M. Briquet 1 - C. Aerts 2
1 - Institut d'Astrophysique et de Géophysique de Liège, Université
de Liège, allée du Six Août 17, 4000 Liège, Belgium
2 - Instituut
voor Sterrenkunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001
Leuven, Belgium
Received 26 September 2002 / Accepted 12 November 2002
Abstract
We present a numerical version of the moment method for the
identification of non-radial pulsation modes. The new version requires less
computation time than the previous one, allowing users to consider all the information
contained in the first three moments of a multiperiodic star and to identify
multiple modes simultaneously. This, together with the use of a new discriminant
that considers the moments calculated at each time of observation, increases
considerably the feasibility and the accuracy of the mode identification for
multiperiodic stars. Moreover, the technique is extended to rotating pulsating
stars. We apply the new version to three B stars showing multiperiodicity.
Key words: line: profiles - stars: oscillations - stars: individual:
Crucis, 16 Lacertae, HD 74195
Recent studies of multiple data-sets of non-radial oscillators have shown the need to improve current mode identification methods. In particular, the outcomes of mode identification based on photometric and spectroscopic data of the same star and for the same mode are often discrepant (e.g. De Cat 2001). This is very unsatisfactory if one wants to use the non-radial oscillations for detailed modelling of the internal structure of the stars. The current paper contributes to the improvement of one of the identification methods which is developed for the interpretation of time-series of high-resolution spectroscopic data.
The natural way of mode identification from line-profile variations is the
method of line-profile fitting. By comparing the observed line-profile
variations with theoretically calculated ones for many wavenumbers and for a large grid of the other continuous parameters that occur in the
velocity expression of a non-radial mode, one chooses the modes which best fit
the observations. Unfortunately, this technique suffers from a major drawback: the
unrealistic computation time for multiple modes.
The idea of the moment method is to replace each line profile by its first three
moments, which describe respectively the centroid velocity of the line, the line
width and the line skewness. The wavenumbers
and the other continuous
velocity parameters are then determined in such a way that the theoretically
computed moment variations best fit the observed ones. The moment method was
first introduced by Balona (1986a,b, 1987)
and was further developed by
Aerts et al. (1992) and Aerts (1996).
They derived analytical expressions of
the first three moment variations, which are valid for stars with a long
rotational period compared to the pulsational periods (
). The major drawback of the moment method in the
formulation by Aerts (1996) is the fact that all the modes are identified
separately in the case of multiple modes. With such an approach, one cannot
force there to be only one unique solution for the inclination angle of the star
and the stellar rotational velocity. This is rather unsatisfactory, as
conflicting values of these parameters are quite often encountered in practical
applications of the method. The reason of Aerts (1996) for not performing one
unified mode identification for all occurring modes simultaneously was of
numerical origin, as it resulted in unrealistic computation times. This is no
longer the case with the efficiency of current computers and with some clever
re-arrangements of the theoretical moment expressions. In this paper, we
present a numerical version of the moment method, which improves considerably
the efficiency of the technique by performing the simultaneous identification of
all the modes that are present in the data.
The paper is organized as follows. In Sect. 2, we describe the new version of
the method. Tests on synthetic data are performed in Sect. 3. Section 4 provides
a summary of the best optimal mode identification strategy. In Sect. 5, we
apply our new method to three multiperiodic B-type stars:
Crucis, 16
Lacertae and HD 74195. Finally, we give conclusions in Sect. 6.
In order to be able to outline our new application of the moment method, we repeat here its basic ingredients, adapted to our numerical approach. The reader is referred to Aerts et al. (1992) and Aerts (1996) for a more in-depth description of the moment method.
Let v be the observed velocity (in km s-1) of a point on the stellar
surface. The nth normalized moment of a line profile
I(v,t) is defined as
A line profile is the convolution of an intrinsic profile g and the component
of the velocity field f. We assume that g is time-independent and symmetric,
which is a good approximation. The first three moments can then be written as
Let
and
be respectively the velocity in the line
of sight due to rotation and pulsation. We use a system of spherical coordinates
whose polar axis coincides with the axis of rotation and we
consider a star which rotates uniformly. We assume also a constant intensity
over the stellar surface and we use a limb-darkening law of the form
where
is the angle
between the local radial vector and the line of sight. We divide the stellar
surface into surface elements by taking steps of d
in
and
d
in
.
One then has the following approximation
We note that we neglect temperature effects on moment variations. For pulsating
B stars this is justified, as the moment variations are very well approximated
with a pulsational velocity field only. Indeed, Dupret et al. (2002) and De
Ridder et al. (2002a) showed that the line-profile variations of Silicon lines
in slowly-rotating non-radially pulsating
Cephei stars and SPB stars are
very little affected by temperature variations at the surface of the star.
For linear theory, the general form for the temporal dependence of the
projected pulsation velocity field due to N modes is given by
In Eqs. (3) and (4), the quantities
,
,
and
are given by
For the identification of the modes, we only use the first three moments <v>, <v2> and <v3> because the higher order observed moments are often too noisy. Moreover, the first three moments suffice to determine correctly the pulsation modes (Aerts et al. 1992).
From Eqs. (1) and (2) and after deleting the terms that equal zero due to symmetry properties, we have
We point out that <v> varies with ;
<v2> varies with
and
and also with the coupling frequencies
and
;
<v3> varies with
,
,
and also with coupling frequencies
,
,
,
,
,
for
.
All these coupling characteristics were also already given in Mathias et al. (1994), but were never implemented so far.
From Eqs. (3) and (4) and by denoting
The great advantage of such a writing is that the integrals I[x] can be
computed and memorized in files once and for all for chosen wavenumbers
and inclinations i. This leads to an important gain of computation
time in the computation of the moments for different values of the other
parameters
,
K,
and the projected rotational velocity
.
We point out that, for chosen
,
the amplitude
is no
longer a free parameter. We limit its range by imposing that the theoretical
first moment amplitude
must be equal
to the observed one
.
This condition, which was not considered by
Aerts (1996), allows us to reduce greatly the grid of tested parameters. Such an
approach is fully justified, as the relative standard error of
is
always much smaller than the ones of any of the other amplitudes of the
higher-order moments.
In the version of Aerts (1996), the mode identification is achieved by comparing
the theoretically calculated amplitudes of <v>, <v2> and <v3> with the
observed ones through a discriminant. In general, the observed amplitudes of the
first moment as well as the constant term of the second moment are determined
accurately. However, the other observed moment amplitudes can have large
uncertainties. For this reason, we prefer using the moment values calculated at
each time of observation tk (
)
instead of the
amplitudes of their fit.
We propose to choose the modes and the parameters for which the following
new discriminant attains the lowest value
The moment method is particularly suited to identify modes whose frequency is
clearly present in the observed moments. Consequently, the degrees of the modes
corresponding to these frequencies are expected to be relatively low
(
).
In what follows, we give a simple criterion allowing to derive an upper limit
for
using the observed first moment amplitude
and the
constant term of the observed second moment
,
which can both be
very accurately determined from the observed line-profile variations. By
imposing the theoretical positive constant E12 to be smaller than
,
we have the following condition:
The technique described above is no more restricted to slow rotators as was the
case for the method by Aerts (1996). Here we extend the application to rotating
pulsating stars by using the theory derived by Lee & Saio (1987). We remind
that the velocity field depends on the ratio
between the angular frequency of rotation and the angular corotating frequency
of pulsation. Note that taking
leads to the same
description as the one of a non-rotating pulsating star.
We implemented this version of the moment method by using Townsend's code (1997) BRUCE, which computes the pulsation velocity field for this theory. A nice feature of our current version of the method is that this numerical version can be easily generalized to an improved formalism for the pulsational velocity, e.g. one that would take into account the effects of the centrifugal forces, should this become available.
We applied the moment method in our present new version to a large number of synthetic data sets in order to test its efficiency and our implemented version. Sets of artificial data were generated in such a way that they resemble real data as much as possible. For each set, we computed 254 line profiles at times of real observations. We added gaussian noise corresponding to a signal-to-noise of about 200 and finally we computed the first three moment variations.
We first tested the method for a monoperiodic star for all the sectoral,
tesseral, and axisymmetric modes with
3
(
). It performs the identification without any problem,
as it was already the case with the previous version (Aerts 1996). We note that,
as expected, axisymmetric modes may be confused with axisymmetric modes of
another degree for low values of the inclination, because of their very similar
visible configuration.
We then tested 144 combinations of two modes including all kinds of modes until
.
The value for
was randomly chosen between 10 and 30 km s-1 and
was taken equal to 5.5 km s-1.
was chosen
so that the value of the radial velocity amplitude is compatible with observed
values for
Cephei stars and SPB stars. Each time, the velocity amplitude
of the second mode was chosen so that its first moment amplitude is smaller than
the one of the first dominant mode. The K-values were also taken according to
typical values of these two kinds of B-type oscillators. In Tables 1-4, which are only available electronically, we
show the values of the input parameters for several sets, together with the five
best solutions of the mode identification. We can conclude that the method
performs the identification very well. However, sometimes, the real combination
of the two modes does not correspond to the lowest value of the
discriminant but appears in the list of the few best solutions. In general, the
estimates for the continuous parameters
are
good, although large deviations do sometimes occur. Such failing of estimating
these parameters well was already put forward by De Ridder et al. (2002b).
