Revised ab initio models for H2–H2 collision-induced absorption at low temperatures

Icarus 189 (2007) 544–549 www.elsevier.com/locate/icarus

Revised ab initio models for H2 –H2 collision-induced absorption at low temperatures Glenn S. Orton a,∗ , Magnus Gustafsson b , Martin Burgdorf c , Victoria Meadows d a MS 169-237, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA b Department of Chemistry and Biochemistry, University of Colorado, Boulder, CO 80309, USA c Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead, CH41 1LD, UK d Spitzer Science Center, California Institute of Technology, MS 220-6, Pasadena, CA 91125, USA

Received 21 November 2006; revised 25 January 2007 Available online 1 March 2007

Abstract A revised ab initio calculation of the H2 –H2 collision-induced absorption results in significant differences compared with the work of J. Borysow et al. [Borysow, J., Trafton, L., Frommhold, L., Birnbaum, G., 1985. Astrophys. J. 296, 644–654] for wavenumbers greater than 600 cm−1 and temperatures below 120 K. The revision has significant influence on the spectra of Uranus and Neptune, and essentially removes the need for models with “super-solar” helium abundances or stratospheric hazes to explain the spectrum of Uranus. © 2007 Elsevier Inc. All rights reserved. Keywords: Atmospheres, structure; Uranus; Neptune; Infrared observations; Spectroscopy

1. Introduction Because the bulk of the atmospheres of the giant planets is composed of H2 , the opacity provided by the collision-induced absorption of H2 is important in their far-infrared spectra. Besides playing a pivotal role in the radiative energy control of the lower stratospheres of the giant planets, the dominance of the opacity of H2 , a well-mixed constituent, allows retrieval of temperatures from remote sensing of mid- and far-infrared thermal radiation. The primary model of this absorption is based on ab initio studies of H2 –H2 collisions by Borysow et al. (1985). The predictions, comparing favorably with most data, were approximated by semi-empirical functions which facilitated their efficient use over a broad spectral range. Many of these results were reviewed and tabulated by Birnbaum et al. (1996). The FORTRAN code for these functions was also made available as a web-based download by A. Borysow (http://www.astro.ku.dk/aborysow/programs/index.html). * Corresponding author.

E-mail address: [email protected] (G.S. Orton). 0019-1035/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2007.02.003

There is evidence that this absorption model is inadequate at temperatures less than 100 K and wavenumbers greater than 600 cm−1 , however. The 700–1000 cm−1 (10–14 µm) spectrum of Uranus, whose brightness temperature in this region ranges from 55 to 85 K, is free of the emission lines which dominate the spectrum of Neptune in this region (Orton et al., 1987, 1990). Orton (1986) attempted to fit the spectral continuum in this region with H2 collision-induced opacity alone, using models for the temperature structure that were consistent with the spectrum over a broad range which included the bulk of its emission at longer wavelengths. He was unable to do so without increasing the He volume mixing ratio to ∼40%, a value precluded by a combined analysis of remote sensing results by Voyager infrared and radio occultation experiments (Conrath et al., 1987). Orton et al. (1987, 1990) thus postulated a spectrally neutral haze emitting from the warm stratosphere of Uranus to match the data (cf. Fig. 2a of Orton et al., 1987). However, recent results from the more sensitive Spitzer Space Telescope (Orton et al., 2005) show that this spectral region is at least qualitatively more consistent with the shape of H2 collision-induced absorption than a spectrally neutral opacity source. Thus we

