70 | 7.6E-2 | 2.10 | 2.3E-2 |
LkCa 15a | 7.43 | 0.70 | 3.04 | 1.20 | 4.19 | 7.E-2 |
a D'Alessio et al. (1999) model. b Bell (1999) model. c Chiang & Goldreich (1997) model. d The observed values, taken from Qi (2000), have an estimated uncertainty of 20%; all values refer to the original beam size of the observations (see Table 1). |
The line profiles are calculated using the full 2D radiative
transfer code for the range of depletions derived in Sect. 5.2. The depletions are further constrained by the absolute
intensities. Specifically, for LkCa 15
and 10 with
is
taken, and for TW Hya the same
was assumed but with
and 200. As a reference, an extra run was performed for LkCa 15 with
no depletions.
A general turbulent width of 0.2 kms-1 is assumed
and the only structured velocity distribution is taken
to be the Keplerian rotation of the disk. This velocity component is
important for the comparison with observations of sources at non-zero
inclination. For these calculations, an inclination of 60
for
LkCa 15 and 0
for TW Hya is used. The results are convolved
with the appropriate beam as given in Table 1.
A model with no depletion was also run for LkCa 15 with an inclination of 0
to check the effect of inclination. Although the total integrated
line intensities changed significantly, their ratios changed only up to 7%.
This justifies the approximate radiative transfer approach used in Sect. 5.2 for
a first estimate of the depletions.
The resulting integrated intensities are presented in Table 3 for the high-J rotational lines and in Table 4 for the lower-Jtransitions. For six high-J rotational lines, the observed profiles are plotted in Fig. 8 with the three calculated model emission profiles superposed. In the left-hand figures, three lines are shown for LkCa 15 whose clear double peaks are due to the Keplerian rotation in the disk. On the right, the single peaks for a face-on disk such as that around TW Hya are seen. The optically thick lines from the latter source show that the disk can be fitted with a turbulent velocity of 0.2 kms-1.
![]() |
Figure 8: Observed line profiles for LkCa 15 (left) and TW Hya (right) with the three models superposed (solid: D'Alessio et al.; dashed: Chiang & Goldreich; dash-dotted: Bell). The different transitions are indicated with the adopted depletions compared to standard molecular cloud values. |
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The absolute intensities in Tables 3 and 4 indicate that refinements of the inferred depletions are required, since different molecules favor different amounts of depletion. Note that the intensities computed for the cold Bell model are always smaller compared to the other two models, for both the high and low rotational lines. The reason for this is twofold. First, for a cold isothermal disk structure the level populations of any molecule at densities above the critical density are the same at each position. This means that the optical depth becomes directly proportional to the column of gas. For a model with an increasing temperature, the optical depth will not be directly proportional to the column but smaller. A model with a step function in its temperature will give results in between these two cases. Second, the colder disk will have slightly narrower, more optically thick lines due to the lower thermal motions in the gas, thereby trapping radiation more effectively.
Both effects are visible in the 13CO and CO lines (Fig. 8). The two peaks in the 13CO intensity for the inclined LkCa 15 disk are reduced significantly compared to the emission at line center in the Bell model. In the CO 3-2 line for the face-on TW Hya disk, the emission predicted by the Bell model is extremely optically thick, shown by the flat-topped emission profile. Also, the total linewidth is somewhat smaller compared to the other two disk models due to the low temperatures. Thus, the observed line profiles argue for a rising temperature structure in the vertical direction to prevent the high optical depths found in the cold isothermal model.
To counteract the low intensities found in the Bell model for the
TW Hya disk, additional calculations were performed for less severe
depletions (Table 3:
,
). The integrated intensities
increase to just above the observed values in this case; however, the
lines are extremely optically thick and show nearly square emission
profiles, which is not observed for the CO 3-2 and HCO+ 4-3
lines. Thus, the line profiles speak against small depletions. In
the two warm disk models, there is no significant difference between
the two assumed depletions and only a slight preference can be given
to
.
