THE ASTROPHYSICAL JOURNAL, 487:L165–L169, 1997 October 1
© 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.
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Rotational Excitation of CO in the Diffuse Interstellar Medium: Effects of Line Emission from Dense Molecular Clouds

 PETER WANNIER, 1 BRYAN E. PENPRASE, 2 AND B-G ANDERSSON 1

Received 1997 January 23; accepted 1997 July 21; published 1997 September 2


ABSTRACT

     We discuss a somewhat neglected interstellar rotational excitation mechanism that may be significant for CO in diffuse gas (n$\mathstrut{_{{\rm tot}}}$≤300 cm-3). Where such gas is found in the vicinity of a dense interstellar cloud, we show that molecular excitation in the diffuse gas may be dominated by spectral line photons emitted by the denser material. We illustrate the mechanism analytically for a simple two-level case and present the numerical results from a multilevel calculation.

     In all six of six sight lines that we analyze, it is plausible that the observed rotational excitation results from radiative excitation, so there is no compelling evidence that collisions play an important role. Therefore, densities derived under the assumption of purely collisional excitation would give incorrect (too large) values. Line-of-sight analyses of CO excitation can be (and have been) misinterpreted because of neglect of this contribution to radiative excitation. Similar situations can arise whenever there is a small continuum opacity and a large contrast of gas density within, or around, an interstellar cloud. It is a situation in which the often-used Sobolev (or large-scale velocity gradient) approximation simply does not apply.

Subject headings: ISM: clouds—ISM: kinematics and dynamics—ultraviolet: ISM

CONTENTS

FOOTNOTES

     1 Jet Propulsion Laboratory, Caltech, Pasadena, CA 91109.

     2 Pomona College, Claremont, CA 91711.

§1.  INTRODUCTION: THE ROTATIONAL EXCITATION OF CO

     The UV systems of CO (A-X, B-X, and C-X) contain many bands, each separated into lines from individual rotational levels. Data from any one of the bands can be used to evaluate the rotational excitation, and the fact that the bands have different intrinsic line strengths ensures that optical depth effects can be minimized along virtually any sight line. For many years CO excitation has been used to make inferences about gas density and temperature in the diffuse interstellar medium (ISM) (see Snow 1977; Smith, Krishna Swamy, & Stecher 1978; Lambert et al. 1994). These analyses assume that excitation is due to collisions with hydrogen in atomic and/or molecular form, balanced by radiative de-excitation. Treatment of the ≈3 K cosmic background radiation (CBR) varies.

     The sight lines toward hot stars generally exhibit very modest excitation temperatures (3–6 K), yet the derived gas densities are seldom much smaller than 100 cm-3 and range up to 3000 cm-3. These high derived densities are required to affect significant excitation in the face of J=1→0 spontaneous emission. During the course of evaluating recent Goddard High Resolution Spectrograph (GHRS) observations toward sight lines in Perseus, we reexamined the usual approach and realized that millimeter-wave spectral line emission may, in fact, provide an important, perhaps dominant, source of rotational excitation (Wannier et al. 1997a, 1997b). We then examined other published analyses and found that this source of rotational excitation is often significant, so that derived gas pressures and/or densities may be reported as too large.

     What degree of excitation can be provided by millimeter-wave emission? The answer to this question depends, in part, on the assumed physical interaction between the diffuse and the dense components of the molecular gas. Several lines of evidence indicate that the two components are often causally connected, so that dense and diffuse components with similar velocities and positions are likely to lie at the same radial distance (Andersson, Wannier, & Morris 1991; Federman et al. 1994). In such a circumstance, the radiative excitation is easily estimated from the CO brightness temperature of the dense CO cloud. Since CO emission lines are reported as the intensity in excess of the CBR, the excitation temperature in the absence of collisions will exceed the CBR temperature by an amount roughly equal to f times TR, where f is the fraction of the total solid angle subtended by the dense cloud, as viewed by a molecule in the absorbing gas, and TR is the brightness temperature of the CO J = 1 → 0 emission line. Because the radiative excitation results from absorption of spectral line radiation, the relative velocities of the diffuse and the dense gas must be considered: specifically, the radial velocity of one as viewed by the other. Some uncertainty must inevitably result from our lack of knowledge about motions in the plane of the sky. Spectral line coupling will occur whenever the velocities of the absorbing and emitting material are the same, as projected along the line of sight connecting them. We can check that both parcels of gas have the same radial velocity along the line of sight to Earth, but strictly speaking, that is neither necessary nor sufficient to prove that radiative coupling actually exists. For our examples below, we assume that coupling exists if and only if we see agreement in VLSR between the emitting and the absorbing gas. This assumption enables us to estimate the radiative coupling based on actual maps of CO brightness temperature, although the coupling may be either over- or underestimated.

