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Improved Dunham coefficients for CO from infrared solar lines of high rotational excitation
JOURNAL OF MOLECULAR SPECTROSCOPY
149,375-390 (1991)
Improved Dunham Coefficients for CO from Infrared Solar Lines of High Rotational Excitation R. FARRENQ AND G. GUELACHVILI Laborutoire
de Physique
Molkulaire et Applications. C.N. R.S.. B&t. 3.50. 91405 Orsup Cede. Fruncr
1C’niversitk Puris-Sud.
A. J. SAUVAL Observatoire
Royal de Be/gique,
ilvenur Circulairr.
3. B-I 180 Bruxek,
Belgium
N. GREVESSE Institut dXrtrophysique.
VniversitP de LiSge, Avenue de Cointr. 5. B-4000 Cointe-LiPge,
Belgium
AND C. Jet Propulsion
B. FARMER’
Laborutory,
Pa.rodena,
Califixnia
91109
About 4500 unblended CO lines have been selected and their wavenumbers accurately measured on high resolution solar spectra obtained from space with the ATMOS Fourier transform spectrometer. Half of these lines are of high rotational excitation energy and have never been observed before in the laboratory. Line positions of the fundamental bands of ‘*CI60 have been measured up to J = 133, those of “Cn’O and of “C “0 up to J = IO3 and 9 I, respectively. The first overtone bands of ‘*CJ60 have been measured up to J = I 10. These new solar CO wavenumbers. with an additional selected set of about 14 000 accurate laboratory measurements, have been simultaneously fitted to the Dunham expression utilizing 10 recently published relations between isotopically invariant parameters V,. The present set of coefficients reproduces all accurate laboratory positions and our solar measurements of high rotational excitation with a standard deviation of about 1Om5cm-’ ( 300kHz). This new set is particularly recommended for all high resolution studies of infrared laboratory and stellar spectra showing CO lines of high J-values. cl 1991 Academic Press. Inc. 1.
INTRODUCTION
Carbon monoxide which is the most stable diatomic molecule is among the best known molecules both in the laboratory and in astrophysics. Numerous investigations have been devoted to its electronic ground state X12 ’ which were largely based on infrared spectra. The available laboratory data extend up to very high vibrational levels (v G 4 1) but remain limited as far as the rotational energy is concerned (the ’ Laboratoire associi aux Universites P. et M. Curie et Paris-Sud. 2 It is with deep sorrow that we report the demise of R. H. Norton on March 22, 199 I. His participation in this research was much appreciated. 375
0022-2852/9 I $3.00 C‘opynght !C I991 by Academic Press. Inc. All rights of reproducuon ,n any form rewvcd
376
FARRENQ
ET AL.
temperature of most laboratory sources is restricted to a few thousand degrees Kelvin). The existing data set only includes about a hundred lines having J larger than 60. All previous determinations of accurate Dunham coefficients were based on experimental vibration-rotation and pure rotation spectra of several isotopic species of CO. In one of the last analyses ( 1)) a total of 35 parameters (25 parameters iJij and 10 parameters A,) were calculated in order to fit all observed data. More recent accurate measurements of ‘*Ci60 laser frequencies (2-4) have also enabled us to determine new Dunham coefficients YiJfor this species (j-.5), which better fit the laser frequencies but do not predict more accurately the high J transitions. Carbon monoxide plays an important role in astrophysics, being one of the most abundant molecules (just after Hz) in most cool astrophysical sources: vibrationrotation spectra, pure rotation lines, and ultraviolet transitions have been detected there so far. Vibration-rotation bands of CO were observed in the Earth’s atmosphere in 1949 (6), in the solar photosphere in 1952 ( 7)) in planetary atmospheres in 1963 (8)) and then in sunspots, in cool stars, in circumstellar envelopes, and in molecular clouds, and more recently in comet Halley (9) and in a supernova (IO) .3 Solid CO has also been detected lately in interstellar grains through its distinctive infrared spectrum (II). In the far infrared and in the microwave regions pure rotation lines of CO are present in interstellar clouds (12), in circumstellar regions, and in planetary atmospheres. In 1953, Goldberg and Mtiller ( 13) derived an improved set of spectroscopic constants from newly detected solar lines of rather high rotational excitation (up to J = 70). Their set, which fitted observed line positions in solar and stellar spectra, was then adopted in spectroscopic compilations (14) and in most astrophysical investigations till the seventies when lines of higher J-values (J =S 94) were first observed in the laboratory ( 15, 16). We also note that CO lines of high rotational excitation (with J up to 100-l 10) have been detected in the past, in ground-based spectra of sunspots (17) and of cool stars ( 18). In solar spectra obtained from balloon-borne spectrometers, such as in a recent investigation (19), lines up to J = 103 were measured. In spite of rather large disagreements which were noted between observed and predicted positions (20, 21) the perturbing presence of strong telluric absorptions ( HzO, CO*, N20, * . *) drastically reduced the number of unblended lines which could be added to the laboratory data and prevented any significant improvement of the spectroscopic constants. The aim of this work is to carry out a new determination of Dunham coefficients for the X ‘Z+ state based on recent infrared solar observations from space obtained by the ATMOS instrument (22). Accurately measured positions of 3700 solar lines of 12C160, i3C 160, 12C“0, and 12C“0 have been included in the data bank of line frequencies; about 2000 lines out of a total of 3700 have never been observed before in the laboratory and correspond to high J-values (up to 133) but to moderate ovalues (up to 20). A plot of the deviation between observed and predicted (using constants of Refs. (1, 3)) line positions against J clearly revealed a gradual increase starting from J around 80-90 (with reverse signs for R and P branches) reaching 0.1 3 See IAU circular 4457 ( 1987).
DUNHAM
Filter
1000
1500
COEFFICIENTS
377
FROM SOLAR CO LINES
2 1
2000 2500 WAVENUMBER
3000 3500 (cm-l )
4000
4500
5
FIG. I. ATMOS spectra obtained with three optical filters. The “wavy” nature of the background, due to the optical transmission of the filters, is not a problem on an expanded frequency scale. The three spectral regions in which CO line positions have been measured are noted by arrows.
cm-’ near J equal to 120. These deviations prompted us to improve the previous sets of spectroscopic constants for the ground electronic state. II. SOLAR DATA
During its first mission aboard Spacelab 3 (April 29-May 6, 1985) the ATMOS (atmospheric trace molecule spectroscopy) instrument also recorded, for the first time, a pure solar spectrum from 600 to 4700 cm-‘, nearly free from any telluric absorption (22) .4 The recorded spectra were obtained in four spectral ranges limited by optical filters. The CO transitions were observed in three regions shown in Fig. 1. The ATMOS equipment is a fast Fourier transform spectrometer yielding an unapodized spectral resolution of 0.0 1 cm-’ . Its circular field-of-view covers a diameter of about one-tenth of the solar disc and the signal-to-noise ratio is around 1000 (see Fig. 1) . In our high resolution and low noise solar spectra, more than 7000 individual lines of four isotopic species of carbon monoxide (“C 160, 13C160, 12C“0, and “C “0) are detected in the spectral ranges 1350-2328 cm-’ (fundamental bands: from 1-O to 20- I9 ) and 34 1O-4360 cm-’ (first overtone bands: from 2-O to 14- 12). Table I shows an overview of the 4546 unblended solar CO lines which were selected, i.e., about half the total number of CO lines, noting that, among them, around 850 lines were measured both in filters 2 and 3 in the overlapping spectral region 1600-2000 cm-‘. Table I also gives the number of bands and of lines of the four isotopic species measured in each filter and the maximum v values. Table II gives some information about the 2000 additional CO lines which were as yet not observed in the laboratory. Most of the solar lines belong to 12C160, the most abundant isotopic molecule: the 4 This is NASA publication 1224 ( 1989). For a copy write to the J.P.L. A few copies are also on deposit in the Editorial office of this journal.
378
FARRENQ
ET AL.
TABLE I Overview of Measured Solar CO Lines ATMOS filter
Spectral range for co spectra (cm-‘)
Number of meaSYrCmC”ts
Number bands (highest
of
2
1350-2000
1711
45 (v’=ZO)
3
1600-2328
1859
32
4
3410-4360
976
Total
12+50
(v’=lB)
] 2679
13 (v’=14)
4546
13~160
Number
v’)
of
1%$80
12~170
mcasurcments
641
243
7
641
243
7
916
90
3655
other species are about 90 times ( 13C160), 500 times (“C 180), and 2500 times ( 1’C’70) less abundant than “C 160 in the sun. Line positions of the fundamental bands of 12C160weremeasuredupto J= 133,andthoseof13C160and12C180upto J= 103 and 9 1, respectively, whereas the first overtone bands of 12C160 were measured up to J = 110. The excitation potential of our solar lines ranges from 0 to 40 000 cm-’ . The strongest lines show a central depth of around 30% and a total width at halfmaximum (FWHM) of about 0.05 cm-’ whereas the faintest lines have an extremely small central depth (less than 0.5%) and a width of 0.03 cm-‘. In spite of their large Doppler width (as compared to laser lines) and of rather large shifts of various origins (see hereafter), solar lines, originating from a rather hot source in local thermodynamic equilibrium, offer the great advantage of showing high rotational lines, far above the usual experimental limit reached in laboratory sources. Figure 2 shows the observed spectral regions obtained by the fourth ATMOS filter around the first overtone bands. Most of the lines in our solar spectra are due to carbon monoxide, the remaining lines being attributed to atoms and also to other molecules like CH, NH, and OH. Telluric lines are also visible. Figure 3 shows the range of J observed in a given u level for the 4 isotopic species which benefit from the new solar observations. It clearly gives account of the important contribution of the new available data. We proceeded as follows for the identification of our solar lines and for the designation of the CO lines. We used a computer program TABLE II New Data of Solar CO Lines at High J-Values
isotopic Observed
bands
species
12cl6(,
lines
Number of different
P lines
R line!
r
nlai
lines a
1-O
to
20-19
1840
1395
809
586
133
2-O
to
14-12
224
224
84
140
110
1-o
to
14-13
363
296
170
126
103
1-O
to
8-7
131
117
63
54
91
1-O
to
3-2
6
6
6
0
52
1 1132
906
I
Total ‘526
Number cf mCas”remc”ts
1
are measured
I 2564
both in filters
2038
2 and 3 in the overlapping
region
1600-2000
cm-’
DUNHAM
COEFFICIENTS
FROM SOLAR CO LINES
379
FIG. 2. Example of ATMOS spectra shown at moderate resolution. Some bandheads of first overtone bands of CO are clearly visible in ATMOS filter 3. The remaining (atomic or molecular) lines are either
telluric or solar.
which accurately determines the positions of all line centers, interpolating our point spacings with a curve of the fifth degree and adopting a method of first differences, in order to calculate the minimum in every absorption line. We applied this program to our observations and also to a synthetic solar spectrum (generated with interim Dunham coefficients from preliminary calculations) including all CO and other identified lines yielding two lists of numbered and designated lines (see Fig. 4). With the help of a graphic terminal, the operator has to search for all CO lines, especially the faintest ones, and then select any couple of observed - calculated lines (line by line), entering the relevant numbers and a weight (which has been attributed to each observed line according to its quality and to its intensity). Our program reproduces to better than -+O.OOOlcm-’ all theoretical line positions: for the strong and medium solar lines the expected accuracy is around a0.0005 cm-’ whereas the position of faint lines is calculated with an accuracy of a few 10e3 cm-’ (reaching kO.0 1 cm-’ for the worst lines). In any case all our measurements of high J isolated CO lines have been included in our set owing to the complete lack of such data in the existing laboratory data. Correction
qf Solar Wavenumbers
All our solar lines are affected by a resultant Doppler shift of about 7 km/set arising from the combined effect of several radial motions: motion of the spacecraft with respect to the sun, rotation of the solar surface at the observed position (near the center of the disc), convective motions in different solar layers (essentially due to the
380
FARRENQET AL.
