# Frequency analysis of $\delta$ Scuti stars towards the Galactic bulge

H. Netzel1,2,3, P. Pietrukowicz4, I. Soszyński4, M. Wrona4
1Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka
18, 00-716 Warszawa, Poland
2Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Eötvös
Loránd Research Network (ELKH),
H-1121 Konkoly Thege Miklós út 15-17, Budapest, Hungary
3MTA CSFK Lendület Near-Field Cosmology Research Group, H-1121 Konkoly Thege
Miklós út 15-17, Budapest, Hungary
4Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland
E-mail<EMAIL_ADDRESS>
(Accepted . Received ; in original form )

###### Abstract

We have performed a frequency analysis of 10,092 $\delta$ Scuti-type stars
detected in the fourth phase of the Optical Gravitational Lensing Experiment
(OGLE) towards the Galactic bulge, which is the most numerous homogeneous
sample of $\delta$ Scuti stars observed so far. The main goal was to search
for stars pulsating in at least two radial modes simultaneously. We have found
3083 candidates for such stars, which is the largest set obtained to date.
Among them, 2655 stars pulsate in two radial modes, 414 stars pulsate in three
radial modes, and 14 stars pulsate in four radial modes at the same time. We
report the identification of 221 $\delta$ Scuti stars pulsating in the
fundamental mode, first overtone, and third overtone simultaneously. We show
the most populated Petersen and Bailey diagrams and discuss statistical
properties of the identified frequencies based on this numerous sample.
Additionally, we present theoretical predictions of period ratios for $\delta$
Scuti stars pulsating in overtones from the fourth to the seventh.

###### keywords:

stars: oscillations – stars: variable: $\delta$ Scuti

††pagerange: Frequency analysis of $\delta$ Scuti stars towards the Galactic
bulge–A††pubyear: 2021

##  1 Introduction

$\delta$ Scuti-type stars are intermediate-mass pulsating stars located in the
Hertzsprung-Russell (HR) diagram on and close to the main sequence and within
the classical instability strip. This location corresponds to spectral types
from A0 to F6 and covers luminosity classes from III to V. This diverse group
consists of main-sequence stars as well as pre-main-sequence (PMS) and post-
main sequence stars. Maroni & Palla (1998) calculated theoretical instability
strip for radial pulsations in PMS stars and concluded that stars with masses
above 1.5 M⊙ are situated within the instability strip before they reach the
zero-age main sequence (ZAMS). Observationally, first PMS stars of $\delta$
Scuti type were detected by Breger (1972). In principle, such stars should be
rare. However, searches resulted in several discoveries (e.g. Díaz-Fraile et
al., 2014). Even less frequent objects with $\delta$ Scuti-type pulsations
should be stars evolving off the main sequence through the Hertzsprung gap.
Niu et al. (2017) analyzed photometric and spectroscopic data for AE UMa and
concluded that this particular star belongs to Population I $\delta$ Scuti
stars evolving through the Hertzsprung gap.

$\delta$ Scuti stars cover a mass range from about 1.0 to 2.5 M⊙ that includes
an interesting transition from stars with radiative cores to stars with
convective cores. Pulsation periods are shorter than 0.3 d, which is a good
criterion to distinguish $\delta$ Scuti stars from nearby groups of pulsating
stars such as RR Lyrae-type stars, rapidly oscillating Ap (roAp) stars, and
$\gamma$ Doradus-type stars in the HR diagram. However, the instability strip
of $\gamma$ Doradus stars overlap with the instability strip of $\delta$ Scuti
stars, which results in so-called hybrid stars. In these stars, we observe
simultaneous pulsations in pressure modes as well as in gravity modes. For
instance, based on time-series multicolor photometry Handler (2009) confirmed
the existence of this type of pulsations in object HD 8801.

The majority of $\delta$ Scuti stars pulsate in many non-radial pressure modes
excited in the $\kappa$ mechanism. Usually, these modes are of low degree and
low order. However, spectroscopic studies confirmed the presence of high-
degree modes as well (Kennelly et al., 1998). Some $\delta$ Scuti stars
pulsate in one or several radial modes with large amplitudes. This subgroup of
variables is called High-Amplitude Delta Scuti stars (HADS). The incidence
rate of HADS is low. Lee et al. (2008) estimated that this group constitutes
only about 0.24 per cent of all Galactic $\delta$ Scuti stars. Several studies
also suggest that besides radial pulsations there may be present also low-
amplitude non-radial modes. HADS, in contrary to typical $\delta$ Scuti stars,
have slow rotation velocities ($v\sin i$ below 30 km/s) which, according to
Breger (2000), is a requirement for high-amplitude radial pulsations. HADS are
considered to be the transition objects from large-amplitude Cepheids to
small-amplitude pulsating main-sequence stars.

The presence of objects at different evolutionary stage and with different
internal structures among $\delta$ Scuti stars makes this group very
interesting for asteroseismology and Milky Way studies. Unfortunately, the
variety of excited low-amplitude non-radial modes, which are often difficult
to identify, limits the usage of $\delta$ Scuti stars. HADS give us an
opportunity to derive stellar parameters, because in principle it is easier to
identify radial modes based on period ratios. Popielski et al. (2000)
discussed the usage of period ratio as a probe of metallicity and applied the
idea to double-mode RR Lyrae stars.

Pietrukowicz et al. (2013) searched for pulsating stars in OGLE-III Galactic
disc fields and identified 57 bona fide $\delta$ Scuti stars. Among them, 22
stars pulsate in two radial modes and seven stars pulsate in at least three
radial modes. Having at least three radial modes, one can determine crucial
stellar parameters such as mass, luminosity, and metallicity. This approach
was applied to seven $\delta$ Scuti stars by Pietrukowicz et al. (2013). In
our work, we use the OGLE-IV photometry for over 10,000 recently detected
Galactic bulge $\delta$ Scuti pulsators (Pietrukowicz et al., 2020) to perform
a frequency search and to select stars that presumably pulsate in at least two
radial modes simultaneously. In Sec. 2, we describe the dataset and method of
the analysis. In Sec. 3, we present results of the analysis. In Sec. 4 we
present theoretical predictions for period ratios which are formed by higher
overtones. Discussion of our results are presented in Sec. 5. Sec. 6
summarizes our findings.

##  2 Data and analysis

###  2.1 Time-series Data

The Optical Gravitational Lensing Experiment (OGLE) is a long-term variability
survey that started in 1992 and is ongoing. In March 2010 it entered its
fourth phase with a new mosaic CCD camera installed to the 1.3-m Warsaw
telescope located at Las Campanas Observatory, Chile (OGLE-IV, Udalski et al.,
2015). OGLE regularly monitors the brightness of about two billion stars
towards the Galactic bulge, Galactic disc, and Magellanic Clouds. We used the
OGLE-IV photometry for the Galactic bulge fields covering six seasons from
2010 to 2015. We decided to use only those seasons out of all observations
carried out in the years 2010–2019 to minimize possible effects of changing
period, mean brightness, and amplitude. Firstly, such changes increase the
noise level in the power spectrum and would hamper the detection of additional
signals. Secondly, during such a long period of observations, proper motions
affect the obtained light curves. Removal of the proper motion effect might
increase the noise as well. Since in this work, we focus on searches for real
additional signals, we did not want to introduce more sources of noise into
the data.