This is largely based on the HCO+/H13CO+ratio, which is ill fitted by a depletion of 200 and only moderately
well for
.
For LkCa 15, CO is best fitted with little depletion: all three models
indicate
close to 1 for the lower rotational lines. The upper
limit on the C18O 3-2 line and the C18O 1-0 emission would
favor some depletion, which, in the case of C18O, can be
explained by enhanced photodissociation in the upper layers due to the
lack of self shielding. The CO 3-2 intensity is too low in all three
models, which may be a sign of extended emission beyond 200 AU since
this line is optically thick. The HCO+ and HCN lines are best
fitted by a moderate depletion of
and
.
For HCN, this
could again be a sign of a lack of shielding against photodissociation
compared to CO. The observed H13CN 1-0 is slightly too high for
all three models. Together, the HCN and H13CN data indicate that
the HCN abundance needs to be lowered primarily in the surface layer
to both bring the main isotope HCN emission down but keep a high
H13CN intensity.
Line ratio | Observed | D'Alessio et al. | Bell | Chiang & Goldreich | |||
LkCa 15c | ![]() ![]() |
Ratio | ![]() ![]() |
Ratio | ![]() ![]() |
Ratio | |
CO
![]() |
0.38 + 0.19- 0.13 | 5, 10a | 0.66 | 1, 1a | 0.36 | 10, 10 | 0.55 |
13CO
![]() |
0.05 + 0.03- 0.02 | 1, 1a | 0.06 | 10, 10a | 0.05 | 1, 1a | 0.05 |
HCO+
![]() |
0.06 + 0.03- 0.02 | 1, 1 | 0.11 | 1, 1a | 0.06 | 1, 1 | 0.08 |
H13CO+
![]() |
<1.9 | 10, 10b | 1.44 | 5, 10 | 1.00 | 10, 10b | 1.78 |
HCN
![]() |
0.08 + 0.04- 0.03 | 5, 10a | 0.09 | 5, 10a | 0.07 | 1, 1a | 0.07 |
TW Hyac | |||||||
CO
![]() |
<1.62 | 100, 10 | 0.48 | 100/10, 10d | 0.12/0.27 | 100, 10 | 0.41 |
CO
![]() |
2.53 +1.26-0.84 | 200, 10 | 1.28 | 200/10, 10d | 1.01/1.13 | 100, 10 | 1.29 |
HCN
![]() |
1.09 +0.54-0.36 | 100, 10 | 1.64 | 100/10, 10d | 1.38/1.13 | 100, 10 | 1.92 |
a Ratios for all three combinations of ![]() ![]() b Ratios for ![]() ![]() c The observed values and all ratios refer to the original beam sizes in which the lines were observed (see Table 1); thus, the beam size differs between species and lines. The error bars correspond to a 20% uncertainty. d Ratios for high and low values of ![]() |
The calculated line ratios which are sensitive to the temperature and density distribution are summarized in Table 5. The temperature of the upper layer in the LkCa 15 models, as probed by the CO 6-5/3-2 ratio, fits within the errors to all three models, confirming the rather cold upper layer of this disk. However, as explained in the previous section, the absolute intensities are too low in the cold Bell model. To reproduce the observed CO 6-5 intensity, the CO abundance would have to be increased well above the cosmic carbon abundance in the cold isothermal model.
For the TW Hya disk, the modeled CO 4-3/3-2 ratios are on the low side, even in the D'Alessio et al. and Chiang & Goldreich models, indicating that the temperature in the surface layers would need to be higher. However, the calibration uncertainties in the older Kastner et al. (1997) data make this conclusion less firm. Further observations of high-J CO lines are needed to constrain the temperature structure of this disk.
The density is probed by the different HCO+ and HCN ratios. All three models are consistent with the observed 4-3/1-0 ratio for the main isotopes, indicating that the density in the layer above the midplane is in the correct range. The upper limits on the H13CN and H13CO+ prevent any conclusions for the midplane. As Table 5 shows, the models make different predictions for these optically thin species and future observations may be used to distinguish them.