For evident computational reasons, we made less tests for three modes than for two. Some examples are given in Tables 5-8, which are again only available electronically. Also for these testcases the identifications are very conclusive, while the same remark as above for the continous parameters applies here.
We strongly encourage users of our mode identifcation method to study Tables 1-8 in order to obtain a feeling of the accuracy and the power, but also of the limitations, of the identification method.
A large number of tests indicates that the first three moments suffice to identify the modes correctly. In Appendix B, we show that it is indeed not necessary to add higher order moments. The computation time required to calculate the discriminant depends very much on the number of modes. It ranges from about half an hour for a monoperiodic star to several days for stars with three modes (with a Pentium 4, 2.4 GHz / 512 Mb RAM). We note that, with the previous version of the moment method, the identification of one mode required several days of computation, and the identification of three modes simultaneously would have required several months or more.
It is evident from Tables 1-8 that, in many cases, one clear combination of different wavenumbers does not occur from the discriminant, as several solutions are almost equivalent in fitting the moment variations. This situation is inherent to the problem of mode identification, with whatever method, as several combinations of the velocity parameters result in almost the same line-profile variations. The strength of the moment method is precisely that it allows one to severely restrict the number of possible solutions.
One would hope that a statistically justified test would inform the user how
many of the candidate modes can be rejected with safety from comparison of the
different values of .
At present, such a significance test is still
lacking. The main difficulty lies in defining a suitable test for the
combination of discrete and continuous parameters. De Ridder et al. (2002b)
have taken the first steps in this direction for a monoperiodic pulsation.
In view of the lack of a significance test, one needs to evaluate the "few''
best solutions resulting from the moment method, by constructing theoretical
line-profile variations and by comparing these with the observed ones. One can
do this for the profiles themselves, or else for their variation of amplitude
and phase across the profile (for a definition of the latter diagnostic values,
we refer to Schrijvers et al. 1997). The value of a "few'' depends on the
complexity of the pulsation. We advise the user to look at at least the 5 best for a monoperiodic star, the 10 best combinations of the wavenumbers for a
biperiodic oscillator and so on. A recent example of such a procedure is
provided in Aerts et al. (2002) for the
Cep star EN Lac - see also
below. We strongly urge users of our method not to omit this last step. We also
stress, however, that, even after such an additional test, ambiguity among the
solutions that survive the test will still remain, i.e. one single outcome will
seldomly be reached. However, the number of possibilities will have decreased
significantly, to such an extent that seismic modelling can be tried on the
basis of the remaining accepted combinations of the wavenumbers.
Aerts et al. (1998) presented numerous high signal-to-noise spectroscopic data
of the
Cephei star
Crucis with a total time span of 13 years. They found three frequencies in the moments of the Si III 4553 Å line:
f1=5.2305468 c/d,
f2=5.958666 c/d and
f3=5.472165 c/d. The
corresponding observed first moment amplitudes are
km s-1,
km s-1 and
km s-1.
Note that only the first frequency was known and detected in
photometric observations before their study. A mode identification with the
1996-version of the moment method was performed. It pointed towards
non-axisymmetric and non-radial modes. The mode corresponding to f1 was found
to be a low-degree sectoral mode with
while f2 and f3 clearly
correspond to higher degrees (
or
), explaining why the two
newly found frequencies were not detected photometrically. Meanwhile, these two
additional modes have also been detected clearly in space photometry gathered by
the WIRE satellite (Cuypers et al. 2002), as well as two additional
candidates. It is then clear that this star exhibits multiple non-radial modes
and is hence an interesting asteroseismic target.
(![]() |
(1, 0) | (1, -1) | (0, 0) | (1, -1) | (0, 0) | (1, 0) | (0, 0) | (1, 0) | (0, 0) | (2, 0) |
(![]() |
(4, 2) | (4, 3) | (4, 2) | (4, 2) | (3, 3) | (3, 3) | (3, 2) | (3, 2) | (4, 3) | (4, 2) |
(![]() |
(3, 3) | (3, 3) | (3, 3) | (3, 2) | (2, 2) | (2, 2) | (3, 1) | (3, 1) | (3, 3) | (2, 2) |
![]() |
6.51 | 8.32 | 7.24 | 10.66 | 7.24 | 5.88 | 7.24 | 5.41 | 7.24 | 11.21 |
![]() |
104.70 | 108.18 | 104.70 | 102.16 | 165.87 | 165.87 | 172.12 | 172.12 | 108.18 | 120.19 |
![]() |
36.35 | 9.21 | 36.35 | 7.93 | 15.92 | 15.92 | 13.25 | 13.25 | 9.21 | 11.39 |
i | 35 | 65 | 35 | 45 | 25 | 25 | 10 | 10 | 65 | 30 |
![]() |
15 | 16 | 15 | 19 | 23 | 23 | 13 | 13 | 17 | 14 |
![]() |
19 | 19 | 19 | 19 | 19 | 19 | 11 | 11 | 19 | 17 |
![]() |
5.43 | 5.43 | 5.44 | 5.44 | 5.45 | 5.45 | 5.45 | 5.46 | 5.46 | 5.47 |
In order to validate and/or improve the mode identification done by Aerts et al. (1998), we performed a mode identification by our new optimized version of
the moment method, using the same K-values as adopted by Aerts et al. (1998):
K1=0.028, K2=0.021 and K3=0.025. In doing so, we force the amplitudes of the first moment to be equal to the observed values mentioned above. The outcome, which is given in Table 9, is compatible with the previous one. In particular, we recover the
nature of the main mode and the
higher-degree nature of the two lower-amplitude modes. While Aerts et al. (1998) found
and 4 for respectively f2 and f3, we find the
reverse here. This is not too surprising as these two modes resemble each other
for a view at moderate inclination. The continuous parameter estimates
we find here are intermittent to the three different
values for the three modes in Aerts et al. (1998).
Note that we also performed a mode identification by using Lee & Saio's
formalism (1987). As we could expect, we obtained the same outcome since the
ratio
is only about 0.015 for this star.
Data with a better time spread are needed to check the validity of the best solutions as outlined in Sect. 4, as the beating between the three modes is very badly covered. We therefore cannot yet perform seismic modelling of the star, but the studies by Aerts et al. (1998) and by Cuypers et al. (2002) have pushed this star upward in the list of potential asteroseismic targets for future space missions.
EN (16) Lacertae is one of the most studied Cep stars ever. Lehmann et al. (2001) made for the first time a detailed spectroscopic study of EN (16)
Lacertae, which is an eclipsing and spectroscopic binary. Besides providing
accurate orbital parameters, they recovered and refined the three intrinsic
frequencies known for this star from photometry in their radial-velocity data:
f1=5.91128 c/d,
f2=5.85290 c/d and
f3=5.50279 c/d. The corresponding
observed first moment amplitudes are
km s-1,
km s-1 and
km s-1 (Aerts
et al. 2002). Chapellier et al. (1995) summarized all photometric mode
identification efforts so far and concluded that
,
and
.
The first spectroscopic mode identification for this star was recently done by
Aerts et al. (2002) who considered a subset of 940 high-resolution high S/Nspectra gathered by Lehmann et al. (2001). Our mode identification presented
here, which was done using
K1=0.0824,
K2=0.0841 and
K3=0.0951, is one
of the identification results adapted by Aerts et al. (2002), who considered
also other spectroscopic diagnostics for identification. Our result
is given in Table 10.
(![]() |
(0, 0) | (0, 0) | (0, 0) | (1, 0) | (0, 0) | (1, 0) | (0, 0) | (0, 0) | (0, 0) | (0, 0) |
(![]() |
(2, 0) | (1, 0) | (2, 0) | (2, 0) | (2, 0) | (2, 0) | (2, 0) | (2, 0) | (1, 0) | (1, 0) |
(![]() |
(1, 0) | (2, 0) | (2, -2) | (2, -2) | (2, 2) | (2, 2) | (2, -1) | (2, 1) | (1, -1) | (1, 1) |
![]() |
13.45 | 13.45 | 12.10 | 8.38 | 12.10 | 8.38 | 12.10 | 12.10 | 13.45 | 13.45 |
![]() |
27.71 | 29.21 | 16.80 | 16.80 | 16.80 | 16.80 | 27.71 | 27.71 | 26.29 | 26.29 |
![]() |
12.38 | 11.98 | 39.99 | 39.99 | 39.99 | 39.99 | 12.69 | 12.69 | 4.82 | 4.82 |
i | 75 | 70 | 25 | 25 | 25 | 25 | 75 | 75 | 70 | 70 |
![]() |
38 | 39 | 1 | 1 | 1 | 1 | 6 | 6 | 7 | 7 |
![]() |
5 | 2 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 |
![]() |
3.86 | 3.86 | 3.87 | 3.87 | 3.87 | 3.87 | 3.88 | 3.89 | 3.90 | 3.90 |
Note that, as for
Crucis, a mode identification with the rotating
formalism does not change anything to the outcome.
