A&A 387, 595-604 (2002)
C. R. Cowley1 - F. Castelli2,3
1 - Astronomy Department, University of Michigan, Ann Arbor, MI 48109-1090 USA
2 - Istituto di Astrofisica Spaziale e Fisica Cosmica, CNR, via del Fosso del Cavaliere, 00133 Roma, Italy
3 - Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34131 Trieste, Italy,
Received 9 January 2002 / Accepted 18 March 2002
We compare the results of Balmer-line calculations using recent theory and improved computational algorithms with those from the widely-used SYNTHE and BALMER9 routines. The resulting profiles are mostly indistinguishable. Good fits to the normalized solar Balmer lines H through Hare obtained (apart from the cores) using the recent unified-broadening calculations by Barklem and his coworkers provided that some adjustment for the continuum is performed. We discuss a surprising linearity with temperature of the Balmer line profiles in dwarfs.
Key words: stars: atmospheres - stars: fundamental parameters - line: formation - line: profiles
Balmer line strengths are highly sensitive to the temperature in cool stars because of the 10.2 eV excitation of the n=2 level from which they arise. Figure 151 from Unsöld's (1955) classic text illustrates this for H equivalent widths. We show the effect in a different way in Fig. 1, based on more recent line-broadening theory. The figure is for points on the Hprofile 4 Å from the line center, but is characteristic of much of the line profile.
|Figure 1: H wing strength vs. for several values of . The profiles are taken from the BP00K2NOVER grid available in http://kurucz.harvard.edu|
An extensive investigation of Balmer lines in cool dwarfs (Fuhrmann et al. 1993; Fuhrmann et al. 1994) concluded these lines provide a more consistent guide to effective temperatures than broad-band colors or b - y. Nevertheless, Balmer line profiles are not regularly used to fix the effective temperature of cool stars. The reasons for this are numerous, but have not been explicitly addressed. Some insight may be gained from the papers by van't Veer-Menneret & Mégessier (1996) or Castelli et al. (1997, henceforth, CGK). A recent paper which does discuss use of H in the determination of effective temperatures is by Peterson et al. (2001). In addition to the uncertainties in placing the continuum level, uncertainties, both in the theory of stellar atmospheres (l/H, convection) and line formation remain unresolved.
The absorption coefficient of neutral hydrogen takes into account the effects due to the natural absorption (natural broadening), the velocity of the absorbing hydrogen atoms (thermal Doppler and microturbulent broadening), the interactions with charged perturbers (linear Stark broadening), with neutral perturbers different from hydrogen (van der Waals broadening), and with neutral hydrogen perturbers (resonance and van der Waals broadening). Each effect is represented by a profile and the total effect requires a convolution. Thermal Doppler and microturbulent broadenings are described by Gaussian functions while natural, resonance, and van der Waals broadenings have Lorentz profiles. These two profiles are combined into a Voigt function. The convolution of the Voigt profile with the Stark profile or Stark plus thermal Doppler effect then gives the total absorption profile.
Most of the damping constants and Stark profiles are computed from complex theories based on several approximations, while the complete convolution of all the above profiles is a very time consuming algorithm.
In this paper we describe our attempts to evaluate several aspects of the calculations of Balmer line profiles.
Most work on stellar atmospheres makes use of codes provided by Kurucz (http://kurucz.harvard.edu). For computing hydrogen lines the codes are either BALMER9 (Kurucz 1993a) which produces profiles for H, H, H, and H or the SYNTHE code (Kurucz 1993b) which produces profiles for any hydrogen line. In the first case Stark profiles are interpolated in the Vidal et al. (1973, henceforth VCS) tables, while in the second case the Stark profiles are based on the quasi-static Griem theory with parameters adjusted in such a way that profiles from Griem theory fit the VCS profiles of the first members of the Lyman and Balmer series.
Only the most recent work on the Balmer lines (e.g. Barklem et al. 2000, henceforth, BPO) has included the new Stark profiles of Chantal Stehlé (henceforth CS) and her coworkers. They are available from a link on her website: http://dasgal.obspm.fr/stehle/. A recent reference is Stehlé & Hutcheon (1999).