-1
L
El
2c1*o
'*C"O
25 V 20
lab.
15
n s~fl
10
I
0
20
40
60
0
20
40
60
J
’
r
7
80
100 120 140
0
20
40
60
80
100 120 140
0
20
40
60
80
100 120 140
80
100 120 140
J
FIG.3. Illustration of the large contribution of solar data for the four isotopic species observed. The highest J level which takes part in a transition measured in the laboratory (lab.) and in the solar spectrum (sun) is
shown for each vibrational state v.
granulation), and the Einstein gravitational redshift. If some of these shifts (such as the relative motion of the spacecraft and the gravitational redshift) are known, others are rather badly determined. Moreover, as shown by Dravins et al. (23) for atomic lines, one physical effect, the nonthermal solar motions, does vary with height and thus depends on the line intensity: weak lines are formed deeper than strong lines and are thus more affected by convection (a velocity difference of around 0.4 km/ set does exist between strong and weak CO lines). The measured line positions have to be corrected line by line in order to convert them to rest wavenumbers. Because of these uncertainties we have preferred to derive, for each filter, a correction curve from accurate laboratory frequencies of CO lines covering a large extent in intensity. Accurate positions of “C I60 lines belonging to the fundamental bands,
DUNHAM
I? (1211 1-o
COEFFICIENTS
(601
R
1-o
1 ,cY*,
I
1-o 10878 ?
THEORY
I
I I I I
I I
I I
Y
I
R (611 1-o x)879 I,
I
I
-I
R (1201
1-o
\I'
I 80
12c 160
10877
10876
381
FROM SOLAR CO LINES
I
I 1 I I I
b
80 -
V L
1
2306
1
r
sun
(cm-11
2307
FIG. 4. Example of measurements of CO solar lines with a graphic terminal. From a comparison of observed (bottom) and calculated (top) solar spectra in a small region around 2306 cm-‘. it appears that four ‘2C’60 lines are unblended in our ATMOS spectrum. The positions of all numbered lines in both spectra which have been calculated in our program are easily included in our data set. Other weak lines (probably of atomic origin) in our observations are unidentihed. The wavenumber scale of the observed spectrum has been slightly shifted ( mean correction of the Doppler effect) sothat there is a good coincidence of CO line positions in both spectra.
measured by Brown and Toth (25) and Schneider et al. (3, 4), have been used to calibrate the wavenumber scale of our ATMOS spectra (filter 2: 1350-2000 cm .’ : filter 3: 1600-2330 cm-‘). Similarly accurate line positions of the 2-O band from Pollock et al. (26) have been used for filter 4 ( 3400-4360 cm-’ ). Unfortunately, from a total of 199 laboratory data, there remain only a limited number of useful solar lines which are unblended and well measurable: 30 lines for filter 2, 28 lines for filter 3, and 11 lines for filter 4. That is due to the fact that most of the 167 laser lines of Schneider et ai. (3. 4) are of very high vibrational excitation and so appear too weak to be observed in the solar spectrum. For each filter we plotted the relative observed Doppler shift of every line, ( gsu,, - ulab)/cpabr as a function of the mean height of formation of the line center which was calculated in our program of synthetic solar spectrum. As expected, there is a gradual increase of the blue shift of line centers from strong to weak CO lines, except for filter 4 where all CO lines are formed practically in the same solar layers. A simple linear correction curve is found (see Fig. 5 corresponding to filter 3) and has been applied to all CO lines of every ATMOS filter. The linear correction derived from Fig. 5 could appear rather uncertain: we checked that the scatter of the representative
382
FARRENQ
-2.20
I
ET AL.
I
I
I 1
I
ATMOS
Filter
High
-2.50 _
3
I 12C I60
lines
Fundamental
bands
layers
Deep
I
1
560 Geometrical
I
layers
I
430 300 150 height in the solar atmosphere
(km1
0
FIG. 5. Observed solar Doppler shift, ( osun- ql~))qab. for 28 laboratory lines of ‘2C’60 measured in ATMOS filter 3 as a function of the geometrical height in the solar atmosphere. For each line, the mean height of formation of the center has been calculated in our program: weak lines, which are formed deeper than strong lines, show a larger blue shift. Accurate laboratory line positions from Refs. (3,4) are represented by crosses and from (25) by open circles. The straight line is a rather good fit of the representative points: this linear correction has been applied to all CO lines in filter 3. We checked that the rather large scatter can be much reduced if we plot the best solar lines giving very small residuals in our calculations.
points is much reduced if we plot the best solar lines which have also been measured in the laboratory (27). Let us illustrate the correction for a CO line at 2000 cm-’ for filter 2: -the correction for the spacecraft motion is -0.045 cm-‘, -the combined effect of rotation and convection in the solar layers is a blue shift ranging from 0.002 cm-’ (for the strongest lines) to 0.005 cm-’ (for the faintest lines), -the solar gravitational redshift is -0.00424 cm-‘. The resultant redshift for such a CO line ranges thus from -0.044 to -0.047 cm-’ when going from a weak to a strong line. III. LABORATORY DATA Almost all the data used in our previous determination of Dunham coefficients ( 1) have been included in this work. In addition to these laboratory data, we have included the high accurate measurements on ‘*C I60 lines recently published: -7 microwave transitions (24)) - 12 P lines belonging to the 1-O band, recorded by FTS at Kitt Peak National Observatory ( 25 ) .
DUNHAM
COEFFICIENTS
FROM
SOLAR
CO LINES
383
-20 lines belonging to the 2-O band, measured by heterodyne frequency technique ( 26 ) , -167 laser lines also measured by heterodyne frequency technique ( 2-4), 48 of them sub-Doppler stabilized ( 3). We have also made use of an emission spectrum recorded by FTS at the laboratoire d’brfrarouge. numbered 1578. This spectrum, described elsewhere (27). covers the 1600-5900 cm-’ region. It shows fundamental, first, and second harmonic bands of “C”O and 13CI’%, up to V = 40. Almost 4200 lines, 1100 of which belong to the AV = 3 bands, were measured. The total resulting amount of laboratory data is of the order of 14 000. Recalibration c$ FT Spectra The previous work (1) was performed on homogeneous data. Most of the spectra had been recorded with our interferometer, or calibrated by reference to the same standard wavenumbers. The data obtained by Schneider and co-workers, Nolt et al., Brown and Toth, and Pollock et al. (2-4, 24-26) change this situation. They all agree perfectly with each other but show a systematic shift, of the order of 12 MHz, with respect to our FT measurements. In the least-squares calculation, these high-weighted data behave as standards for “C’60 lines, and then induce a relative distortion with regard to other isotopic species. The source of the systematic errors have been identified (28) though it is only partly understood. These errors are not linear vs frequency, but we are not able, at this time. to carry out a rigorous correction. Then. we have decided to make the new calibration of all our laboratory spectra by the usual, linear, procedure in FTS, and to overvalue the uncertainties by 9 MHz. The standard wavenumbers are those of Refs. (2-4. -IS) for fundamental bands. and those of Ref. (26) for harmonic bands, The total effect is to lower the wavenumbers by about 4 X lop4 cm-‘. IV. NUMERICAL
TREATMENT
Though a fully quantum-mechanical procedure was recently carried out (8 ), the diatomic molecular line positions are, in general, calculated from the Dunham representation. In this semiclassical frame, several methods have been accounted for ( 30). Wavenumbers can be directly fitted to the coefficients of the potential energy function, or to the expression of the energy levels:
(2) where the notations are identical to those given in Ref. ( I), and -1 = M,’ + MG’. CL The numerical values are in Table III.
384
FARRENQ
ET AL.
TABLE III Masses (in a.m.u.) and Velocity of Light (in m/set)
0.548580228
ma x lo3 M(12C) M(13C) M(14C) M(160) M(“0) M(‘*O) C
12.00000000 13.00335440 14.00324197 15.99491502 16.99913290 17.99916002 299 792 458
These two methods are equivalent if we restrict the independent parameters of Eq. (2) to the { IJi.0, Vi,, } and { A$, A;} groups. The other parameters, U,,j k 2, can be obtained analytically from the first group (31, 32). Tyuterev and Velichko (32) have published 10 relations, forj 2 2, i + j =G5, of the form Ulj = Uij( UWl.1, Um,o).