De Cat (2001) studied the slowly pulsating B star HD 74195, among 12 other such
stars. Based on multicolour Geneva photometry and high-resolution spectroscopy,
he found four frequencies:
f1=0.35475 c/d,
f2=0.35033 c/d,
f3=0.34630 c/d and
f4=0.39864 c/d. The corresponding observed first moment amplitudes
are
km s-1,
km s-1,
km s-1 and
km s-1. The
mode identification from the moments of the 4128 Å Si II line based on the
moment method in the version by Aerts (1996) attributed the first three
frequencies to
modes and f4 to an
mode. The identification
from photometry, however, was found to be incompatible with this result by De Cat (2001) since it points towards
modes for f1, f2 and f3 and
an
mode for f4.
In an attempt to resolve the issue, we performed a new identification with our
version of the moment method. We point out that the values of the observed first
moment amplitudes and of the observed constant term of the second moment clearly
impose
for the four frequencies, directly excluding a high degree
mode for f4. By eliminating degrees greater than 3 in the photometric
outcome, the new candidate degree from Geneva data is also
for the
fourth frequency.
The result with the new moment method by taking K1=41, K2=42, K3=43 and
K4=32 is given in the upper part of Table 11.
(![]() |
(1, 0) | (1, 0) | (2, 0) | (2, 0) | (2, -1) | (2, 0) | (2, -1) | (1, 0) | (1, 0) | (2, 0) |
(![]() |
(2, -2) | (2, -2) | (2, -2) | (3, -1) | (3, -1) | (2, -2) | (3, -1) | (3, -1) | (3, -1) | (3, -1) |
(![]() |
(2, 1) | (2, 2) | (2, 2) | (2, 1) | (2, 2) | (2, 1) | (3, 1) | (2, 1) | (2, 2) | (2, 2) |
(![]() |
(1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) |
![]() |
0.27 | 0.34 | 0.30 | 0.30 | 0.24 | 0.16 | 0.20 | 0.28 | 0.26 | 0.18 |
![]() |
1.04 | 0.36 | 0.44 | 0.25 | 0.23 | 1.04 | 0.23 | 0.23 | 0.23 | 0.22 |
![]() |
0.11 | 0.17 | 0.21 | 0.09 | 0.48 | 0.11 | 0.11 | 0.10 | 0.48 | 0.34 |
![]() |
0.62 | 0.37 | 0.41 | 0.41 | 0.62 | 0.62 | 0.46 | 0.53 | 0.62 | 0.52 |
i | 25 | 45 | 40 | 40 | 25 | 25 | 35 | 30 | 25 | 30 |
![]() |
17 | 23 | 23 | 20 | 15 | 15 | 17 | 15 | 12 | 14 |
![]() |
5 | 3 | 1 | 6 | 8 | 6 | 8 | 9 | 9 | 9 |
![]() |
5.50 | 5.50 | 5.51 | 5.52 | 5.52 | 5.53 | 5.54 | 5.54 | 5.54 | 5.55 |
(![]() |
(1, 0) | (1, 0) | (2, 0) | (2, 0) | (2, 0) | (2, -1) | (2, -1) | (1, 0) | (1, 0) | (1, 0) |
(![]() |
(2, -2) | (2, -2) | (2, -2) | (3, -1) | (2, -2) | (3, -1) | (3, -1) | (2, -2) | (3, -1) | (3, -1) |
(![]() |
(2, 1) | (2, 2) | (2, 2) | (2, 1) | (2, 1) | (2, 2) | (3, 1) | (1, 0) | (2, 1) | (2, -2) |
(![]() |
(1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) |
![]() |
0.38 | 0.26 | 0.23 | 0.23 | 0.22 | 0.18 | 0.15 | 0.37 | 0.22 | 0.22 |
![]() |
1.49 | 0.28 | 0.34 | 0.19 | 1.49 | 0.18 | 0.18 | 1.46 | 0.17 | 0.20 |
![]() |
0.16 | 0.13 | 0.16 | 0.07 | 0.16 | 0.38 | 0.08 | 0.18 | 0.08 | 0.42 |
![]() |
0.88 | 0.28 | 0.31 | 0.32 | 0.88 | 0.47 | 0.35 | 0.86 | 0.41 | 0.51 |
K1 | 28 | 52 | 52 | 52 | 28 | 52 | 52 | 28 | 52 | 49 |
K2 | 29 | 53 | 53 | 53 | 29 | 53 | 53 | 29 | 53 | 50 |
K3 | 29 | 55 | 55 | 55 | 29 | 55 | 55 | 29 | 55 | 51 |
K4 | 22 | 41 | 41 | 41 | 22 | 41 | 41 | 22 | 41 | 39 |
i | 25 | 45 | 40 | 40 | 25 | 25 | 35 | 25 | 30 | 25 |
![]() |
27 | 23 | 23 | 20 | 17 | 15 | 17 | 17 | 15 | 12 |
![]() |
5 | 3 | 1 | 6 | 5 | 8 | 8 | 5 | 9 | 9 |
![]() |
5.49 | 5.50 | 5.50 | 5.51 | 5.51 | 5.52 | 5.54 | 5.54 | 5.54 | 5.55 |
(![]() |
(2, 0) | (1, 0) | (1, 0) | (2, 0) | (2, 0) | (2, 0) | (1, 0) | (1, 0) | (2, 0) | (2, 0) |
(![]() |
(3, -1) | (3, -1) | (2, -1) | (3, -1) | (2, -1) | (3, -1) | (2, -1) | (3, -1) | (3, -1) | (3, -1) |
(![]() |
(3, 1) | (3, 1) | (3, 1) | (2, 2) | (2, 2) | (2, 1) | (2, 2) | (2, 2) | (1, 0) | (2, 0) |
(![]() |
(1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) | (1, -1) |
![]() |
0.23 | 0.30 | 0.27 | 0.23 | 0.18 | 0.23 | 0.28 | 0.30 | 0.23 | 0.23 |
![]() |
0.03 | 0.03 | 0.02 | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 |
![]() |
0.16 | 0.16 | 0.17 | 0.82 | 1.21 | 0.15 | 1.21 | 0.82 | 0.14 | 0.11 |
![]() |
0.25 | 0.25 | 0.24 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
K1 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 |
K2 | 139 | 139 | 281 | 139 | 180 | 139 | 180 | 139 | 139 | 139 |
K3 | 19 | 19 | 15 | 11 | 9 | 19 | 9 | 11 | 42 | 42 |
K4 | 88 | 88 | 152 | 88 | 108 | 88 | 108 | 88 | 88 | 88 |
i | 25 | 45 | 40 | 40 | 25 | 25 | 35 | 25 | 30 | 25 |
i | 35 | 35 | 25 | 35 | 30 | 35 | 30 | 35 | 35 | 35 |
![]() |
20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 |
![]() |
6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
![]() |
5.51 | 5.52 | 5.52 | 5.53 | 5.55 | 5.55 | 5.55 | 5.56 | 5.56 | 5.56 |
One can doubt the reliability of a mode identification by fixing K-values of SPBs. Indeed, because there are large uncertainties on the mass and radius of most of these stars, K-values can have uncertainties up to 30%, which is very large. In order to test our mode identification as done above, we performed a new one by varying the parameter from 0.7 K to 1.3 K with a step of 0.1 K and we obtained the outcome given in the middle part of Table 11.
The identification of the modes leads to exactly the same outcome as before,
except that the values of
differ. One can conclude that the uncertainty on
the K-value does not affect the identification of
.
This result is not
too surprising since line-profile variations of an SPB are mainly due to the
horizontal velocity field variations and slightly to the vertical ones. The
relevant amplitude is then the horizontal one
,
which is indeed
found to be about the same for each of the modes in the two different
identifications.
One can also doubt the reliability of a mode identification for SPB stars by
using the non-rotating theory since their observed ratios of the rotational
frequency to the pulsational frequency are in general larger than 0.1. Again to
test our mode identification, we performed a new one using the formalism of Lee
& Saio (1987). The K-values are then computed using the corotating angular
frequency related to the observed one by
.
The identification is given in the lower
part of Table 11.
This again leads to a very similar outcome for the wavenumbers .
Basically, the same combinations for
occur each time. The
K-values change considerably, but this is compensated by the values for
and does not affect the mode identification appreciably. We do point out that
the formalism of Lee & Saio (1987) does not apply to modes with m=0 so that
axisymmetric modes mentioned in the lower part of Table 11 were computed using
the non-rotating formalism.
