THE ASTROPHYSICAL JOURNAL, 536:481È493, 2000 June 10 ( The American Astronomical Society. All rights reserved. Printed in U.S.A.

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1 THE ASTROPHYSICAL JOURNAL, 536:481È493, 2000 June 10 ( The American Astronomical Society. All rights resered. Printed in U.S.A. THE CO FUNDAMENTAL VIBRATION-ROTATION LINES IN THE SOLAR SPECTRUM. II. NON-LTE TRANSFER MODELING IN STATIC AND DYNAMIC ATMOSPHERES H. UITENBROEK Harard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138; huitenbroek=cfa.harard.edu Receied 1999 Noember 8; accepted 2000 January 26 ABSTRACT We present a numerical method for soling radiatie transfer in molecular ibration-rotation bands that allows for departures from local thermodynamic equilibrium (LTE) while accurately including a large number of lines. The method is applied to the formation of the CO fundamental ibration-rotation bands in seeral plane-parallel hydrostatic models and in a sequence of 20 snapshots from a radiationhydrodynamics simulation of chromospheric dynamics. Calculations for the hydrostatic models performed with di erent alues of the collisional coupling between di erent ibrational states conðrm earlier results in the literature showing that the CO lines hae LTE source functions in the solar atmosphere, so emergent CO intensities reñect actual temperatures therein. Only if the canonical collisional strengths are too large by more than 2 orders of magnitude would it be possible to explain the low temperatures deried from CO line core intensities at the solar limb by scattering in an atmosphere with much higher temperatures, consistent with the alues deried from UV line and continuum and Ca II resonance line diagnostics. An interesting feature in the waelength structure of the CO ibration-rotation bands is pointed out, in which pairs of lines can be found in di erent bands but of similar strength and waelength. In principle such pairs proide a diagnostic for departures from LTE in the CO lines. CO line core intensity ariations computed from the sequence of dynamical snapshots, which represent a typical episode in the chromospheric dynamics simulation, hae an amplitude that is 2.5 times higher than obsered. It is shown that this large amplitude is due in part to the up and down shift of the CO line formation region during the eolution of the atmosphere and is related to the assumption of instantaneous chemical equilibrium that was assumed to calculate CO concentrations. This suggests that the CO concentration is not in equilibrium, may be lower than would be expected on the basis of chemical equilibrium at the time-aeraged mean temperature of the atmosphere, and may hae reduced ariations compared to instantaneous chemical equilibrium alues at the local temperatures. Subject headings: line: formation È methods: numerical È radiatie transfer È Sun: atmosphere È Sun: infrared È waes 1. INTRODUCTION The processes that lead to the production of enhanced emission in the solar chromosphere and transition region are poorly understood. In part this reñects the difficulty of obsering these tenuous layers, as they are optically thick only in UV lines and continua, which cannot be obsered from the ground, and microwae continua, which can be obsered only with low spatial and temporal resolution. As more and more detailed obserations become aailable, howeer, it is clear that high-resolution obserations are essential for improing our understanding of the outer layers of the Sun, which are not quiet and homogeneous but are ery dynamic and appear to change constantly on all obserable scales. Analyzing such a complicated enironment requires detailed models including proper treatment of optically thick radiation. Traditionally, the chromosphere and transition region hae been represented through one-dimensional hydrostatic models, most of which include a sophisticated radiatie transfer treatment and some of which hae been ery successful in reproducing the spatially and temporally aeraged spectra of di erent solar surface components (quiet-sun, bright network, actie plage; e.g., Fontenla, Arett, & Loeser 1990, 1991, 1993). Obiously, these models lack the rich spatial forms and ariability of the real Sun, but, considering how well they reproduce (spatially and temporally) aeraged spectra, it may seem reasonable to expect that they at least represent the aerage 481 properties of the atmosphere. If this could be demonstrated reliably, it would greatly simplify the analysis of solar spectra in particular and stellar spectra in general by requiring only computationally inexpensie onedimensional hydrostatic models based on aeraged spectra. An important question that has to be answered is whether hydrostatic models (or static models with prescribed mechanical heating, e.g., Anderson & Athay 1989) deried from aerage spectra correctly account for the energy balance of the atmosphere, i.e., whether they accurately predict the amount of mechanical heating required to produce the obsered chromospheric and transition region emission spectra, in particular in the presence of mass motions and/or spatial inhomogeneities. An indication that this may not be the case came from recent radiationhydrodynamic simulations of chromospheric dynamics presented by Carlsson & Stein (1992, 1995, 1997, 1999). These authors were ery successful in reproducing the dynamic behaior of the chromospheric Ca II H line core emission with ab initio calculations and were able to closely match the obsered appearance of so-called H grains. As a striking example of the dependence of the 2V results of spectral analysis on the chosen underlying model, they show that the obsered chromospheric emission does not necessarily require a chromospheric rise in the mean gas temperature in their dynamic model, een when the semiempirical hydrostatic model deried from the same obserations does. An