(3)
Each U, is related to the first (2i + j) terms of the series obtained from the two where m takes all the integer values. For instance, elements IC.J~_~,~, Um,O, u13
=
Ul3(UOl,
UlO,
Ul,,
u20,
U21)
(three transcription mistakes found in their paper are reported in appendix). Indeed, there was no evidence that these relations, though they are exact, do not lead to distortions in the data reproduction, calculated from Eqs. ( 1) and (2) which are inevitably truncated (31). Then, we have made our computer program suitable for this last method, so that Eqs. (3) could be introduced one by one, in the increasing order of (2i + j) values. Thus, we are able to control the influence of each constraint on the quality of the computed frequencies. V. RESULTS
Starting from the set of constants obtained in Ref. (I), preliminary fits, where all the U, parameters are freely adjusted, have shown that terms of order (i + j) = 4, 5, and 6 were required to take solar data into account. The relation (3) between the Uv has been introduced in the fit without any loss of precision, as soon as two mass scaling parameters A$z and A% were added. Though they are not well defined, they are necessary to balance the constraint on the centrifugal distortion constant Uo2. Due to the lack of published relations for parameters of the order (i + j) = 6, Ud2, U24, Uls, and Uo6 have been determined as independent parameters. The final weighted nonlinear least-squares calculation required 33 independent pa-
DUNHAM
COEFFICIENTS
FROM SOLAR CO LINES
385
TABLE IV Isotopically Invariant Dunham Parameters Free Dunham lJ(l,O)
UC&O1
Parameters
5681.3676166 (3535)
A'(l,O)
0.700930 (1131)
-91.1055149 (1287)
A'(l,O,
-0.171324 (1586)
U(3,O)
0.18703287 13326)
A'12.0)
0.42815 (1384)
U(4,O) xld
0.3260748 (9465)
A"(2,0)
-0.91035 (36651
U(5,O) x104
0.20671 (1700)
AcC3,0,
-12.2868 (3442)
'j(6,O)x106
0.6637 (1842)
A"(3,0J
-3.534 (2091)
U(7,O) x106
-0.71632 (1185)
A'(O,l,
-2.057208 (6005)
A'(O,l,
-2.119921 (8720) -1.2452 (13361
U(8,O) x10'
0.273954 (4155)
U(9,O) x109
-0.476404 (61181
A'Ll,
U(O,l)
13.243473898 (4559)
A"oll,l,
-3.0206 (1777)
U&l)
-0.3142988561 (7182)
Ac(0,2,
-7.353 (1530)
U(2,l) x104 LJ(3,l)x105 U(4,l) xl05 U(5,l) x106 U(6,l) x10* U(7,l) x109
0.337229 (3179) -0.26419 (1582) 0.142950 (3586) -0.114861 (4071) 0.27528 (2264) -0.122960 (4886)
A"(0,2)
1.350 (2032)
U(4,2) x10' U(2,4) xl016 U(1,5) xl018 IJ(O,6)x1021
ConstrainedDunham U(O,2) x103 U(1,2) x106 Uv.,2) x10' U(3,2) x10' U(O,3) xl08
-0.2878458471 130001 0.1273849 (1242) -0.5961686 (8060) 0.205163 (5224) 0.189666901 (2861)
-0.230581 (3473) -0.7658 (14181 -0.23494 (1094) -0.15785 (1251)
Parameters
U(1,3) x109 U(2,3) x10" U(O,4) XlOB U(1,4) xl014 U(O,5) x1018
-0.12056492 (45261 -0.270763 (5899) -0.79898533 (24681 -0.4284994 (75191 -0.6901397 (4704)
~ The UCi,j) constants are in units of ~K~x(a.rn.u.1~'~ +? . The Acco(i,j) are dirwnsionless. The standard deviation is given in parentheses in alit of the last digit.
rameters (2 1 U, and 12 A,) and 10 calculated Vi, parameters. They are reported in Table IV. By means of Eq. (2), we can deduce a group of 3 1 Yij constants for each isotopic species. The constants relating to the four main species are listed in Table V. The statistical parameters of the fit are: -Root-mean-squares
of the residuals (RMS)
yo;4y2 -root-mean-squares
cm-‘,
of the weighted residuals (standard deviation )
c w(0 i
-estimated
= 8.6 X IO-’
-
c M’
cy
“7
= 9.7 X 10-O cm-‘,
1
variance c2
c No [
N-P
-
cl2 = 1
o.25
386
FARRENQ ET AL. TABLE V Dunham Coefficients Yti(and Their Standard Errors) for Four Isotopic Species of CO (in cm-’ ) Cl2016
Y(l,O)
0.216981267OD+O4 (O.l33D-04)
Y(2,0)-O.l328787634D+O2 (0.7740-05)
Cl2017
Cl2018
Cl3016
0.2142164929D+04 (O.l46D-04)
0.2117397251D+O4 (O.l79D-04)
0.2121439026D+04 (O.l62D-04)
-O.l295142054D+02 (O-7551)-05)
-O.l265367652D+02 (0.748D-05)
-O.l270204643D+OZ (0.747D-05)
Y(3,O)
O.l041106647D-01 (O.l72D-05)
O.l001820399D-01 (O.l65D-05)
0.9674767670D-02 (O.l59D-05)
0.9730658140D-02 (O.l60D-05)
Y(4,O)
0.6936640756D-04 (0.201D-06)
0.6589785167D-04 (O.l91D-06)
0.6290258358D-04 (O.l82D-06)
0.6338503741D-04 (O.l84D-06)
Y(5,O)
O.l679352306D-06 (O.l38D-07)
O.l575050132D-06 (O.l30D-07)
0.14860756401)-06 (O.l22D-07)
O.l500336759D-06 (O.l23D-07)
Y(6,O)
0.2059251576D-08 (0.572D-09)
0.19067444320-08 (0.529D-09)
O.l778231615D-08 (0.494D-09)
O.l798728967D-08 (0.499D-09)
-0.7759369245D-09 (0.128D-10)
-0.7152725631D-09 (O.l18D-10)
-0.7249007258D-09 (O.l20D-10)
O.l118886245D-10 (O.l69D-12)
0.10194839410-10 (O.l54D-12)
O.l035182507D-10 (O.l57D-12)
-0.7336207812D-13 (0.941D-15)
-0.6607168549D-13 (0.847D-15)
-0.6721736565D-13 (0.862D-15)
0.1882384581DtOl (0.349D-07)
O.l839113877D+Ol (0.607~-07)
O.l846151725D+Ol (0,39OD-07)
Y(7,0)-O.E488145707D-09 (O.l40D-10) Y(8,O)
O.l239772013D-10 (O.l88D-12)
Y(9,0)-O.S233737278D-13 (O.l06D-14) Y(O,l)
0.1931280985DtOl (O.l78D-07)
Y(l,l)-O.l750439229D-01 (O.l30D-07) Y(2,l)
0.7173917007D-06 (0.676D-08)
Y(3,1)-0.21463545860-07 (O.l29D-08) Y(4,l)
0.4435403909D-08 (O.lllD-09)
Y(5,1)-O.l36106945OD-09 (0.482D-11)
-O.l684384498D-01 (O.l36D-07) 0.6815196800D-06 (0.643D-08) -0.2013047567D-07 (O.l21D-08)
-0.16266402281)-01 (O.l59D-07) 0.6505424311D-06 (0.613D-08) -0.1899330625[3-07 (O.l15D-08)
-O.l635976671D-01 (O.l42D-07) 0.6555319987D-06 (0.618D-08) -O.l917557543D-07 (O.l16D-08)
0.4106920109D-08 (O.l03D-09)
0.3830117481D-08 (0.959D-10)
0.3874266547D-08 (0.970D-10)
-O.l244210549D-09 (0.440D-11)
-O.l146935582D-09 (0.406D-11)
-O.l162374288D-09 (0.411D-11)
where o and c are observed and calculated values (cm-‘), w = (6~))’ is the weight related to the experimental uncertainty 6a (cm-‘) of the measurement, N = 18 523 is the number of spectral data, and p = 33 is the number of free parameters. The correlation coefficient and dispersion matrices cannot be given here. They may be obtained from the authors. Figure 3 helps to visualize the range of u and J for which calculated wavenumbers can be obtained with the accuracy reported here. VI. DISCUSSION
The quality of this new set of constants has been tested for its ability to reproduce the high J solar lines as well as the new accurate laboratory measurements mentioned in Section III (2-4, 24-26). For the first set (solar data), no significant shifts were
DUNHAM
COEFFICIENTS
FROM
SOLAR
387
CO LINES
TABLE V-Continued Cl2016
Cl2017
Cl2018
Cl3016
O.l124313572D-11 (0.923D-13)
O.l024429101D-11 (0.8410-13)
O.l040203815D-11 (0.854D-13)
Y(7,1)-O.Z125123415D-13 (0.843D-15)
-O.l893471515D-13 (0.751D-15)
-O.l705306851D-13 (0.677D-15)
-O.l734876798D-13 (0.688D-15)
Y(O,Z)-0.6121615183D-05 (0.89213-11)
-0_581549?725D-05 (0.265D-10)
-0.5551151313D-05 (0.449D-10)
-0_5593901826D-05 (0.299D-10)
0.97064536560-09 (0.657D-12)
0.9158136642D-09 (0.620D-12)
0.9246022661D-09 (0.626D-12)
-O.l712774161D-09 (O.l53D-12)
-O.l597334762D-09 (0.142D-12)
-O.l615746949D-09 (O.l44D-12)
0.2222380202D-11 (0.356D-13)
0.2048629898D-11 (0.328D-13)
0.20762061580-11 (0.332D-13)
-0.9417416975D-13 (O.l42D-14)
-0.8580769858D-13 (O.l29D-14)
-0.8712901196D-13 (O.l31D-14)
0.5449071903D-11 (0.716D-16)
0.5081809481D-11 (0.668D-16)
0.5140386572D-11 (0.676~-16)
Y(l,3)-0.14286532770-12 (0.519D-16)
-O.l305991754D-12 (0.475D-16)
-O.l203886604D-12 (0.4380-16)
-O.l220091918D-12 (0.443D-16)
Y(2,3)-0.12253284990-14 (O.l85D-16)
-O.l105851067D-14 (O.l67D-16)
-O.l007606812D-14 (O.l52D-16)
-O.l023122487D-14 (O.l54D-16)
Y(O,4)-0.36157857450-16 (0.855D-21)
-0.3263223311D-16 (0.772D-21)
-0.2973317238D-16 (0.703D-21)
-0.3019101986D-16 (0.714D-21)
Y(1,4)-0.7405801298D-18 (0.9381)-21)
-0.6598522094D-18 (0.836D-21)
-0.5942790712D-18 (0.7530-21)
-0.6045838445D-18 (0.766D-21)
Y(2,4)-0.5054376594D-20 (0.935D-21)
-0.4446033496D-20 (0.822D-21)
-0.3957908937F20 (0.732b21)
-0.4034237544D-20 (0.746D-21)
Y(O,5)-0.4555298526D-22 (0.297P25)
-0.40070243001)-22 (0.261D-25)
-0.3567098022D-22 (0.232D-25)
-0.3635889807D-22 (0.237D-25)
Y(1,5)-O.S922257515D-23 (0.275D-24)
-0.5143075905D-23 (0.239D-24)
-0.4525486860D-23 (0.210D-24)
-0.4621580733D-23 (0.215D-24)
Y(O,6)-O.l519641502D-26 (0.12013-27)
-O.l302888664D-26 (O.l03D-27)
-O.l133180223D-26 (0.897D-28)
-O.l159454710D-26 (0.917D-28)
Y(6,l)
Y(1,2)
0.12457857151)-11 (0.1020-12)
O.l034922952D-08 (0.700D-12)
Y(2,2)-O.l84976698lD-09 (O.l65D-12) Y(3,2)
0.24311108770-11 (0.389D-13)
Y(4,2)-O.l043488564D-12 (O.l57D-14) Y(O,3)
0.5884905033D-11 (0.774D-16)
found taking the estimated uncertainties into account (from 1 to 5.10 _’ cm ’ for most of the lines). For the second set, only 17 measurements out of 2 18 are not calculated within the limits of their uncertainty, but all of the residuals are lower than twice this uncertainty. In order to give synthetic but more detailed information, all the data has been classified into five groups in Table VI. Laboratory measurements are distributed into the first four groups according to their experimental accuracy. The fifth one is devoted to new solar measurements. These five groups are divided into subgroups related to the spectral range of the observation. For each subgroup, the isotopic species. the number of data, and their relative weights (in parts of 10 000) in the global fit are shown. The mean experimental accuracy ( 1 6a/n) is in column 6. In column 7. for each subgroup. the mean deviation ( C (o - c)/n) shows possible systematic shifts between
388
FARRENQ ET AL.