It is clear that, particularly in the case of multiple g-modes, we need
additional observational information to pinpoint definitely the wavenumbers
of all the different modes. Our method, however, implies a serious
improvement for the spectroscopic mode identification in such cases. In the
particular case of HD 74195, De Cat et al. (2002) are currently using Dupret et al.'s (2002) method in order to improve the photometric mode identification.
Line-profile fitting and an attempt at seismic modelling will be performed
after further elimination of combinations from Table 11 according to the
multicolour mode identification in progress and is beyond the scope of our
current paper.
The study of the oscillations of a pulsating star allows us to probe its internal structure. A successful application of asteroseismic techniques requires the identification of many pulsation modes. Therefore, high quality data as well as powerful mode identification methods are needed. Among the techniques of mode identification from line-profile variations, only the line-profile fitting method and the moment method derive the full pulsational information. However, even with current computers, a simultaneous identification of multiple modes is not possible by direct line-profile fitting. Moreover, mode identification with the moment method was still difficult for multiperiodic stars. Indeed, because of large computation time, the previous version of the moment method did not take into account coupling terms appearing in the second and third moments of a multiperiodic star so that multiple modes were determined independently, often leading to inconsistent values of the continuous velocity parameters.
We presented a new numerical version of the moment method, which is efficient in
computation time and which identifies all the modes by requiring that the
rotational velocity, the inclination angle and the intrinsic line width have one
unique value. Consequently, all observed terms of the first three moments can be
used, in particular the constant term of the second moment which is an important
constraint. By means of a new discriminant which compares theoretical to
observed first three moments, the new version identifies the wavenumbers of the
multiple modes simultaneously, leading to only one derived value for ,
i, and
.
We performed a large number of tests on artificial data representing the
presence of respectively one, two and three modes. It appears that the method
performs very well on synthetic data sets. An application to two Cephei
stars and one SPB star was done. We subsequently plan to use the new version of
the moment method in the near future to several data sets of pulsating B stars
of different kinds. Our new method is also relevant to obtain reliable mode
identification in multiperiodic bright
Scuti and
Doradus
stars.
Acknowledgements
We thank the referee, Dr. G. Handler, for his pertinent comments which helped to improve our manuscript. This work was supported by the P.A.I. (Pôle d'Attraction Interuniversitaire) and was carried out within the Belgian Asteroseismology Group.
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In all versions of the moment method for slowly rotating stars, one limits the discriminating function for mode identification to the first three moments (Balona 1986a, 1987; Aerts et al. 1992; Aerts 1996). We provide here a justification for this.
For one mode, we rewrite the first three moments as follows:
We search for different
and
so that
<v>(t)=<v>'(t),
<v2>(t)=<v2>'(t) and
<v3>(t)=<v3>'(t) for all t, i.e. so that A=A',
C=C', F=F' for appropriate values of the parameters. We use the fact that
the K-value is determined for a given star and for a given frequency
(K=K'). For chosen
and
,
we then have the
following system of three equations for
and K to solve:
From (1) and (2) (
), one obtains
an equation in K which can be solved analytically for chosen
and
:
The coefficients of this equation are not simultaneously equal to zero. Consequently, the equation has four solutions or less.
From (1) and (3) as well as from (2) and (3), one obtains two equations in K, which must also be satisfied by the solutions.
We tested numerically
the feasibility of more than one solution
by varying
from 0
to 4 with
.
We varied i from 0
to 90
with
a step of 1
.
For each combination
and
,
we
searched for solutions which satisfy the equations in K. Note that we did not
confront an axisymmetric mode with a non-axisymmetric mode since they are
discriminated by the behaviour of the second moment. We did not confront the
same couples of wavenumbers with different i either.
The outcome is that the system of three Eqs. (1)-(3) has no solution for each tested case. So the first three moments discriminate the different modes whatever the values of the velocity parameters are.
In practice, because of too few and/or
too noisy data, and because of the uncertainty of K, it obviously happens that
several couples
fit the moment variations with almost the same
quality. However, the message we want to bring here is that it in principle
suffices to consider the three moments to discriminate between the wavenumbers.
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K1 | K2 | i | ![]() |
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(1, -1) | (1, 0) | 2.25 | 0.53 | 65.54 | 12.03 | 6 | 25.86 | 5.5 | |
(1, -1) | (1, 0) | 2.86 | 0.58 | 56.64 | 10.39 | 5 | 22 | 4 | 1.68 |
(1, -1) | (2, 0) | 2.86 | 0.32 | 56.64 | 10.39 | 5 | 22 | 4 | 1.77 |
(1, -1) | (3, 0) | 2.86 | 0.58 | 56.64 | 10.39 | 5 | 22 | 4 | 2.85 |
(1, 1) | (1, 0) | 2.86 | 0.58 | 56.64 | 10.39 | 5 | 22 | 4 | 4.99 |