     The numerical calculations are complicated by two facts: (1) molecular clouds are not uniformly bright, and (2) brightness temperatures are generally reported as if the Rayleigh-Jeans limit applies, which is not true for our case. Nonetheless, it is straightforward to evaluate the excitation temperature for a molecule, irradiated by the CBR and a nearby dense cloud. We further assume that the absorbing CO is embedded in a gas of temperature Tk and density n, consisting primarily of H2.

     The case of a two-level system is easy to express analytically, and we have done so below. However, for CO, more levels are useful to include when collisions become important, because of the importance of the ΔJ=2 collisional excitation rate, which can even produce an inversion in the J=1→0 transition (Goldsmith 1972; McKee, Storey, & Watson 1982). The excitation temperature for a two-level system (J=0 and J=1 CO in our case) is



where T$\mathstrut{_{*}}$=hν&solm0;k=5.53 K for the J=1→0 CO line, T$\mathstrut{^{e}_{R}}$ is the total effective J=1→0 CO line intensity as viewed by the absorbing gas, and C10 and A10 are the usual de-excitation rates due to collisions and spontaneous radiation. T$\mathstrut{^{e}_{R}}$ is the total radiation intensity as viewed by the absorbing gas, so



where TR is the conventional measure of millimeter-wave spectral line intensity and is most accurately understood as the intensity in excess of the CBR, reported in temperature units (see Kutner & Ulich 1981). For an emitting cloud of uniform brightness temperature, we can write



where f is the fraction of 4π sr subtended by the cloud as viewed by the absorbing gas. In most applications, equation (2) cannot be evaluated directly because the line-of-sight CO distribution is simply not known. Said another way, we generally have insufficient information to evaluate the effective solid angle, f. Therefore, some additional assumption is needed in order to derive Tex even if a complete CO emission-line map is available. For our calculations we assume (1) that absorbing gas and emitting gas at the same velocity lie at the same distance, and (2) that the emitting gas in any cloud or cloud fragment is circularly symmetric as viewed by the absorbing gas (see Fig. 1). Although either (or both) of these assumptions may be invalid in a specific case, this geometry is at least a plausible one and as such can be used to test the usual assumption made when evaluating CO absorption-line observations: namely, that radiative coupling is safely ignorable. There is, in fact, some general evidence for assuming a physical proximity (equal distance from the observer), which comes from observations suggesting that the neutral gas is commonly found to be the extended neutral halos of molecular clouds (Andersson et al. 1991; Federman et al. 1994). The second assumption (of circular symmetry) is easier to understand and can lead to either an overestimate or underestimate of T$\mathstrut{^{e}_{R}}$. It is accurate for the case in which the emitting gas is in a spherical cloud and the absorbing material, for example, might lie in the extended periphery of the cloud at a tangent point. Then the double integral in equation (2) becomes a single integral along &phis;, the angle measured from the axis of symmetry, lying in the plane of the sky. For a cloud of uniform surface brightness at the J=1→0 line frequency, equation (2) then becomes



where &phis;1 is half the opening angle of the dense cloud as viewed by the absorbing material.