observed and calculated values; such a shift appears in the results of the first row of the second group (+ 1.2 X 10P5 cm-’ or 360 kHz), though it is lower than the experimental accuracy. The root-mean-squares of the residuals are given in column 8. The overall RMS appears rather poor (0.8 X 10-3 cm-’ ). This is clearly due to the less accurate measurements involved in the fit. Nevertheless, in order to obtain the most reliable set of spectroscopic constants, it seemed important to us to not, a posteriori, sort out any data. The standard deviation, in column 9, is a more significant statistical parameter to show the quality of the fit. Very recently, Zink et al. (33) have reported new precise measurements on the rotational spectrum of 13C160, after our results were obtained. As an illustration of the ability of our set of constants to predict unobserved lines, Table VII shows the excellent agreement between the frequencies measured with their tunable far infrared spectrometer (the most accurate among their measurements) and our calculated values. A brief comparison with our previous results ( I ) shows the real progress performed, essentially owing to the new solar and laboratory data: about twice as much data are reproduced with a better standard deviation which has been decreased from 1.3 X 10M5 to 0.97 X 10m5cm-‘, though the number of free parameters has decreased from 35 to 33. The U, constants calculated from the set { U,,O , U,,- ,, 1} ( 1 =Gm G 4)) are perfectly integrated in the calculation. This guarantees the physical significance of this set with respect to the determination of the potential energy function. In this semiclassical frame, a new improvement could be obtained if measurements of other isotopic species as accurate as those on 12C160 were performed and if all the remaining systematic errors in FTS data could be removed.
TABLE VI Details on the Different Data Sets (All the Values Are in 10e3 cm-‘)
-
Accuracy 60
AV
Isotopic
species
I
Number of data
Relative weight
r _PRF_.---.SENT
Meal’ 1 Mean eviation ccuracy
RESU -7 .TS :eferences 1 R.M.S. Standard jeviation residua, Is 4
I
sa < 0.01 L A B 0 R A T 0 R Y
80 < 0.01 I,
0.010 0.004
la, 24 _.....___.. 2, 3. 4, 25 26b
0.56 0.82 0.90 __.___. 5.0 2.2 3.8
-0.014 -0.012 +0.151 __._.___. +0.19 +0.23 -0.35
0.16 0.39 0.34 . . .._. 2.6 1.7 1.6
0.13 0.31 0.33
la, 27 la, 26b, 27 la, 27
2.6 0.9 1.0
la la 1”
1.1 2.0
0.9 1.5
This work This work
0.86
0.0097
9700
0.002
179 11
100 190
7024 4447 1079
11 2.4 0.3 _ _ 0.002 0.07 0.01
O.lI I, I,
2 3
s u N -
0.003 _._._. 0.037 0.008 . ..___
0.0005
0.069 0.013 __.....
+0.0003 ____..... +0.012 +0.0032 __.._....
19 __..._____
‘w60,‘w0,‘3c’60 1
‘3C’%
I
I
191 5.50 477
I
&>I
0.34 0.05
I,
W LABORATI
AND SOLAR DATA
1 18523
10000
B Data previously used and quoted in ref. (1) b Lines measured by Pollock et al. (26) have been distributed
1.8 2.3
+0.07 a.25
among the second and third groups
DUNHAM
COEFFICIENTS
FROM
TABLE Rotational
Transition
SOLAR
CO
Frequencies
of “CL60
6-5
661
067.276
0 050
0.00
-0 02
1 211
329.636
0.050
-0.03
-0.06
15-14
1 650
767.344
0.055
0 03
0.01
19-18
2 089
240.033
0.055
-0 06
-0.08
25-24
2 744
579.059
0.060
-0 02
-0 03
26-25
2 853
474.444
0.060
0.04
0.05
28-27
3 070
948.140
0.070
0.05
30-29
3 287
972.525
0.100
a From
Zink et al. (33). constants of Table
389
VII
11-10
b From
LINES
-0
10
0 07 -0 06
IV
VII. CONCLUSION
Accurate wavenumbers of about 4500 solar lines have been measured for the fundamental and the first overtone bands of “C 160, 13C160, “C 180, and “C “0; half of these lines have never been observed before in the laboratory, being of high rotational excitation (up to J = 133 ) . These new solar data were fitted to the Dunham formula, simultaneously taking into account 14 000 accurate laboratory measurements and adopting 10 specific relations between the U,, parameters. A new set of 3 1 Y, constants has been calculated. This new set of Dunham coefficients reproduces perfectly all laboratory data (with an accuracy better than 50.0005 cm-’ ) as well as the new solar lines of high J-values (accuracy generally better than kO.002 cm-’ for the fundamental bands and kO.004 cm-’ for the first overtone bands). For only about 300 very faint solar lines out of a total of 4500 the calculated and measured wavenumbers differ by twice or more the quoted deviations: these solar lines are of bad quality and very difficult to measure. The present set also reproduces much better than all previous observed CO spectra in sunspots ( 17) and in stars ( 28) and is especially recommended for generating CO synthetic spectra at relatively high temperature. We hope that our improved set of constants partly based on solar lines will play in future astrophysical applications of CO the same role as that played by the spectroscopic constants derived by Goldberg and Mtiller ( 13) 40 years ago. APPENDIX
In Eqs. ( 10) of Ref. (31), one must read: U23 = (* * ~-12U:ou:‘u~(‘+ u’3=
instead of ( * * * -12ou:ou:,
* - *> {..-}
u,, = {* - a+21 312UloU”U&
u,, + * * f ‘i
instead of U,, = { * - . > + - - -}
insteadof
(. - .+21 312UloU~, + - -. )
ACKNOWLEDGMENTS We are grateful to R. H. Tipping, L. R. Brown, and M. Morillon-Chapey for fruitful discussions and comments, and to V. G. Tyuterev who has corrected the mistakes in Ref. (32). All measurements of solar
FARRENQ
390
ET AL.
lines were performed at the Observatoire Royal de Belgique (Brussels) using a Univac 1100 computer and a Tektronix 4 I 14 terminal, and the fittings of CO frequencies were made at Orsay, utilizing a H.P. 9000 / 350 computer. We thank W. Nijs (Brussels) and J. Collet (Orsay) for their help in programs involved in this investigation. RECEIVED:
April 22, 199 1 REFERENCES
1. G. GUELACHVILI, D. DE VILLENEUVE,R. FARRENQ, W. URBAN, AND J. VERGES, J, Mol. Spectrosc.
98,64-79 (1983). 2. J. S. WELLS, A. HINZ, AND A. G. MAKI, J. Mol. Spectrosc. 114, 84-96 ( 1985). 3. M. SCHNEIDER, K. M. EVENSON, M. D. VANEK, D. A. JENNINGS,J. S. WELLS, A. STAHN, AND W. URBAN, J. Mol. Spectrosc. 135, 197-206 ( 1989). 4. M. SCHNEIDER,J. S. WELLS, AND A. G. MAKI, J. Mol. Spectrosc. 139, 432-438 ( 1990). 5. A. LE FLOCH, Mol. Phys. 72, 133-144 (1991). 6. M. MIGEOTTE, Phys. Rev. 75, 1108-l 109 ( 1949). 7. L. GOLDBERG, R. R. MCMATH, 0. C. MOHLER, AND A. K. PIERCE, Phjs. Rev. 85, 140, 481-482 (1952). 8. W. M. SINTON, J. Quant. Spectrosc. Radial. Transfer 3, 551-558 ( 1963). 9. M. COMBES, V. 1. MOROZ, J. CROVISIER,T. ENCRENAZ, J.-P. BIBRING,A. V. GRIGORIEV, N. F. SANKO, N. CORON, J. F. CRIFO, R. GISPERT, D. BOCKELEE-MORVAN, Yu. POLSKY, C. EMERICH, J. M. LAMARRE, AND F. ROCARD, Icarus
V. NIKOLSKY, V. A. KRASNO-
76,404-436 ( 1988).