We analyze only $I$-band observations, as these are numerous, hence more
suitable for the frequency search than observations in the $V$ band. Our input
sample consists of 10,092 $\delta$ Scuti stars published in Pietrukowicz et
al. (2020). The original sample was reduced by nineteen sources that doubled
other objects. The number of observations for each star varies from field to
field. The most frequently observed fields are those located in the densest
regions of the Galactic bulge. The largest number of measurements taken over
the years 2010–2015 is 12,906, whereas the smallest number is 35. Some fields
were observed for a shorter period of time. A histogram of the number of
$I$-band measurements is presented in Fig. 1. We did not exclude any stars
from our analysis based on the number of observations. However, the number of
data points affects noise level in the power spectrum. To estimate the noise
level for each star, we prewhitened the Fourier spectrum of each star with the
dominant frequency, $f$, and its four harmonics and used the mean noise level
in the frequency range from $6f$ to $7f$. This relation is presented in Fig.
2. Stars were divided into three subsets based on the number of points: $N\leq
500$, $500<N\leq 5000$, and $N>5000$. For all groups, the noise level
increases with the fading brightness. The lowest noise level is for frequently
observed stars and the highest noise level is for stars with the smallest
number of data points.

Figure 1: Histogram of the number of $I$-band observations of the Galactic
bulge stars.

Figure 2: Noise level for $\delta$ Scuti stars in the function of mean
$I$-band brightness. $\delta$ Scuti stars are divided into three subgroups
depending on the number of observations ($N$) as indicated in the bottom right
corner of the figure.

###  2.2 Frequency Analysis

The analysis was divided into two steps. The first step was to determine
frequencies present in each star. The second step was to analyze the obtained
frequencies and to select stars with period ratios falling into known groups
of multimode $\delta$ Scuti stars.

The first step was based on Fourier analysis and the consecutive prewhitening
method. Due to the large number of stars we used an automatic procedure for
this analysis. First, the procedure fitted the dominant frequency and its
harmonics to the data in the form of Fourier series:

$I(t)=A_{0}+\sum A_{k}\sin(2\pi f_{k}t+\phi_{k}),$ (1)

where $A_{k}$, $f_{k}$, and $\phi_{k}$ are the amplitude, frequency, and phase
of the $k$-th harmonic. Only frequencies fulfilling the criterion
$A_{k}/\sigma_{k}>4$ were fitted to the data. After prewhitening with the
dominant frequency and harmonics, residuals were checked for additional
signals in a range 0–80 cycles per day. If any signal in this range was
detected, the procedure checked whether its signal-to-noise ratio (S/N) was
higher than 4.0. If it was higher, the signal was included in the fit and the
program checked again the residuals for the presence of additional
frequencies. The procedure stopped once no frequency in the analyzed range
fulfilled the criterion $S/N>4.0$ or when the number of independent
frequencies exceeded sixteen. Fitting frequencies to the data was done with
the help of a non-linear least-squares solver which uses modified Levenberg-
Marquardt algorithm. All amplitudes, frequencies, and phases were adjusted
while fitting to the data.

We note, that because the OGLE data is ground-based photometry, we had to deal
with additional signals that are not connected to the variability of the
objects (e.g. daily aliases, instrumental signals). Our procedure took into
account the presence of such signals and did not include them in the resultant
frequency list. However, we note that the higher priority was to detect
signals that form period ratios characteristic for HADS. Therefore, we were
not extremely strict with excluding signals and rather checked suspicious
signals later during the analysis. In some stars, we observed long-term trends
that manifest as signals of low frequencies in the power spectrum. Together
with daily aliases, they increase the noise level and hence hamper the
detection of low-amplitude additional signals. These trends were removed using
a third-order polynomial before the search for additional frequencies.

The program was run on the whole sample of variable stars and provided us a
list of frequencies for each star. An example of such list is shown in Tab. 1.
The automatic procedure provided also a simple interpretation of frequencies,
whether they are true independent signals, combination frequencies, or
harmonics of signals found previously.

Signal |  $f$ [c/d] |  $A$ [mag] | $\phi$
---|---|---|---
$f_{1}$ | 15.972768(5) | 0.048(1) | 5.6(2)
$2f_{1}$ | 31.94554(1) | 0.016(1) | 1.2(4)
$f_{2}$ | 20.56591(2) | 0.012(1) | 5.0(8)
$f_{2}-f_{1}$ | 4.59314(2) | 0.008(1) | 3.4(8)
$f_{1}+f_{2}$ | 36.53868(2) | 0.007(1) | 0.4(8)
$f_{3}$ | 46.91555(5) | 0.006(1) | 6(2)
$f_{4}$ | 18.61421(5) | 0.006(1) | 0(2)
Table 1: Example of frequency list generated by the automatic procedure for
star OGLE-BLG-DSCT-03647. Consecutive columns provide interpretation of the
detected signals, their frequencies, amplitudes, and phases.

The second step of the analysis was to select stars for which frequencies fit
to the multimode pattern. This was done with the help of the Petersen diagram,
which is a diagram of shorter to longer period ratio versus the logarithm of
the longer period. Because the main goal of the analysis was to find stars
pulsating in radial modes, we narrowed down our sample to stars for which
period ratio of two detected periodicities fell into the range 0.75–0.82. In
this period ratio, we expect either $\delta$ Scuti stars pulsating in the
fundamental mode (F) and first overtone (1O) or the first and second overtone
(2O) simultaneously as well as stars with more than two radial modes, two of
which form period ratio that falls into this range. This resulted in a sample
of 3286 stars. For each star, we plotted all frequencies in the Petersen
diagram. For reference, we used stars with identified radial modes from the
Galactic disc (Pietrukowicz et al., 2013). All these diagrams were inspected
visually. Examples of such diagrams for one double-mode star, two triple-mode
stars, and one star with four radial modes are presented in Fig. 3. Stars were
classified based on their position in the Petersen diagram. In some stars,
besides frequencies forming the period ratios corresponding to defined
sequences for radial pulsations, we detected other signals that do not fit the
pattern. Those might be non-radial modes, as they can be present together with
radial ones. We assume that all points falling at or close to the sequences
are due to radial modes.

Figure 3: Petersen diagrams for selected $\delta$ Scuti stars from our sample.
Period ratios formed by detected frequencies are marked with black open
circles. Filled circles correspond to multimode $\delta$ Scuti stars from the
Galactic disc (Pietrukowicz et al., 2013). On the left side of each sequence
it is indicated which modes form the period ratio corresponding to the
sequence. Panel A: a double-mode star pulsating in the fundamental mode,
$f_{1}$, and in the first overtone, $f_{2}$. Panel B: a triple-mode star
pulsating in the fundamental mode, $f_{1}$, in the first overtone, $f_{2}$,
and in the second overtone, $f_{3}$. Panel C: a triple-mode star pulsating in
the first overtone, $f_{1}$, in the second overtone, $f_{2}$, and in the third
overtone, $f_{3}$. Panel D: a star pulsating in four modes simultaneously:
fundamental mode $f_{1}$, first overtone $f_{2}$, second overtone $f_{5}$, and
third overtone $f_{3}$.

Additionally, we checked period ratios corresponding to pulsations F+2O, F+3O,
and 1O+3O in order to detect stars that might be candidates for such double-
mode radial pulsators. We verified the location of the stars in the Petersen
diagram, in the period ratio range corresponding to these multimode
pulsations. Then, we selected only objects showing signals being combinations
of these modes in the power spectra.