Revised ab initio H2 –H2 collision-induced absorption

thought it imperative to re-examine the collision-induced absorption model. 2. Model Our H2 –H2 collision-induced absorption profiles, α(ω, T ), were computed according to methods described in detail elsewhere (Frommhold, 1993) with a quantum scattering computer code over frequencies from 0.1 to 2400 cm−1 and temperatures from 40 to 400 K. The collision dynamics were determined using an ab initio potential energy surface (Schäfer and Köhler, 1989) and the radiative matrix elements were evaluated using the ab initio dipole surface by Meyer et al. (1989a). Convergence with respect to all parameters has been verified to be better than 1%. Furthermore, as is illustrated in Fig. 1, agreement with laboratory measurement is generally confirmed. Most notably the new calculation provides a significant correction to the (λ1 λ2 λL =) 2233 dipole component. Our calculations show that this component is roughly a factor of two too large in the line shape calculations of Borysow et al. (1985). The required correction results in weaker double transition features around 941 and 1174 cm−1 compared with the absorption profile re-

Fig. 1. The absorption coefficient, α, for equilibrium hydrogen normalized by the square of gas density, ρ (in units of amagat), as function of frequency at 77 K. The measurement by Birnbaum is shown by dots. The solid line represents the total calculated absorption and the dashed and dotted curves are for individual dipole components as indicated. The result of Borysow et al. (1985) is also shown for comparison (gray dash–dot) and with the 2233 component divided by two (gray dotted).

545

ported previously (Meyer et al., 1989b). The influence of the anisotropy of the potential was investigated and had a small effect on the spectrum, which we deemed not to be essential for the purposes of this work. Calculations including the anisotropy are very time consuming but may be important in some cases (Gustafsson et al., 2000, 2003). Table 1 presents the zeroth spectral moments, defined as (Frommhold, 1993) ∞ γ0 =

α(ω, T ) coth(h¯ ω/kB T ) dω, n2 ω

(1)

0

of the five most important dipole components for the spectra considered in this work. The density, n, in Eq. (1) is in units of cm−3 . Spectral moments may be obtained without first computing the spectrum (Frommhold, 1993), and such a semiclassical formula with a Wigner–Kirkwood quantum correction (Nienhuis, 1970) to the classical pair distribution was used to compute the values in the first column of Table 1. The data reported as “profile integral” in Table 1 are obtained by numerical integration of α(ω, T ) according to Eq. (1). For the sake of checking the calculations of α(ω, T ) we compared the zeroth and first spectral moments, γ0 and γ1 (Frommhold, 1993), from the profile integral with the semi-classical result over the whole temperature range (40 to 400 K). Good agreement was observed except at the lowest temperatures, where the semi-classical formula breaks down. This may be observed for the two selected temperatures in Table 1. The discrepancies between the 2021 and 0221 moments, as well as the 2023 and 0223 moments stem from the asymmetry of the rotational correction to the dipole moments of Meyer et al. (1989a). The results of Borysow et al. (1985) in Table 1 were reproduced using Eqs. (1) and (10) and Table 3 of their work. Again, the zeroth moment γ0 of the 2233 dipole component calculated by Borysow et al. (1985) appears to be too big by a factor of two. Fig. 1 also demonstrates the effect on the total absorption of dividing the γ0 (or S = 3h¯ cγ0 /(2π 2 )) value of the 2233 dipole component in A. Borysow’s FORTRAN code by a factor of two. This effect accounts for most, but not all, of the difference between the results of J. Borysow et al. and our model. The remainder of the difference is likely due to that our absorption was computed on a very fine frequency grid and not fitted to any semi-empirical line shape formula. The calculated absorption at high frequencies (>2000 cm−1 ) has previously been found to be significantly stronger than laboratory measurements (Gustafsson et al., 2003). This may stem

Table 1 Zeroth spectral moment, γ0 , in units of cm5 for the most important dipole components at two sample temperatures λ1 λ2 λL 2021 0221 2023 0223 2233

77.4 K

239.794 K

Semi-classical

Profile integral

BTFB

Semi-classical

Profile integral

BTFB

0.685E−46 0.669E−46 0.298E−44 0.297E−44 0.578E−46

0.637E−46 0.622E−46 0.283E−44 0.282E−44 0.549E−46

0.557E−46 0.557E−46 0.298E−44 0.298E−44 0.127E−45

0.158E−45 0.152E−45 0.384E−44 0.381E−44 0.694E−46

0.157E−45 0.151E−45 0.378E−44 0.374E−44 0.682E−46

0.126E−45 0.126E−45 0.368E−44 0.368E−44 0.146E−45

Note. The results labeled BTFB are those of Borysow et al. (1985).