(1, 1) | (2, 0) | 2.86 | 0.32 | 56.64 | 10.39 | 5 | 22 | 4 | 5.07 |
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K1 | K2 | i | ![]() |
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(2, 0) | (1, 1) | 10.38 | 21.92 | 0.082 | 0.086 | 16 | 23.52 | 5.5 | |
(2, 0) | (1, 1) | 9.99 | 23.27 | 0.094 | 0.099 | 15 | 21 | 7 | 1.53 |
(1, 0) | (3, 3) | 9.06 | 93.26 | 0.094 | 0.099 | 35 | 10 | 7 | 1.68 |
(1, 0) | (2, 1) | 7.92 | 12.82 | 0.094 | 0.099 | 20 | 25 | 3 | 1.87 |
(2, 0) | (2, 1) | 10.82 | 12.82 | 0.094 | 0.099 | 20 | 24 | 4 | 1.95 |
(2, 0) | (3, 1) | 9.99 | 24.59 | 0.094 | 0.099 | 15 | 13 | 10 | 1.95 |
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K1 | K2 | i | ![]() |
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(3, 2) | (1, -1) | 1.69 | 0.77 | 6.48 | 17.40 | 52 | 19.33 | 5.5 | |
(3, 2) | (1, -1) | 1.45 | 0.52 | 8.20 | 22.02 | 60 | 19 | 5 | 2.56 |
(1, -1) | (2, 2) | 4.90 | 2.19 | 8.20 | 22.02 | 20 | 14 | 1 | 2.87 |
(3, 2) | (2, 2) | 2.64 | 0.61 | 4.41 | 11.86 | 60 | 19 | 5 | 2.99 |
(1, -1) | (1, -1) | 3.37 | 0.90 | 8.20 | 22.02 | 30 | 17 | 6 | 3.17 |
(2, 2) | (1, -1) | 10.00 | 1.90 | 4.41 | 11.86 | 25 | 16 | 3 | 3.29 |
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K1 | K2 | i | ![]() |
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(3, -3) | (2, 0) | 32.97 | 9.60 | 0.085 | 0.095 | 87 | 10.14 | 5.5 | |
(3, -3) | (2, 0) | 38.49 | 10.44 | 0.051 | 0.057 | 80 | 9 | 5 | 2.39 |
(2, -2) | (2, 0) | 17.16 | 11.88 | 0.051 | 0.057 | 75 | 18 | 2 | 2.40 |
(3, -3) | (1, 1) | 38.49 | 5.42 | 0.051 | 0.057 | 80 | 9 | 5 | 2.53 |
(3, -3) | (2, 1) | 49.42 | 10.12 | 0.051 | 0.057 | 65 | 8 | 3 | 2.57 |
(3, -2) | (1, 1) | 40.01 | 6.17 | 0.051 | 0.057 | 60 | 11 | 5 | 2.64 |
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K1 | K2 | i | ![]() |
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(2, 2) | (1, -1) | 4.02 | 0.91 | 5.61 | 19.74 | 37 | 19.14 | 5.5 | |
(1, -1) | (1, -1) | 2.88 | 0.66 | 7.05 | 24.79 | 45 | 23 | 2 | 2.19 |
(2, 2) | (1, -1) | 4.30 | 1.18 | 3.79 | 13.35 | 45 | 23 | 1 | 2.20 |
(1, -1) | (2, 2) | 6.09 | 1.67 | 4.34 | 15.25 | 30 | 21 | 3 | 2.23 |
(3, 1) | (1, -1) | 2.10 | 0.46 | 7.05 | 24.79 | 90 | 22 | 1 | 3.04 |
(2, 2) | (3, 1) | 2.16 | 0.85 | 3.79 | 13.35 | 85 | 22 | 2 | 3.19 |
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K1 | K2 | i | ![]() |
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(2, -1) | (1, -1) | 30.45 | 21.69 | 0.085 | 0.099 | 17 | 10.84 | 5.5 | |
(3, -1) | (2, -1) | 40.09 | 11.17 | 0.054 | 0.062 | 30 | 16 | 2 | 2.30 |
(3, -1) | (1, -1) | 42.16 | 15.91 | 0.054 | 0.062 | 25 | 14 | 3 | 2.52 |
(2, -1) | (2, -1) | 29.18 | 15.16 | 0.054 | 0.062 | 20 | 14 | 6 | 2.53 |
(3, -2) | (2, -2) | 46.16 | 12.86 | 0.054 | 0.062 | 60 | 14 | 4 | 2.54 |
(2, -1) | (1, -1) | 29.18 | 19.72 | 0.054 | 0.062 | 20 | 15 | 6 | 2.56 |
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K1 | K2 | i | ![]() |
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(2, 0) | (1, 1) | 0.45 | 6.98 | 11.45 | 4.91 | 16 | 23.52 | 5.5 | |
(2, 0) | (1, 1) | 0.30 | 1.84 | 16.99 | 5.21 | 15 | 19 | 8 | 1.88 |
(1, 0) | (1, 1) | 0.58 | 1.84 | 16.99 | 5.21 | 15 | 19 | 8 | 1.91 |
(3, 0) | (1, 1) | 0.69 | 1.84 | 9.14 | 12.82 | 10 | 16 | 9 | 2.44 |
(2, -1) | (1, 1) | 0.70 | 1.84 | 16.99 | 3.97 | 20 | 18 | 8 | 3.72 |
(1, 0) | (3, 2) | 2.93 | 1.84 | 9.14 | 2.67 | 70 | 25 | 1 | 3.74 |
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K1 | K2 | i | ![]() |
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(3, -2) | (2, 2) | 35.19 | 13.21 | 0.086 | 0.091 | 52 | 15.82 | 5.5 | |
(3, -2) | (2, 2) | 36.24 | 12.33 | 0.071 | 0.075 | 55 | 16 | 4 | 2.39 |
(3, -3) | (3, 2) | 37.90 | 30.83 | 0.071 | 0.075 | 75 | 13 | 2 | 2.57 |
(2, -2) | (2, 2) | 30.80 | 16.56 | 0.071 | 0.075 | 45 | 19 | 2 | 2.79 |
(3, -3) | (2, 2) | 37.90 | 8.85 | 0.071 | 0.075 | 75 | 13 | 5 | 2.87 |
(3, -3) | (3, 1) | 34.14 | 23.50 | 0.071 | 0.075 | 90 | 14 | 4 | 3.43 |
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K1 | K2 | i | ![]() |
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(3, 3) | (2, 1) | 2.04 | 1.88 | 5.14 | 8.81 | 81 | 24.33 | 5.5 | |
(3, 3) | (2, 1) | 1.35 | 1.10 | 7.84 | 13.45 | 80 | 26 | 4 | 2.25 |
(3, 3) | (3, 0) | 2.22 | 0.98 | 5.43 | 9.31 | 70 | 23 | 7 | 2.82 |
(3, 3) | (2, 0) | 2.46 | 0.92 | 4.22 | 7.24 | 80 | 25 | 6 | 2.85 |
(3, 3) | (2, -1) | 1.35 | 1.10 | 7.84 | 13.45 | 80 | 26 | 4 | 3.53 |
(3, 3) | (3, 1) | 4.32 | 0.72 | 6.64 | 11.38 | 45 | 17 | 8 | 3.68 |
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K1 | K2 | i | ![]() |
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(3, 2) | (1, -1) | 36.20 | 7.24 | 0.078 | 0.097 | 52 | 19.33 | 5.5 | |
(3, 2) | (1, -1) | 36.61 | 6.96 | 0.076 | 0.095 | 55 | 20 | 5 | 1.79 |
(2, 2) | (1, -1) | 37.95 | 8.88 | 0.076 | 0.095 | 40 | 21 | 5 | 2.07 |
(3, 2) | (2, -1) | 36.61 | 8.36 | 0.076 | 0.095 | 55 | 20 | 5 | 2.16 |
(2, 2) | (2, -1) | 37.95 | 7.99 | 0.076 | 0.095 | 40 | 21 | 5 | 2.45 |
(3, 2) | (1, 0) | 39.91 | 5.69 | 0.076 | 0.095 | 45 | 16 | 7 | 2.93 |
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K1 | K2 | i | ![]() |
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(2, 1) | (1, 0) | 16.10 | 4.80 | 0.080 | 0.095 | 37 | 11.91 | 5.5 | |
(2, 1) | (1, 0) | 22.63 | 4.48 | 0.054 | 0.064 | 25 | 10 | 6 | 1.95 |
(1, 1) | (1, 0) | 34.83 | 4.33 | 0.054 | 0.064 | 20 | 10 | 6 | 2.01 |
(2, 1) | (2, 0) | 27.09 | 6.17 | 0.054 | 0.064 | 20 | 8 | 6 | 2.07 |
(3, 1) | (1, 0) | 44.34 | 4.33 | 0.054 | 0.064 | 20 | 6 | 2 | 2.08 |
(3, 1) | (2, 0) | 44.34 | 6.17 | 0.054 | 0.064 | 20 | 6 | 1 | 2.13 |
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K1 | K2 | i | ![]() |
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(3, -1) | (2, -2) | 0.84 | 0.21 | 16.40 | 27.36 | 86 | 24.62 | 5.5 | |