Fig. 1

     Equation (1) is valid only for a two-level system, but the analysis is easy to generalize to a multilevel calculation, which we have done by writing the transition matrix and evaluating it for its meaningful eigenvectors. We have used the tabulated collision rates of Flower (1989), scaled appropriately for the mixture of He, ortho-H2, and para-H2 toward each sight line (Wannier et al. 1997b). Figure 2 shows plots of Tex as a function of density for a cloud with a CO brightness temperature, TR, of 10 K, and for several values of f and with Tk. In order to answer the question of whether or not radiative excitation alone can account for a given, observed rotational excitation, we only need to look at the left-hand axes (all plots yield the same Tex value when n=0). Additionally, one can use Figure 2 to contrast the effects of radiative and collisional excitation. The plotted curves include radiation for the upper rotational transitions (J=1→2, J=2→3, etc.) as if they followed a blackbody law. The symbols are similar but are for models deficient in radiation for the upper levels, simulating the case of a cool, subthermally excited CO cloud. The symbols are for a deficiency of a factor 3 for each level above (J=0→1, so that J=1→2 is reduced by a factor 3, J=2→3 by a factor 9, etc. The effect of reduced radiation in the upper levels is actually to increase the J=1 excitation, because there is a reduced dilution among the upper energy levels. We conclude that CO excitation does not provide a reliable measure of space densities in the diffuse gas (n≤1000 cm-3). Such inferences have, however, been made. In the well-studied line of sight toward ζ Oph, for example, Smith et al. (1978) have used CO excitation to infer that the total mean density of hydrogen lies in the range from 48 to 360 cm-3, assuming that excitation is due to collisions with H2 molecules and to the CBR. In fact, that sight line has been shown to lie near a molecular cloud, so that values of f of 0.1 and larger are quite likely (Barrett, Solomon, & Mooney 1989; Liszt 1993).

Fig. 2

§2.  CO EXCITATION IN OPHIUCHUS AND PERSEUS

     Is radiative excitation actually sufficient to explain observed CO excitation? We briefly examine six sight lines in Ophiuchus and Perseus (three in each), paying particular attention to the well-known one toward ζ Oph (Table 1). We use available CO line emission maps to estimate an excitation temperature, T$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$, which would result from radiative excitation alone, in the absence of any collisions (n=0). In both regions, there are sufficient CO emission-line maps to estimate the spectral line irradiation of the line-of-sight CO, and in both cases the agreement in radial velocity between the absorbing gas and the emitting gas indicates that the two components are located at nearly the same radial distance. In Perseus, this association is further substantiated by spectral line maps of CO and H I, which suggests that the CO may be part of an outflow from the molecular cloud (Andersson, Roger, & Wannier 1992), and a similar suggestion associating the gas in the ζ Oph sight line with nearby molecular clouds has been made by several authors (see Barrett et al. 1989; Liszt 1993).

     To determine a plausible radiative excitation for the CO, we have assumed that agreement in velocity between the absorbing and the emitting gas implies that they are equidistant, and we have looked for agreement in velocity using a single, narrow-velocity (Barrett, Solomon, & Mooney 1997, hereafter BSM) map. The assumption of spatial correlation between dense and nearby diffuse gas has independent support from observations that imply that molecular clouds generally have extended neutral halos (Andersson et al. 1991; Federman et al. 1994).