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from an inaccuracy in the highest (λ = 4) dipole components. Despite the vast success in computing absorption spectra using the interaction-induced dipole of Meyer et al. (1989a) for the past two decades, further improvement of those components is desirable for accurate simulation of spectra in the 5-µm region. Based on the estimate of 3% accuracy of the ab initio dipole (Meyer et al., 1989a) we estimate that the absorption spectra computed in this work are accurate to within 5% up to 1200 cm−1 . Above 1200 cm−1 the accuracy may be worse due to the reason stated above. 3. Application The portion of the H2 –H2 collision-induced absorption spectrum most significantly revised by these calculations is for wavenumbers greater than 600 cm−1 (Fig. 1). For Jupiter and Saturn, the spectral range over which the H2 collisioninduced opacity is important in the mid-infrared extends beyond 600 cm−1 only to ∼780 cm−1 where the opacity is dominated by C2 H6 emission and, in the case of Jupiter, NH3 absorption. The temperature regime of the thermal emission in both cases is sufficiently high that the outgoing thermal emission spectra are almost negligibly different (less than 1% in radiance) from those computed using the absorption of Borysow et al. (1985). On the other hand, for the spectra of Uranus and Neptune, the revision is significant. The spectrum of Uranus provides the best example. Fig. 2 shows alternative models of the midinfrared H2 –H2 collision-induced absorption continuum spectra of Uranus, compared with observations of Uranus made by the Spitzer IRS instrument. Unlike Neptune’s spectrum, which is

filled with emission features from stratospheric hydrocarbons, a large section of the spectrum of Uranus shows the H2 continuum. The revised absorption coefficients shift the spectrum upward, obviating the need to invoke additional opacity sources. For the same spectral region in Neptune, the H2 collisioninduced absorption is nowhere the dominant opacity source, but the revised absorption model produces an upward shift of some 2 K between 1000 and 1120 cm−1 underneath emission lines of C2 H4 , CH3 D, and CH4 . Table 2 lists the coefficients in a coarse spectral grid of 50 cm−1 for a range of temperatures logarithmically spaced at equal intervals between 40 and 400 K. The coefficients show a smooth behavior in linear-cm−1 , log-temperature, and Fig. 2 have been computed using this as the basis of an interpolation. A finer spectral grid is available from the authors and will be submitted to the Atmospheres Node of NASA’s Planetary Data System (PDS). Note that Table 2 is for H2 with para vs ortho states in equilibrium at the local temperature, and that Table 3 reports corresponding results for normal H2 , corresponding to the high-temperature asymptotic 1:3 ratio of para vs ortho states. So far as we know, there are no laboratory data at low temperatures between those shown in Fig. 1 and ∼2000 cm−1 , and verification of these calculations is certainly recommended. In the absence of such measurements, we recommend our absorption model results. Their consistency with available laboratory measurements, significant correction of the 2233 term calculated by Borysow et al. (1985) and demonstrable improvement in fitting astronomical spectra for low-temperature environments commend them as a significant improvement in radiative-transfer modeling of Uranus and Neptune (and sim-

Fig. 2. Disk-averaged brightness temperature spectrum of Uranus, with temperature structure taken from the Moses et al.’s (2005) Model A. The observed spectrum (thin line and filled circles) is taken from the short-wavelength, low-resolution observations by the Spitzer Infrared Spectrometer (IRS) team (cf. Burgdorf et al., 2006). The thick gray lines show two models for the spectral continuum arising from the H2 collision-induced absorption. The dashed line is based on a model using the coefficients of Borysow et al. (1985), and the solid line is based on our revised absorption model. Other spectral features are indicated: emission by acetylene, ethane and the H2 S(3) dipole, and absorption by methane. Our revised model for the continuum comes closest to reproducing the shape and even the absolute brightness of the spectrum, although no attempt to adjust the temperature structure or composition given by Moses et al. (2005) was made to improve the fit.