##  3 Results

###  3.1 Additional signals in $\delta$ Scuti stars

The majority of the analyzed $\delta$ Scuti stars have more than one
periodicity. In Tab. 2, we list how many stars were found with a various
number of frequencies. In the entire sample of 10,092 stars, 2872 objects are
single-periodic $\delta$ Scuti pulsators. They can either pulsate in a single
radial mode, most likely in the fundamental mode or first overtone, or pulsate
in a non-radial mode. Single-periodic $\delta$ Scuti stars are slightly less
numerous than stars pulsating in two modes simultaneously. The group of
double-mode pulsators is the most abundant in the whole sample and consists of
2947 objects. Although the maximum number of independent frequencies was set
to 16, we did not detect any star with this number of signals. Fifteen
independent frequencies were detected in two stars, but none of the frequency
patterns fits to any radial mode sequence in the Petersen diagram. We did not
investigate the nature of additional frequencies except for period ratios
corresponding to those of radial modes. Some other detected signals may come
from non-radial modes, blending, or other physical phenomena, e.g. eclipses.
Signals corresponding to binarity were detected in 14 stars as reported by
Pietrukowicz et al. (2020).

Number of signals | $N_{\rm stars}$ | $N_{\rm stars}/N_{\rm tot}$
---|---|---
1 | 2872 | 0.28 %
2 | 2947 | 0.29 %
3 | 2007 | 0.20 %
4 | 1177 | 0.12 %
5 | 565 | 0.06 %
6 | 267 | 0.03 %
7 | 116 | 0.01 %
8 | 54 |  $<$1 %
9 | 36 |  $<$1 %
10 | 19 |  $<$1 %
11 | 14 |  $<$1 %
12 | 10 |  $<$0.1 %
13 | 4 |  $<$0.1 %
14 | 2 |  $<$0.1 %
15 | 2 |  $<$0.1 %
16 | 0 | –
Table 2: Number of detected independent frequencies in 10,092 analyzed
$\delta$ Scuti variables. Consecutive columns provide the following
information: the number of frequencies, number of stars, and incidence rate.

In the top panel of Fig. 4, we present a period-amplitude diagram for all
analyzed $\delta$ Scuti stars, while in the bottom panel, such a diagram for
only single-mode stars. Both groups cover the same range of amplitudes. In the
case of single-mode $\delta$ Scuti stars, we observe a slightly shorter range
of periods, up to 0.3 d, whereas for all $\delta$ Scuti Scuti stars, there are
several objects with periods between 0.3 and 0.35 d. No distinct structures
are visible in the diagrams.

Figure 4: Period–amplitude diagram for all $\delta$ Scuti stars (blue points
in the top panel) and single-mode $\delta$ Scuti stars (red crosses in the
bottom panel). Stars with very high amplitudes are faint and blended objects.

Figure 5: Histogram of frequency separations for all multimode $\delta$ Scuti
stars in the sample.

We were curious whether there are any characteristic equidistant frequency
separations between the detected signals in the power spectrum. Specific
separations are expected in the case of non-radial pulsations. We calculated
separations between frequencies detected in all multimode $\delta$ Scuti stars
from our sample. In Fig. 5, we plot histogram of the obtained separations. For
the majority of stars, the separations are very small, but they extend to over
40 c/d. There is no clumping of the values apart from an excess observed
around 3–5 c/d.

###  3.2 Multimode radial pulsations in $\delta$ Scuti stars

The main goal of this work was to select candidates for multimode radial
pulsators. Results are collected in Tab. 3, where we provide the number of
stars together with the incidence rate for various groups of modes. We
detected 3083 stars pulsating in at least two radial modes. Among them, 2655
stars are candidates for objects pulsating in two radial modes simultaneously.
This corresponds to over 26 per cent of the total sample. The most abundant
group is formed of stars pulsating in the fundamental mode and the first
overtone (F+1O). This group consists of 2333 objects. The second most numerous
group are double-mode stars pulsating in the first and second overtones
(1O+2O). Besides the two groups, we found candidates for stars pulsating in
the fundamental mode and second overtone (F+2O), fundamental mode and third
overtone (F+3O), and first and third overtones (1O+3O). There are 18, 23, and
25 stars in each group, respectively. We note, however, that stars classified
as F+2O, F+3O, and 1O+3O pulsators had to fulfill an additional criterion,
i.e. the presence of combination frequencies between the radial modes. Hence,
the numbers of F+2O, F+3O and 1O+3O stars are the lower limit on the number of
these types of double-mode pulsating stars. Four-hundred-and-fourteen $\delta$
Scuti variables are classified as stars pulsating in three radial modes and 14
objects as stars pulsating in four radial modes simultaneously. All stars
classified as double-mode radial pulsators are shown in the Bailey diagram in
Fig. 6, where we plotted period and amplitude of the dominant variability.
With different symbols, we mark F+1O and 1O+2O stars. Stars from both groups
cover a similar range of periods. Except for one 1O+2O pulsator, of an
amplitude of 0.6 mag, the remaining F+1O and 1O+2O pulsators have amplitudes
below 0.5 mag. The amplitudes are lower for stars with longer pulsation
periods for both groups.

Figure 6: Period-amplitude (Bailey) diagram for $\delta$ Scuti stars classified as objects pulsating in two radial modes simultaneously. With blue points we mark stars pulsating in the fundamental mode and first overtone. Stars classified as objects pulsating in the first and second overtones are marked with red crosses. Modes | $N_{\rm stars}$ | $N_{\rm stars}/N_{\rm tot}$
---|---|---
F+1O | 2333 | 23.1 %
1O+2O | 256 | 2.5 %
F+1O+3O | 221 | 2.2 %
F+1O+2O | 145 | 1.4 %
1O+2O+3O | 48 | 0.5 %
1O+3O | 25 | 0.2 %
F+3O | 23 | 0.2 %
F+2O | 18 | 0.2 %
F+1O+2O+3O | 14 | 0.1 %
Table 3: Census of stars pulsating in at least two radial modes. Consecutive
columns provide the type of multiple radial pulsations, number of stars, and
fraction of all stars falling into each group.

As mentioned above, in the analyzed sample, 414 stars pulsate in three radial
modes simultaneously. We classified 145 stars as objects pulsating in the
fundamental mode, first, and second overtones (F+1O+2O), 221 stars as objects
pulsating in the fundamental mode, first, and third overtones (F+1O+3O), and
48 stars as objects pulsating in the first, second, and third overtones
(1O+2O+3O). This corresponds to incidence rates of 1.4, 2.2, and 0.5 per cent
for F+1O+2O, F+1O+3O, and 1O+2O+3O pulsators, respectively. Interestingly, the
most common type of triple-mode radial pulsations is F+1O+3O. This mode
combination was not detected in $\delta$ Scuti stars from the OGLE-III
Galactic disc fields. Very recently, Yang et al. (2021) reported the detection
of a $\delta$ Scuti variable pulsating in three radial modes corresponding to
the fundamental mode, first, and third overtones. This star was detected in
the data from the Kepler space mission, and so far was the only candidate for
such type of radial pulsations. Here, we report additional 221 candidates for
triple-mode F+1O+3O stars. An interesting group of objects, however, are those
pulsating in four radial modes simultaneously. We found 14 candidates for
stars pulsating in the fundamental mode, first, second, and third overtones.
Tab. 4 lists periods of the signals identified as radial modes for all the
quadruple-mode $\delta$ Scuti pulsators. Analogous information on triple-mode
stars is provided in Tab. A1, sample of which is provided in Tab. 5.