     For Ophiuchus we have used two CO emission line maps to estimate the CO spectral line irradiation: one by de Geus, Bronfman, & Thaddeus (1990, hereafter GBT) and one by BSM. The GBT map is extensive and low-resolution (5600 spectra from the Columbia University Sky Survey Telescope), covering a region more or less 20° × 25° in extent with a resolution of 9&arcmin;. The BSM map (1500 spectra from the FCRAO) covers 1&fdg;25 × 2&fdg;75 with a resolution of 3&arcmin;. It is more useful for the ζ Oph sight line, while the larger GBT map provides valuable information about more distant CO features, enables evaluations of ρ Oph A and χ Oph, and provides confirmation of all the CO features seen by BSM. Near the ζ Oph sight line several bright and extensive CO clouds are seen by GBT that lie outside the BSM map, but we have not included these in our evaluation because their radial velocities leads to doubt about their coupling to the ζ Oph sight line. To determine a plausible radiative excitation for the CO, we have used a single-channel, narrow-velocity BSM map, covering V$\mathstrut{_{{\rm LSR}}}$=-1 to 0 km s-1 (V$\mathstrut{_{{\odot}}}$=13 to 14 km s-1), and with contours from 1 to 4 K at 1 K intervals (Solomon 1997). For the 1 K contour the filling factor f is uncertain but lies between 0.15 and 0.5, depending on the line-of-sight cloud geometry. A single cloud to the northwest of the ζ Oph sight line subtends 80°, implying f=0.12. Another to the southwest yields f=0.07, and there are additional, smaller clouds. It is easier to get a value of f for the brighter contours since they are more compact, yielding f=0.10 for the sum of all cloudlets enclosed by the 2 K contour, f=0.03 for the 3 K contour, and f=0.02 for the 4 K contour. Since the radiative excitation adds linearly, and the contour intervals are separated by 1 K, we can add these filling factors to yield an effective value of f for a single source of brightness temperature of 1 K, yielding f=0.30 to 0.65 at TR = 1 K. This implies T$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$=3.4±0.2 K. Comparing this to observations, we look to Lambert et al. (1994), who derive T$\mathstrut{_{{\rm ex}}}$=3.4±0.4 K, using their high signal-to-noise ratio GHRS data and their preferred set of oscillator strengths from Chan, Cooper, & Brion 1993. The excellence of this agreement is a bit fortuitous, given the uncertainties involved, but one conclusion is robust: that there is no evidence for significant collisional excitation of CO toward ζ Oph! The ρ Oph A and χ Oph sight lines yield similar results. These were recently observed in the UV lines of CO by Federman et al. (1997), yielding T$\mathstrut{_{{\rm ex}}}$=8.0±0.8 and 3.7±0.4, respectively, at heliocentric velocities of -7 km s-1. We used the CO (J=1→0) maps of GBT at the equivalent VLSR of 4 km s-1 to derive T$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$=7.3±1.0 K for ρ Oph A and 3.3±0.4 for χ Oph. Again, there is no evidence for significant collisional excitation.

     For Perseus we have used a 12CO map made with the AT&T Bell Laboratories 7 m antenna (Moriarty-Schieven et al. 1997). As with the Ophiuchus sight lines, we made conservative and liberal assessments of the filling factors for each contour level and have given the average resulting radiative excitation temperatures in Table 1. The uncertainty is set just to include the full range of values (conservative to liberal). In all three Perseus sight lines, we see that T$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$ always exceeds Tex, reinforcing the idea that collisional excitation may not be significant and certainly cannot be assumed to be dominant! On the contrary, the estimated illumination by J=1→0 CO line emission is more than sufficient to produce the observed rotational excitation, suggesting that in-plane motions might be decoupling the gas seen in absorption from that in the larger molecular clouds. In one sight line (HD 23180), there are indications that the star is a member of the IC 348 cluster, in which case the UV-sampled gas might be foreground (e.g., Bachiller et al. 1987). That could decrease our estimates of f and make T$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$ smaller by several K, depending on whether the star is closer than the CO emission-line gas or they are equidistant. We have checked our estimates using the 13CO map of Bachiller & Cernicharo (1986) to estimate the J=1→0 brightness temperature, scaling up by a factor 4 to obtain a 12CO brightness temperature, consistent with a lower resolution 12CO map by Ungerechts & Thaddeus (1987). Resulting values of T$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$ from this second estimate always agreed within the uncertainty limits. These Perseus results will be discussed in more detail in a subsequent publication (Wannier et al. 1997b).