2 482 UITENBROEK Vol. 536 important role in these dynamical simulations is played by the hydrogen ionization rate, which cannot keep up with the atmospheric dynamics during the passage of an acoustic shock wae through the chromosphere. As a result wae energy is turned into thermal energy, which produces a sharp temperature spike and gies rise to strong UV radiation losses rather than being turned into ionization energy as would be the case with instantaneous ionization. Directly behind the shock the onset of hydrogen ionization cools the shock-heated atmosphere, while een farther behind the shock recombination occurs, which releases additional energy in the form of radiatie cooling. The sharply peaked temperature behind the shock and concurrent radiatie losses cause the time-aeraged atmospheric mean temperature to be below that expected from hydrostatic semiempirical modeling. This explanation of chromospheric emission has been challenged by Kalkofen, Ulmschneider, & Arett (1999), who claim that the success of the (Carlsson & Stein 1997) modeling of H grains does not guarantee the alidity of their calculations 2V in the higher layers and contend that substantial power is missing from the calculations at higher frequencies. Indeed, it still remains to be shown that obserations of Carlsson, Judge, & Wilhelm (1997) showing persistent emission in UV lines are consistent with dynamical models that hae no permanent chromospheric temperature rise. Another major challenge to the use of hydrostatic models to represent the solar atmosphere is posed by the dark cores of the infrared CO ibration-rotation lines that are obsered close to the limb. When modeled with a onedimensional hydrostatic atmosphere these lines imply the presence of material with temperatures as low as 3700 K (Noyes & Hall 1972; Ayres & Testerman 1981; Ayres, Testerman, & Brault 1986; Ayres & Wiedemann 1989; Ayres & Brault 1990) at altitudes where the inner wings of the Ca II H and K lines and UV and microwae continua indicate that temperatures cannot be below the classical minimum of approximately 4400 K. Hence, the CO lines proide another indication that the aerage properties of the inhomogeneous dynamic solar atmosphere cannot be represented ery well by hydrostatic models. This paper is the second in a series inestigating the formation characteristics of these infrared CO ibrationrotation lines, which, apart from being important temperature diagnostics, dominate the infrared spectrum of cool stars like the Sun and proide an important path for radiatie cooling to their atmospheres. The aim of this paper is to present forward modeling of the CO lines in a sample chromospheric dynamics simulation in order to improe understanding of the formation characteristics of these lines in such an enironment and to better understand their diagnostic alue. By forward modeling we mean soling the molecular transfer problem with enough sophistication, including many lines and departures from local thermodynamic equilibrium (LTE), that computed intensities can be compared with obsered alues but not accounting for the back coupling of the CO lines on the dynamics. Eentually, a better understanding of the formation of the releant diagnostics in models that are more physically realistic than hydrostatic models should lead to a uniðed model that explains CO, as well as Ca II and UV intensities, at the same time. In the Ðrst paper of this series (Uitenbroek 2000, hereafter Paper I) spatially and temporally resoled obserations obtained at the McMath-Pierce telescope on Kitt Peak are described and compared with LTE computation of the formation of CO lines in a three-dimensional snapshot of a solar granulation simulation. The obserations show that at disk center the CO line core intensity exhibits an inerted granular contrast: dark granules surrounded by bright intergranular lanes. Theoretical radiatie transfer calculations through a three-dimensional granulation simulation snapshot explain the inerted obsered contrast in terms of the strong adiabatic cooling that hot rising granular matter undergoes when it runs into the steep drop in density with height of the oerlaying stable layer and has to expand to adapt to the low-density enironment. Oer the intergranular lanes the return Ñow is decelerated and heated by compression, giing rise to an inerted temperature contrast in the layer in which the CO line cores form. The computed contrast in the CO line cores is, howeer, much higher than obsered, and the calculated spatially aeraged line core intensities are lower than obsered. In Paper I it is suggested that the main reason for these discrepancies is the assumption of instantaneous chemical equilibrium at the local temperature that was used to calculate CO number densities. If the concentration of CO due to a slow rate of association in the hot upñowing matter is less than would be expected on the basis of instantaneous equilibrium, the formation height of CO lines would be lower, intensities would be higher, and contrast would be lower, just as is obsered. Unfortunately, the granulation simulation used in the modeling for Paper I does not extend to high enough altitudes to model the formation of CO lines reliably toward the limb. Therefore, the question of where the dark limb cores originate still remains. Ayres & Wiedemann (1989) hae demonstrated that it is unlikely that they are caused by scattering in the CO lines themseles (nor are they caused by scattering in the background continuum; see Paper I). In the present paper the alidity of LTE is further inestigated in calculations through di erent onedimensional hydrostatic models ( 4.1) and a sequence of snapshots from a chromospheric dynamics simulation by Carlsson & Stein (1999) ( 4.3). A brief discussion of the ATMOS spectrum, which is used to compare calculated intensities with obserations for a large number of lines, is gien in 2. Section 3 discusses the employed model atmospheres, molecular data, and numerical method for solution of the molecular non-lte problem. A discussion of the results is gien in ATMOS OBSERVATIONS Most CO ibration-rotation lines are hard to obsere from the ground because they are blocked out by molecular absorption in the terrestrial atmosphere; only a few spectral windows that are relatiely free of telluric contamination remain. By contrast the Spacelab 3 ATMOS experiment (Farmer & Norton 1989), which was designed to measure the composition of the upper earth atmosphere, was able to record an uncontaminated solar infrared atlas stretching from 2.1 to 15.5 km (Farmer 1994). The atlas, with a resolution of 0.01 cm~1, was obtained as a reference spectrum and contains 6478 identiðed fundamental and 2394 Ðrst-oertone CO transitions (amounting to more than onehalf the total number of its spectral features) and many more lines from atomic and other molecular species. Unfortunately, the ATMOS atlas represents intensities integrated