A problem arises when a given Stark profile is interpolated either in the VCS or in the CS tables by using the interpolation method taken from the BALMER9 code. This is a bilinear interpolation in and , followed by a linear interpolation in the parameter . Here, F0 is the normal field strength in Gaussian cgs units, , so the interpolation in is not independent of the previous one which involves the electron density . We find this introduces a small error that shows up as an oscillation in a plot of the Stark profile vs. depth in the solar atmosphere for a small range of displacements from the line center as shown in Fig. 2.
|Figure 2: Normalized Stark width at Å for H vs. 137 depths in an Holweger-Müller (1974) solar model. Each depth step is 0.05 in . The vertical lines mark depths corresponding to boundaries of the tables giving for a fixed value of the electron density.|
We were able to remove the oscillations by rewriting the CS tables with as the third (independent) variable, and using essentially the same interpolation scheme as BALMER9. Fortunately, it has resulted that the improved interpolation leads to no perceptible changes in the resulting line profiles.
Neither the BALMER9 code nor the SYNTHE code perform profile convolutions, but all the profiles are simply added. In the BALMER9 code, for separations larger than 0.2 Å from the line center, a Lorentz profile (representing the natural broadening and the resonance broadening) is added linearly to the Stark-thermal Doppler profile interpolated in the VCS tables. For separations smaller than 0.2 Å no Lorentz profile was considered.
In the SYNTHE code, the Doppler profile, the Stark profile, and the Lorentz profile (for natural broadening, resonance broadening, and van der Waals broadening from He I and H2) are still summed together. The very inner core is that of the profile (Doppler, Stark, or Lorentz) with the largest full width at half maximum FWHM.
This method due to Peterson (1993), which we shall call the PK approximation, would be rigorously true for the wings of two Lorentzians. Since the wing-dependence of the Stark profile differs from that of a Lorentzian only by , one might expect the approximation to be good, as we verified that it is.
|4760||1.3||2.3||CS22892-052 (cf. Sneden et al. 1996)|
|8000||3.5||2.0||like cool Ap or Am|
|8000||1.5||12.0||test of large|
|12 000||3.0||2.0||hot star|
|Figure 3: H profiles for a model with , = 1.5. The lower curve is for a CSII calculation with an assumed microturbulence km s-1. The upper curve, displaced upward for purposes of illustration, was made using BALMER9, the older interpolation scheme for VCS tables, and the PK approximation. There is no perceptible difference in the two profiles beyond the line core.|
Replacing the sum of the Stark and Lorentz profile in BALMER9 by a convolution takes a large amount of computing time in that the step of the convolution has to be very small (less than 0.001 Å) in order to account for the narrow full width at half maximum FWHM of the Lorentz profile. This problem can be overcome by including a microturbulent velocity in the computations.
Both the VCS and CS tables include thermal Doppler, but not microturbulent broadening. The BALMER9 code makes no provision for the inclusion of microturbulence in the line profiles owing to the sum of the Stark-thermal-Doppler profile, interpolated in the VCS tables, with the Lorentz profile. The SYNTHE code does allow for a microturbulence in that it adds the Stark profile to a Doppler-microturbulence Gaussian profile.
The only way to rigorously include all broadening mechanisms is to do a convolution of the Stark-thermal Doppler profile, interpolated in the VCS or CS tables, with a profile which includes both the Lorentz broadening and turbulent motions. If we assume a Gaussian distribution of microturbulent velocities, the VCS or CS profiles need to be convolved with a Voigt profile.
To check BALMER9 and SYNTHE profiles we did calculations using the new CS profiles with improved interpolation, and a full convolution including a microturbulent velocity. We shall refer to such profiles and to the corresponding code with the abbreviation CSII (Convolution, Stehle, improved interpolation). Table 1 shows models parameters for which we made calculations of an H profile in order to test the effects of the various approximations and improvements mentioned above. All models were generated with the ATLAS9 code (Kurucz 1993a). Solar abundances were assumed for all but CS22892-052, for which abundances were chosen to roughly match those of Sneden et al. (1996).
We find, with one exception, that the BALMER9 profiles computed with no convolutions and no microturbulent velocity are in excellent agreement with CSII calculations. The only exception occurs for the supersonic microturbulent velocity 12 km s-1. In this case the line core of the profile computed for 12 km s-1 is larger than that computed without microturbulence, as is shown in Fig. 3. However, the H profile computed by SYNTHE with no convolutions, but by assuming 12 km s-1 agrees well with the CSII profile.