Revised ab initio H2 –H2 collision-induced absorption

547

Table 2 The absorption coefficient, α/ρ 2 , in units of cm−1 amagat−2 for equilibrium hydrogen as a function of frequency (in cm−1 ) and temperature [cm−1 ] 50.000 100.000 150.000 200.000 250.000 300.000 350.000 400.000 450.000 500.000 550.000 600.000 650.000 700.000 750.000 800.000 850.000 900.000 950.000 1000.000 1050.000 1100.000 1150.000 1200.000 1250.000 1300.000 1350.000 1400.000 1450.000 1500.000 1550.000 1600.000 1650.000 1700.000 1750.000 1800.000 1850.000 1900.000 1950.000 2000.000 2050.000 2100.000 2150.000 2200.000 2250.000 2300.000 2350.000 2400.000

Temperature [K] 40.000

51.662

66.724

86.177

111.302

143.753

185.664

239.794

309.705

400.000

4.45E−08 3.85E−08 2.59E−08 1.96E−08 5.48E−08 8.33E−07 9.81E−06 9.26E−06 3.84E−06 1.85E−06 1.32E−06 2.12E−06 1.20E−06 7.42E−07 5.00E−07 2.65E−07 1.59E−07 1.10E−07 1.16E−07 7.31E−08 4.39E−08 3.13E−08 3.25E−08 4.16E−08 3.71E−08 2.82E−08 2.05E−08 1.47E−08 1.06E−08 7.76E−09 5.84E−09 5.21E−09 5.51E−09 4.83E−09 3.71E−09 2.74E−09 2.01E−09 1.48E−09 1.10E−09 8.23E−10 6.23E−10 4.76E−10 3.66E−10 2.84E−10 2.22E−10 1.75E−10 1.38E−10 1.11E−10

8.62E−08 8.66E−08 6.19E−08 5.01E−08 1.15E−07 1.09E−06 8.12E−06 7.48E−06 3.40E−06 1.75E−06 1.77E−06 3.64E−06 2.15E−06 1.13E−06 6.65E−07 3.73E−07 2.29E−07 1.66E−07 1.76E−07 1.14E−07 6.66E−08 4.60E−08 4.48E−08 5.01E−08 3.98E−08 2.96E−08 2.16E−08 1.57E−08 1.15E−08 8.55E−09 6.93E−09 7.28E−09 8.58E−09 7.96E−09 6.32E−09 4.71E−09 3.46E−09 2.53E−09 1.86E−09 1.38E−09 1.03E−09 7.76E−10 5.89E−10 4.51E−10 3.48E−10 2.70E−10 2.11E−10 1.66E−10

1.22E−07 1.44E−07 1.12E−07 9.50E−08 1.97E−07 1.21E−06 5.83E−06 5.66E−06 2.85E−06 1.69E−06 2.52E−06 5.16E−06 3.29E−06 1.68E−06 9.38E−07 5.56E−07 3.54E−07 2.53E−07 2.39E−07 1.60E−07 9.74E−08 6.81E−08 6.63E−08 6.91E−08 4.91E−08 3.44E−08 2.47E−08 1.81E−08 1.34E−08 1.03E−08 9.04E−09 1.06E−08 1.24E−08 1.18E−08 9.66E−09 7.39E−09 5.51E−09 4.07E−09 3.01E−09 2.24E−09 1.69E−09 1.29E−09 9.80E−10 7.48E−10 5.74E−10 4.43E−10 3.43E−10 2.68E−10