3 No. 1, 2000 CO LINES IN SOLAR SPECTRUM. II. 483 oer a large circular area extending from disk center out to 0.28 R (corresponding to a cosine of the iewing angle k \ 0.96) _ for j\4.97 km and out to 0.58 R (k \ 0.81) for longer waelengths and was produced by aeraging _ many separate spectra. Neertheless, because it contains so many lines uncontaminated by telluric absorptions, the atlas is ery useful for studying the spatially aeraged behaior of CO lines. Temperature [K] FAL C XCO COOLC FLUXT 3. NON-LTE MODELING 3.1. Static and Dynamical Model Atmospheres Seeral one-dimensional hydrostatic models were used to inestigate the importance of departures from LTE for the solar CO ibration-rotation lines. Since these models lack the obsered dynamical ariability and spatial diersity of the real Sun, one could argue that they are not truly representatie of solar atmosphere conditions. Neertheless, they proide self-consistent models with appropriate density stratiðcations that can be used to study certain aspects of the formation of spectral diagnostics without being distracted by the complexities introduced by ariability and spatial di erentiation. Of course, in order to use these diagnostics to measure properties of the atmosphere, i.e., to extract physical properties from obsered spectra, more sophisticated models are needed. Figure 1 shows the temperature stratiðcation as a function of column mass for the four hydrostatic models used in this paper. Model FAL C was constructed by Fontenla et al. (1993) to reproduce the time- and space-aeraged spectrum of the quiet Sun. It matches the aerage intensities of important diagnostics such as UV and microwae continua, the Ca II line wings, the hydrogen Lya line, and their respectie center-to-limb behaiors but does not match infrared CO ibration-rotation line intensities (e.g., see 4.2). The latter can be reproduced with model XCO, a model in hydrostatic equilibrium deeloped by Arett (1995) to match speciðcally a large number of CO lines from the ATMOS atlas. This CO model, howeer, seerely underestimates intensities in UV lines and UV and microwae continua. The other two models, FLUXT and COOLC, were constructed as a pair by Ayres et al. (1986) to represent a hot Ñux-tube atmosphere and its cool nonmagnetic surroundings. A linear combination of 7.5% of the emergent Column Mass [kg m 2 ] FIG. 1.ÈTemperature stratiðcation of one-dimensional hydrostatic models FAL C, XCO, COOLC, and FLUXT as a function of column mass. spectrum of the FLUXT model and 92.5% of that of COOLC matches both the aerage obsered spectrum of the calcium resonance lines and that of a large number of CO lines obsered from the ground with the Fourier Transform Spectrometer (FTS) at Kitt Peak. The COOLC and FLUXT combination, howeer, oerestimates UV continuum intensities by as much as a factor of 20 (at 140 nm, see Arett 1995). Model COOLC closely follows the aerage quiet-sun model FAL C up to the middle of the photosphere (at a column mass of 10 kg m~2). Higher up its temperature drops o to reach a constant alue of 3500 K in the outermost layers. The temperature of model XCO drops slightly below that of FAL C in the deep photosphere, runs hotter in the middle of the photosphere to cross oer again at a column mass of 1 kg m~2, drops to a minimum alue of 3800 K, and then rises to chromospheric temperatures. The radiation-hydrodynamical simulations of chromospheric dynamics by Carlsson & Stein (1995) and similar e orts by the Heidelberg group (e.g., Ulmschneider et al. 1978; Schmitz, Ulmschneider, & Kalkofen 1985; Ulmschneider, Muchmore, & Kalkofen 1987; Rammacher & Ulmschneider 1992) seem to proide a class of models that are more appropriate to the obsered dynamic nature of the solar atmosphere than hydrostatic models. In particular, the fact that Carlsson & Stein (1992, 1997) were able to accurately reproduce such intricate dynamical behaior as the Ca II H grains is indicatie of a high degree of realism. 2V The dynamical models, howeer, hae not been tested extensiely against detailed obserations, and the e ects of the dynamics on the emergent spectrum hae been detailed only for a ery limited number of diagnostics, not including the CO infrared ibration-rotation lines. This paper makes a Ðrst attempt at studying the formation of these lines in a sample dynamical model. To this end non-lte emergent intensities of a large number of CO lines were calculated from a sequence of 20 snapshots, each 10 s in real time after the preious, from a recent simulation by Carlsson & Stein (1999). The total duration of 190 s of the sequence is comparable to the period of a typical chromospheric oscillation. In the simulations, acoustic waes, which are drien by a piston at the bottom of the atmosphere, deelop into shock waes when they run upward into the low-density regions of the outer solar atmosphere. The temperature stratiðcation in each of the 20 snapshots is plotted as a function of column mass in Figure 2. For comparison the Ðgure also gies the temperature stratiðcation of hydrostatic model XCO. The results in this paper suggest that the temporal eolution of the CO concentration has a signiðcant inñuence on the formation height of the CO lines and, therefore, on the temperature-height relation deried from these lines CO Data In an important paper, Ayres & Wiedemann (1989, hereafter AW) hae shown that, because of collisional quenching by neutral hydrogen, the CO ibration-rotation transitions are most likely in LTE, so intensities at di erent waelengths in the lines reliably map kinetic temperatures at unit optical depth for those waelengths. It is, therefore, unlikely that the dark CO cores at the solar limb are the result of line scattering in the low-density chromosphere. Preiously, the alidity of LTE for CO formation had been considered by Hinkle & Lambert (1975) and Thompson (1973), and non-lte calculations of CO formation in the

4 484 UITENBROEK Vol. 536 Temperature [K] XCO Column Mass [kg m 2 ] FIG. 2.ÈTemperature stratiðcation of a sequence of 20 snapshots from a dynamical simulation ( Carlsson & Stein 1999; solid cures) and of hydrostatic model XCO (dashed cure). For clarity each of the solid cures is o set upward by 500 K relatie to the preious one. Sun and the red giant Arcturus hae been presented by Carbon, Milkey, & Heasley (1976), Heasley & Milkey (1976), and Heasley et al. (1978). Results in these papers are uncertain mainly because of the uncertainties in the collisional rates for the excitation and de-excitation of CO ibration-rotation transitions by neutral hydrogen, H mol- 2 ecules, and electrons. In calculations presented below ( 4.1) it is shown that, een if collisional strengths are oerestimated by 1 order of magnitude, the conclusions reached by AW still hold. CO line depths do not increase by more than a few percent when collisional strengths of only 10% of the canonical alues quoted by AW are used. These canonical alues are employed as the default collisional strengths in this paper. The line list and energy leels for the CO molecule were taken from Gooritch (1994), who proides tables of fundamental, Ðrst- and second-oertone rotation-ibration, and pure rotational transitions. For the results presented in this paper, only the fundamental ibration-rotation transitions of the 12C16O isotope were taken into account. The oertone bands are weaker, and because their waelengths are shorter and lie closer to the H~ opacity minimum at 1.6 km, they form deeper in the atmosphere, where population numbers are een more likely to be in LTE. The calculation of chemical equilibrium accounted for the molecules H,H C,N,O, CH, CO, CN, NH, NO, and H O and their 2 2`, constituent 2 2 atoms. 