OGLE-BLG- |  $P_{\rm F}$ [d] |  $P_{\rm 1O}$ [d] |  $P_{\rm 2O}$ [d] |  $P_{\rm 3O}$ [d]
---|---|---|---|---
-DSCT- |  |  |  |
00751 | 0.054775 | 0.042734 | 0.034395 | 0.028151
01345 | 0.054168 | 0.041257 | 0.032238 | 0.026991
03522 | 0.074613 | 0.058328 | 0.047497 | 0.039045
03652 | 0.144371 | 0.111366 | 0.088956 | 0.071905
04220 | 0.134911 | 0.106412 | 0.082682 | 0.072704
04712 | 0.106594 | 0.082204 | 0.064173 | 0.054827
04958 | 0.076545 | 0.059235 | 0.046955 | 0.038854
05749 | 0.161760 | 0.122848 | 0.094629 | 0.082316
06799 | 0.072748 | 0.056377 | 0.045357 | 0.038473
07380 | 0.075060 | 0.058121 | 0.046853 | 0.039223
07587 | 0.071205 | 0.055139 | 0.045207 | 0.037168
07793 | 0.070645 | 0.054875 | 0.042816 | 0.036100
08124 | 0.050560 | 0.038931 | 0.031649 | 0.026535
08774 | 0.116664 | 0.08935 | 0.072157 | 0.061194
Table 4: $\delta$ Scuti stars pulsating in four radial modes simultaneously. Consecutive columns provide the name, period of the fundamental mode, period of the first overtone up to the period of the third overtone. OGLE-BLG- |  $P_{\rm F}$ [d] |  $P_{\rm 1O}$ [d] |  $P_{\rm 2O}$ [d] |  $P_{\rm 3O}$ [d]
---|---|---|---|---
-DSCT- |  |  |  |
00011 | – | 0.05727 | 0.04661 | 0.03945
00018 | 0.07775 | 0.06058 | 0.04904 | –
00075 | 0.10713 | 0.08376 | 0.06761 | –
00078 | 0.15776 | 0.11854 | – | 0.07758
00121 | 0.06908 | 0.05344 | – | 0.03749
00136 | 0.09938 | 0.07705 | – | 0.05021
00143 | 0.07965 | 0.06176 | – | 0.04026
00181 | 0.07628 | 0.06013 | 0.04782 | –
00193 | 0.06612 | 0.05110 | – | 0.03562
00336 | – | 0.15557 | 0.11981 | 0.09938
⋮ | ⋮ | ⋮ | ⋮ | ⋮
Table 5: $\delta$ Scuti stars pulsating in three radial modes simultaneously.
Consecutive columns provide star’s ID and periods of the fundamental mode,
first, second and third overtones. Mode that is not present is omitted with
‘–’. Full table is provided in the Appendix available online.

Galactic bulge $\delta$ Scuti stars pulsating in two or more radial modes are
plotted in the Petersen diagram in Fig. 7 together with Galactic disc $\delta$
Scuti stars analyzed in Pietrukowicz et al. (2013). Colors indicate period
ratios corresponding to various combinations of radial modes. A diagram of
solely double-mode radial pulsators is shown in Fig. 8. In both figures, the
scatter for bulge stars seems to be larger than for disc stars. This can be
explained by two facts. First, the total number of disc stars is much smaller
than bulge stars. For example, there are only two stars in the sequence
corresponding to the period ratio of 3O/F. The same problem is for the
sequence corresponding to the period ratio of 3O/1O. Second, $\delta$ Scuti
stars observed towards the Galactic bulge represent various populations, hence
they have a wide metallicity range. Metal content has a significant influence
on the observed period ratios. This is the reason why we did not exclude stars
with relatively small departures from the sequences defined by the Galactic
disc stars.

Figure 7: Petersen diagram for $\delta$ Scuti stars pulsating in at least two
radial modes simultaneously. Stars from the Galactic bulge sample are marked
with filled circles. Black open circles represent stars detected in the
Galactic disc by Pietrukowicz et al. (2013).

Figure 8: Petersen diagram for $\delta$ Scuti stars pulsating in two radial
modes simultaneously. Stars from the bulge sample are marked with filled
circles. Black open circles represent disc stars reported in Pietrukowicz et
al. (2013).

###  3.3 Light curve decomposion of F+1O pulsators

Pietrukowicz et al. (2020) discussed extensively light curve shapes of
$\delta$ Scuti stars based on Fourier coefficients. They studied only single-
mode pulsators. Here, we analyze light curve shapes of F+1O pulsators. Thanks
to the characteristic period ratio we were able to indicate which mode is the
fundamental one and which one is the first overtone. We disentangled the light
curves by removing one of the modes. Then, we fitted two separate light curves
with a sine series of the same length in the form described with Eq. 1.
Amplitude ratios are defined as

$R_{k1}=\frac{A_{k}}{A_{1}}$

and phase differences are defined as

$\phi_{k1}=\phi_{k}-k\phi_{1}.$

In Fig. 9, we plot peak-to-peak amplitudes and Fourier coefficients in the
function of the period for disentangled light curves of F+1O pulsators. Even
though for describing light curve shapes we used sine series of the same
length, we plotted only those coefficients that are made by significant terms
of the sine series ( $A_{k}/\sigma_{k}>4$). As expected, the first-overtone
components have significantly smaller amplitudes. Except for a few stars, they
generally cover half of the range represented by the fundamental mode
components. The amplitude ratios $R_{21}$ are on average smaller for the first
overtone than for the fundamental mode. Phase differences $\phi_{21}$ and
$\phi_{31}$ for fundamental-mode components form clear sequences. There are
not many points in the case of $\phi_{31}$ for the first overtone. This is
because light curves of the first-overtone components can be described with a
shorter Fourier series. In the case of $\phi_{21}$, we observe a slightly
larger scatter. Based on the Fourier coefficients alone it is impossible to
identify the pulsation modes unambiguously.

Figure 9: Fourier parameters of disentangled light curves of the fundamental-
mode (in black) and first-overtone components (in red) in F+1O stars. Top
panel shows peak-to-peak amplitude in the function of period. Middle and
bottom panels show the amplitude ratios and phase differences in the function
of period.

##  4 Predicted period ratios

We did not find any good candidates for stars pulsating in overtones higher
than 3O. During the classification, we, however, compared the results of the
frequency analysis with theoretically predicted period ratios for the fourth
(4O), fifth (5O), sixth (6O), and seventh overtones (7O). To obtain
approximated period ratios we calculated a grid of models for a range of
masses and metallicities using the Warsaw evolution and pulsation codes
(Paczyński, 1969; Dziembowski, 1977). We generated models for masses from 1.0
to 2.5 M⊙ with a step of 0.1 M⊙ and metal content $log~{}Z$ from $-3.4$ to
$-1.1$ with a step of 0.1. We did not include rotation and we set the mixing
length parameter to $\alpha_{\rm MLT}=1.5$. We selected models in which
certain radial modes are unstable: the fundamental mode, fourth, fifth, sixth,
or seventh overtone. Due to the frozen-in description of convection, the code
did not allow us to determine the position of the red edge of the instability
strip but was able to approximately predict the position of the blue edge. The
position of the blue edge also changes due to the frozen-in convection
approximation, but only slightly. To limit the number of models, we used the
predicted position of the red edge based on observations (Pamyatnykh, 2000).
We plot the predicted period ratios of unstable radial modes in Fig. 10.

Figure 10: Theoretically predicted period ratios for radial modes in $\delta$
Scuti stars.

##  5 Discussion

In this work, we analyzed a large sample of 10,092 $\delta$ Scuti stars
towards the Galactic bulge from the OGLE-IV survey. Two decades earlier, a
catalog of $\delta$ Scuti stars from different sources was prepared by
Rodríguez et al. (2000) and contained 636 objects. Unfortunately, the authors
did not provide information on the number of HADS and multiperiodicity of
stars. Balona & Dziembowski (2011) analyzed 1568 $\delta$ Scuti stars from the
Kepler mission. They focused on the mode content and comparison between the
theory and observations. The authors did not provide the number of stars
classified as HADS, but the number of stars with one dominant signal. Thanks
to excellent quality of the Kepler photometry, the stars are classified as
objects showing one signal with significantly higher amplitude than others,
not necessarily as those having only one signal in the power spectrum. Balona
& Dziembowski (2011) identified 160 stars with one dominant frequency in the
power spectrum, which corresponds to about 10 per cent of their sample. In our
sample, we identified 28 per cent of single-periodic stars. The difference
stems from a significantly higher noise level in ground-based photometry.