The effect of a microturbulent velocity will be small until approaches the sound speed. It is not surprising, therefore, that the only case we have found where plots of H obtained using BALMER9 with the PK approximation and CSII differed significantly is that for of the order of the sound speed. Even in this situation, only the deepest parts of the core were affected. The line wings still matched beautifully.
The calculations of Fuhrmann et al. (1993, 1994) included Lorentz broadening by a full convolution, while BPO used the PK approximation. The above comparisons led us to conclude that any differences between their results and other calculations (e.g. CGK or Gardiner et al. 1999) cannot be attributed to the PK approximation or to different Stark profiles (VCS or CS) - the immediate line core excepted.
The BALMER9 and SYNTHE codes allow for the broadening of the hydrogen lines due to the collisions with other neutral H I atoms through the resonance broadening based on the Ali & Griem theory (1965, 1966). Actually the van der Waals effect due to H I should also be included, but it can not be simply added to the resonance broadening (Lortet & Roueff 1969) and therefore it was always neglected in the hydrogen profile calculations. Only recently BPO (Barklem et al. 2000) presented a unified theory of the H I-H I collisions in the stellar atmospheres. The differences in Balmer profiles computed with only resonance broadening and with both resonance and van der Waals broadenings are fully discussed in BPO.
We have included in our hydrogen synthetic spectra (BALMER9, SYNTHE and
CSII) the BPO broadening.
The line half half-width HWHM per unit hydrogen atom density w/N(H) is
computed according to Anstee & O'Mara (1995):
In the CSII code, HWHM was computed in according to BPO for each given temperature of the atmospheric layers. For Hthe broadening by neutrals was obtained by extrapolating BPO's Table 3, but the profile is dominated by Stark broadening, and is nearly independent of the broadening by neutrals. In BALMER9 and in SYNTHE, HWHM was obtained for each temperature of the atmospheric layers from a function HWHM=HWHM0 (T/10 000)y where HWHM0is the value of HWHM for T= 10 000 K and y was derived from the best fit of the above function to the HWHM,T points for T ranging from 2000 K to 11 500 K at steps of 500 K (Fig. 3 in BPO). The parameter y is 0.15 for H, 0.275 for H, and 0.30 for H.
For the calculation of the solar Balmer profiles we adopted the Holweger-Müller model (1974, henceforth, HM) to avoid additional complications from various solar models, already discussed, for example, by CGK. We started from the HM T- relation given for 29 layers, and extrapolated-interpolated to suit the depth ranges used by our respective codes.
There are differences in the optical depth coverage of the Michigan and Trieste codes. In the first case, the T- relation was interpolated-extrapolated to 135 layers, while in the second case it was interpolated for 50 layers before using it in the Kurucz codes. While the Michigan code performs integrations directly in terms of , the use of the Kurucz codes requires a conversion from the depth scale to a RHOX (or ) depth scale, where is the density of the stellar gas and x is the geometrical height in the atmosphere. The conversion was obtained by computing the continuous opacity at 5000 Å by means of the ATM code from Holweger, Steffen & Steenbock (1992, private communication) and by deriving RHOX from the relation = . The original HM model was made more than a quarter of a century ago. Since that time, abundances and the continuous opacity routines have been modified, presumably for the better. This means that the current relation between and is no longer the same as in the HM paper. The latter is inconsistent with the RHOX scale of the modern Kurucz codes.
We adopted as solar abundances the meteoritic values from Grevesse & Sauval (1998) and a constant microturbulent velocity 1 km s-1.
The HM model used in the Kurucz codes is given in the Appendix A.
|Figure 4: Comparison of the solar intensity from the center of the sun predicted by the HM model (full line) with the observations from Neckel & Labs (1984) (dashed line). The line opacity in this low-resolution calculation is entirely from the ODFs.|
For clarity, we first list several categories of opacity relevant to the current problems:
When the opacity of both classified and unclassified lines is considered in the calculations, the agreement of the low resolution observations with the low resolution predictions seems to be rather good at the first glance. However, a closer inspection shows that the observed and computed pseudo-continuum levels agree well in the regions 4200-4500 Å and 5700-6600 Å, but that elsewhere the computed intensity is systematically larger than the observed one, with differences of the order of 5-10%. This disagreement may indicate that either the observed low-resolution central intensity is affected by uncertainties larger than the estimated limit of 1% (Neckel & Labs 1984), or that the HM model should be refined, or that the problem of the missing opacity has not been completely solved.