(3, -1) | (2, -2) | 0.88 | 0.23 | 15.54 | 25.93 | 90 | 27 | 3 | 1.83 |
(1, 1) | (2, -2) | 1.04 | 0.37 | 19.43 | 32.42 | 45 | 19 | 10 | 1.94 |
(1, 1) | (3, -1) | 0.89 | 0.19 | 25.26 | 42.14 | 40 | 16 | 11 | 2.28 |
(1, 1) | (2, -1) | 1.13 | 0.34 | 13.60 | 22.69 | 65 | 28 | 2 | 2.64 |
(1, 1) | (3, -2) | 0.59 | 0.31 | 25.26 | 42.14 | 75 | 14 | 11 | 3.16 |
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K1 | K2 | i | ![]() |
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(1, 1) | (1, 0) | 25.54 | 4.34 | 0.080 | 0.095 | 27 | 11.78 | 5.5 | |
(1, 1) | (1, 0) | 74.02 | 4.41 | 0.053 | 0.063 | 10 | 4 | 4 | 2.07 |
(1, 1) | (2, 0) | 29.54 | 7.27 | 0.053 | 0.063 | 25 | 10 | 6 | 2.12 |
(2, 1) | (2, 0) | 23.82 | 7.27 | 0.053 | 0.063 | 25 | 9 | 6 | 2.12 |
(2, 1) | (1, 0) | 18.44 | 5.56 | 0.053 | 0.063 | 40 | 14 | 4 | 2.16 |
(3, 1) | (1, 0) | 41.23 | 4.71 | 0.053 | 0.063 | 25 | 7 | 3 | 2.31 |
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K1 | K2 | i | ![]() |
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(3, -2) | (2, -1) | 0.54 | 0.35 | 21.86 | 16.32 | 47 | 15.094 | 5.5 | |
(3, -2) | (2, -1) | 0.94 | 0.62 | 12.30 | 9.18 | 50 | 15 | 6 | 1.88 |
(3, -3) | (2, -1) | 0.86 | 0.35 | 22.84 | 17.06 | 55 | 11 | 6 | 2.10 |
(3, -3) | (2, -2) | 0.47 | 0.33 | 22.84 | 17.06 | 90 | 13 | 7 | 2.22 |
(3, -1) | (2, -1) | 0.44 | 0.38 | 22.84 | 17.06 | 30 | 19 | 2 | 2.59 |
(3, -2) | (2, -2) | 0.55 | 0.41 | 22.84 | 17.06 | 65 | 18 | 3 | 2.60 |
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K1 | K2 | i | ![]() |
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(3, -1) | (2, 1) | 34.05 | 7.83 | 0.084 | 0.093 | 42 | 12.49 | 5.5 | |
(3, -1) | (1, 1) | 33.13 | 9.81 | 0.058 | 0.065 | 35 | 12 | 6 | 2.27 |
(3, -1) | (2, 1) | 33.13 | 8.60 | 0.058 | 0.065 | 35 | 12 | 6 | 2.29 |
(3, -2) | (1, 1) | 58.29 | 5.81 | 0.058 | 0.065 | 75 | 11 | 2 | 2.55 |
(2, -1) | (2, 1) | 17.56 | 9.31 | 0.058 | 0.065 | 60 | 20 | 1 | 2.65 |
(2, -1) | (1, 1) | 15.47 | 8.75 | 0.058 | 0.065 | 40 | 17 | 5 | 2.67 |
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K1 | K2 | i | ![]() |
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(3, -3) | (2, 1) | 0.57 | 0.64 | 40.24 | 8.95 | 52 | 18.29 | 5.5 | |
(3, -3) | (2, 1) | 0.51 | 1.89 | 26.24 | 5.83 | 75 | 19 | 5 | 2.77 |
(3, -2) | (2, 1) | 0.26 | 0.53 | 48.73 | 10.84 | 50 | 22 | 2 | 2.99 |
(3, -3) | (2, 0) | 0.26 | 0.71 | 48.73 | 10.84 | 80 | 18 | 6 | 3.50 |
(3, -3) | (3, 0) | 0.55 | 1.75 | 26.24 | 5.83 | 70 | 18 | 5 | 3.56 |
(3, -3) | (3, 1) | 1.31 | 1.57 | 26.24 | 5.83 | 45 | 14 | 1 | 3.60 |
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K1 | K2 | i | ![]() |
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(2, 0) | (2, 2) | 12.62 | 44.62 | 0.082 | 0.082 | 26 | 29.84 | 5.5 | |
(2, 0) | (2, 2) | 10.02 | 68.16 | 0.11 | 0.11 | 20 | 25 | 9 | 2.01 |
(2, 0) | (3, 2) | 16.22 | 25.03 | 0.11 | 0.11 | 35 | 30 | 5 | 2.02 |
(1, 0) | (2, 2) | 7.53 | 68.16 | 0.11 | 0.11 | 20 | 25 | 9 | 2.22 |
(1, 0) | (3, 2) | 7.36 | 108.36 | 0.11 | 0.11 | 15 | 13 | 2 | 2.31 |
(3, 0) | (2, 2) | 20.27 | 120.87 | 0.11 | 0.11 | 15 | 17 | 10 | 3.18 |
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K1 | K2 | i | ![]() |
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(3, 0) | (3, 2) | 1.52 | 1.93 | 11.45 | 7.49 | 32 | 10.35 | 5.5 | |
(3, 1) | (3, 2) | 3.42 | 2.44 | 7.03 | 4.60 | 70 | 16 | 1 | 3.78 |
(3, 0) | (3, 2) | 3.29 | 2.09 | 7.03 | 4.60 | 45 | 14 | 2 | 4.04 |
(3, 0) | (2, 2) | 1.74 | 5.31 | 7.03 | 4.60 | 30 | 19 | 1 | 4.31 |
(2, 0) | (3, 2) | 3.81 | 1.95 | 7.03 | 4.60 | 50 | 14 | 6 | 4.50 |
(2, 0) | (2, 2) | 3.81 | 2.25 | 7.03 | 4.60 | 50 | 20 | 1 | 4.78 |
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K1 | K2 | i | ![]() |
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(3, 1) | (2, -1) | 44.57 | 43.88 | 0.080 | 0.097 | 86 | 24.54 | 5.5 | |
(3, 1) | (2, -1) | 42.84 | 33.76 | 0.095 | 0.11 | 85 | 25 | 6 | 2.59 |
(3, 1) | (1, 0) | 42.84 | 35.41 | 0.095 | 0.11 | 85 | 22 | 9 | 3.29 |
(3, 1) | (2, 1) | 42.84 | 33.76 | 0.095 | 0.11 | 85 | 27 | 3 | 3.30 |
(2, 2) | (3, -1) | 25.31 | 19.44 | 0.095 | 0.11 | 50 | 13 | 13 | 3.43 |
(2, 2) | (2, -1) | 14.94 | 33.76 | 0.095 | 0.11 | 85 | 16 | 12 | 3.57 |
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K1 | K2 | i | ![]() |
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(2, 1) | (1, 0) | 1.11 | 0.68 | 7.81 | 12.03 | 16.62 | 32 | 5.5 | |
(2, 1) | (2, 0) | 1.43 | 0.31 | 8.78 | 13.52 | 20 | 11 | 8 | 2.47 |
(2, 1) | (1, 0) | 2.15 | 1.02 | 4.73 | 7.28 | 25 | 12 | 8 | 2.48 |
(3, 1) | (1, 0) | 2.22 | 1.02 | 4.73 | 7.28 | 25 | 5 | 9 | 2.49 |
(3, 1) | (2, 0) | 1.22 | 0.35 | 8.78 | 13.52 | 25 | 4 | 9 | 2.49 |
(3, 3) | (1, 0) | 5.03 | 1.44 | 4.73 | 7.28 | 50 | 3 | 7 | 2.64 |
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(3, 0) | (1, -1) | (3, 1) | 1.45 | 0.67 | 0.57 | 11.45 | 14.11 | 13.75 | 73 | 10.72 | 5.5 | |
(3, 0) | (1, -1) | (3, 1) | 1.17 | 0.53 | 0.38 | 14.88 | 18.34 | 17.87 | 75 | 10 | 6 | 2.07 |
(3, 0) | (2, 2) | (3, 1) | 1.17 | 0.30 | 0.38 | 14.88 | 18.34 | 17.87 | 75 | 12 | 5 | 2.11 |
(2, 1) | (1, -1) | (3, 1) | 4.03 | 0.65 | 0.33 | 11.45 | 14.11 | 13.75 | 85 | 7 | 2 | 2.28 |
(3, 0) | (2, 2) | (1, -1) | 2.16 | 0.55 | 0.46 | 8.01 | 9.87 | 9.62 | 75 | 11 | 6 | 2.29 |
(3, 0) | (1, -1) | (1, -1) | 2.16 | 0.93 | 0.46 | 8.01 | 9.87 | 9.62 | 75 | 11 | 6 | 2.30 |
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(2, -1) | (1, -1) | (2, 1) | 25.13 | 14.05 | 10.05 | 0.084 | 0.086 | 0.092 | 64 | 19.87 | 5.5 | |
(2, -1) | (1, -1) | (2, 1) | 28.31 | 15.86 | 11.51 | 0.084 | 0.085 | 0.092 | 60 | 16 | 7 | 2.22 |
(1, -1) | (1, -1) | (1, 1) | 17.00 | 15.15 | 6.91 | 0.084 | 0.085 | 0.092 | 65 | 19 | 6 | 2.44 |