2 In the hydrostatic models 2 as well as in the snapshots from the chromospheric dynamics simulation it was assumed that molecular populations were gien by chemical equilibrium according to Saha at the instantaneous local particle temperature. The partition function and equilibrium constant for CO were calculated from polynomial Ðts by Kurucz (1985). A table with coefficients for these Ðts is proided in Paper I Numerical Method Where a few strong transitions usually dominate radiatie transfer in atoms or ions, a ast number of weak transitions characterize transfer in molecular species. There are, for instance, 2(N [ 1)(N [ 1) allowed fundamental ibration-rotation transitions J between the Ðrst N ibra- tional states with N rotational states each. At the typical J alues of 10 for N and 121 for N used in this paper this J results in seeral thousand releant transitions. To sole simultaneously for radiatie transfer in all allowed transitions and for statistical equilibrium in all rotationibration leels separately while allowing for departures from LTE seems a daunting task. Howeer, two assumptions brought forward by AW and described in detail below facilitate the computational e ort. First, if a Boltzmann distribution can be assumed for the N rotational leels within each of the N indiidual ibrational J states, only the total of radiatie and collisional rates (summed oer J) between the ibrational states and their departures from LTE as a whole need to be considered. This reduces the number of statistical equilibrium equations to be soled at each location in the atmosphere from N N to N. Second, because the CO collisional excitation and J de-excitation are dominated by neutral hydrogen collisions, the rates follow the exponential density increase inward dictated by a graitationally strati- Ðed atmosphere. The exponential increase in collisional excitation rates with depth implies that scattering e ects can propagate only a few optical depths inward and that a simple " iteration starting from an LTE solution is an efficient method of solution. By contrast, the density of electrons, which are mainly responsible for collisional coupling of atomic transitions, remains almost constant throughout the temperature minimum region in a typical hydrostatic model of the solar atmosphere. Within a gien electronic conðguration the energy leels E of a diatomic molecule are gien by J E J(J ] 1) (1) J /hc \ u 0 ] B (e.g., Weissbluth 1978 p. 606), where is the ibrational and J the rotational quantum number. For the case of CO the alue for the ibrational constant is u \ ] 105 m~1, 0 and for the rotational constant of the ibrational ground state, B \ m~1. Since u? B, it follows that suc- 0 0 cessie rotational leels within one ibrational state are much closer in energy than similar rotational leels in successie ibrational states. Moreoer, purely rotational transitions hae smaller radiatie de-excitation rates than ibration-rotation transitions by a factor of 5È10 (e.g., see Gooritch 1994). As a consequence it is a ery good approximation, een at low densities, to assume that the relatie occupation numbers of rotational leels within one ibrational state are determined by collisions and thus gien by Boltzmann statistics. In this case the relatie population n /n of two rotational leels J and J@ within ibrational J J { state is gien by (g /g where J J { ) exp [[(E J [ E J {)/kt ], the statistical weight g \ 2J ] 1 only depends on the rota- tional J and not on the J ibrational quantum number. If the ibrational state populations too are distributed according to Boltzmann statistics, then the LTE population number of a leel (, J) is gien by nlte \ g J exp ([E J /kt ) n, (2) J U (T ) CO CO where U (T ) 4 g exp ([E /kt ) is the CO partition function CO and n the,j total J CO number J density. Here it is assumed that all CO CO molecules are in the X 1&` electronic ground state, neglecting contributions from other electronic conðgurations to the partition function U. Writing the total number of molecules in ibrational CO state as n \ n, deðning the partition function for as U (T ) 4 J g J exp ([E /kt ), and assuming that within state the J J J rotational leels obey Boltzmann statistics, the population

5 No. 1, 2000 CO LINES IN SOLAR SPECTRUM. II. 485 number n may be written as J A n \ n J U (T ) g J exp [ E J ktb, (3) where the n are now not necessarily gien by their LTE alues. Under these assumptions it is possible to set up a set of linear equations dn dt \ ; { P { [ n ; P { (4) for the population numbers n, where the coefficients P { \ R are gien by the total of the rates R of all { ] C allowed radiatie { transitions plus the rates C of possible collisional transitions between ibrational states This rate equation holds een when, in addition to the fundamental transitions [ \ 1, oertone transitions [ \ 2, 3... are taken into account in the radiatie rates R, in contrast to the method described by AW, which only accounts for fundamental transitions. To derie expressions for the rates of equation (4) in terms of the population numbers n we Ðrst write down the expressions for absorption coefficient aj,j { and emission coefficient jj,j { (see for instance Mihalas l 1978) under the assumption l of complete frequency redistribution: aj, { J { \ hl l 4n r l J,{ J { B J, {J{A nj [ g J n g@ J {J{B, jj, { J{ \ hl l 4n r l J,{ J{ A J, {J { n {. (5) The quantities B and are the Einstein coeffi- J, {J { A cients for absorption and spontaneous J, {J { emission, respectiely. Using equations (3), the familiar Einstein relations, and nlte/nlte the set of equations (5) can { \ U (T )/U {(T ), be written in terms of the ibrational populations n and n { as follows: aj, { J { \ hl l 4n r l J,{ J { B J, {J { A B ] n [ n LTE nlte e~hl@ktn e~ej@kt gj { {, U (T ) jj, { J { \ hl l 4n r l J, {J { 2hl3 B J, {J{ c2 And, Ðnally, deðning ] e~hl@ktn { g J e~ej@kt U (T ) nlte nlte {. (6) gj, { J{ 4 n LTE l nlte e~hl@kt, and { V J, { J { 4 hl l 4n r l J, {J { B J, {J { g e~ej@kt, (7) J U (T ) the set (5) simpliðes to the desired result aj, { J { \ V J, { J { (n [ gj, { J { n l l l { ), jj, { J { \ 2hl3 l c2 g l J,{ J { V J, { J { n l {. (8) The total excitation rate per second due to radiatie transitions from ibrational state is the sum oer all allowed rotation-ibration transitions (using eqs. [3] and [7]): d) ; n R J J, {J{ \P P dl 4n JJ@ A B ] ; B J, {J JJ@ { r l J,{ J { I n l J P \ n d) P dl A ; B V J,{J{ I hl l l JJ@ \ n R {. (9) Likewise, the radiatie de-excitation rate can be written as P ; n R J J, {J{ \ n d) P dl { hl C ] ; gj, { J { V J, { J { A2hl3 l l c2 ] I l \ n { R {. (10) The solution to the molecular