As far as observations are concerned we would like to remark that the absolute integrals of the solar disk-center intensity measured by Burlov-Vasiljev et al. (1995) are higher by about 6% than that of Neckel & Labs (1984) at H, 4% at H, 2% at H, while it is about 2% lower at H. Burlov-Vasiljev et al. (1995) estimated errors from 2.5% at 3100 Å to 2.2% at 6800 Å. This implies that the different levels of the observations at the position of H and H are outside the error limits.
In Sect. 5.3 we will show that the HM model produces almost the same discrepancy as the theoretical solar Kurucz model does when high-resolution observed and computed Balmer profiles, unnormalized to the continuum level are compared.
Section 5.4 deals with the effects of the missing opacity on the Balmer profiles. Its nature is somewhat controversial, and will not be argued here. A recent reference, with citations to earlier discussion, is Peterson et al. (2001).
Limb darkening predictions from the HM model are compared in Fig. 5 with those from Neckel & Labs (1994). In this case, opacity from lines is not included in the computations in accordance with the assumption of Neckel & Labs (1994) of observations made at wavelengths free from lines contaminating the continuum. The departure of the computations from the observations in the violet can be explained with the poor chance to have regions free from lines in this part of the solar spectrum. Except for the violet wavelengths, the agreement is satisfactory.
|Figure 5: Comparison between observed (points) and computed (full line) solar limb-darkening curves ( )/ (0). Observations are from Neckel & Labs (1994) and computed curves are based on the HM model.|
Figure 6 shows the observed and computed Balmer profiles for the disk center in absolute intensity. We have adopted the Kitt Peak observations available at the Hamburg site (ftp.hs.uni-hamburg.de; pub/outgoing/FTS-Atlas) and described by Neckel (1999, henceforth, KPN). The files include absolute intensities, as well as continuum estimates at each wavelength. The resolution of the observations is about 350 000.
The synthetic Balmer profiles were computed with the SYNTHE code and the HM model. Two different spectra were computed, the first only with the relevant Balmer line, the second one with all classified and unclassified lines. For both spectra standard continuous opacity sources were used. The second synthetic spectrum is computed with the same line opacity adopted for computing ODFs, so that it can be directly compared with the intensity from the center of the sun predicted by the ATLAS9 code and the HM model. Each synthetic spectrum was degraded at the observed resolution and it was broadened by assuming a macroturbulent velocity 1.5 km s-1, although Balmer profiles are independent of instrumental and macroturbulence broadenings of the order of those here adopted.
Figure 6 shows that, in agreement with Fig. 4, the observations fall below the calculated profiles, especially for H and H. The differences are very small for H, i.e. less than 1%, but they are of the order of 5% for H, 4% for H, and 8% for H. This result is very similar to that obtained by CGK from the theoretical solar Kurucz model (Fig. 7 in Castelli et al. 1997), indicating that the discrepancy is rather independent of the specific solar model adopted for the computations.
The two synthetic spectra plotted in Fig. 6 indicate that the high points of the calculation including all lines generally reach the profile where only the Balmer line is included. Therefore the difference between the observed and computed intensity levels is not resolved by the inclusion of all classified and unclassified lines in the calculation. A reasonable interpretation is that the majority of the opacity from the unclassified lines is seen as relatively strong features that appear as absorption lines rather than a smooth pseudo-continuum or veil of weak features. We conclude that a direct comparison of theory and observation in absolute units cannot be made unless this discrepancy is taken into account. We do this in a crude way in the following section, where we used Balmer profiles normalized to the continuum levels in order to avoid all the uncertainties related with absolute calibration of the observed solar intensity from the disk center.
|Figure 6: Two unnormalized to the continuum level (0) computed spectra (thin lines) are shown in each panel, with (1) only the relevant Balmer line, and (2) all classified and unclassified lines. The unnormalized observed (0) spectrum (KPN, thick line) generally falls below the computed spectra. The y scale gives (0) intensities in units of 106 erg cm-2 s-1 stear-1 Å-1, which have to be multiplied by 3.2 for H, 5.0 for for H and H, and 5.5 for H.|
|Wavelength (Å)||This work||KPN|
In the current work, one of us (CRC)
attempted new estimates of the continuum for the observed spectrum - less as an attempt to
improve on the KPN values, as to gain some insight into the
uncertainties in this endeavor. We began with spectral high
points within 10 Å intervals plotted vs. wavelength, and
smoothed the "envelope'' by selectively deleting points, in an
obviously subjective way, to achieve an overall smooth plot.