1.39E−07 1.91E−07 1.65E−07 1.51E−07 2.79E−07 1.23E−06 4.30E−06 4.18E−06 2.37E−06 1.75E−06 3.28E−06 6.28E−06 4.31E−06 2.29E−06 1.29E−06 8.15E−07 5.48E−07 3.79E−07 3.12E−07 2.13E−07 1.37E−07 9.91E−08 1.02E−07 9.57E−08 6.53E−08 4.40E−08 3.12E−08 2.33E−08 1.75E−08 1.38E−08 1.26E−08 1.44E−08 1.64E−08 1.57E−08 1.33E−08 1.04E−08 7.95E−09 5.99E−09 4.51E−09 3.42E−09 2.64E−09 2.06E−09 1.60E−09 1.23E−09 9.47E−10 7.31E−10 5.67E−10 4.41E−10

1.37E−07 2.20E−07 2.11E−07 2.08E−07 3.46E−07 1.09E−06 3.20E−06 3.12E−06 2.06E−06 1.95E−06 4.34E−06 6.83E−06 5.05E−06 2.87E−06 1.71E−06 1.17E−06 8.39E−07 5.69E−07 4.21E−07 2.95E−07 2.05E−07 1.52E−07 1.44E−07 1.29E−07 8.92E−08 6.03E−08 4.32E−08 3.35E−08 2.57E−08 2.03E−08 1.84E−08 1.96E−08 2.09E−08 1.98E−08 1.70E−08 1.37E−08 1.08E−08 8.33E−09 6.43E−09 5.03E−09 4.02E−09 3.23E−09 2.57E−09 2.03E−09 1.58E−09 1.23E−09 9.62E−10 7.56E−10

1.22E−07 2.29E−07 2.47E−07 2.61E−07 4.02E−07 9.97E−07 2.25E−06 2.40E−06 1.90E−06 2.21E−06 4.44E−06 6.85E−06 5.45E−06 3.37E−06 2.17E−06 1.63E−06 1.24E−06 8.49E−07 6.05E−07 4.48E−07 3.45E−07 2.59E−07 2.18E−07 1.81E−07 1.28E−07 8.94E−08 6.60E−08 5.26E−08 4.10E−08 3.23E−08 2.82E−08 2.78E−08 2.75E−08 2.53E−08 2.17E−08 1.79E−08 1.44E−08 1.14E−08 9.11E−09 7.36E−09 6.08E−09 5.04E−09 4.13E−09 3.33E−09 2.66E−09 2.11E−09 1.68E−09 1.36E−09

1.04E−07 2.22E−07 2.69E−07 3.05E−07 4.43E−07 8.98E−07 1.74E−06 1.95E−06 1.86E−06 2.47E−06 4.66E−06 6.57E−06 5.58E−06 3.78E−06 2.66E−06 2.15E−06 1.73E−06 1.24E−06 9.14E−07 7.47E−07 6.41E−07 4.94E−07 3.76E−07 2.89E−07 2.08E−07 1.49E−07 1.13E−07 8.99E−08 7.08E−08 5.56E−08 4.70E−08 4.32E−08 4.01E−08 3.52E−08 2.97E−08 2.46E−08 2.02E−08 1.65E−08 1.34E−08 1.11E−08 9.40E−09 7.96E−09 6.66E−09 5.50E−09 4.50E−09 3.68E−09 3.04E−09 2.56E−09

8.57E−08 2.07E−07 2.77E−07 3.35E−07 4.70E−07 8.28E−07 1.38E−06 1.66E−06 1.86E−06 2.64E−06 4.56E−06 6.05E−06 5.47E−06 4.06E−06 3.13E−06 2.67E−06 2.26E−06 1.73E−06 1.39E−06 1.27E−06 1.18E−06 9.47E−07 7.04E−07 5.19E−07 3.77E−07 2.76E−07 2.09E−07 1.65E−07 1.30E−07 1.02E−07 8.45E−08 7.46E−08 6.57E−08 5.54E−08 4.58E−08 3.79E−08 3.15E−08 2.60E−08 2.14E−08 1.80E−08 1.54E−08 1.31E−08 1.11E−08 9.32E−09 7.80E−09 6.57E−09 5.63E−09 4.94E−09