Poleski et al. (2010) detected 2786 $\delta$ Scuti stars in the OGLE-III data
from the Large Magellanic Cloud (LMC). In their sample, they found 92
multimode pulsators. Sixty-seven stars were classified as F+1O pulsators,
three stars as 1O+2O pulsators, three stars as 2O+3O pulsators, and three
stars as 1O+3O pulsators. There are two triple-mode pulsators. Fourteen
objects did not fit the radial mode pattern in the Petersen diagram. This
gives an incident rate for F+1O pulsators of 2.4 per cent, which is by an
order of magnitude lower than inferred from our analysis. In total, Poleski et
al. (2010) classified 78 objects as multimode radially pulsating $\delta$
Scuti stars. This corresponds to a rate of 3 per cent. It is significantly
smaller than the 31 per cent obtained for our sample. This might be due to the
difference in observed brightness of $\delta$ Scuti stars located in the LMC
fields and Galactic bulge fields. The noise level is higher in more distant
LMC objects.

Pietrukowicz et al. (2013) identified 57 $\delta$ Scuti stars in the OGLE-III
Galactic disc fields. They detected a single frequency in 28 stars from the
analyzed 57 stars, which corresponds to about 49 per cent. This is higher than
the 28 per cent obtained here. The incidence rate of stars with two radial
modes was 39 per cent and stars with three radial modes was 12 per cent for
the disc stars. The incidence rate of double-mode radial pulsations is higher
than 26 per cent in the sample analyzed here. In the Galactic bulge sample, we
detected triple-mode radial pulsations in 4 per cent of stars, which is three
times smaller than in the case of the disc sample. Only one star was
identified as a quadruple-mode pulsator in the Galactic disc. This corresponds
to the incidence rate of about 2 per cent. The rate of such stars is
significantly lower in our analysis (below 1 per cent), but the detected
number of stars is 14. Hence, our analysis significantly increased the total
number of known stars with four radial modes.

We detected a group of stars pulsating in three radial modes F+1O+3O
simultaneously. Interestingly, the second overtone is not excited in these
stars or has an amplitude below the detection limit. A candidate for such star
was reported only recently by Yang et al. (2021), who analyzed Kepler
photometry for star KIC 10975348. In this object, they detected three
independent signals. In KIC 10975348 two of the signals form a period ratio
around 0.758, which is close to the sequence corresponding to F and 1O
pulsations. It is slightly lower than typical period ratios of 1O/F which
implies a high metal content for this star. The third detected signal forms a
period ratio of 0.539 with the fundamental mode and a period ratio of 0.71
with the first overtone. These period ratios are close to 3O/F and 3O/1O
sequences. In Fig. 11, we show F+1O+3O $\delta$ Scuti pulsators from the OGLE-
IV Galactic bulge sample together with variables from the OGLE-III Galactic
disc sample (Pietrukowicz et al., 2013) and KIC 10975348 (Yang et al., 2021).
In the Galactic disc sample, there are only two stars in the 3O/1O and 3O/F
sequences. Bulge stars selected in our work form sequences of significant
width compared to the sequences formed by the Galactic disc sample. The period
ratios of KIC 10975348 are within the scatter observed for the bulge stars.
However, two observed period ratios of 3O/1O and 3O/F are higher than the
typical values and one period ratio of 1O/F is lower than the typical values.
Similarly, the period ratios of KIC 10975348 differ from the sequences defined
by the Galactic disc sample. The period ratio of 1O/F is smaller, whereas the
period ratios of 3O/1O or 3O/F are higher than observed for the Galactic disc
stars. The effect of metallicity should shift all the observed period ratios
in the same direction from the main trends. Therefore, the observed period
ratios seem inconsistent with the proposed interpretation and it remains
uncertain whether KIC 10975348 is a truly F+1O+3O pulsator.

Figure 11: Petersen diagram for $\delta$ Scuti stars classified as F+1O+3O.
Red, lime, and orange points correspond to the period ratios of 1O/F, 3O/1O,
and 3O/F, respectively. Black open circles represent radially pulsating
$\delta$ Scuti stars detected in the Galactic disc (Pietrukowicz et al., 2013)
forming those period ratios. Blue squares correspond to object KIC 10975348.

##  6 Conclusions

We analyzed the sample of 10,092 $\delta$ Scuti stars observed towards the
Galactic bulge during the fourth phase of the OGLE project (Pietrukowicz et
al., 2020), which is the largest sample of $\delta$ Scuti stars known so far.
We searched for multi-periodic pulsators with period ratios characteristic for
radial modes. In 2872 stars, we detected only one period. We classified 3083
objects as candidates for stars pulsating in at least two radial modes
simultaneously. Among them, 2655 stars were classified as objects pulsating in
two radial modes. The most common type of pulsations is the fundamental mode
and first overtone, which was detected in 23.1 per cent of the analyzed
sample. Another 256 stars were classified as objects pulsating in the first
and second overtones. Additionally, we found 25, 18, and 23 stars, which are
candidates for 1O+3O, F+2O, and F+3O pulsators, respectively. Another 414
objects were classified as candidates for stars pulsating in three radial
modes. In 145 stars, we detected signals that correspond to the fundamental
mode, first and second overtones. Forty-eight stars were classified as objects
pulsating in the first, second, and third overtones. We recognized 221 stars
which are candidates for stars pulsating in the fundamental mode, first
overtone, and third overtone, simultaneously. We selected 14 candidates for
stars with four radial modes, F+1O+2O+3O. We note here, that besides signals
identified as radial modes, in some of the stars we also detected other
frequencies that did not fit any radial modes. Some of them are located close
to those identified as radial modes. The signals can be either due to
additional pulsations in non-radial modes or of other origin. Additional study
of selected multi-mode radially pulsating candidates is required to confirm
their classification. Such study focused on theoretical modeling of the
selected sample is ongoing.

For stars classified as F+1O pulsators, we verified light curve shapes after
disentangling signals from the two modes. We did not notice any characteristic
clumping in the Fourier space, which would enable unambiguous mode
identification based on Fourier decomposition in future studies. We prepared
theoretical predictions for period ratios corresponding to pulsations in the
fourth, fifth, sixth, and seventh overtones. We did not detect any good
candidates for such pulsators in the analyzed sample.

Based on the huge sample of OGLE-IV $\delta$ Scuti stars we tried to find any
characteristic pulsation properties of this type of variables. We presented
Bailey, Petersen, and Fourier diagrams with the best statistics. There are no
specific features in the period-amplitude diagrams. We also searched for
frequency grouping in the power spectra. There is only a slight excess of
frequency separations around 3-5 c/d, which is difficult to interpret. To
conclude, $\delta$ Scuti stars seem to form a very diverse group of pulsators
without evident properties.

## Acknowledgments

HN is supported by the Polish Ministry of Science and Higher Education under
grant 0192/DIA/2016/45 within the Diamond Grant Program for years 2016–2021
and by the Foundation for Polish Science (FNP). This project has been
supported by the Lendület Program of the Hungarian Academy of Sciences,
project No. LP2018-7/2020. This work has also been supported by the National
Science Centre, Poland, grant MAESTRO 2016/22/A/ST9/00009 to IS.

## Data availability

Photometric data used for this study was published as a part of the OGLE
project (Pietrukowicz et al., 2020) and are available at the OGLE On-line Data
Archive.