(1, -1) | (1, -1) | (2, 1) | 18.81 | 16.77 | 10.61 | 0.084 | 0.085 | 0.092 | 55 | 17 | 7 | 2.46 |
(2, -1) | (1, -1) | (1, 1) | 28.31 | 15.86 | 7.23 | 0.084 | 0.085 | 0.092 | 60 | 18 | 6 | 2.47 |
(1, -1) | (1, -1) | (2, 1) | 17.00 | 15.15 | 13.01 | 0.084 | 0.085 | 0.092 | 65 | 20 | 5 | 2.53 |
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(3, 2) | (2, -2) | (1, 0) | 1.55 | 0.50 | 0.31 | 7.6 | 20.46 | 16.31 | 44 | 11.86 | 5.5 | |
(3, 2) | (2, -2) | (2, -1) | 1.34 | 0.51 | 0.15 | 9.12 | 24.55 | 19.57 | 40 | 14 | 5 | 2.06 |
(3, 2) | (2, -2) | (1, 0) | 1.46 | 0.56 | 0.27 | 8.36 | 22.51 | 17.94 | 40 | 14 | 5 | 2.07 |
(3, 2) | (2, -2) | (2, 0) | 2.26 | 0.87 | 0.41 | 5.32 | 14.32 | 11.41 | 40 | 14 | 5 | 2.21 |
(3, 2) | (3, -1) | (2, 0) | 1.82 | 0.24 | 0.13 | 9.88 | 26.60 | 21.20 | 30 | 11 | 5 | 2.26 |
(3, 2) | (3, -1) | (1, 0) | 1.82 | 0.24 | 0.20 | 9.88 | 26.60 | 21.20 | 30 | 11 | 5 | 2.30 |
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(1, 1) | (3, -2) | (2, 0) | 3.01 | 0.21 | 0.21 | 7.35 | 49.73 | 16.31 | 37 | 22.43 | 5.5 | |
(1, 1) | (3, -2) | (1, 0) | 2.85 | 0.19 | 0.26 | 7.35 | 49.73 | 16.31 | 40 | 25 | 2 | 1.87 |
(1, 1) | (3, -2) | (2, 0) | 3.12 | 0.21 | 0.28 | 6.61 | 44.76 | 14.68 | 40 | 25 | 1 | 1.92 |
(1, 1) | (3, -2) | (2, -1) | 3.45 | 0.24 | 0.20 | 5.88 | 39.78 | 13.05 | 40 | 25 | 2 | 2.28 |
(1, 1) | (3, -3) | (2, -1) | 1.77 | 0.21 | 0.13 | 9.56 | 64.65 | 21.20 | 55 | 22 | 5 | 2.48 |
(1, 1) | (3, -3) | (1, 0) | 2.05 | 0.33 | 0.22 | 9.56 | 64.65 | 21.20 | 45 | 19 | 6 | 2.61 |
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(2, -1) | (1, 0) | (2, 2) | 0.85 | 1.54 | 0.29 | 13.93 | 12.03 | 10.90 | 68 | 13.00 | 5.5 | |
(2, -2) | (1, 0) | (1, -1) | 0.56 | 2.06 | 0.42 | 15.33 | 13.23 | 11.99 | 75 | 9 | 7 | 2.20 |
(2, -2) | (1, 0) | (2, 2) | 0.62 | 2.25 | 0.27 | 13.93 | 12.03 | 10.90 | 75 | 9 | 7 | 2.20 |
(2, -1) | (1, 0) | (1, -1) | 0.82 | 1.56 | 0.44 | 15.33 | 13.23 | 11.99 | 70 | 12 | 6 | 2.22 |
(2, -1) | (1, 0) | (2, 2) | 0.90 | 1.70 | 0.29 | 13.93 | 12.03 | 10.90 | 70 | 15 | 4 | 2.23 |
(2, -1) | (2, 0) | (1, -1) | 0.58 | 1.14 | 0.63 | 13.93 | 12.03 | 10.90 | 45 | 9 | 7 | 2.28 |
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(1, 0) | (1, -1) | (1, 1) | 0.91 | 0.77 | 1.39 | 11.45 | 28.75 | 8.19 | 27 | 23.57 | 5.5 | |
(1, 0) | (1, -1) | (1, 1) | 1.13 | 0.87 | 1.48 | 9.16 | 23.00 | 6.55 | 30 | 26 | 3 | 2.15 |
(2, 0) | (1, -1) | (1, 1) | 0.54 | 0.83 | 1.44 | 11.45 | 28.75 | 8.19 | 25 | 22 | 7 | 2.24 |
(1, 0) | (2, 2) | (1, 1) | 1.17 | 2.40 | 2.42 | 8.01 | 20.12 | 5.73 | 20 | 21 | 2 | 2.68 |
(2, 0) | (2, 2) | (1, 1) | 0.67 | 2.40 | 2.42 | 8.01 | 20.12 | 5.73 | 20 | 21 | 2 | 2.77 |
(2, 0) | (1, -1) | (2, 2) | 0.43 | 1.34 | 5.55 | 11.45 | 28.75 | 8.19 | 15 | 14 | 8 | 2.93 |
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(3, -2) | (1, 1) | (3, 2) | 0.60 | 1.72 | 0.70 | 26.05 | 8.74 | 8.29 | 34 | 14.14 | 5.5 | |
(3, -2) | (1, 1) | (3, 2) | 0.72 | 2.09 | 0.88 | 20.84 | 6.99 | 6.63 | 35 | 16 | 5 | 2.34 |
(3, -2) | (1, -1) | (3, 2) | 0.72 | 2.09 | 0.88 | 20.84 | 6.99 | 6.63 | 35 | 16 | 5 | 3.13 |
(3, -3) | (1, 1) | (3, 2) | 0.72 | 1.38 | 0.63 | 20.84 | 6.99 | 6.63 | 60 | 18 | 3 | 3.43 |
(3, -2) | (1, 1) | (2, 2) | 1.38 | 3.17 | 2.74 | 18.24 | 6.12 | 5.80 | 25 | 13 | 1 | 3.69 |
(3, -3) | (1, -1) | (3, 2) | 1.07 | 1.39 | 0.54 | 26.05 | 8.74 | 8.29 | 45 | 14 | 2 | 3.71 |
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(2, 0) | (1, 0) | (2, 2) | 0.49 | 0.55 | 6.00 | 11.45 | 12.03 | 3.91 | 22 | 24.72 | 5.5 | |
(2, 0) | (1, 0) | (2, 2) | 0.49 | 0.61 | 15.52 | 10.30 | 10.82 | 3.52 | 15 | 17 | 8 | 2.06 |
(2, 0) | (1, 0) | (3, 2) | 0.63 | 0.76 | 11.75 | 8.01 | 8.42 | 2.74 | 15 | 7 | 10 | 2.12 |
(2, 0) | (2, 0) | (2, 2) | 0.49 | 0.32 | 15.52 | 10.30 | 10.82 | 3.52 | 15 | 17 | 8 | 2.15 |
(1, 0) | (1, 0) | (2, 2) | 0.93 | 0.61 | 15.52 | 10.30 | 10.82 | 3.52 | 15 | 17 | 8 | 2.24 |
(1, 0) | (2, 0) | (1, -1) | 0.68 | 0.24 | 3.41 | 14.88 | 15.64 | 5.08 | 20 | 28 | 2 | 2.24 |
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(1, 0) | (3, 3) | (1, 1) | 1.44 | 1.94 | 0.49 | 11.45 | 6.33 | 12.40 | 56 | 15.29 | 5.5 | |
(1, 0) | (3, 3) | (1, 1) | 0.93 | 1.96 | 0.41 | 14.88 | 8.23 | 16.12 | 50 | 13 | 6 | 2.15 |
(2, -1) | (3, 3) | (1, 1) | 2.00 | 1.32 | 0.45 | 10.30 | 5.70 | 11.16 | 80 | 16 | 4 | 2.54 |
(2, -1) | (3, 3) | (2, 0) | 1.06 | 1.52 | 0.48 | 10.30 | 5.70 | 11.16 | 70 | 16 | 5 | 2.78 |
(1, 0) | (3, 3) | (3, 0) | 2.49 | 2.15 | 0.54 | 8.01 | 4.43 | 8.68 | 65 | 16 | 5 | 2.81 |
(2, -1) | (3, 3) | (3, -1) | 2.54 | 1.68 | 0.67 | 8.01 | 4.43 | 8.68 | 80 | 16 | 4 | 2.81 |
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(3, 3) | (2, 0) | (1, 0) | 1.47 | 0.65 | 0.86 | 7.67 | 12.03 | 16.31 | 75 | 10.71 | 5.5 | |
(3, 3) | (2, 0) | (1, 0) | 2.05 | 0.88 | 1.11 | 5.37 | 8.42 | 11.41 | 75 | 11 | 6 | 2.26 |
(3, 3) | (3, 0) | (1, 0) | 2.49 | 0.97 | 0.68 | 5.37 | 8.42 | 11.41 | 65 | 10 | 6 | 2.28 |
(3, 3) | (2, 0) | (2, -2) | 2.23 | 1.08 | 0.26 | 5.37 | 8.42 | 11.41 | 70 | 11 | 6 | 2.33 |
(3, 3) | (3, 0) | (2, -1) | 2.49 | 0.97 | 0.30 | 5.37 | 8.42 | 11.41 | 65 | 10 | 6 | 2.34 |
(3, 3) | (2, 0) | (2, -1) | 1.12 | 0.48 | 0.25 | 9.97 | 15.64 | 21.20 | 75 | 11 | 6 | 2.36 |
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(1, 1) | (2, 1) | (2, -2) | 1.88 | 0.61 | 0.09 | 8.38 | 8.74 | 45.84 | 57 | 21.28 | 5.5 | |
(1, 1) | (2, 1) | (2, -2) | 1.35 | 0.68 | 0.05 | 10.89 | 11.36 | 59.60 | 70 | 23 | 3 | 2.04 |
(1, 1) | (2, 1) | (3, -1) | 1.80 | 0.44 | 0.08 | 10.89 | 11.36 | 59.60 | 45 | 17 | 8 | 2.21 |
(1, 1) | (2, 1) | (2, -1) | 1.93 | 0.47 | 0.05 | 10.05 | 10.48 | 55.01 | 45 | 19 | 7 | 2.27 |
(3, -1) | (2, 0) | (2, -2) | 2.41 | 0.99 | 0.09 | 5.86 | 6.11 | 32.09 | 85 | 22 | 3 | 2.60 |