points are shown in Table 2, along with those
from KPN. We make no claim that the current continuum is
superior in any way to that chosen in KPN. It was simply used
in the Michigan work for normalization purposes. We employed a
four-point Lagrange interpolation scheme to normalize observations
between the chosen points.
|Figure 7: H profile for the center of the solar disk normalized to the continuum level. The thin curve is the observed KPN spectrum, and the solid the CSII calculation with an assumed microturbulence km s-1. In this calculation no allowance for missing opacity has been made, and the continuum has been adopted as described.|
Our independent evaluation of the continuum based on the points shown in Table 2 is in excellent agreement with KPN, with the exception of the region near H. The value shown in Col. 2 for 4861 interpolated with the four-point Lagrange formula, from the surrounding points, is 1.2% higher than the KPN continuum. This region appears depressed for reasons that are unclear and deserve investigation.
The continuous specific intensity using the HM model and Michigan codes matches the interpolated continuum from Table 2 at H to within 1%. For H through H, the calculated continua fall above the measured (as interpolated in Table 2) continua by 2.4, 3.9, and 7.8% respectively. These results agree well with those discussed in the previous section of the comparison of the observed and computed absolute intensities.
If we assume the "missing opacity'' as cause for these disagreements as well as for those shown in Fig. 6, there is at present no obviously correct way to account for it. For these calculations, we assumed this opacity has the same depth dependence as standard continuous opacity sources. We have simply scaled them by constant factors until the calculated specific continuous intensities agree with the observed chosen continuum.
When spectra normalized to the continuum levels are compared, we find an excellent agreement for H (Fig. 7). The results are the same both from the CSII and the SYNTHE code, and are to be compared with BPO's Fig. 8 (upper), done for the solar flux. We see good agreement in all cases. The agreement of the CSII profiles with BPO profiles is expected, since the only basic difference is the use in BPO of the PK approximation while CSII uses a full numerical convolution, a distinction we have found thus far to be unimportant.
As far as the three higher, normalized Balmer lines are concerned, the best fits to the wings are obtained when the "observed'' continua are adjusted downward from values obtained by interpolation in Table 2 - the sense is that the continuum there is too high. For H and H, the downward adjustment is 2%. The observed continuum at H needed a downward adjustment of 3%; problems with the continuum in this region were mentioned earlier in this section. Figure 8 shows the fit for H. The other two Balmer line fits may be seen at the url: http://www.astro.lsa.umich.edu/users/cowley/balmers.html/
In principle, the adjustment of the continuum requires an iteration with a new continuous opacity to the new continuum. Fortunately, the normalized Balmer profiles are not very sensitive to small adjustments for the missing opacity.
For perhaps a century we have known that the spectrum of the solar photosphere varies from one point on the disk to another. The first high-resolution spectra obtained from the McMath-Hulbert Observatory showed striking spatial variations that came to be known as "wiggley lines''. The solar line profiles vary markedly, both in time and space, and while we have understood the general the nature and cause of these variations for decades, recent numerical calculations by Nordlund, Stein, and their collaborators have provided a detailed description (cf. Nordlund & Stein 2001).
In spite of its origin in a turbulent roil, the average line spectrum of the sun is remarkably constant. This is particularly surprising in the case of the Balmer lines, where the large Boltzmann factor ( ) suggests huge local non-linear effects. Naively, one would not expect them to average out, and the extent to which they do average out remains to be fixed.
In the 1950's, de Jager (1952) attempted to fix the temperature fluctuations in the solar atmosphere by making use of the putative nonlinearities of the Balmer lines. His conclusions, of temperature differences of a thousand degrees from hot to cool columns agrees remarkably with modern numerical models. Surely, he was guided by physical insight into what the answer needed to be. The Stark-broadening theory of that time was rudimentary, and the influence of collisions with neutral hydrogen were entirely neglected.
We have found that reasonable matches to the four lower Balmer lines can be achieved using modern Stark profiles provided recent parameters for broadening by neutral hydrogen by BPO and the HM model are used. In fact, the fits illustrated in Figs. 7 and 8, were all based on the empirical plane-parallel Holweger-Müller model, and include no attempts to improve the fits by plausible adjustments of the line-broadening parameters. Other studies have explored the sensitivity of the Balmer lines to different theoretical model atmospheres and to variations in the convective mixing length to the pressure scale height (l/H).