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##  Appendix A Data tables

Table 6: $\delta$ Scuti stars classified as pulsating in three radial modes. Consecutive columns provide star’s ID and periods of the fundamental mode, first, second and third overtones. Mode that is not present is omitted with ‘–’. OGLE-BLG-DSCT- |  $P_{\rm F}$ [d] |  $P_{\rm 1O}$ [d] |  $P_{\rm 2O}$ [d] |  $P_{\rm 3O}$ [d] |  OGLE-BLG-DSCT- |  $P_{\rm F}$ [d] |  $P_{\rm 1O}$ [d] |  $P_{\rm 2O}$ [d] |  $P_{\rm 3O}$ [d]
---|---|---|---|---|---|---|---|---|---
00011 |  – |  0.05727 |  0.04661 |  0.03945 |  05730 |  0.07188 |  0.05563 |  0.04485 |  –
00018 |  0.07775 |  0.06058 |  0.04904 |  – |  05765 |  0.07040 |  0.05469 |  – |  0.03669
00075 |  0.10713 |  0.08376 |  0.06761 |  – |  05797 |  0.07310 |  0.05662 |  – |  0.03886
00078 |  0.15776 |  0.11854 |  – |  0.07758 |  05804 |  0.06494 |  0.05027 |  0.04015 |  –
00121 |  0.06908 |  0.05344 |  – |  0.03749 |  05809 |  – |  0.07316 |  0.05810 |  0.04728
00136 |  0.09938 |  0.07705 |  – |  0.05021 |  05813 |  0.09405 |  0.07289 |  – |  0.04900
00143 |  0.07965 |  0.06176 |  – |  0.04026 |  05836 |  0.24507 |  0.18685 |  0.14868 |  –
00181 |  0.07628 |  0.06013 |  0.04782 |  – |  05855 |  0.11330 |  0.08729 |  0.07008 |  –
00193 |  0.06612 |  0.05110 |  – |  0.03562 |  05867 |  0.05254 |  0.04050 |  – |  0.02818
00336 |  – |  0.15557 |  0.11981 |  0.09938 |  05869 |  0.10177 |  0.07911 |  – |  0.05106
00358 |  0.06059 |  0.04685 |  – |  0.03099 |  05885 |  0.05687 |  0.04411 |  – |  0.02843
00394 |  0.14252 |  0.11055 |  0.08915 |  – |  05907 |  0.06509 |  0.05042 |  – |  0.03377
00434 |  – |  0.20614 |  0.16369 |  0.13566 |  05937 |  0.11495 |  0.08946 |  – |  0.06153
00648 |  0.06346 |  0.04906 |  0.03972 |  – |  06029 |  0.08555 |  0.06613 |  0.05524 |  –
00758 |  0.08913 |  0.06973 |  – |  0.04800 |  06039 |  0.05964 |  0.04612 |  – |  0.03074
00830 |  0.10817 |  0.08351 |  – |  0.05675 |  06079 |  0.06112 |  0.04717 |  – |  0.03206
00871 |  0.06834 |  0.05328 |  – |  0.03644 |  06089 |  0.10008 |  0.07756 |  0.06253 |  –
00880 |  0.06209 |  0.04802 |  0.04049 |  – |  06091 |  0.08801 |  0.06877 |  – |  0.04483
00899 |  0.10475 |  0.08104 |  – |  0.05428 |  06099 |  0.06772 |  0.05269 |  – |  0.03590
00906 |  – |  0.19362 |  0.15366 |  0.12716 |  06122 |  0.05885 |  0.04551 |  0.03625 |  –
00922 |  0.10210 |  0.07944 |  – |  0.05380 |  06154 |  0.05466 |  0.04238 |  0.03491 |  –
00929 |  0.05251 |  0.04086 |  – |  0.02688 |  06210 |  0.09311 |  0.07395 |  – |  0.04812
00953 |  0.06857 |  0.05302 |  – |  0.03550 |  06211 |  0.07828 |  0.06047 |  0.04862 |  –
00957 |  0.06735 |  0.05140 |  0.03982 |  – |  06281 |  0.07145 |  0.05619 |  – |  0.03705
00980 |  0.06786 |  0.05274 |  0.04309 |  – |  06293 |  0.10367 |  0.08017 |  – |  0.05359
01013 |  0.09848 |  0.07630 |  – |  0.05229 |  06309 |  0.05787 |  0.04397 |  – |  0.02858
01050 |  0.05904 |  0.04595 |  – |  0.03117 |  06323 |  0.10086 |  0.08108 |  – |  0.05312
01065 |  0.07498 |  0.05799 |  0.04809 |  – |  06348 |  0.04971 |  0.03845 |  0.03135 |  –
01092 |  0.06326 |  0.04972 |  – |  0.03373 |  06349 |  0.09000 |  0.07025 |  0.05731 |  –
01119 |  0.12605 |  0.09747 |  0.07842 |  – |  06369 |  – |  0.07804 |  0.06264 |  0.05233
01189 |  0.11759 |  0.09126 |  – |  0.05993 |  06373 |  0.04787 |  0.03676 |  – |  0.02453
01198 |  – |  0.14808 |  0.11770 |  0.09712 |  06375 |  0.05569 |  0.04295 |  – |  0.02877
01223 |  0.09873 |  0.07757 |  – |  0.05097 |  06396 |  – |  0.07133 |  0.05581 |  0.04804
01299 |  – |  0.08599 |  0.06847 |  0.05961 |  06429 |  0.11077 |  0.08570 |  – |  0.05715
01474 |  0.05758 |  0.04461 |  0.03773 |  – |  06458 |  0.08837 |  0.06903 |  – |  0.04490
01534 |  0.05364 |  0.04155 |  0.03323 |  – |  06474 |  0.06890 |  0.05242 |  – |  0.03413
01573 |  0.04709 |  0.03640 |  – |  0.02523 |  06522 |  0.06877 |  0.05318 |  – |  0.03455
01697 |  0.09457 |  0.07302 |  – |  0.04878 |  06615 |  0.06345 |  0.04952 |  – |  0.03156
01778 |  0.11052 |  0.08388 |  0.06928 |  – |  06624 |  0.04888 |  0.03817 |  – |  0.02506
01853 |  0.09553 |  0.07466 |  0.06186 |  – |  06675 |  – |  0.13158 |  0.10484 |  0.08664
01870 |  0.12981 |  0.10021 |  0.08058 |  – |  06712 |  0.06177 |  0.04810 |  – |  0.03184
01893 |  0.10118 |  0.07762 |  – |  0.05465 |  06748 |  0.07702 |  0.05955 |  0.04926 |  –
01896 |  0.12843 |  0.10240 |  – |  0.06758 |  06755 |  0.04686 |  0.03624 |  – |  0.02527
02004 |  0.09091 |  0.07059 |  – |  0.04759 |  06758 |  0.06008 |  0.04637 |  0.03747 |  –
02034 |  0.06156 |  0.04759 |  0.03788 |  – |  06774 |  0.04605 |  0.03565 |  0.02914 |  –
02060 |  0.06962 |  0.05435 |  – |  0.03607 |  06775 |  0.11581 |  0.08965 |  – |  0.06147
02064 |  0.06121 |  0.04771 |  0.03889 |  – |  06784 |  0.11692 |  0.09240 |  0.07416 |  –
02109 |  0.06708 |  0.05202 |  0.04457 |  – |  06798 |  0.08098 |  0.06284 |  – |  0.04070
02156 |  0.13567 |  0.10455 |  0.08731 |  – |  06808 |  0.13316 |  0.10270 |  0.08184 |  –
