A&A 387, 595-604 (2002)
DOI: 10.1051/0004-6361:20020421
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
Abstract
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 H
are 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 H
profile 4 Å from the line center, but is characteristic of
much of the line profile.
![]() |
Figure 1:
H![]() ![]() ![]() |
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
![]() ![]() ![]() ![]() |
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.
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Comment |
4500 | 1.5 | 3.0 | solar abundances |
4760 | 1.3 | 2.3 | CS22892-052 (cf. Sneden et al. 1996) |
5770 | 4.4 | 1.0 | Sun |
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 |
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
![]() ![]() ![]() |
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.
Wavelength (Å) | This work | KPN |
3298.973 | 0.3235 | 0.3231 |
3355.431 | 0.3269 | 0.3272 |
3782.919 | 0.4083 | 0.4093 |
4020.705 | 0.4589 | 0.4591 |
4279.262 | 0.4652 | 0.4666 |
4419.404 | 0.4598 | 0.4609 |
4504.079 | 0.4540 | 0.4545 |
4861.000 | 0.4230 | 0.4179 |
5102.095 | 0.3999 | 0.3990 |
5203.252 | 0.3906 | 0.3902 |
5801.460 | 0.3435 | 0.3424 |
6109.561 | 0.3200 | 0.3189 |
6202.178 | 0.3146 | 0.3144 |
6409.847 | 0.2990 | 0.2972 |
6500.584 | 0.2907 | 0.2899 |
6802.324 | 0.2660 | 0.2663 |
6850.076 | 0.2619 | 0.2627 |
6950.356 | 0.2546 | 0.2553 |
6972.875 | 0.2536 | 0.2540 |
7000.000 | 0.2524 | 0.2524 |
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.
The adopted
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.
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![]() ![]() |
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 H
for
= 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.
Acknowledgements
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.
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RHOX | T | Pgas | Ne |
-6.54 | 6.133 ![]() |
4.956 ![]() |
4.7137 ![]() |
3900 | 1.274 ![]() |
2.448 ![]() |
-6.39 | 6.606 ![]() |
6.959 ![]() |
6.6340 ![]() |
3910 | 1.791 ![]() |
3.394 ![]() |
-6.23 | 7.200 ![]() |
9.569 ![]() |
9.1603 ![]() |
3924 | 2.472 ![]() |
4.605 ![]() |
-6.08 | 7.927 ![]() |
1.294 ![]() |
1.2441 ![]() |
3939 | 3.357 ![]() |
6.146 ![]() |
-5.93 | 8.833 ![]() |
1.723 ![]() |
1.6643 ![]() |
3960 | 4.487 ![]() |
8.096 ![]() |
-5.77 | 9.899 ![]() |
2.262 ![]() |
2.1990 ![]() |
3988 | 5.943 ![]() |
1.055 ![]() |
-5.62 | 1.119 ![]() |
2.933 ![]() |
2.8759 ![]() |
4022 | 7.762 ![]() |
1.366 ![]() |
-5.47 | 1.278 ![]() |
3.772 ![]() |
3.7234 ![]() |
4052 | 1.007 ![]() |
1.747 ![]() |
-5.31 | 1.457 ![]() |
4.085 ![]() |
4.7816 ![]() |
4084 | 1.291 ![]() |
2.218 ![]() |
-5.16 | 1.673 ![]() |
6.072 ![]() |
6.0935 ![]() |
4120 | 1.648 ![]() |
2.793 ![]() |
-5.01 | 1.931 ![]() |
7.617 ![]() |
7.7163 ![]() |
4159 | 2.084 ![]() |
3.507 ![]() |
-4.85 | 2.239 ![]() |
9.523 ![]() |
9.7130 ![]() |
4188 | 2.624 ![]() |
4.365 ![]() |
-4.70 | 2.599 ![]() |
1.183 ![]() |
1.2160 ![]() |
4220 | 3.289 ![]() |
5.415 ![]() |
-4.55 | 3.022 ![]() |
1.463 ![]() |
1.5156 ![]() |
4255 | 4.102 ![]() |