10. R. CATCHP~LE AND I. GLOSS. 11. W. HAGEN, L. J. ALLAMANDOLA, AND J. M. GREENBERG,Astron. Astrophys. 86, L3-L6 (1980). 12. R. W. WILSON, K. B. JEFFERTS,AND A. A. PENZIAS,Astrophys. J. 161, L43-L44 ( 1970). 13. L. GOLDBERG AND E. A. MOLLER, Astrophys. J. 118, 397-411 ( 1953). 14. W. S. BENEDICT,R. HERMAN, G. E. MOORE, AND S. SILVERMAN,Astrophys. J. 135.277-297 ( 1962). 15. J. M. WEINBERG, E. S. FISHBURNE,AND K. NARAHARI RAO, J. Mol. Spectrosc. 18, 428-442 ( 1965). 16. A. W. MANTZ AND J. P. MAILLARD, J. Mol. Specfrosc. 53,466-478 ( 1974). 17. D. N. B. HALL, “An Atlas of Infrared Spectra of the Solar Photosphere and of Sunspot Umbrae in the
Spectral Intervals 4040-5095, 5550-6700 and 7400-8790 cm-‘,” Kitt Peak National Observatory, 1973. 18. J. P. MAILLARD, in “Highlights of Astronomy,” G. Contopoulos, Ed.), Vol. 3, pp. 269-284, Reidel, Dordrecht, 1974. 19. M. J. HARRIS, D. L. LAMBERT, AND A. GOLDMAN, Mon. Not. R. Astron. Sot. 224,237-255 ( 1987). 20. D. N. B. HALL, Astrophys. J. 182, 977-982 (1973). 21. A. W. MANTZ, J. P. MAILLARD, W. B. ROH, AND K. NARAHARI RAO, J. Mol. Spectrosc. 57, 155-159 (1975). 22. C. B. FARMER AND R. H. NORTON, “A High Resolution Atlas of the Infrared Spectrum of the Sun and
the Earth Atmosphere from Space. A Compilation of ATMOS Spectra of the Region from 650 to 4800 cm-’ (2.3 to 16 pm). Volume I: The Sun.” NASA, R.P. 1224 (1989). DRAVINS, L. LINDEGREN,AND A. NORDLUND, Astron. Astrophys. 96, 345-364 ( 198 I ). 24. I. G. NOLT, J. V. RADOSTITZ, G. DI LONARDO, K. M. EVENSON, D. A. JENNINGS,K. R. LEOPOLD, M. D. VANEK, L. R. ZINK, A. HINZ, AND K. W. CHANCE, J. Mol. Specirosc. 125,274-287 ( 1987). 25. L. R. BROWN AND R. A. TOTH, J. 0p1. Sot. Am. B 2,842-856 ( 1985). 26. C. R. POLLOCK, F. R. PETERSEN,D. A. JENNINGS,J. S. WELLS, AND A. G. MAKI, J. Mol. Specwosc.
23. D.
99,357-368 (1983). 27. R. FARRENQ, C. ROSSETTI,G. GUELACHVILI,AND W. URBAN, Chem. fhvs. 92,389-399 (1985). 28. Q. L. KOU, Thesis 3442, Universite de Paris-Sud, Orsay. 1990. 29. J. A. COXON AND P. G. HAJIGEORGIOU,J. Mol. Spectrosc. 139,84-106 ( 1990). 30. J. F. OGILVIE. Spectrochim. Acfa Part A 46, 43-46 ( 1990). 31. J. K. WATSON, J. Mol. Spectrosc. 80, 411-42 1 ( 1980). 32. V. G. TYUTEREV AND T. J. VELICHKO, Chem. Phy.p. Letf. 104,596-604 (1984). 33. L. R. ZINK, P. DE NATALE, F. S. PAVONE, M. PREVEDELLI,K. M. EVENSON, AND M. INGUSCIO, J. Mol. Spectrosc. 143, 304-310 ( 1990).
149,375-390 (1991)
Improved Dunham Coefficients for CO from Infrared Solar Lines of High Rotational Excitation R. FARRENQ AND G. GUELACHVILI Laborutoire
de Physique
Molkulaire et Applications. C.N. R.S.. B&t. 3.50. 91405 Orsup Cede. Fruncr
1C’niversitk Puris-Sud.
A. J. SAUVAL Observatoire
Royal de Be/gique,
ilvenur Circulairr.
3. B-I 180 Bruxek,
Belgium
N. GREVESSE Institut dXrtrophysique.
VniversitP de LiSge, Avenue de Cointr. 5. B-4000 Cointe-LiPge,
Belgium
AND C. Jet Propulsion
B. FARMER’
Laborutory,
Pa.rodena,
Califixnia
91109
About 4500 unblended CO lines have been selected and their wavenumbers accurately measured on high resolution solar spectra obtained from space with the ATMOS Fourier transform spectrometer. Half of these lines are of high rotational excitation energy and have never been observed before in the laboratory. Line positions of the fundamental bands of ‘*CI60 have been measured up to J = 133, those of “Cn’O and of “C “0 up to J = IO3 and 9 I, respectively. The first overtone bands of ‘*CJ60 have been measured up to J = I 10. These new solar CO wavenumbers. with an additional selected set of about 14 000 accurate laboratory measurements, have been simultaneously fitted to the Dunham expression utilizing 10 recently published relations between isotopically invariant parameters V,. The present set of coefficients reproduces all accurate laboratory positions and our solar measurements of high rotational excitation with a standard deviation of about 1Om5cm-’ ( 300kHz). This new set is particularly recommended for all high resolution studies of infrared laboratory and stellar spectra showing CO lines of high J-values. cl 1991 Academic Press. Inc. 1.
INTRODUCTION
Carbon monoxide which is the most stable diatomic molecule is among the best known molecules both in the laboratory and in astrophysics. Numerous investigations have been devoted to its electronic ground state X12 ’ which were largely based on infrared spectra. The available laboratory data extend up to very high vibrational levels (v G 4 1) but remain limited as far as the rotational energy is concerned (the ’ Laboratoire associi aux Universites P. et M. Curie et Paris-Sud. 2 It is with deep sorrow that we report the demise of R. H. Norton on March 22, 199 I. His participation in this research was much appreciated. 375
0022-2852/9 I $3.00 C‘opynght !C I991 by Academic Press. Inc. All rights of reproducuon ,n any form rewvcd
376
FARRENQ
ET AL.
temperature of most laboratory sources is restricted to a few thousand degrees Kelvin). The existing data set only includes about a hundred lines having J larger than 60. All previous determinations of accurate Dunham coefficients were based on experimental vibration-rotation and pure rotation spectra of several isotopic species of CO. In one of the last analyses ( 1)) a total of 35 parameters (25 parameters iJij and 10 parameters A,) were calculated in order to fit all observed data. More recent accurate measurements of ‘*Ci60 laser frequencies (2-4) have also enabled us to determine new Dunham coefficients YiJfor this species (j-.5), which better fit the laser frequencies but do not predict more accurately the high J transitions. Carbon monoxide plays an important role in astrophysics, being one of the most abundant molecules (just after Hz) in most cool astrophysical sources: vibrationrotation spectra, pure rotation lines, and ultraviolet transitions have been detected there so far. Vibration-rotation bands of CO were observed in the Earth’s atmosphere in 1949 (6), in the solar photosphere in 1952 ( 7)) in planetary atmospheres in 1963 (8)) and then in sunspots, in cool stars, in circumstellar envelopes, and in molecular clouds, and more recently in comet Halley (9) and in a supernova (IO) .3 Solid CO has also been detected lately in interstellar grains through its distinctive infrared spectrum (II). In the far infrared and in the microwave regions pure rotation lines of CO are present in interstellar clouds (12), in circumstellar regions, and in planetary atmospheres. In 1953, Goldberg and Mtiller ( 13) derived an improved set of spectroscopic constants from newly detected solar lines of rather high rotational excitation (up to J = 70). Their set, which fitted observed line positions in solar and stellar spectra, was then adopted in spectroscopic compilations (14) and in most astrophysical investigations till the seventies when lines of higher J-values (J =S 94) were first observed in the laboratory ( 15, 16). We also note that CO lines of high rotational excitation (with J up to 100-l 10) have been detected in the past, in ground-based spectra of sunspots (17) and of cool stars ( 18). In solar spectra obtained from balloon-borne spectrometers, such as in a recent investigation (19), lines up to J = 103 were measured. In spite of rather large disagreements which were noted between observed and predicted positions (20, 21) the perturbing presence of strong telluric absorptions ( HzO, CO*, N20, * . *) drastically reduced the number of unblended lines which could be added to the laboratory data and prevented any significant improvement of the spectroscopic constants. The aim of this work is to carry out a new determination of Dunham coefficients for the X ‘Z+ state based on recent infrared solar observations from space obtained by the ATMOS instrument (22). Accurately measured positions of 3700 solar lines of 12C160, i3C 160, 12C“0, and 12C“0 have been included in the data bank of line frequencies; about 2000 lines out of a total of 3700 have never been observed before in the laboratory and correspond to high J-values (up to 133) but to moderate ovalues (up to 20). A plot of the deviation between observed and predicted (using constants of Refs. (1, 3)) line positions against J clearly revealed a gradual increase starting from J around 80-90 (with reverse signs for R and P branches) reaching 0.1 3 See IAU circular 4457 ( 1987).
DUNHAM
Filter
1000
1500
COEFFICIENTS
377
FROM SOLAR CO LINES
2 1
2000 2500 WAVENUMBER
3000 3500 (cm-l )
4000
4500
5
FIG. I. ATMOS spectra obtained with three optical filters. The “wavy” nature of the background, due to the optical transmission of the filters, is not a problem on an expanded frequency scale. The three spectral regions in which CO line positions have been measured are noted by arrows.
cm-’ near J equal to 120. These deviations prompted us to improve the previous sets of spectroscopic constants for the ground electronic state. II. SOLAR DATA
During its first mission aboard Spacelab 3 (April 29-May 6, 1985) the ATMOS (atmospheric trace molecule spectroscopy) instrument also recorded, for the first time, a pure solar spectrum from 600 to 4700 cm-‘, nearly free from any telluric absorption (22) .4 The recorded spectra were obtained in four spectral ranges limited by optical filters. The CO transitions were observed in three regions shown in Fig. 1. The ATMOS equipment is a fast Fourier transform spectrometer yielding an unapodized spectral resolution of 0.0 1 cm-’ . Its circular field-of-view covers a diameter of about one-tenth of the solar disc and the signal-to-noise ratio is around 1000 (see Fig. 1) . In our high resolution and low noise solar spectra, more than 7000 individual lines of four isotopic species of carbon monoxide (“C 160, 13C160, 12C“0, and “C “0) are detected in the spectral ranges 1350-2328 cm-’ (fundamental bands: from 1-O to 20- I9 ) and 34 1O-4360 cm-’ (first overtone bands: from 2-O to 14- 12). Table I shows an overview of the 4546 unblended solar CO lines which were selected, i.e., about half the total number of CO lines, noting that, among them, around 850 lines were measured both in filters 2 and 3 in the overlapping spectral region 1600-2000 cm-‘. Table I also gives the number of bands and of lines of the four isotopic species measured in each filter and the maximum v values. Table II gives some information about the 2000 additional CO lines which were as yet not observed in the laboratory. Most of the solar lines belong to 12C160, the most abundant isotopic molecule: the 4 This is NASA publication 1224 ( 1989). For a copy write to the J.P.L. A few copies are also on deposit in the Editorial office of this journal.