non-lte problem then proceeds as follows. Gien an initial solution for the population numbers n (LTE alues in this case), equations (8) are used to compute the absorption and emission coefficients for each ibration-rotation line so that the equation of transfer can be soled for all releant frequencies. Using the calculated intensities (integrated oer angle to get mean intensities) and using the already computed alues for V J,J { and gj,j {, radiatie rates are calculated according l l to equations (9) and (10). New populations n are computed using the statistical equilibrium equation (4), and the whole process can be iterated until conergence (i.e., until relatie changes in n become smaller than a predeðned limit of 10~2, which is congruous with the condition of n/*n > as recommended by Auer 1991, since the collisional destruction parameter is of order unity). Usually, under typical solar conditions and with the canonical alues for the collision strengths, conergence was reached after two or three iterations. It is worth noting that, if plain " iterations would result in oerly slow conergence, the aboe equations can be adapted easily to the multileel accelerated " iteration (MALI) scheme formulated by Rybicki & Hummer (1991, 1992) for the solution of non-lte problems in atoms and ions with oerlapping radiatie transitions. To allow for such an extension the equations in this paper were cast in precisely the same form as employed by Rybicki & Hummer (1992), using the quantities g and V (eq. [7]) to ealuate both the absorption and emission coefficients, as well as the radiatie rates, in an efficient way. The quantities n used in their paper to denote the atomic leel population numbers i are to be replaced in this case with the ibrational state populations n. Figure 3 shows the departure coefficients b (deðned as b 4 n /nlte, the ratio of the actual population numbers oer their LTE alues) of the Ðrst 10 CO ibration states in hydrostatic model XCO. The Ðgure shows the familiar pattern of underpopulation in the higher excitation and oerpopulation in the lower excitation states that is due to photon losses, in this case in the ibration-rotation lines. At a column mass of 10~1 kg m~2 (corresponding to a height of about 600 km aboe the photosphere) photons escaping from the atmosphere in the CO lines start to depopulate the BD

6 486 UITENBROEK Vol. 536 n / n LTE = Height [km] XCO Column mass [kg m 2 ] FIG. 3.ÈDeparture coefficients b \ n /nlte of the Ðrst 10 CO ibration leels as a function of column mass in hydrostatic model XCO. higher excitation ibration states with respect to LTE. Populations in the lowest excitation states start to depart from LTE seeral density scale heights further upward, only at column masses below 10~2 kg m~2. This is because lower excitation states are more densely populated because of their larger Boltzmann factors, so any shift in population numbers has a relatiely small impact on the low excitation and a relatiely high impact on the high excitation state populations. Most notably, going upward the departure coefficient b of the ground state remains close to unity for 0 at least another 200 km, or almost 2 density scale heights, compared to the coefficient b for the highest excitation 9 state in this sample. The high excitation state departure coefficients in model XCO show no sign of an upturn in the upper layers of the atmosphere as was reported by AW, nor was such behaior found for model COOLC (not shown here); at any gien depth the departure coefficients b decrease monotonically with increasing in both these models W aelength Structure of the CO Rotation-V ibration Bands The ibration-rotation lines of a diatomic molecule like CO fall into two waelength-separated branches according to the selection rule *J \^1. In the following, the primed and J@ are used to denote the upper leel, and unprimed quantities denote the lower leel of a transition. The longer waelength P-branch is then deðned by J@ \ J [ 1 and the shorter waelength R-branch by J@ \ J ] 1. Within a gien electronic conðguration the energy of a ibration-rotation leel is gien by equation (1), where the rotational constant B \ h/8n2i c decreases with increasing quantum number because it is inersely proportional to the moment of inertia I, which increases with (e.g., Weissbluth 1978). For the CO molecule, for instance, the B and B hae alues of and m~1, respectiely. 0 The 1 energy di erences between leels that are coupled by radiatie transitions in the two branches are *E /hc \ u ] 2B ] (3B [ B )J J,`1J`1 0 `1 `1 ] (B [ B )J2, R-branch, `1 *E /hc \ u [ (B ] B )J J,`1J~1 0 `1 ] (B [ B )J2, P-branch. `1 Together these equations can be grouped as follows: *E/hc \ 1/j \ u ] (B ] B )m ] (B [ B )m2 0 `1 `1 Gm \ 1, 2, 3,... R-branch, ] (11) m \[1, [2, [3,... P-branch. Since B \ B for CO, the R-branch has a minimum waelength `1 at m \ (B ] B )/2(B [ B ); this deðnes the rotation-ibration `1 band head for a gien `1. The shift in waelength of the di erent ibration bands introduced by the decrease of B with creates a potentially interesting CO line diagnostic that may, howeer, be difficult to use from the ground because so many CO lines are blocked by telluric absorptions. Figure 4 shows a plot of the strength s of lines in the \ 0 and 2 fundamental bands as a function of waelength and for a gien temperature of 4000 K. Line strength s, deðned as Line strength s J, {J{ e2 \A4m e 0 cb gj f e~ej@kt(1 [ e~hl@kt), (12) J, {J{ is related to the LTE absorption coefficient as follows. The oscillator strength f can be expressed in terms of the J, {J Einstein coefficient for absorption { through the familiar relation (e2/4m c) f e 0 J, {J { \ B J, { J {(hl/4n) (MKSA units). (13) Thus, when the molecular leels are populated according to Boltzmann statistics, the opacity due to the line is gien by aj, { J { \ s l J, {J { r l J,{ J { n /U (T ). CO CO The cure of strength s of the \ 2 band in Figure 4 (squares) is shifted downward with respect to the groundstate cure (triangles) because of the increase of ibrational energy with and to the right, toward longer waelengths, because of the decrease of rotational energy constant B with (cf. eq. [11]). This property of the ibration-rotation transitions is ery interesting, as it allows us to Ðnd line pairs of ery similar strength and waelength (and thus with almost exactly the same background opacities) in separate ibration bands. Such pairs are deðned by intersections of the cures drawn in Figure 4. This should proide us with a diagnostic for departures from LTE in the CO lines that is Bandheads R branch 1 0 transitions 3 2 T = 4000 K P branch Waelength [nm] FIG. 4.ÈLine strength s (see deðnition in the text) s. waelength for transitions of the fundamental \ 0(triangles) and \ 2(squares) bands.