We remark here on the surprising linearity of the Balmer profiles with the temperature of plane-parallel models. This may be illustrated in several ways. In Fig. 1 we can see that for about 4000 K to 6250 K the wing strengths plot nearly linearly with temperature for the three higher gravities. This near linearity holds for most points on the line profiles, apart from the most central portions. If one takes an equally weighted average of H fluxes from Kurucz models with = 5500 K and 6500 K, the resulting mean differs imperceptibly from that for a = 6000 K model. Means for = 5000 K and 7000 K models differ only by 2% from the = 6000 K model beyond 3 Å from the line center. Even for the mean of = 4500 K and 7500 K models the difference is of the order of 5% (see Fig. 9).
|Figure 9: Percentage differences in H profiles for 6000 K model and average profiles for three pairs of models as indicated (H profiles from Kurucz 1993a).|
The same effect may be seen in the left panel of Fig. 3 of Fuhrmann et al. (1993). They show a series of Balmer profiles from H through Hfor = 4, with effective temperatures running from 5000 K to 6700 K, in steps of 100 K. It can be seen that the different profiles are, for the most part, quite evenly spaced.
The simple means of Fig. 9 are certainly not equivalent to the detailed calculation performed, for example, by Asplund et al. (1999), based on the 3-dimensional numerical models of the solar convection zone. Nevertheless, they demonstrate that the non-linearities that one might expect from the very large Boltzmann factors of the n = 2 level are not realized in the resultant Balmer profiles of cool stars. This, in turn, supports endeavors to use theoretical profiles from simplified stellar models to help fix fundamental stellar parameters.
We have explored recent techniques for computing Balmer line profiles in the sun, and H profiles in several models with effective temperatures ranging from 4500 K to 12 000 K. We find that new Stark profiles, rigorous convolution, and improved interpolation techniques make almost no difference in the resulting calculated profiles, compared with algorithms used in the Kurucz codes for several decades.
Good fits to normalized disk center solar profiles for the H through H are obtained from the Holweger-Müller (HM) model, provided that some adjustment of the computed continuum is performed according to the hypothesis of missing opacity.
The H profile can also be reasonably fitted in absolute intensity, but the calculated continua for H through H are too high. This may reasonably be attributed to missing UV opacity, perhaps also to inadequacies of the HM model used here, as well as to uncertainties in the absolute solar calibration.
In spite of severe temperature inhomogeneities in the solar atmosphere, the plane-parallel model appears remarkably robust.
Numerous scientific colleagues have kindly consulted with us on various parts of this project, and we will doubtless omit some unintentionally. For this we apologize. Explicit thanks are due to Drs. P. Barklem, N. Grevesse, K. Fuhrmann, R. L. Kurucz, M. Lemke, H. Neckel, J. Sauval, B. Smalley, and C. Stehlé.
|-6.54||6.133 10-4||4.956 10-11||4.7137 10-4||3900||1.274 101||2.448 109|
|-6.39||6.606 10-4||6.959 10-11||6.6340 10-4||3910||1.791 101||3.394 109|
|-6.23||7.200 10-4||9.569 10-11||9.1603 10-4||3924||2.472 101||4.605 109|