02195 |  0.26725 |  0.20304 |  0.16371 |  – |  06812 |  0.08976 |  0.06941 |  0.05852 |  –
02486 |  0.08391 |  0.06520 |  0.05489 |  – |  06840 |  0.10395 |  0.08028 |  0.06450 |  –
02575 |  0.10966 |  0.08485 |  – |  0.05646 |  06844 |  0.09028 |  0.07047 |  – |  0.04834
02578 |  – |  0.14517 |  0.11540 |  0.09507 |  06851 |  0.09580 |  0.07375 |  – |  0.04951
02605 |  0.06617 |  0.05116 |  – |  0.03508 |  06896 |  0.05651 |  0.04366 |  – |  0.02908
02606 |  0.12185 |  0.09389 |  0.07293 |  – |  06898 |  0.13387 |  0.10307 |  0.08007 |  –
02684 |  0.05794 |  0.04513 |  0.03696 |  – |  06904 |  0.17478 |  0.14164 |  – |  0.09121
02686 |  0.07814 |  0.06039 |  – |  0.03860 |  06933 |  0.10220 |  0.07898 |  – |  0.05388
02708 |  – |  0.20608 |  0.16343 |  0.13538 |  06948 |  0.08205 |  0.06360 |  – |  0.04293
02709 |  0.10624 |  0.08191 |  0.06802 |  – |  06952 |  0.13768 |  0.10592 |  0.08289 |  –
02713 |  0.07463 |  0.05780 |  – |  0.03876 |  06956 |  0.09580 |  0.07383 |  – |  0.05031
02769 |  0.15639 |  0.12003 |  0.09856 |  – |  06958 |  0.05079 |  0.03928 |  – |  0.02703
02798 |  0.06850 |  0.05294 |  – |  0.03474 |  06985 |  – |  0.06078 |  0.04896 |  0.04015
02812 |  0.05441 |  0.04241 |  – |  0.02825 |  07039 |  0.05902 |  0.04577 |  – |  0.03041
02817 |  0.07201 |  0.05581 |  – |  0.03761 |  07094 |  0.07686 |  0.05984 |  – |  0.04024
02854 |  0.10973 |  0.08715 |  – |  0.05591 |  07102 |  0.06770 |  0.05234 |  0.04215 |  –
02867 |  0.12697 |  0.09813 |  0.08195 |  – |  07160 |  0.04694 |  0.03627 |  – |  0.02553
02885 |  – |  0.11563 |  0.09216 |  0.07525 |  07186 |  0.06209 |  0.04800 |  – |  0.03308
02988 |  – |  0.22673 |  0.17976 |  0.14909 |  07200 |  0.06801 |  0.05256 |  – |  0.03416
02989 |  0.07705 |  0.05957 |  – |  0.03863 |  07241 |  0.07839 |  0.06083 |  0.04914 |  –
03015 |  0.07167 |  0.05534 |  – |  0.03928 |  07269 |  – |  0.14140 |  0.11325 |  0.09550
03035 |  0.05668 |  0.04385 |  – |  0.02994 |  07289 |  0.07248 |  0.05677 |  0.04639 |  –
03046 |  0.09154 |  0.07094 |  – |  0.04809 |  07300 |  0.11836 |  0.09180 |  – |  0.05843
03115 |  0.14686 |  0.11352 |  0.09254 |  – |  07301 |  0.06424 |  0.04981 |  – |  0.03445
03119 |  0.13806 |  0.10462 |  0.08640 |  – |  07324 |  0.09674 |  0.07479 |  – |  0.05114
03153 |  0.10055 |  0.07937 |  0.06378 |  – |  07345 |  0.07916 |  0.06114 |  0.04761 |  –
03168 |  0.08820 |  0.06998 |  – |  0.04685 |  07348 |  0.08531 |  0.06576 |  0.05276 |  –
03171 |  0.09488 |  0.07326 |  0.06127 |  – |  07358 |  0.06729 |  0.05270 |  – |  0.03591
03183 |  0.08973 |  0.06959 |  – |  0.04858 |  07362 |  – |  0.11709 |  0.09490 |  0.07978
03232 |  0.06119 |  0.04742 |  0.03960 |  – |  07369 |  0.05617 |  0.04360 |  – |  0.02886
03241 |  0.05991 |  0.04754 |  – |  0.03236 |  07383 |  0.04927 |  0.03809 |  – |  0.02644
03261 |  0.14088 |  0.10670 |  – |  0.07456 |  07388 |  0.11305 |  0.08723 |  0.06995 |  –
03274 |  0.07935 |  0.06128 |  – |  0.04138 |  07398 |  0.10770 |  0.08372 |  – |  0.05826
03336 |  0.14052 |  0.10742 |  0.08574 |  – |  07427 |  0.19371 |  0.14798 |  0.11607 |  –
03373 |  0.06384 |  0.04950 |  0.04010 |  – |  07449 |  0.11371 |  0.08789 |  – |  0.06138
03390 |  – |  0.09956 |  0.07939 |  0.06969 |  07513 |  0.04744 |  0.03662 |  – |  0.02419
03392 |  – |  0.19607 |  0.15544 |  0.12863 |  07564 |  – |  0.15997 |  0.12756 |  0.10810
03399 |  0.05706 |  0.04430 |  – |  0.02961 |  07589 |  0.06986 |  0.05396 |  0.04338 |  –
03425 |  0.12352 |  0.09430 |  – |  0.06397 |  07618 |  0.19114 |  0.14612 |  0.11715 |  –
03444 |  0.09624 |  0.07447 |  – |  0.05085 |  07656 |  0.07766 |  0.06020 |  0.04827 |  –
03473 |  0.07178 |  0.05610 |  0.04406 |  – |  07669 |  0.10213 |  0.07946 |  – |  0.05264
03515 |  0.05828 |  0.04536 |  – |  0.03157 |  07692 |  0.08985 |  0.06933 |  0.05552 |  –
03516 |  – |  0.19780 |  0.15709 |  0.12836 |  07697 |  0.06460 |  0.05058 |  0.04099 |  –
03553 |  0.05846 |  0.04536 |  – |  0.02934 |  07714 |  0.08552 |  0.06692 |  – |  0.04391
03561 |  0.06733 |  0.05224 |  – |  0.03447 |  07723 |  0.04660 |  0.03607 |  – |  0.02458
03581 |  – |  0.11175 |  0.08901 |  0.07349 |  07757 |  0.06951 |  0.05464 |  – |  0.03583
03598 |  0.08250 |  0.06429 |  – |  0.04298 |  07775 |  0.08084 |  0.06263 |  – |  0.04216
03603 |  0.05610 |  0.04343 |  – |  0.03041 |  07798 |  0.09323 |  0.07234 |  – |  0.04907
03713 |  0.07120 |  0.05515 |  0.04373 |  – |  07821 |  0.06219 |  0.04705 |  – |  0.03283
03749 |  0.10116 |  0.07848 |  0.06079 |  – |  07833 |  0.12028 |  0.09411 |  0.07554 |  –
03751 |  0.05127 |  0.03911 |  – |  0.02463 |  07836 |  0.13291 |  0.10089 |  0.08230 |  –
03766 |  0.06597 |  0.05200 |  – |  0.03411 |  07844 |  0.06462 |  0.05011 |  0.04031 |  –
03771 |  0.13997 |  0.11151 |  0.08993 |  – |  07868 |  0.07111 |  0.05582 |  – |  0.03707
03773 |  0.11241 |  0.08472 |  – |  0.05930 |  07885 |  0.11299 |  0.08742 |  – |  0.05901