6.89E−08 1.85E−07 2.73E−07 3.50E−07 4.80E−07 7.46E−07 1.14E−06 1.46E−06 1.82E−06 2.64E−06 4.14E−06 5.32E−06 5.08E−06 4.14E−06 3.46E−06 3.13E−06 2.75E−06 2.30E−06 2.03E−06 2.08E−06 1.96E−06 1.66E−06 1.28E−06 9.76E−07 7.51E−07 5.74E−07 4.38E−07 3.39E−07 2.64E−07 2.07E−07 1.68E−07 1.42E−07 1.21E−07 9.95E−08 8.14E−08 6.74E−08 5.63E−08 4.67E−08 3.87E−08 3.25E−08 2.78E−08 2.38E−08 2.01E−08 1.70E−08 1.45E−08 1.25E−08 1.09E−08 9.75E−09

5.48E−08 1.61E−07 2.59E−07 3.51E−07 4.77E−07 6.93E−07 9.95E−07 1.32E−06 1.73E−06 2.49E−06 3.67E−06 4.51E−06 4.54E−06 4.02E−06 3.62E−06 3.43E−06 3.17E−06 2.88E−06 2.77E−06 2.89E−06 2.88E−06 2.55E−06 2.07E−06 1.66E−06 1.33E−06 1.06E−06 8.24E−07 6.44E−07 5.06E−07 4.02E−07 3.25E−07 2.70E−07 2.24E−07 1.84E−07 1.51E−07 1.25E−07 1.05E−07 8.76E−08 7.30E−08 6.17E−08 5.30E−08 4.53E−08 3.85E−08 3.27E−08 2.81E−08 2.44E−08 2.14E−08 1.91E−08

ilar low-temperature atmospheres of extrasolar planets) in the middle infrared. These revised absorption coefficients will subsequently be used in the detailed modeling of the Spitzer IRS spectra of both Uranus and Neptune which has heretofore been problematical. Acknowledgments G.S.O. acknowledges support from grants to the Jet Propulsion Laboratory, California Institute of Technology, from NASA’s Research and Analysis program, Planetary Atmos-

pheres discipline, as well as the Spitzer mission. M.G. would like to thank Lothar Frommhold for initiating the collaboration between M.G. and the other authors of this work. The radiativetransfer calculations that created the models in Fig. 2 were performed on JPL supercomputer facilities which were provided by funding from the JPL Office of the Chief Information Officer. We thank J. Moses for digital versions of her models. This work was compared with observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.

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G.S. Orton et al. / Icarus 189 (2007) 544–549

Table 3 The absorption coefficient, α/ρ 2 , in units of cm−1 amagat−2 for normal hydrogen as a function of frequency (in cm−1 ) and temperature [cm−1 ] 50.000 100.000 150.000 200.000 250.000 300.000 350.000 400.000 450.000 500.000 550.000 600.000 650.000 700.000 750.000 800.000 850.000 900.000 950.000 1000.000 1050.000 1100.000 1150.000 1200.000 1250.000 1300.000 1350.000 1400.000 1450.000 1500.000 1550.000 1600.000 1650.000 1700.000 1750.000 1800.000 1850.000 1900.000 1950.000 2000.000 2050.000 2100.000 2150.000 2200.000 2250.000 2300.000 2350.000 2400.000

Temperature [K] 40.000

51.662

66.724

86.177

111.302

143.753

185.664

239.794

309.705

400.000

2.98E−07 2.58E−07 1.72E−07 1.16E−07 9.10E−08 3.05E−07 2.86E−06 2.65E−06 1.11E−06 6.46E−07 2.43E−06 1.04E−05 5.54E−06 2.33E−06 1.15E−06 6.39E−07 3.84E−07 2.56E−07 2.46E−07 1.53E−07 9.14E−08 6.27E−08 8.19E−08 1.29E−07 6.26E−08 3.37E−08 2.10E−08 1.41E−08 9.93E−09 7.30E−09 6.43E−09 1.08E−08 1.89E−08 1.88E−08 1.46E−08 1.05E−08 7.46E−09 5.27E−09 3.75E−09 2.69E−09 1.95E−09 1.43E−09 1.05E−09 7.86E−10 5.90E−10 4.47E−10 3.41E−10 2.62E−10