|-6.08||7.927 10-4||1.294 10-10||1.2441 10-3||3939||3.357 101||6.146 109|
|-5.93||8.833 10-4||1.723 10-10||1.6643 10-3||3960||4.487 101||8.096 109|
|-5.77||9.899 10-4||2.262 10-10||2.1990 10-3||3988||5.943 101||1.055 1010|
|-5.62||1.119 10-3||2.933 10-10||2.8759 10-3||4022||7.762 101||1.366 1010|
|-5.47||1.278 10-3||3.772 10-10||3.7234 10-3||4052||1.007 102||1.747 1010|
|-5.31||1.457 10-3||4.085 10-10||4.7816 10-3||4084||1.291 102||2.218 1010|
|-5.16||1.673 10-3||6.072 10-10||6.0935 10-3||4120||1.648 102||2.793 1010|
|-5.01||1.931 10-3||7.617 10-10||7.7163 10-3||4159||2.084 102||3.507 1010|
|-4.85||2.239 10-3||9.523 10-10||9.7130 10-3||4188||2.624 102||4.365 1010|
|-4.70||2.599 10-3||1.183 10-9||1.2160 10-2||4220||3.289 102||5.415 1010|
|-4.55||3.022 10-3||1.463 10-9||1.5156 10-2||4255||4.102 102||6.684 1010|
|-4.39||3.528 10-3||1.803 10-9||1.8818 10-2||4286||5.093 102||8.221 1010|
|-4.24||4.118 10-3||2.214 10-9||2.3278 10-2||4317||6.295 102||1.006 1011|
|-4.09||4.814 10-3||2.713 10-9||2.8713 10-2||4349||7.762 102||1.229 1011|
|-3.93||5.630 10-3||3.313 10-9||3.5330 10-2||4382||9.572 102||1.497 1011|
|-3.78||6.572 10-3||4.038 10-9||4.3379 10-2||4415||1.175 103||1.824 1011|
|-3.63||7.703 10-3||4.914 10-9||5.3175 10-2||4448||1.439 103||2.212 1011|
|-3.47||9.024 10-3||5.975 10-9||6.5074 10-2||4477||1.762 103||2.679 1011|
|-3.32||1.059 10-2||7.256 10-9||7.9518 10-2||4506||2.153 103||3.237 1011|
|-3.17||1.241 10-2||8.797 10-9||9.7037 10-2||4536||2.630 103||3.911 1011|
|-3.02||1.453 10-2||1.065 10-8||1.1833 10-1||4568||3.206 103||4.723 1011|
|-2.86||1.706 10-2||1.290 10-8||1.4414 10-1||4597||3.899 103||5.682 1011|
|-2.71||2.000 10-2||1.561 10-8||1.7550 10-1||4624||4.753 103||6.822 1011|
|-2.56||2.347 10-2||1.888 10-8||2.1346 10-1||4651||5.781 103||8.192 1011|
|-2.40||2.749 10-2||2.281 10-8||2.5958 10-1||4681||7.031 103||9.854 1011|
|-2.25||3.221 10-2||2.753 10-8||3.1560 10-1||4716||8.551 103||1.184 1012|
|-2.10||3.776 10-2||3.321 10-8||3.8363 10-1||4754||1.040 104||1.429 1012|
|-1.94||4.418 10-2||3.998 10-8||4.6626 10-1||4799||1.262 104||1.729 1012|
|-1.79||5.171 10-2||4.814 10-8||5.6680 10-1||4846||1.535 104||2.092 1012|
|-1.64||6.053 10-2||5.782 10-8||6.8889 10-1||4903||1.866 104||2.544 1012|
|-1.48||7.103 10-2||6.942 10-8||8.3714 10-1||4964||2.270 104||3.105 1012|
|-1.33||8.318 10-2||8.311 10-8||1.0172 100||5040||2.754 104||3.824 1012|
|-1.18||9.787 10-2||9.934 10-8||1.2355 100||5122||3.350 104||4.726 1012|
|-1.02||1.157 10-1||1.184 10-7||1.4988 100||5217||4.064 104||5.909 1012|
|-0.87||1.374 10-1||1.410 10-7||1.8150 100||5308||4.920 104||7.396 1012|
|-0.72||1.651 10-1||1.669 10-7||2.1921 100||5416||5.957 104||9.425 1012|
|-0.56||2.054 10-1||1.950 10-7||2.6321 100||5567||7.145 104||1.263 1013|
|-0.41||2.756 10-1||2.225 10-7||3.1174 100||5781||8.472 104||1.875 1013|
|-0.26||4.009 10-1||2.470 10-7||3.6118 100||6032||9.817 104||3.037 1013|
|-0.10||6.179 10-1||2.667 10-7||4.0810 100||6315||1.109 105||5.255 1013|
|0.05||9.745 10-1||2.812 10-7||4.5088 100||6617||1.227 105||9.274 1013|
|0.20||1.471 100||2.933 10-7||4.9028 100||6902||1.334 105||1.545 1014|
|0.35||2.392 100||2.988 10-7||5.2624 100||7266||1.432 105||2.810 1014|
|0.51||3.978 100||2.988 10-7||5.5724 100||7679||1.517 105||5.172 1014|
|0.66||6.175 100||2.979 10-7||5.8448 100||8059||1.592 105||8.643 1014|
|0.81||8.426 100||3.007 10-7||6.1102 100||8335||1.663 105||1.225 1015|
|0.97||1.018 101||3.084 10-7||6.4090 100||8500||1.746 105||1.512 1015|