7 No. 1, 2000 CO LINES IN SOLAR SPECTRUM. II. 487 fairly model independent in the following way. Since the departure coefficients of the \ 0 and 1 states remain close to unity throughout most of the region of line formation, the intensities of the \ 0 fundamental band should proide an accurate temperature diagnostic, een close to the limb. Once the temperature is determined from \ 0 band lines, graphs like the ones plotted in Figure 4 allow us to Ðnd combinations of \ 0 lines with lines in other bands with equal strength and similar waelength, as described aboe. If LTE populations preail for all ibrational states, these pairs should hae equal line depths. If, on the other hand, scattering in the CO lines is important enough to force the departure coefficients in the higher excitation states to drop substantially below unity in the line formation region, then the opacity of lines in higher excitation bands is reduced, and these lines form deeper in the atmosphere. Howeer, their line source Sl also depends on b through the ratio of departure coefficients of lower ibration state and upper Sl J, {J { \ 2hl3 Ab ehl@kt [ 1 B~1. (14) c2 b { The line source function, therefore, is reduced compared to its LTE alue when b [ b as appears to be the case in the {, CO lines. Depending on how the shift in line formation height and change in source function together modify the proðle of the high-excitation line, it may hae higher or lower core intensities. Howeer, if its intensities are di erent from the \ 0 line in the pair, this can only be due to departures from LTE. Unfortunately, it is ery difficult to Ðnd pairs of lines that are obserable from the ground because of blocking by the terrestrial atmosphere. Moreoer, the ATMOS spectrum, despite all its unobscured lines, represents intensities at disk center, where departures from LTE are smallest. The method described aboe could, therefore, probably be employed only when a space- or balloon-based instrument with sufficient spatial resolution to obsere the solar limb became aailable or when the terrestrial spectrum could be remoed from a sufficient number of line pairs by careful analysis of ground-based FTS obserations at di erent air masses. 4. RESULTS 4.1. Departures from L T E To test the inñuence of departures from LTE on the CO ibration-rotation proðles, line core intensities were calculated with and without assuming LTE populations for the ibrational states. In Figure 5 the ratio of non-lte oer LTE line core intensities is plotted for three di erent iewing angles corresponding to k 4 cos (h) \ 0.95, 0.23, and For none of the lines is the e ect of departures from LTE on the core intensities larger than 2%, een ery close to the limb (k \ 0.05) where the lines sample lower densities. For small the e ect of non-lte departures on the line core intensities increases with, reaches a maximum for \ 2È4, and then decreases again for larger alues of the ibrational quantum number. This is due to the inñuence of the ratio b /b of departure coefficients of the `1 lower and upper ibration states of the lines on the line source function (see eq. [14]). For the \ 0 band both b 0 and b are close to unity as is their ratio, causing Sl to be 1 close to the Planck function. With increasing the b drop progressiely further below unity in the line-forming region and, since b /b [ 1, the line source drops below the `1 Planck function. For een larger alues of the ratio b /b decreases again as b and b drop o together. In `1 `1 addition, lines in the higher excitation bands are generally weaker because the higher excitation ibration states are less populated (een without departures from LTE) so that these lines form deeper in the atmosphere where all the I / I LTE µ = = 0 XCO = I / I LTE = 4 = 8 FIG. 5.ÈRelatie CO line core intensities I/ILTE for hydrostatic model XCO with standard collisional excitation for three di erent iewing angles

8 488 UITENBROEK Vol. 536 departure coefficients are close to unity. These two combined e ects make the inñuence of departures from LTE on the line core intensities decrease again for [ 4. When collisional excitation and de-excitation strengths are reduced by 1 order of magnitude eerywhere, the e ect of departures from LTE on the emergent proðles increases. Figure 6 shows that line core intensities in the strongest lines of the medium-excitation \ 2 and 4 bands are reduced by maximally 8% compared with their LTE alues close to the limb (k \ 0.05), while the e ect on disk center proðles is less than 2% in all bands. This deepening of the line cores at the limb would, howeer, still not be large enough in magnitude to explain the di erence between the 3700 and 4400 K deried from CO and UV and Ca II line diagnostics, respectiely, as caused by scattering alone. Een when collisional strengths are reduced by another order of magnitude to 1% of their canonical alues, the maximum reduction in line core intensities is of the order of 8% at k \ 0.23, where typical limb measurements are taken (e.g., see Ayres & Testerman 1981; Ayres & Brault 1990), and 15% at the extreme limb (k \ 0.05, corresponding to about 1A from the limb proper, which would be ery difficult to obsere), still not enough to increase temperatures determined in CO to alues obsered in the UV within the framework of hydrostatic modeling. It follows that scattering in the lines is unlikely to be responsible for the UV-CO discrepancy at the solar limb, unless the estimates used by AW of collisional coupling between the CO ibrational states are o by more than 2 orders of magnitude Hydrostatic Model Intensities Compared with the AT MOS Spectrum Figure 7 shows the disk center CO line core intensities (relatie to the surrounding continuum) computed from model FAL C in four di erent fundamental ibrationrotation bands, compared with line core intensities from the ATMOS spectrum. The lines were calculated while allowing for departures from LTE, using the method described aboe and the standard collisional excitation strengths gien by AW. This Ðgure, in particular its upper left panel with the \ 0 band intensities, clearly demonstrates the incompatibility of aerage quiet-sun model FAL C with obsered CO intensities, een at the center of the disk. The strongest lines in the \ 0, 2 bands all hae much shallower line cores than obsered, while the opposite is true for the weaker lines in those bands. This implies that the temperatures in model C are too high to represent the aeraged CO lines in the higher layers of the atmosphere and too low in the deeper layers. In the \ 8 band (lower right panel) computed intensities fall below obsered alues at all waenumbers because the higher temperatures in the higher layers of model C preferentially populate higher excitation ibration states (by increasing their Boltzmann factors), causing the strongest lines in these ibration states to form farther outward at lower temperatures. Intensities occasionally fall well below the general trend in each of the panels of Figure 7 because of blending of lines. When two or more lines combine their opacities they form higher in the atmosphere at a lower temperature than they would by themseles. Since the computed spectrum has only the fundamental ibration-rotation lines of the 12C16O isotope, this happens more often in the obsered spectrum than in the calculated one. The hydrostatic equilibrium model XCO (Arett 1995) was speciðcally designed to match the ATMOS obserations at disk center. Figure 8 clearly shows the resulting good match between obsered and computed core intensities in most bands, in particular those with lower excitation. The slightly lower temperatures of model XCO in the low photosphere (see Fig. 1) explain the better Ðt of weak lines in this model compared with model FAL C. The calculated spectrum here was not broadened to the instrumental I / I LTE µ = = 0 XCO = I / I LTE = 4 = 8 FIG. 6.ÈRatio of CO line core intensities computed with reduced collisional strength (10% of canonical) oer LTE alues