(1, 1) | (2, 1) | (2, 0) | 3.69 | 0.69 | 0.03 | 10.89 | 11.36 | 59.60 | 20 | 10 | 10 | 2.60 |
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(2, 2) | (3, 2) | (1, 0) | 110.38 | 100.95 | 4.34 | 0.078 | 0.080 | 0.095 | 26 | 10.44 | 5.5 | |
(3, 2) | (3, 2) | (2, 0) | 82.39 | 75.46 | 14.96 | 0.048 | 0.049 | 0.058 | 40 | 12 | 7 | 3.62 |
(3, 2) | (3, 2) | (1, 0) | 82.39 | 75.46 | 5.38 | 0.048 | 0.049 | 0.058 | 40 | 12 | 7 | 3.67 |
(3, 2) | (3, 2) | (1, -1) | 86.03 | 78.79 | 6.21 | 0.048 | 0.049 | 0.058 | 40 | 11 | 8 | 4.39 |
(3, 2) | (3, 2) | (2, -1) | 82.39 | 75.46 | 9.45 | 0.048 | 0.049 | 0.058 | 40 | 11 | 8 | 4.87 |
(3, 2) | (3, 2) | (1, 1) | 86.03 | 78.79 | 6.21 | 0.048 | 0.049 | 0.058 | 70 | 17 | 4 | 4.88 |
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(2, 0) | (3, 1) | (1, 1) | 0.42 | 4.90 | 7.28 | 11.45 | 2.74 | 3.15 | 8 | 23.57 | 5.5 | |
(2, 0) | (2, 1) | (1, 1) | 0.58 | 13.17 | 14.55 | 8.01 | 1.92 | 2.21 | 5 | 22 | 5 | 2.26 |
(1, 0) | (2, 1) | (1, 1) | 1.03 | 13.17 | 14.55 | 8.01 | 1.92 | 2.21 | 5 | 22 | 5 | 2.32 |
(3, 0) | (2, 1) | (1, 1) | 0.93 | 11.76 | 13.20 | 9.16 | 2.19 | 2.52 | 5 | 22 | 3 | 3.11 |
(2, 0) | (2, -1) | (1, -1) | 0.58 | 13.17 | 14.55 | 8.01 | 1.92 | 2.21 | 5 | 22 | 5 | 3.29 |
(2, 0) | (3, 1) | (1, 1) | 0.43 | 3.97 | 5.83 | 11.45 | 5.83 | 2.64 | 10 | 27 | 1 | 3.32 |
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(3, -3) | (3, -1) | (2, 1) | 1.14 | 0.43 | 0.24 | 34.30 | 16.43 | 11.95 | 38 | 13.00 | 5.5 | |
(3, -3) | (3, -1) | (2, 1) | 1.41 | 0.64 | 0.31 | 24.01 | 11.50 | 8.37 | 40 | 15 | 7 | 2.76 |
(3, -3) | (2, -2) | (2, 1) | 0.49 | 0.56 | 0.40 | 24.01 | 11.50 | 8.37 | 65 | 22 | 7 | 2.77 |
(3, -3) | (2, -2) | (3, 0) | 0.57 | 0.62 | 0.54 | 24.01 | 11.50 | 8.37 | 60 | 20 | 8 | 2.81 |
(3, -3) | (2, -2) | (2, 0) | 0.44 | 0.52 | 0.59 | 24.01 | 11.50 | 8.37 | 70 | 21 | 8 | 2.82 |
(3, -2) | (2, -1) | (2, 1) | 0.70 | 0.53 | 0.36 | 24.01 | 11.50 | 8.37 | 30 | 19 | 8 | 2.94 |
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(2, -1) | (3, 1) | (3, -2) | 0.38 | 0.88 | 0.04 | 21.71 | 7.33 | 135.78 | 33 | 22.43 | 5.5 | |
(2, -1) | (3, 1) | (3, -2) | 0.53 | 1.27 | 0.05 | 15.20 | 5.13 | 95.05 | 35 | 23 | 5 | 1.78 |
(2, -1) | (3, 1) | (3, -3) | 0.50 | 1.38 | 0.14 | 15.20 | 5.13 | 95.05 | 40 | 21 | 6 | 2.88 |
(2, -2) | (3, 1) | (3, -2) | 0.54 | 0.88 | 0.02 | 28.22 | 9.53 | 176.52 | 45 | 24 | 1 | 3.00 |
(2, -2) | (3, 1) | (3, -3) | 1.21 | 1.38 | 0.14 | 15.20 | 2.30 | 95.05 | 40 | 20 | 6 | 3.16 |
(1, 0) | (3, 1) | (3, -2) | 0.72 | 1.25 | 0.07 | 15.20 | 5.13 | 95.05 | 30 | 20 | 8 | 3.38 |
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(2, 2) | (1, 1) | (2, 1) | 26.12 | 14.77 | 8.92 | 0.077 | 0.081 | 0.091 | 58 | 28.15 | 5.5 | |
(2, 2) | (1, 1) | (1, 1) | 25.32 | 14.33 | 6.66 | 0.102 | 0.107 | 0.121 | 70 | 28 | 6 | 1.86 |
(2, 2) | (1, 1) | (2, 1) | 23.96 | 13.94 | 19.93 | 0.102 | 0.107 | 0.121 | 75 | 29 | 4 | 1.99 |
(2, 2) | (2, 1) | (1, 1) | 27.23 | 27.97 | 6.91 | 0.102 | 0.107 | 0.121 | 65 | 26 | 7 | 2.57 |
(2, 2) | (2, 1) | (2, 1) | 38.16 | 21.77 | 10.13 | 0.102 | 0.107 | 0.121 | 50 | 23 | 9 | 3.05 |
(3, 2) | (1, 1) | (1, 1) | 67.14 | 14.86 | 6.91 | 0.102 | 0.107 | 0.121 | 70 | 22 | 1 | 3.20 |
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(1, -1) | (2, 2) | (3, 2) | 1.95 | 8.23 | 2.29 | 20.64 | 5.18 | 6.30 | 18 | 11.86 | 5.5 | |
(1, -1) | (2, 2) | (3, 2) | 1.50 | 5.82 | 1.55 | 24.77 | 6.22 | 7.56 | 20 | 14 | 7 | 2.34 |
(1, -1) | (2, 2) | (2, 2) | 2.03 | 6.24 | 2.82 | 14.45 | 3.63 | 4.41 | 25 | 20 | 6 | 3.58 |
(1, -1) | (3, 2) | (3, 2) | 1.72 | 4.44 | 2.00 | 14.45 | 3.63 | 4.41 | 30 | 23 | 4 | 3.68 |
(1, -1) | (1, 1) | (3, 2) | 2.21 | 4.53 | 2.30 | 16.51 | 4.14 | 5.04 | 20 | 14 | 9 | 3.74 |
(1, -1) | (1, -1) | (3, 2) | 0.83 | 1.81 | 0.57 | 26.84 | 6.74 | 8.19 | 35 | 24 | 2 | 3.75 |
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(1, 0) | (1, 1) | (3, 1) | 13.79 | 18.09 | 20.30 | 0.08 | 0.081 | 0.092 | 43 | 18.72 | 5.5 | |
(1, 0) | (1, 1) | (3, 1) | 14.30 | 20.86 | 27.55 | 0.078 | 0.077 | 0.087 | 40 | 17 | 6 | 2.08 |
(1, 0) | (1, 1) | (2, 1) | 19.09 | 16.34 | 11.00 | 0.078 | 0.077 | 0.087 | 55 | 21 | 4 | 2.51 |
(1, 0) | (1, 1) | (1, 1) | 15.49 | 18.95 | 9.19 | 0.078 | 0.077 | 0.087 | 45 | 19 | 6 | 2.89 |
(1, 0) | (2, 1) | (2, 1) | 15.49 | 21.32 | 10.34 | 0.078 | 0.077 | 0.087 | 45 | 17 | 7 | 3.01 |
(1, 0) | (2, 1) | (1, 1) | 12.66 | 24.70 | 13.03 | 0.078 | 0.077 | 0.087 | 30 | 11 | 9 | 3.15 |
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(2, 1) | (2, 2) | (3, 1) | 2.21 | 13.34 | 0.64 | 5.22 | 3.05 | 6.58 | 17 | 21.28 | 5.5 | |
(2, 1) | (2, 2) | (3, 1) | 1.48 | 8.42 | 0.45 | 6.79 | 3.96 | 8.56 | 20 | 25 | 1 | 2.42 |
(3, 1) | (3, 2) | (3, 3) | 2.40 | 3.08 | 2.83 | 3.65 | 2.13 | 4.61 | 40 | 18 | 9 | 2.92 |
(2, 1) | (1, -1) | (3, 1) | 1.91 | 5.87 | 0.57 | 6.79 | 3.96 | 8.56 | 15 | 22 | 7 | 3.10 |
(3, 1) | (3, 2) | (3, -3) | 1.48 | 4.47 | 0.45 | 6.79 | 3.96 | 8.56 | 20 | 27 | 1 | 3.19 |
(2, 1) | (2, 2) | (3, -1) | 1.59 | 9.06 | 0.49 | 6.27 | 3.66 | 7.90 | 20 | 25 | 1 | 3.36 |
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(3, 3) | (2, 2) | (1, 1) | 164.32 | 43.79 | 8.68 | 0.075 | 0.079 | 0.092 | 42 | 18.72 | 5.5 | |
(3, 3) | (2, 2) | (1, 1) | 123.82 | 34.74 | 7.49 | 0.071 | 0.075 | 0.087 | 55 | 20 | 8 | 4.02 |
(3, 3) | (3, 2) | (1, 1) | 123.82 | 63.13 | 10.40 | 0.071 | 0.075 | 0.087 | 55 | 19 | 3 | 4.30 |
(3, 3) | (3, 2) | (2, 1) | 123.82 | 63.13 | 10.40 | 0.071 | 0.075 | 0.087 | 55 | 19 | 3 | 4.30 |
(3, 3) | (2, 2) | (2, 1) | 151.52 | 39.75 | 9.92 | 0.071 | 0.075 | 0.087 | 50 | 17 | 1 | 4.50 |
(3, 3) | (2, 2) | (1, 0) | 151.52 | 39.75 | 6.74 | 0.071 | 0.075 | 0.087 | 50 | 18 | 1 | 4.89 |