Overall, the absolute line intensities and ratios are consistent with the models of D'Alessio et al. and Chiang & Goldreich for reasonable values of the depletions. The current data cannot distinguish between these two flared disk models. There is some evidence, however, both from the line ratios and from the line profiles that the surface of the disks needs to be warmer than that of a shielded isothermal outer disk such as computed by Bell (1999).
The high depletions derived for TW Hya are in agreement with the
conclusions by Kastner et al. (1997). The depletion can be due to two
reasons: the first is a change in the gas- to dust-mass ratio to a
value lower than 100 due to removal of gas. The second possibility is
a large depletion of CO and other molecules, but not H2. The
former possibility could be partly tested by searches for the pure
rotational H2 lines (Thi et al. 2001). Regarding the second
option, the present analysis indicates that the depletion cannot
simply be due to the freezing out of molecules resulting in a large
value of :
the molecules also need to be depleted in the warmer
upper layers probed by
.
This latter conclusion could be
consistent with the fact that TW Hya is a very active UV and X-ray
producing star (Kastner et al. 1999), capable of destroying CO due to
dissociation or ionization. This can be tested by measuring the CO
ionization or dissociation products, such as C+, C and CO+.
TW Hya seems well described by a flared disk model where the
ultraviolet radiation is capable of heating the upper layer (see Sect. 5.3).
The modeling suggests freezing out with a common value of
of at
least 10. The depletion of molecules onto grains can be an important
chemical sink in these disks since most of the mass is cold. This
makes mass determinations of disks using CO or any of its isotopomers
highly uncertain.
New models have recently been fitted to the SEDs of LkCa 15 and TW Hya
by Chiang et al. (2001) and Chiang (2000). These models use as one
of the main parameters the dust settling toward the midplane. For
LkCa 15, the SED modeling indicates that the dust should have settled
within the disk scale height H to explain the observations, with a
high dust temperature in that region (
K at 100 AU). In these high density regions, the gas and dust temperature
should be coupled; however at heights above the scale height H the
lack of grains will reduce the heating of the gas due to the
photo-electric effect, although some small grains may still be
present. Together with enhanced cooling due to [O I] and [C II], this
may cause a second temperature inversion with a cool upper layer free
of dust. The lack of grains in the upper layer would be consistent
with the non-detection of LkCa 15 by HST optical images (K.
Stapelfeldt, private communication). This suggests a lack of
scattering which should have been readily seen for a flared disk at an
inclination of 60
with grains well mixed with the gas and an
albedo of 0.5. In addition, the models used were calculated for a
stellar temperature of 4000 K, whereas LkCa 15 has a higher effective
temperature (
K) which would result in higher
disk dust-temperatures. The relatively low gas temperature in the disk,
indicated by the line observations, strengthen
the conclusions derived from the scattering and SED observations that
dust settling has taken place in LkCa 15. Self-consistent models of
the gas temperature and abundances of LkCa 15 are needed, taking
dust-settling into account.
As noted above, it is not yet possible to distinguish between the
D'Alessio et al. (1999) and the Chiang & Goldreich (1997) models with
the current observations. The data lack spatial resolution and have
insufficient sensitivity to observe the optically thin isotopic
lines. In addition to higher spatial resolution and sensitivity,
better calibration of the data is needed, all of which will be
provided by the Atacama Large Millimeter Array.
The main conclusions from this work are:
Acknowledgements
The authors are very grateful to P. D'Alessio, R. Bell and E. Chiang for sending and discussing the models used in the paper. They thank M. Hogerheijde and F. van der Tak for useful discussions and providing their radiative transfer code, and are grateful to the staff of the CSO and JCMT for their support. Astrochemistry in Leiden is supported by a SPINOZA grant from The Netherlands Organization for Scientific Research (NWO). This paper is dedicated to Fred Baas, who died on April 4, 2001. His expert, generous support at the JCMT was essential to make these observations possible.