9 No. 1, 2000 CO LINES IN SOLAR SPECTRUM. II ATMOS FALC = 0 = = 4 = 8 FIG. 7.ÈNon-LTE disk center line core intensities relatie to the continuum for the CO fundamental \ 0, 2, 4, and 8 ibration-rotation bands computed in non-lte from hydrostatic model FAL C (diamonds) compared to relatie line core intensities in the ATMOS spectrum (crosses). resolution, nor was it aeraged oer the range of iewing angles that is appropriate for the ATMOS obserations as was done by Arett, explaining the fact that the calculated proðles are slightly deeper than the obsered ones. Since the temperature structure of the COOLC model closely follows that of model FAL C, it is not surprising that the weak-line intensities shown in Figure 9, which plots core intensities for the COOLC]FLUXT combination, behae similarly to those in Figure 7. The weaker lines in the higher excitation bands are too dark in the calculations by a larger factor than in the lower excitation bands. As in the case of model FAL C this is due to the temperature being too high for these lines, which causes oerpopulation of the higher excitation states just aboe the photosphere ATMOS XCO = 0 = = 4 = 8 FIG. 8.ÈSame as Fig. 7 but for model XCO

10 490 UITENBROEK Vol ATMOS COOLC + FLUXT = 0 = = 4 = 8 FIG. 9.ÈSame as Fig. 7 but for the linear combination of intensities computed from models COOLC (92.5%) and FLUXT (7.5%) 4.3. Dynamic Snapshots We now turn to the calculation of non-lte CO line intensities in a sequence of 20 snapshots from a onedimensional radiation-hydrodynamics simulation of chromospheric dynamics (Carlsson & Stein 1999; see Fig. 2 aboe). Since, in contrast with hydrostatic models, these snapshots hae a nonzero macroscopic elocity Ðeld, the e ort to compute the opacities for all lines increases by a factor of 2 ] N, where N is the number of points in the k k angle discretization (Ðe in this case). To somewhat reduce the computational e ort that would be required by including seeral thousand lines for these dynamic calculations, a smaller set of 294 lines was chosen comprising all fundamental ibration-rotation lines between the Ðrst four ibration leels with 50 rotation leels each. Figure 10 shows the time-aeraged mean core intensities from the 20 simulation snapshots (diamonds) compared with obsered alues (crosses) from the ATMOS experiment, and the computed obsered dynamic aerage XCO 0.65 = 0 = FIG. 10.ÈTemporal mean of fundamental CO line core intensities relatie to the continuum in the \ 0 and 2 bands from snapshots of Carlsson & Stein chromospheric dynamics simulation (diamonds) compared with the ATMOS spectrum (crosses) and calculated XCO intensities (triangles).

11 No. 1, 2000 CO LINES IN SOLAR SPECTRUM. II. 491 intensities from hydrostatic model XCO (triangles). Departures from LTE were allowed for the CO line source functions; howeer, the CO chemical equilibrium was calculated according to Saha equilibrium at the local instantaneous temperatures in the snapshots. The alues for model XCO plotted in Figure 10 were also calculated with the smaller set of 294 fundamental lines employed for the dynamical model. Comparison of Figures 8 and 10 shows that the limited size of the set has little inñuence on the computed line intensities in hydrostatic model XCO; the same will probably hold for the dynamical calculation. The calculated mean intensities from the snapshot sequence drawn in Figure 10 are much higher than the obsered alues, which represent spatial and temporal aerages. In addition, inspection of the indiidual line proðles from the time series shows that the amplitude of calculated brightness ariations in strong CO lines such as the 3È2 R14 line is approximately 1000 K, which is 2.5 times higher than obsered peak-to-peak ariations in that line (Uitenbroek, Noyes, & Rabin 1994; Uitenbroek 2000). Thus, important parts are missing from either the chromospheric simulations or the way CO intensities were calculated from the simulation snapshots. A possible indication for the latter and cause for the discrepancy between obsered and calculated amplitudes can be identiðed by looking at the behaior of the CO formation height in the eoling atmosphere. Figure 11 shows the total source function (CO plus background continuum; heay solid cure) at the core waelength of the strong CO 3È2 R14 line, at four di erent times in the snapshot sequence (t [ t \ 20, 60, 100, and 140 s). 0 The Ðgure also shows the Planck function at the local temperature (dashed cure), the angle-aeraged mean intensity J (dot-dashed cure), and cures for the CO line and background continuum source functions separately (light solid and dotted cures, respectiely). In the main graph the downward pointing arrows indicate the position of optical depths q \ 0.3, 1.0, and 3.0 in the total opacity, and the upward pointing j arrow marks the position of optical depth unity in the continuum alone. The arrows in the main graph thus gie an indication of the location in the atmosphere from where the CO line core intensity and the neighboring continuum predominantly escape. The fact that the line and total source functions start to decouple from the Planck function at a column mass of approximately 0.1 kg m~2, similar to the situation in the hydrostatic models, is an indication that the latter models are a alid proxy to study the e ects of departures from LTE on the CO line proðles. Also in the dynamic case LTE seems to hold mostly for the regions where the CO lines form. In the upper layers of the atmosphere (starting at column masses of 3 ] 10~4 kg m~2) the total source function falls much further below the Planck function because of Thomson scattering o free electrons. During moments at which the lowest temperatures occur in the sequence of snapshots (at t [ t \ 20 s; upper left 0 panel in Fig. 11) the formation height of the CO lines shifts outward to een lower temperatures as the assumption of instantaneous chemical equilibrium allows much more CO to form, raising opacity in the lines. CO is destroyed when a shock passes through, causing line formation to shift inward FIG. 11.ÈSource function of the CO 3È2 R14 line core at four di erent times t [ t \ 20, 60, 100, and 140 s in a hydrodynamic simulation by Carlsson & Stein. Downward pointing arrows indicate the location of total (line plus continuum) 0 optical depths 0.3, 1.0, and 3.0 (left to right). Upward pointing arrow in main graph indicates location of unit optical depth in the continuum. The waelength for which the source functions are drawn is marked by the upward pointing arrow on the calculated spectrum at each instant in the upper right corner of the graphs.