2.60E−07 2.61E−07 1.85E−07 1.29E−07 1.17E−07 4.26E−07 2.91E−06 2.54E−06 1.18E−06 8.16E−07 2.88E−06 9.52E−06 5.53E−06 2.48E−06 1.26E−06 7.04E−07 4.26E−07 2.89E−07 2.56E−07 1.66E−07 1.02E−07 7.25E−08 9.19E−08 1.24E−07 6.61E−08 3.70E−08 2.34E−08 1.59E−08 1.13E−08 8.52E−09 8.09E−09 1.24E−08 1.87E−08 1.86E−08 1.49E−08 1.11E−08 8.00E−09 5.76E−09 4.15E−09 3.01E−09 2.20E−09 1.62E−09 1.20E−09 9.01E−10 6.80E−10 5.17E−10 3.95E−10 3.05E−10

2.22E−07 2.62E−07 2.01E−07 1.49E−07 1.61E−07 5.76E−07 2.53E−06 2.45E−06 1.28E−06 1.06E−06 3.47E−06 8.75E−06 5.56E−06 2.67E−06 1.40E−06 8.06E−07 4.98E−07 3.39E−07 2.80E−07 1.88E−07 1.20E−07 8.80E−08 1.04E−07 1.23E−07 7.18E−08 4.21E−08 2.72E−08 1.89E−08 1.36E−08 1.07E−08 1.04E−08 1.47E−08 1.91E−08 1.89E−08 1.56E−08 1.19E−08 8.85E−09 6.49E−09 4.76E−09 3.50E−09 2.60E−09 1.94E−09 1.45E−09 1.10E−09 8.33E−10 6.36E−10 4.89E−10 3.78E−10

1.88E−07 2.59E−07 2.19E−07 1.79E−07 2.25E−07 7.47E−07 2.45E−06 2.38E−06 1.42E−06 1.39E−06 3.90E−06 8.21E−06 5.64E−06 2.93E−06 1.61E−06 9.70E−07 6.25E−07 4.23E−07 3.28E−07 2.24E−07 1.49E−07 1.12E−07 1.31E−07 1.29E−07 8.12E−08 5.02E−08 3.35E−08 2.40E−08 1.78E−08 1.42E−08 1.38E−08 1.70E−08 2.03E−08 1.99E−08 1.69E−08 1.33E−08 1.01E−08 7.61E−09 5.70E−09 4.28E−09 3.25E−09 2.48E−09 1.89E−09 1.45E−09 1.11E−09 8.53E−10 6.59E−10 5.12E−10

1.57E−07 2.52E−07 2.38E−07 2.18E−07 3.00E−07 8.23E−07 2.34E−06 2.28E−06 1.58E−06 1.79E−06 4.74E−06 7.72E−06 5.71E−06 3.22E−06 1.88E−06 1.23E−06 8.51E−07 5.74E−07 4.20E−07 2.96E−07 2.10E−07 1.59E−07 1.59E−07 1.45E−07 9.79E−08 6.40E−08 4.46E−08 3.37E−08 2.57E−08 2.05E−08 1.91E−08 2.10E−08 2.29E−08 2.20E−08 1.89E−08 1.53E−08 1.20E−08 9.25E−09 7.12E−09 5.51E−09 4.33E−09 3.42E−09 2.69E−09 2.10E−09 1.64E−09 1.27E−09 9.94E−10 7.81E−10