378
FARRENQ
ET AL.
TABLE I Overview of Measured Solar CO Lines ATMOS filter
Spectral range for co spectra (cm-‘)
Number of meaSYrCmC”ts
Number bands (highest
of
2
1350-2000
1711
45 (v’=ZO)
3
1600-2328
1859
32
4
3410-4360
976
Total
12+50
(v’=lB)
] 2679
13 (v’=14)
4546
13~160
Number
v’)
of
1%$80
12~170
mcasurcments
641
243
7
641
243
7
916
90
3655
other species are about 90 times ( 13C160), 500 times (“C 180), and 2500 times ( 1’C’70) less abundant than “C 160 in the sun. Line positions of the fundamental bands of 12C160weremeasuredupto J= 133,andthoseof13C160and12C180upto J= 103 and 9 1, respectively, whereas the first overtone bands of 12C160 were measured up to J = 110. The excitation potential of our solar lines ranges from 0 to 40 000 cm-’ . The strongest lines show a central depth of around 30% and a total width at halfmaximum (FWHM) of about 0.05 cm-’ whereas the faintest lines have an extremely small central depth (less than 0.5%) and a width of 0.03 cm-‘. In spite of their large Doppler width (as compared to laser lines) and of rather large shifts of various origins (see hereafter), solar lines, originating from a rather hot source in local thermodynamic equilibrium, offer the great advantage of showing high rotational lines, far above the usual experimental limit reached in laboratory sources. Figure 2 shows the observed spectral regions obtained by the fourth ATMOS filter around the first overtone bands. Most of the lines in our solar spectra are due to carbon monoxide, the remaining lines being attributed to atoms and also to other molecules like CH, NH, and OH. Telluric lines are also visible. Figure 3 shows the range of J observed in a given u level for the 4 isotopic species which benefit from the new solar observations. It clearly gives account of the important contribution of the new available data. We proceeded as follows for the identification of our solar lines and for the designation of the CO lines. We used a computer program TABLE II New Data of Solar CO Lines at High J-Values
isotopic Observed
bands
species
12cl6(,
lines
Number of different
P lines
R line!
r
nlai
lines a
1-O
to
20-19
1840
1395
809
586
133
2-O
to
14-12
224
224
84
140
110
1-o
to
14-13
363
296
170
126
103
1-O
to
8-7
131
117
63
54
91
1-O
to
3-2
6
6
6
0
52
1 1132
906
I
Total ‘526
Number cf mCas”remc”ts
1
are measured
I 2564
both in filters
2038
2 and 3 in the overlapping
region
1600-2000
cm-’
DUNHAM
COEFFICIENTS
FROM SOLAR CO LINES
379
FIG. 2. Example of ATMOS spectra shown at moderate resolution. Some bandheads of first overtone bands of CO are clearly visible in ATMOS filter 3. The remaining (atomic or molecular) lines are either
telluric or solar.
which accurately determines the positions of all line centers, interpolating our point spacings with a curve of the fifth degree and adopting a method of first differences, in order to calculate the minimum in every absorption line. We applied this program to our observations and also to a synthetic solar spectrum (generated with interim Dunham coefficients from preliminary calculations) including all CO and other identified lines yielding two lists of numbered and designated lines (see Fig. 4). With the help of a graphic terminal, the operator has to search for all CO lines, especially the faintest ones, and then select any couple of observed - calculated lines (line by line), entering the relevant numbers and a weight (which has been attributed to each observed line according to its quality and to its intensity). Our program reproduces to better than -+O.OOOlcm-’ all theoretical line positions: for the strong and medium solar lines the expected accuracy is around a0.0005 cm-’ whereas the position of faint lines is calculated with an accuracy of a few 10e3 cm-’ (reaching kO.0 1 cm-’ for the worst lines). In any case all our measurements of high J isolated CO lines have been included in our set owing to the complete lack of such data in the existing laboratory data. Correction
qf Solar Wavenumbers
All our solar lines are affected by a resultant Doppler shift of about 7 km/set arising from the combined effect of several radial motions: motion of the spacecraft with respect to the sun, rotation of the solar surface at the observed position (near the center of the disc), convective motions in different solar layers (essentially due to the
380
FARRENQET AL.
-1
L
El
2c1*o
'*C"O
25 V 20
lab.
15
n s~fl
10
I
0
20
40
60
0
20
40
60
J
’
r
7
80
100 120 140
0
20
40
60
80
100 120 140
0
20
40
60
80
100 120 140
80
100 120 140
J
FIG.3. Illustration of the large contribution of solar data for the four isotopic species observed. The highest J level which takes part in a transition measured in the laboratory (lab.) and in the solar spectrum (sun) is
shown for each vibrational state v.
granulation), and the Einstein gravitational redshift. If some of these shifts (such as the relative motion of the spacecraft and the gravitational redshift) are known, others are rather badly determined. Moreover, as shown by Dravins et al. (23) for atomic lines, one physical effect, the nonthermal solar motions, does vary with height and thus depends on the line intensity: weak lines are formed deeper than strong lines and are thus more affected by convection (a velocity difference of around 0.4 km/ set does exist between strong and weak CO lines). The measured line positions have to be corrected line by line in order to convert them to rest wavenumbers. Because of these uncertainties we have preferred to derive, for each filter, a correction curve from accurate laboratory frequencies of CO lines covering a large extent in intensity. Accurate positions of “C I60 lines belonging to the fundamental bands,
DUNHAM
I? (1211 1-o
COEFFICIENTS
(601
R
1-o
1 ,cY*,
I
1-o 10878 ?
THEORY
I
I I I I
I I
I I
Y
I
R (611 1-o x)879 I,
I
I
-I
R (1201
1-o
\I'
I 80
12c 160
10877
10876
381
FROM SOLAR CO LINES
I
I 1 I I I
b
80 -
V L
1
2306
1
r
sun
(cm-11
2307
FIG. 4. Example of measurements of CO solar lines with a graphic terminal. From a comparison of observed (bottom) and calculated (top) solar spectra in a small region around 2306 cm-‘. it appears that four ‘2C’60 lines are unblended in our ATMOS spectrum. The positions of all numbered lines in both spectra which have been calculated in our program are easily included in our data set. Other weak lines (probably of atomic origin) in our observations are unidentihed. The wavenumber scale of the observed spectrum has been slightly shifted ( mean correction of the Doppler effect) sothat there is a good coincidence of CO line positions in both spectra.
measured by Brown and Toth (25) and Schneider et al. (3, 4), have been used to calibrate the wavenumber scale of our ATMOS spectra (filter 2: 1350-2000 cm .’ : filter 3: 1600-2330 cm-‘). Similarly accurate line positions of the 2-O band from Pollock et al. (26) have been used for filter 4 ( 3400-4360 cm-’ ). Unfortunately, from a total of 199 laboratory data, there remain only a limited number of useful solar lines which are unblended and well measurable: 30 lines for filter 2, 28 lines for filter 3, and 11 lines for filter 4. That is due to the fact that most of the 167 laser lines of Schneider et ai. (3. 4) are of very high vibrational excitation and so appear too weak to be observed in the solar spectrum. For each filter we plotted the relative observed Doppler shift of every line, ( gsu,, - ulab)/cpabr as a function of the mean height of formation of the line center which was calculated in our program of synthetic solar spectrum. As expected, there is a gradual increase of the blue shift of line centers from strong to weak CO lines, except for filter 4 where all CO lines are formed practically in the same solar layers. A simple linear correction curve is found (see Fig. 5 corresponding to filter 3) and has been applied to all CO lines of every ATMOS filter. The linear correction derived from Fig. 5 could appear rather uncertain: we checked that the scatter of the representative
382
FARRENQ
-2.20
I
ET AL.
I
I
I 1
I
ATMOS
Filter
High
-2.50 _
3
I 12C I60
lines
Fundamental
bands
layers
Deep
I
1
560 Geometrical
I
layers
I
430 300 150 height in the solar atmosphere
(km1
0
FIG. 5. Observed solar Doppler shift, ( osun- ql~))qab. for 28 laboratory lines of ‘2C’60 measured in ATMOS filter 3 as a function of the geometrical height in the solar atmosphere. For each line, the mean height of formation of the center has been calculated in our program: weak lines, which are formed deeper than strong lines, show a larger blue shift. Accurate laboratory line positions from Refs. (3,4) are represented by crosses and from (25) by open circles. The straight line is a rather good fit of the representative points: this linear correction has been applied to all CO lines in filter 3. We checked that the rather large scatter can be much reduced if we plot the best solar lines giving very small residuals in our calculations.
points is much reduced if we plot the best solar lines which have also been measured in the laboratory (27). Let us illustrate the correction for a CO line at 2000 cm-’ for filter 2: -the correction for the spacecraft motion is -0.045 cm-‘, -the combined effect of rotation and convection in the solar layers is a blue shift ranging from 0.002 cm-’ (for the strongest lines) to 0.005 cm-’ (for the faintest lines), -the solar gravitational redshift is -0.00424 cm-‘. The resultant redshift for such a CO line ranges thus from -0.044 to -0.047 cm-’ when going from a weak to a strong line. III. LABORATORY DATA Almost all the data used in our previous determination of Dunham coefficients ( 1) have been included in this work. In addition to these laboratory data, we have included the high accurate measurements on ‘*C I60 lines recently published: -7 microwave transitions (24)) - 12 P lines belonging to the 1-O band, recorded by FTS at Kitt Peak National Observatory ( 25 ) .