§3.  CONCLUSIONS

     We have reexamined the rotational populations of CO in six sight lines studied by UV spectroscopy. If the UV-absorbing CO is spatially associated with the CO-emitting gas near the same sight lines, then we conclude that the observed excitation in the absorbing gas is consistent with radiative, rather than collisional, excitation. This result casts doubt on prior analyses that have derived space densities under the implicit assumption that collisional excitation dominates the CO excitation. We suggest that derived space densities yielding n$\mathstrut{_{{\rm tot}}}$≤1000 cm-3 should be reevaluated on a case-by-case basis. In most interstellar sight lines there is probably enough CO emission to preclude reliable derivations yielding n$\mathstrut{_{{\rm tot}}}$≤100 cm-3. There are several possible ways to distinguish more clearly between collisional and radiative excitation, but which require additional observations, involving higher rotational levels and/or isotopically substituted CO. The case for observing 13CO or even C18O is based on the generally lower brightness temperatures of molecular clouds in these spectral lines, which should then lead to lower radiative excitation. The case for higher rotational levels stems from the fact that the selection rules for collisional and radiative excitation are different, leading to differences in the two population ratios. In any event, the CO rotational lines can still provide good upper limits to ntot.

     Wherever strong density contrasts exist, molecular excitation in the less dense gas may be dominated by spectral line photons emitted by the denser material. This situation might also apply to some transitions near dense cores or dense protostars within molecular clouds. In such circumstances it is simply not valid to evaluate molecular excitation using the Sobolev approximation, the approximation underlying commonly used “large-scale velocity gradient” codes.

ACKNOWLEDGMENTS

     We are grateful to S. Federman for providing valuable information about oscillator strengths to use for the UV bands of CO and for sharing the results of his Hubble Space Telescope observations of ρ Oph A and χ Oph prior to publication. We are grateful to W. Langer for using his own excitation code to provide a valuable check of our multilevel excitation calculations. An anonymous referee took the time necessary to examine thoroughly the Letter, and the manuscript is significantly improved as a result. The research described in this Letter was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

REFERENCES

FIGURES


Full image (25kb) | Discussion in text

     FIG. 1.—The geometry is indicated, showing the absorbing molecule lying alongside a dense molecular cloud. The angle &phis; corresponds to that in eqs. (2) and (4).


Full image (72kb) | Discussion in text

     FIG. 2.—Tex is shown for a cloud illuminated by the CBR and by additional millimeter-wave emission (CO brightness temperature T$\mathstrut{_{R}}$=10 K, and filling factor f=0.0, 0.1, 0.2, 0.3, and 0.4). The effects of pure radiative excitation are seen on the left-hand axes (all panels are the same for n=0). The plots include the effects of collisional excitation for a gas with T$\mathstrut{_{k}}$=10, 30, 100, and 300 K. The models include 10 rotational levels and appropriate radiation for the upper rotational transitions (J=1→2, J=2→3, etc.) as if they followed a blackbody law. The symbols are similar but are for models deficient in radiation in the upper levels, simulating the case of a cool, subthermally excited CO cloud (see text). From the plots, it is easy to contrast the effects of radiative and collisional excitation. For example, in the middle panel we see that for T$\mathstrut{_{k}}$=30 K and f=0.1, the excitation from radiation alone (left-hand axis) is equal to that produced by pure collisions (f=0.0) in a gas of density of 160 cm-3. However, for a colder gas (10 K, top) we would need a density of 550 cm-3. We see that densities of 100 cm-3 or less, derived from CO excitation measurements, under the assumption of purely collisional excitation, must be viewed with some suspicion.

TABLES

TABLE 1
CO ROTATIONAL EXCITATION
StarT$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$
(K)		Tex
(K)
ζ Oph (HD 149757)...3.4 ± 0.23.4 ± 0.4
ρ Oph A...7.3 ± 1.08.0 ± 0.8
χ Oph (HD 148184)...3.3 ± 0.43.7 ± 0.4
o Per (HD 23180)...7.1 ± 1.0 a3.6 ± 0.3 b
40 Per (HD 22951)...3.5 ± 0.33.0 ± 0.3
HD 23625...4.9 ± 0.62.7 ± 0.3

     a T$\mathstrut{^{{\rm rad}}_{{\rm ex}}}$ may be smaller for HD 23180 if it is not background to IC 348.
     b T$\mathstrut{_{{\rm ex}}}$(CO) for the dominant velocity component (Wannier et al. 1997b).

Image of typeset table | Discussion in text
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