12 492 UITENBROEK Vol. 536 to higher temperatures. This up and down shift of the line formation region with time contributes signiðcantly to the high amplitude in CO line intensity ariations and is due to the strong temperature sensitiity of the CO concentration. By contrast, the formation height of Ca II lines does not shift up and down much because most calcium is in the singly ionized form oer a wide range of temperatures; intensity ariations in the Ca II line wings, therefore, reñect much more the local ariations in temperature. 5. DISCUSSION This paper presents further eidence that the solar ibration-rotation lines of the CO molecule hae source functions close to LTE in the line-forming region, so their intensities map actual temperatures in the solar atmosphere. Calculations of their formation in di erent hydrostatic models of the solar atmosphere show that, unless the current estimates for collisional excitation in these lines are o by more than 2 orders of magnitude, scattering cannot account for the di erence between the low temperatures deried from the CO line core intensities at the limb and the much higher minimum temperatures deried from UV and Ca II resonance line obserations. This result conðrms earlier calculations by AW. The numerical method for the solution of the molecular non-lte radiatie transfer problem described here accounts for the actual intensities in a large number of CO lines (2160 in the cases considered here) and does not require interpolation of mean intensities to obtain radiatie rates as was performed by AW. It is straightforward to extend the method described here to an accelerated " iteration scheme as described in 3.3, so that it does not hae to rely on the possibly slow conergence of plain " iteration, although the latter method works well in solar conditions and was used for the results presented here. This property will make it straightforward to extend non-lte calculations of CO molecular rotation-ibration lines to lower density atmospheres, such as those of giants and supergiants, or to situations where no exponential density stratiðcation is present, such as interstellar or circumstellar clouds. Since the simulations of Carlsson & Stein (1995, 1997) were so successful in explaining the intricate details of the formation of H grains, it seems natural to erify whether 2V they are also compatible with the obsered CO lines and whether they can proide a clue to the obsered discrepancies with temperatures deried from the calcium resonance line and UV line diagnostics. In this paper the non-lte CO line formation problem was soled in a sequence of 20 snapshots from a one-dimensional simulation of chromospheric dynamics by Carlsson & Stein (1999). The resulting intensities in the CO rotation-ibration line cores are substantially too high compared with the obsered spatially aeraged disk center intensities from the ATMOS experiment. This may be because the (Carlsson & Stein 1999) simulations do not include the e ects of radiatie cooling by the CO lines on the dynamics of the atmosphere, which therefore lacks an important source of cooling. In addition, the amplitude of intensity ariations in the CO cores is 2.5 times larger than the amplitude obsered in spatially resoled spectra obtained with the Near-Infrared Magnetograph at Kitt Peak (Uitenbroek et al. 1994; Uitenbroek 2000). By tracing the CO line formation height in the sequence of dynamical snapshots it is clear that the large amplitude of calculated core intensities is in part due to the change in altitude from which line radiation escapes when ariations in temperature allow more or less CO to form. During cool phases the assumption of instantaneous chemical equilibrium that was used to calculate the CO concentration throughout the atmosphere allows more CO to form, causing the line formation region to shift outward to lower temperatures, while the opposite happens in hot phases. Unless the Carlsson & Stein (1999) simulations grossly oerestimate the actual temperature ariations in the atmosphere, which seems unlikely since they reproduce the intricate behaior of the Ca II H and K lines so accurately, the comparison with obsered intensity amplitudes, therefore, suggests that the assumption of instantaneous chemical equilibrium may not be alid. Indeed, preliminary calculations of CO formation and destruction times by Arett et al. (1996) indicate that typical timescales of CO association may be as long as 1 minute at about 100 km aboe the photosphere to hours at 1000 km, while dissociation is essentially instantaneous, i.e., much faster than the dynamical timescales of the atmosphere. If these estimates are realistic, the CO concentration does not hae time to establish its equilibrium concentration during cooler phases after a shock wae passes through the atmosphere and destroys most CO. This has two e ects on the CO concentration: the aerage concentration is lower than would be expected on the basis of chemical equilibrium at the time-aeraged mean temperature of the atmosphere, and ariations in concentration are reduced with respect to chemical equilibrium at the instantaneous temperatures. It is clear that these two e ects would hae a great inñuence on the way physical information is deried from CO intensities. Een though the temperatures measured from CO line core intensities would be directly related to actual temperatures (since the CO line source functions are most likely close to LTE), the location at which these temperatures are measured would be di erent than expected, because of the nonequilibrium CO concentration. A similar conclusion was reached in Paper I by comparing calculated granulation in the CO line cores with the obsered contrast, which is much lower. A possible cause for this discrepancy might be that, because of its slow association rates, CO cannot form quickly enough in the hot upwelling granular Ñows. The lines therefore should form deeper than expected on the basis of instantaneous chemical equilibrium, and this would reduce their contrast and raise line core intensities in the calculations, bringing both closer to obsered alues. It will be interesting to erify whether indeed the Ðnite CO chemical formation rate causes the CO lines to form at di erent heights than the Ca II resonance lines and whether their respectie intensities thus would reñect physical conditions in di erent parts of the atmosphere. Establishing this Ðrmly with accurate CO formation rate calculations would perhaps allow a natural resolution of the current chromospheric temperature discrepancy. In more general terms, the change in location of the CO line formation region with time in the sequence of dynamical snapshots shows that it is a questionable procedure to determine the mean physical conditions in the atmosphere from a temporally aeraged intensity spectrum of an inherently dynamic atmosphere. This is because ariations in temperature and or density in the atmosphere not only a ect the emissiity but also the opacity of the material (in this particular case through the CO concentration) and,