1.29E−07 2.40E−07 2.57E−07 2.62E−07 3.76E−07 8.80E−07 1.95E−06 2.09E−06 1.71E−06 2.16E−06 4.59E−06 7.18E−06 5.72E−06 3.53E−06 2.24E−06 1.64E−06 1.22E−06 8.34E−07 5.95E−07 4.44E−07 3.46E−07 2.62E−07 2.24E−07 1.87E−07 1.32E−07 9.10E−08 6.64E−08 5.23E−08 4.07E−08 3.22E−08 2.85E−08 2.84E−08 2.84E−08 2.61E−08 2.25E−08 1.86E−08 1.49E−08 1.19E−08 9.42E−09 7.56E−09 6.19E−09 5.07E−09 4.13E−09 3.32E−09 2.64E−09 2.10E−09 1.67E−09 1.35E−09

1.06E−07 2.25E−07 2.71E−07 3.04E−07 4.34E−07 8.63E−07 1.67E−06 1.87E−06 1.80E−06 2.46E−06 4.71E−06 6.67E−06 5.67E−06 3.84E−06 2.68E−06 2.14E−06 1.71E−06 1.23E−06 9.06E−07 7.43E−07 6.41E−07 4.96E−07 3.78E−07 2.91E−07 2.10E−07 1.50E−07 1.13E−07 8.97E−08 7.05E−08 5.55E−08 4.70E−08 4.34E−08 4.04E−08 3.55E−08 3.00E−08 2.49E−08 2.04E−08 1.66E−08 1.35E−08 1.12E−08 9.41E−09 7.94E−09 6.64E−09 5.47E−09 4.47E−09 3.65E−09 3.01E−09 2.54E−09

8.59E−08 2.07E−07 2.78E−07 3.35E−07 4.68E−07 8.23E−07 1.37E−06 1.65E−06 1.86E−06 2.64E−06 4.57E−06 6.07E−06 5.48E−06 4.07E−06 3.13E−06 2.66E−06 2.25E−06 1.73E−06 1.38E−06 1.27E−06 1.18E−06 9.48E−07 7.04E−07 5.20E−07 3.77E−07 2.76E−07 2.09E−07 1.65E−07 1.30E−07 1.02E−07 8.45E−08 7.47E−08 6.58E−08 5.54E−08 4.59E−08 3.80E−08 3.15E−08 2.60E−08 2.14E−08 1.80E−08 1.54E−08 1.31E−08 1.11E−08 9.31E−09 7.79E−09 6.56E−09 5.62E−09 4.93E−09

6.89E−08 1.85E−07 2.73E−07 3.50E−07 4.79E−07 7.45E−07 1.14E−06 1.46E−06 1.82E−06 2.64E−06 4.14E−06 5.32E−06 5.08E−06 4.14E−06 3.46E−06 3.13E−06 2.75E−06 2.30E−06 2.03E−06 2.08E−06 1.96E−06 1.66E−06 1.28E−06 9.76E−07 7.51E−07 5.74E−07 4.38E−07 3.39E−07 2.64E−07 2.07E−07 1.68E−07 1.42E−07 1.21E−07 9.95E−08 8.15E−08 6.74E−08 5.63E−08 4.67E−08 3.87E−08 3.25E−08 2.78E−08 2.38E−08 2.01E−08 1.70E−08 1.45E−08 1.25E−08 1.09E−08 9.75E−09

5.48E−08 1.61E−07 2.59E−07 3.51E−07 4.77E−07 6.92E−07 9.93E−07 1.32E−06 1.73E−06 2.49E−06 3.67E−06 4.51E−06 4.55E−06 4.03E−06 3.62E−06 3.42E−06 3.17E−06 2.88E−06 2.77E−06 2.89E−06 2.88E−06 2.55E−06 2.07E−06 1.66E−06 1.33E−06 1.06E−06 8.24E−07 6.44E−07 5.06E−07 4.01E−07 3.25E−07 2.70E−07 2.24E−07 1.84E−07 1.51E−07 1.25E−07 1.05E−07 8.76E−08 7.30E−08 6.17E−08 5.29E−08 4.53E−08 3.85E−08 3.27E−08 2.80E−08 2.44E−08 2.14E−08 1.91E−08

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