6.684 ![]() |
-4.39 | 3.528 ![]() |
1.803 ![]() |
1.8818 ![]() |
4286 | 5.093 ![]() |
8.221 ![]() |
-4.24 | 4.118 ![]() |
2.214 ![]() |
2.3278 ![]() |
4317 | 6.295 ![]() |
1.006 ![]() |
-4.09 | 4.814 ![]() |
2.713 ![]() |
2.8713 ![]() |
4349 | 7.762 ![]() |
1.229 ![]() |
-3.93 | 5.630 ![]() |
3.313 ![]() |
3.5330 ![]() |
4382 | 9.572 ![]() |
1.497 ![]() |
-3.78 | 6.572 ![]() |
4.038 ![]() |
4.3379 ![]() |
4415 | 1.175 ![]() |
1.824 ![]() |
-3.63 | 7.703 ![]() |
4.914 ![]() |
5.3175 ![]() |
4448 | 1.439 ![]() |
2.212 ![]() |
-3.47 | 9.024 ![]() |
5.975 ![]() |
6.5074 ![]() |
4477 | 1.762 ![]() |
2.679 ![]() |
-3.32 | 1.059 ![]() |
7.256 ![]() |
7.9518 ![]() |
4506 | 2.153 ![]() |
3.237 ![]() |
-3.17 | 1.241 ![]() |
8.797 ![]() |
9.7037 ![]() |
4536 | 2.630 ![]() |
3.911 ![]() |
-3.02 | 1.453 ![]() |
1.065 ![]() |
1.1833 ![]() |
4568 | 3.206 ![]() |
4.723 ![]() |
-2.86 | 1.706 ![]() |
1.290 ![]() |
1.4414 ![]() |
4597 | 3.899 ![]() |
5.682 ![]() |
-2.71 | 2.000 ![]() |
1.561 ![]() |
1.7550 ![]() |
4624 | 4.753 ![]() |
6.822 ![]() |
-2.56 | 2.347 ![]() |
1.888 ![]() |
2.1346 ![]() |
4651 | 5.781 ![]() |
8.192 ![]() |
-2.40 | 2.749 ![]() |
2.281 ![]() |
2.5958 ![]() |
4681 | 7.031 ![]() |
9.854 ![]() |
-2.25 | 3.221 ![]() |
2.753 ![]() |
3.1560 ![]() |
4716 | 8.551 ![]() |
1.184 ![]() |
-2.10 | 3.776 ![]() |
3.321 ![]() |
3.8363 ![]() |
4754 | 1.040 ![]() |
1.429 ![]() |
-1.94 | 4.418 ![]() |
3.998 ![]() |
4.6626 ![]() |
4799 | 1.262 ![]() |
1.729 ![]() |
-1.79 | 5.171 ![]() |
4.814 ![]() |
5.6680 ![]() |
4846 | 1.535 ![]() |
2.092 ![]() |
-1.64 | 6.053 ![]() |
5.782 ![]() |
6.8889 ![]() |
4903 | 1.866 ![]() |
2.544 ![]() |
-1.48 | 7.103 ![]() |
6.942 ![]() |
8.3714 ![]() |
4964 | 2.270 ![]() |
3.105 ![]() |
-1.33 | 8.318 ![]() |
8.311 ![]() |
1.0172 ![]() |
5040 | 2.754 ![]() |
3.824 ![]() |
-1.18 | 9.787 ![]() |
9.934 ![]() |
1.2355 ![]() |
5122 | 3.350 ![]() |
4.726 ![]() |
-1.02 | 1.157 ![]() |
1.184 ![]() |
1.4988 ![]() |
5217 | 4.064 ![]() |
5.909 ![]() |
-0.87 | 1.374 ![]() |
1.410 ![]() |
1.8150 ![]() |
5308 | 4.920 ![]() |
7.396 ![]() |
-0.72 | 1.651 ![]() |
1.669 ![]() |
2.1921 ![]() |
5416 | 5.957 ![]() |
9.425 ![]() |
-0.56 | 2.054 ![]() |
1.950 ![]() |
2.6321 ![]() |
5567 | 7.145 ![]() |
1.263 ![]() |
-0.41 | 2.756 ![]() |
2.225 ![]() |
3.1174 ![]() |
5781 | 8.472 ![]() |
1.875 ![]() |
-0.26 | 4.009 ![]() |
2.470 ![]() |
3.6118 ![]() |
6032 | 9.817 ![]() |
3.037 ![]() |
-0.10 | 6.179 ![]() |
2.667 ![]() |
4.0810 ![]() |
6315 | 1.109 ![]() |
5.255 ![]() |
0.05 | 9.745 ![]() |
2.812 ![]() |
4.5088 ![]() |
6617 | 1.227 ![]() |
9.274 ![]() |
0.20 | 1.471 ![]() |
2.933 ![]() |
4.9028 ![]() |
6902 | 1.334 ![]() |
1.545 ![]() |
0.35 | 2.392 ![]() |
2.988 ![]() |
5.2624 ![]() |
7266 | 1.432 ![]() |
2.810 ![]() |
0.51 | 3.978 ![]() |
2.988 ![]() |
5.5724 ![]() |
7679 | 1.517 ![]() |
5.172 ![]() |
0.66 | 6.175 ![]() |
2.979 ![]() |
5.8448 ![]() |
8059 | 1.592 ![]() |
8.643 ![]() |
0.81 | 8.426 ![]() |
3.007 ![]() |
6.1102 ![]() |
8335 | 1.663 ![]() |
1.225 ![]() |
0.97 | 1.018 ![]() |
3.084 ![]() |
6.4090 ![]() |
8500 | 1.746 ![]() |
1.512 ![]() |