DUNHAM
COEFFICIENTS
FROM
SOLAR
CO LINES
383
-20 lines belonging to the 2-O band, measured by heterodyne frequency technique ( 26 ) , -167 laser lines also measured by heterodyne frequency technique ( 2-4), 48 of them sub-Doppler stabilized ( 3). We have also made use of an emission spectrum recorded by FTS at the laboratoire d’brfrarouge. numbered 1578. This spectrum, described elsewhere (27). covers the 1600-5900 cm-’ region. It shows fundamental, first, and second harmonic bands of “C”O and 13CI’%, up to V = 40. Almost 4200 lines, 1100 of which belong to the AV = 3 bands, were measured. The total resulting amount of laboratory data is of the order of 14 000. Recalibration c$ FT Spectra The previous work (1) was performed on homogeneous data. Most of the spectra had been recorded with our interferometer, or calibrated by reference to the same standard wavenumbers. The data obtained by Schneider and co-workers, Nolt et al., Brown and Toth, and Pollock et al. (2-4, 24-26) change this situation. They all agree perfectly with each other but show a systematic shift, of the order of 12 MHz, with respect to our FT measurements. In the least-squares calculation, these high-weighted data behave as standards for “C’60 lines, and then induce a relative distortion with regard to other isotopic species. The source of the systematic errors have been identified (28) though it is only partly understood. These errors are not linear vs frequency, but we are not able, at this time. to carry out a rigorous correction. Then. we have decided to make the new calibration of all our laboratory spectra by the usual, linear, procedure in FTS, and to overvalue the uncertainties by 9 MHz. The standard wavenumbers are those of Refs. (2-4. -IS) for fundamental bands. and those of Ref. (26) for harmonic bands, The total effect is to lower the wavenumbers by about 4 X lop4 cm-‘. IV. NUMERICAL
TREATMENT
Though a fully quantum-mechanical procedure was recently carried out (8 ), the diatomic molecular line positions are, in general, calculated from the Dunham representation. In this semiclassical frame, several methods have been accounted for ( 30). Wavenumbers can be directly fitted to the coefficients of the potential energy function, or to the expression of the energy levels:
(2) where the notations are identical to those given in Ref. ( I), and -1 = M,’ + MG’. CL The numerical values are in Table III.
384
FARRENQ
ET AL.
TABLE III Masses (in a.m.u.) and Velocity of Light (in m/set)
0.548580228
ma x lo3 M(12C) M(13C) M(14C) M(160) M(“0) M(‘*O) C
12.00000000 13.00335440 14.00324197 15.99491502 16.99913290 17.99916002 299 792 458
These two methods are equivalent if we restrict the independent parameters of Eq. (2) to the { IJi.0, Vi,, } and { A$, A;} groups. The other parameters, U,,j k 2, can be obtained analytically from the first group (31, 32). Tyuterev and Velichko (32) have published 10 relations, forj 2 2, i + j =G5, of the form Ulj = Uij( UWl.1, Um,o).
(3)
Each U, is related to the first (2i + j) terms of the series obtained from the two where m takes all the integer values. For instance, elements IC.J~_~,~, Um,O, u13
=
Ul3(UOl,
UlO,
Ul,,
u20,
U21)
(three transcription mistakes found in their paper are reported in appendix). Indeed, there was no evidence that these relations, though they are exact, do not lead to distortions in the data reproduction, calculated from Eqs. ( 1) and (2) which are inevitably truncated (31). Then, we have made our computer program suitable for this last method, so that Eqs. (3) could be introduced one by one, in the increasing order of (2i + j) values. Thus, we are able to control the influence of each constraint on the quality of the computed frequencies. V. RESULTS
Starting from the set of constants obtained in Ref. (I), preliminary fits, where all the U, parameters are freely adjusted, have shown that terms of order (i + j) = 4, 5, and 6 were required to take solar data into account. The relation (3) between the Uv has been introduced in the fit without any loss of precision, as soon as two mass scaling parameters A$z and A% were added. Though they are not well defined, they are necessary to balance the constraint on the centrifugal distortion constant Uo2. Due to the lack of published relations for parameters of the order (i + j) = 6, Ud2, U24, Uls, and Uo6 have been determined as independent parameters. The final weighted nonlinear least-squares calculation required 33 independent pa-
DUNHAM
COEFFICIENTS
FROM SOLAR CO LINES
385
TABLE IV Isotopically Invariant Dunham Parameters Free Dunham lJ(l,O)
UC&O1
Parameters
5681.3676166 (3535)
A'(l,O)
0.700930 (1131)
-91.1055149 (1287)
A'(l,O,
-0.171324 (1586)
U(3,O)
0.18703287 13326)
A'12.0)
0.42815 (1384)
U(4,O) xld
0.3260748 (9465)
A"(2,0)
-0.91035 (36651
U(5,O) x104
0.20671 (1700)
AcC3,0,
-12.2868 (3442)
'j(6,O)x106
0.6637 (1842)
A"(3,0J
-3.534 (2091)
U(7,O) x106
-0.71632 (1185)
A'(O,l,
-2.057208 (6005)
A'(O,l,
-2.119921 (8720) -1.2452 (13361
U(8,O) x10'
0.273954 (4155)
U(9,O) x109
-0.476404 (61181
A'Ll,
U(O,l)
13.243473898 (4559)
A"oll,l,
-3.0206 (1777)
U&l)
-0.3142988561 (7182)
Ac(0,2,
-7.353 (1530)
U(2,l) x104 LJ(3,l)x105 U(4,l) xl05 U(5,l) x106 U(6,l) x10* U(7,l) x109
0.337229 (3179) -0.26419 (1582) 0.142950 (3586) -0.114861 (4071) 0.27528 (2264) -0.122960 (4886)
A"(0,2)
1.350 (2032)
U(4,2) x10' U(2,4) xl016 U(1,5) xl018 IJ(O,6)x1021
ConstrainedDunham U(O,2) x103 U(1,2) x106 Uv.,2) x10' U(3,2) x10' U(O,3) xl08
-0.2878458471 130001 0.1273849 (1242) -0.5961686 (8060) 0.205163 (5224) 0.189666901 (2861)
-0.230581 (3473) -0.7658 (14181 -0.23494 (1094) -0.15785 (1251)
Parameters
U(1,3) x109 U(2,3) x10" U(O,4) XlOB U(1,4) xl014 U(O,5) x1018
-0.12056492 (45261 -0.270763 (5899) -0.79898533 (24681 -0.4284994 (75191 -0.6901397 (4704)
~ The UCi,j) constants are in units of ~K~x(a.rn.u.1~'~ +? . The Acco(i,j) are dirwnsionless. The standard deviation is given in parentheses in alit of the last digit.
rameters (2 1 U, and 12 A,) and 10 calculated Vi, parameters. They are reported in Table IV. By means of Eq. (2), we can deduce a group of 3 1 Yij constants for each isotopic species. The constants relating to the four main species are listed in Table V. The statistical parameters of the fit are: -Root-mean-squares
of the residuals (RMS)
yo;4y2 -root-mean-squares
cm-‘,
of the weighted residuals (standard deviation )
c w(0 i
-estimated
= 8.6 X IO-’
-
c M’
cy
“7
= 9.7 X 10-O cm-‘,
1
variance c2
c No [
N-P
-
cl2 = 1
o.25
386
FARRENQ ET AL. TABLE V Dunham Coefficients Yti(and Their Standard Errors) for Four Isotopic Species of CO (in cm-’ ) Cl2016
Y(l,O)
0.216981267OD+O4 (O.l33D-04)
Y(2,0)-O.l328787634D+O2 (0.7740-05)
Cl2017
Cl2018
Cl3016
0.2142164929D+04 (O.l46D-04)
0.2117397251D+O4 (O.l79D-04)
0.2121439026D+04 (O.l62D-04)
-O.l295142054D+02 (O-7551)-05)
-O.l265367652D+02 (0.748D-05)
-O.l270204643D+OZ (0.747D-05)
Y(3,O)
O.l041106647D-01 (O.l72D-05)
O.l001820399D-01 (O.l65D-05)
0.9674767670D-02 (O.l59D-05)
0.9730658140D-02 (O.l60D-05)
Y(4,O)
0.6936640756D-04 (0.201D-06)
0.6589785167D-04 (O.l91D-06)
0.6290258358D-04 (O.l82D-06)
0.6338503741D-04 (O.l84D-06)
Y(5,O)
O.l679352306D-06 (O.l38D-07)
O.l575050132D-06 (O.l30D-07)
0.14860756401)-06 (O.l22D-07)
O.l500336759D-06 (O.l23D-07)
Y(6,O)
0.2059251576D-08 (0.572D-09)
0.19067444320-08 (0.529D-09)
O.l778231615D-08 (0.494D-09)
O.l798728967D-08 (0.499D-09)
-0.7759369245D-09 (0.128D-10)
-0.7152725631D-09 (O.l18D-10)
-0.7249007258D-09 (O.l20D-10)
O.l118886245D-10 (O.l69D-12)
0.10194839410-10 (O.l54D-12)
O.l035182507D-10 (O.l57D-12)
-0.7336207812D-13 (0.941D-15)
-0.6607168549D-13 (0.847D-15)
-0.6721736565D-13 (0.862D-15)
0.1882384581DtOl (0.349D-07)
O.l839113877D+Ol (0.607~-07)
O.l846151725D+Ol (0,39OD-07)
Y(7,0)-O.E488145707D-09 (O.l40D-10) Y(8,O)
O.l239772013D-10 (O.l88D-12)
Y(9,0)-O.S233737278D-13 (O.l06D-14) Y(O,l)
0.1931280985DtOl (O.l78D-07)
Y(l,l)-O.l750439229D-01 (O.l30D-07) Y(2,l)
0.7173917007D-06 (0.676D-08)
Y(3,1)-0.21463545860-07 (O.l29D-08) Y(4,l)
0.4435403909D-08 (O.lllD-09)
Y(5,1)-O.l36106945OD-09 (0.482D-11)
-O.l684384498D-01 (O.l36D-07) 0.6815196800D-06 (0.643D-08) -0.2013047567D-07 (O.l21D-08)
-0.16266402281)-01 (O.l59D-07) 0.6505424311D-06 (0.613D-08) -0.1899330625[3-07 (O.l15D-08)
-O.l635976671D-01 (O.l42D-07) 0.6555319987D-06 (0.618D-08) -O.l917557543D-07 (O.l16D-08)
0.4106920109D-08 (O.l03D-09)
0.3830117481D-08 (0.959D-10)
0.3874266547D-08 (0.970D-10)
-O.l244210549D-09 (0.440D-11)
-O.l146935582D-09 (0.406D-11)
-O.l162374288D-09 (0.411D-11)
where o and c are observed and calculated values (cm-‘), w = (6~))’ is the weight related to the experimental uncertainty 6a (cm-‘) of the measurement, N = 18 523 is the number of spectral data, and p = 33 is the number of free parameters. The correlation coefficient and dispersion matrices cannot be given here. They may be obtained from the authors. Figure 3 helps to visualize the range of u and J for which calculated wavenumbers can be obtained with the accuracy reported here. VI. DISCUSSION
The quality of this new set of constants has been tested for its ability to reproduce the high J solar lines as well as the new accurate laboratory measurements mentioned in Section III (2-4, 24-26). For the first set (solar data), no significant shifts were
DUNHAM
COEFFICIENTS
FROM
SOLAR
387
CO LINES
TABLE V-Continued Cl2016
Cl2017
Cl2018
Cl3016
O.l124313572D-11 (0.923D-13)
O.l024429101D-11 (0.8410-13)
O.l040203815D-11 (0.854D-13)
Y(7,1)-O.Z125123415D-13 (0.843D-15)
-O.l893471515D-13 (0.751D-15)
-O.l705306851D-13 (0.677D-15)
-O.l734876798D-13 (0.688D-15)
Y(O,Z)-0.6121615183D-05 (0.89213-11)
-0_581549?725D-05 (0.265D-10)
-0.5551151313D-05 (0.449D-10)
-0_5593901826D-05 (0.299D-10)
0.97064536560-09 (0.657D-12)
0.9158136642D-09 (0.620D-12)
0.9246022661D-09 (0.626D-12)
-O.l712774161D-09 (O.l53D-12)
-O.l597334762D-09 (0.142D-12)
-O.l615746949D-09 (O.l44D-12)
0.2222380202D-11 (0.356D-13)
0.2048629898D-11 (0.328D-13)
0.20762061580-11 (0.332D-13)
-0.9417416975D-13 (O.l42D-14)
-0.8580769858D-13 (O.l29D-14)
-0.8712901196D-13 (O.l31D-14)
0.5449071903D-11 (0.716D-16)
0.5081809481D-11 (0.668D-16)
0.5140386572D-11 (0.676~-16)
Y(l,3)-0.14286532770-12 (0.519D-16)
-O.l305991754D-12 (0.475D-16)
-O.l203886604D-12 (0.4380-16)
-O.l220091918D-12 (0.443D-16)
Y(2,3)-0.12253284990-14 (O.l85D-16)
-O.l105851067D-14 (O.l67D-16)
-O.l007606812D-14 (O.l52D-16)
-O.l023122487D-14 (O.l54D-16)
Y(O,4)-0.36157857450-16 (0.855D-21)
-0.3263223311D-16 (0.772D-21)
-0.2973317238D-16 (0.703D-21)
-0.3019101986D-16 (0.714D-21)
Y(1,4)-0.7405801298D-18 (0.9381)-21)
-0.6598522094D-18 (0.836D-21)
-0.5942790712D-18 (0.7530-21)
-0.6045838445D-18 (0.766D-21)
Y(2,4)-0.5054376594D-20 (0.935D-21)
-0.4446033496D-20 (0.822D-21)
-0.3957908937F20 (0.732b21)
-0.4034237544D-20 (0.746D-21)
Y(O,5)-0.4555298526D-22 (0.297P25)
-0.40070243001)-22 (0.261D-25)
-0.3567098022D-22 (0.232D-25)
-0.3635889807D-22 (0.237D-25)
Y(1,5)-O.S922257515D-23 (0.275D-24)
-0.5143075905D-23 (0.239D-24)
-0.4525486860D-23 (0.210D-24)
-0.4621580733D-23 (0.215D-24)
Y(O,6)-O.l519641502D-26 (0.12013-27)
-O.l302888664D-26 (O.l03D-27)
-O.l133180223D-26 (0.897D-28)
-O.l159454710D-26 (0.917D-28)
Y(6,l)
Y(1,2)
0.12457857151)-11 (0.1020-12)
O.l034922952D-08 (0.700D-12)
Y(2,2)-O.l84976698lD-09 (O.l65D-12) Y(3,2)
0.24311108770-11 (0.389D-13)
Y(4,2)-O.l043488564D-12 (O.l57D-14) Y(O,3)
0.5884905033D-11 (0.774D-16)
found taking the estimated uncertainties into account (from 1 to 5.10 _’ cm ’ for most of the lines). For the second set, only 17 measurements out of 2 18 are not calculated within the limits of their uncertainty, but all of the residuals are lower than twice this uncertainty. In order to give synthetic but more detailed information, all the data has been classified into five groups in Table VI. Laboratory measurements are distributed into the first four groups according to their experimental accuracy. The fifth one is devoted to new solar measurements. These five groups are divided into subgroups related to the spectral range of the observation. For each subgroup, the isotopic species. the number of data, and their relative weights (in parts of 10 000) in the global fit are shown. The mean experimental accuracy ( 1 6a/n) is in column 6. In column 7. for each subgroup. the mean deviation ( C (o - c)/n) shows possible systematic shifts between
388
FARRENQ ET AL.
observed and calculated values; such a shift appears in the results of the first row of the second group (+ 1.2 X 10P5 cm-’ or 360 kHz), though it is lower than the experimental accuracy. The root-mean-squares of the residuals are given in column 8. The overall RMS appears rather poor (0.8 X 10-3 cm-’ ). This is clearly due to the less accurate measurements involved in the fit. Nevertheless, in order to obtain the most reliable set of spectroscopic constants, it seemed important to us to not, a posteriori, sort out any data. The standard deviation, in column 9, is a more significant statistical parameter to show the quality of the fit. Very recently, Zink et al. (33) have reported new precise measurements on the rotational spectrum of 13C160, after our results were obtained. As an illustration of the ability of our set of constants to predict unobserved lines, Table VII shows the excellent agreement between the frequencies measured with their tunable far infrared spectrometer (the most accurate among their measurements) and our calculated values. A brief comparison with our previous results ( I ) shows the real progress performed, essentially owing to the new solar and laboratory data: about twice as much data are reproduced with a better standard deviation which has been decreased from 1.3 X 10M5 to 0.97 X 10m5cm-‘, though the number of free parameters has decreased from 35 to 33. The U, constants calculated from the set { U,,O , U,,- ,, 1} ( 1 =Gm G 4)) are perfectly integrated in the calculation. This guarantees the physical significance of this set with respect to the determination of the potential energy function. In this semiclassical frame, a new improvement could be obtained if measurements of other isotopic species as accurate as those on 12C160 were performed and if all the remaining systematic errors in FTS data could be removed.
TABLE VI Details on the Different Data Sets (All the Values Are in 10e3 cm-‘)
-
Accuracy 60
AV
Isotopic
species
I
Number of data
Relative weight
r _PRF_.---.SENT
Meal’ 1 Mean eviation ccuracy
RESU -7 .TS :eferences 1 R.M.S. Standard jeviation residua, Is 4
I
sa < 0.01 L A B 0 R A T 0 R Y
80 < 0.01 I,
0.010 0.004
la, 24 _.....___.. 2, 3. 4, 25 26b
0.56 0.82 0.90 __.___. 5.0 2.2 3.8
-0.014 -0.012 +0.151 __._.___. +0.19 +0.23 -0.35
0.16 0.39 0.34 . . .._. 2.6 1.7 1.6
0.13 0.31 0.33
la, 27 la, 26b, 27 la, 27
2.6 0.9 1.0
la la 1”
1.1 2.0
0.9 1.5
This work This work
0.86
0.0097
9700
0.002
179 11
100 190
7024 4447 1079
11 2.4 0.3 _ _ 0.002 0.07 0.01
O.l
2 3
s u N -
0.003 _._._. 0.037 0.008 . ..___
0.0005
0.069 0.013 __.....
+0.0003 ____..... +0.012 +0.0032 __.._....
19 __..._____
‘w60,‘w0,‘3c’60 1
‘3C’%
I
I
191 5.50 477
I
&>I
0.34 0.05
I,
W LABORATI
AND SOLAR DATA
1 18523
10000
B Data previously used and quoted in ref. (1) b Lines measured by Pollock et al. (26) have been distributed
1.8 2.3
+0.07 a.25
among the second and third groups
DUNHAM
COEFFICIENTS
FROM
TABLE Rotational
Transition
SOLAR
CO
Frequencies
of “CL60
6-5
661
067.276
0 050
0.00
-0 02
1 211
329.636
0.050
-0.03
-0.06
15-14
1 650
767.344
0.055
0 03
0.01
19-18
2 089
240.033
0.055
-0 06
-0.08
25-24
2 744
579.059
0.060
-0 02
-0 03
26-25
2 853
474.444
0.060
0.04
0.05
28-27
3 070
948.140
0.070
0.05
30-29
3 287
972.525
0.100
a From
Zink et al. (33). constants of Table
389
VII
11-10
b From
LINES
-0
10
0 07 -0 06
IV
VII. CONCLUSION
Accurate wavenumbers of about 4500 solar lines have been measured for the fundamental and the first overtone bands of “C 160, 13C160, “C 180, and “C “0; half of these lines have never been observed before in the laboratory, being of high rotational excitation (up to J = 133 ) . These new solar data were fitted to the Dunham formula, simultaneously taking into account 14 000 accurate laboratory measurements and adopting 10 specific relations between the U,, parameters. A new set of 3 1 Y, constants has been calculated. This new set of Dunham coefficients reproduces perfectly all laboratory data (with an accuracy better than 50.0005 cm-’ ) as well as the new solar lines of high J-values (accuracy generally better than kO.002 cm-’ for the fundamental bands and kO.004 cm-’ for the first overtone bands). For only about 300 very faint solar lines out of a total of 4500 the calculated and measured wavenumbers differ by twice or more the quoted deviations: these solar lines are of bad quality and very difficult to measure. The present set also reproduces much better than all previous observed CO spectra in sunspots ( 17) and in stars ( 28) and is especially recommended for generating CO synthetic spectra at relatively high temperature. We hope that our improved set of constants partly based on solar lines will play in future astrophysical applications of CO the same role as that played by the spectroscopic constants derived by Goldberg and Mtiller ( 13) 40 years ago. APPENDIX
In Eqs. ( 10) of Ref. (31), one must read: U23 = (* * ~-12U:ou:‘u~(‘+ u’3=
instead of ( * * * -12ou:ou:,
* - *> {..-}
u,, = {* - a+21 312UloU”U&
u,, + * * f ‘i
instead of U,, = { * - . > + - - -}
insteadof
(. - .+21 312UloU~, + - -. )
ACKNOWLEDGMENTS We are grateful to R. H. Tipping, L. R. Brown, and M. Morillon-Chapey for fruitful discussions and comments, and to V. G. Tyuterev who has corrected the mistakes in Ref. (32). All measurements of solar
FARRENQ
390
ET AL.
lines were performed at the Observatoire Royal de Belgique (Brussels) using a Univac 1100 computer and a Tektronix 4 I 14 terminal, and the fittings of CO frequencies were made at Orsay, utilizing a H.P. 9000 / 350 computer. We thank W. Nijs (Brussels) and J. Collet (Orsay) for their help in programs involved in this investigation. RECEIVED:
April 22, 199 1 REFERENCES
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