03780 |  0.09660 |  0.07460 |  0.05994 |  – |  07891 |  0.07023 |  0.05468 |  – |  0.03550
03815 |  0.05833 |  0.04529 |  – |  0.03004 |  07918 |  – |  0.13302 |  0.10597 |  0.08765
03820 |  0.05823 |  0.04495 |  – |  0.03061 |  07926 |  0.15753 |  0.12092 |  – |  0.08356
03838 |  0.10761 |  0.08358 |  – |  0.05758 |  07927 |  0.05753 |  0.04481 |  0.03727 |  –
03843 |  0.06157 |  0.04783 |  0.03863 |  – |  07952 |  0.06835 |  0.05307 |  – |  0.03417
03849 |  0.06623 |  0.05174 |  – |  0.03552 |  07956 |  0.04852 |  0.03749 |  – |  0.02482
03857 |  0.08536 |  0.06682 |  0.05502 |  – |  07969 |  0.08779 |  0.06821 |  – |  0.04470
03862 |  0.09835 |  0.07680 |  0.05970 |  – |  07980 |  0.06516 |  0.05067 |  – |  0.03364
03874 |  0.26655 |  0.19914 |  0.16239 |  – |  07981 |  0.07068 |  0.05474 |  – |  0.03448
03887 |  0.07403 |  0.05715 |  0.04589 |  – |  07998 |  0.08542 |  0.06800 |  – |  0.04462
03891 |  0.10296 |  0.08032 |  0.06498 |  – |  08009 |  – |  0.08972 |  0.07222 |  0.06020
03892 |  0.05342 |  0.04147 |  – |  0.02667 |  08010 |  0.08489 |  0.06727 |  0.05422 |  –
03910 |  – |  0.15085 |  0.11984 |  0.09895 |  08021 |  0.08225 |  0.06530 |  – |  0.04175
03943 |  0.07095 |  0.05609 |  – |  0.03711 |  08050 |  0.16804 |  0.12628 |  0.09867 |  –
03949 |  0.14844 |  0.11417 |  – |  0.07407 |  08070 |  – |  0.12953 |  0.10290 |  0.08536
03986 |  0.08931 |  0.06894 |  0.05538 |  – |  08075 |  0.20811 |  0.15851 |  0.12237 |  –
03990 |  0.06111 |  0.04738 |  – |  0.03150 |  08132 |  0.04942 |  0.03814 |  – |  0.02614
04045 |  0.05575 |  0.04341 |  – |  0.02990 |  08141 |  0.09863 |  0.07692 |  0.06085 |  –
04071 |  0.09699 |  0.07647 |  0.06213 |  – |  08155 |  0.06917 |  0.05398 |  – |  0.03727
04075 |  0.05999 |  0.04661 |  – |  0.03247 |  08165 |  0.07966 |  0.06142 |  0.04926 |  –
04116 |  0.10203 |  0.07906 |  – |  0.05255 |  08200 |  – |  0.15760 |  0.12540 |  0.10376
04164 |  0.10324 |  0.07989 |  – |  0.05200 |  08216 |  0.05847 |  0.04516 |  – |  0.03034
04210 |  0.10629 |  0.08414 |  0.06755 |  – |  08235 |  0.06723 |  0.05215 |  – |  0.03479
04212 |  – |  0.04763 |  0.03761 |  0.03147 |  08238 |  0.05503 |  0.04261 |  – |  0.02923
04232 |  – |  0.12723 |  0.10122 |  0.08347 |  08253 |  0.05605 |  0.04332 |  0.03507 |  –
04237 |  0.08320 |  0.06433 |  0.05165 |  – |  08263 |  0.07220 |  0.05497 |  – |  0.03612
04260 |  0.06319 |  0.04930 |  – |  0.03335 |  08266 |  0.09773 |  0.07621 |  0.06160 |  –
04276 |  0.06804 |  0.05348 |  – |  0.03632 |  08279 |  0.05033 |  0.03886 |  0.03049 |  –
04278 |  0.09513 |  0.07325 |  – |  0.04741 |  08319 |  0.07324 |  0.05671 |  0.04555 |  –
04298 |  0.08051 |  0.06234 |  – |  0.04265 |  08325 |  0.06937 |  0.05430 |  0.04283 |  –
04300 |  0.08951 |  0.06927 |  0.05547 |  – |  08329 |  0.09968 |  0.07719 |  – |  0.05183
04311 |  0.06154 |  0.04792 |  – |  0.03175 |  08335 |  0.11280 |  0.08770 |  – |  0.05753
04332 |  0.05080 |  0.03931 |  0.03250 |  – |  08345 |  0.06795 |  0.05273 |  – |  0.03623
04348 |  0.08745 |  0.06776 |  – |  0.04400 |  08348 |  0.08998 |  0.06968 |  – |  0.04607
04349 |  0.05811 |  0.04509 |  0.03720 |  – |  08362 |  – |  0.07645 |  0.06219 |  0.05073
04376 |  0.09283 |  0.07261 |  – |  0.04868 |  08369 |  0.05246 |  0.04059 |  – |  0.02673
04440 |  0.06041 |  0.04661 |  – |  0.03115 |  08372 |  – |  0.08044 |  0.06473 |  0.05377
04458 |  0.08868 |  0.06852 |  0.05511 |  – |  08377 |  0.12473 |  0.09588 |  0.07669 |  –
04473 |  0.05301 |  0.04097 |  0.03218 |  – |  08379 |  0.20068 |  0.15055 |  0.12045 |  –
04542 |  0.08691 |  0.06737 |  0.05402 |  – |  08388 |  0.05971 |  0.04602 |  – |  0.03028
04545 |  0.05004 |  0.03867 |  – |  0.02661 |  08408 |  0.07410 |  0.05874 |  – |  0.03848
04559 |  0.05459 |  0.04235 |  – |  0.02812 |  08463 |  0.05564 |  0.04321 |  – |  0.02739
04561 |  0.12311 |  0.09719 |  0.07806 |  – |  08480 |  0.05516 |  0.04273 |  0.03446 |  –
04568 |  0.08642 |  0.06722 |  – |  0.04664 |  08534 |  0.05342 |  0.04120 |  0.03291 |  –
04603 |  – |  0.14171 |  0.11315 |  0.09391 |  08561 |  0.04966 |  0.03846 |  – |  0.02547
04604 |  0.08640 |  0.06669 |  0.05349 |  – |  08585 |  0.10638 |  0.08245 |  – |  0.05743
04643 |  0.08652 |  0.06719 |  0.05422 |  – |  08588 |  0.06445 |  0.04988 |  0.04153 |  –
04663 |  0.07531 |  0.05813 |  0.04529 |  – |  08589 |  0.07227 |  0.05659 |  – |  0.03710
04664 |  0.10469 |  0.08080 |  0.06524 |  – |  08598 |  0.11178 |  0.08625 |  0.06824 |  –
04683 |  0.07294 |  0.05674 |  – |  0.04002 |  08632 |  0.07083 |  0.05518 |  0.04461 |  –
04720 |  0.08574 |  0.06666 |  – |  0.04355 |  08642 |  0.06710 |  0.05074 |  0.04047 |  –
04750 |  0.07231 |  0.05594 |  – |  0.03761 |  08647 |  0.06388 |  0.04942 |  – |  0.03379
04777 |  0.05455 |  0.04222 |  – |  0.02723 |  08671 |  0.12075 |  0.09306 |  – |  0.06329
04781 |  0.08713 |  0.06757 |  – |  0.04571 |  08687 |  0.05444 |  0.04230 |  – |  0.02746
04783 |  – |  0.15992 |  0.12748 |  0.10574 |  08709 |  0.07300 |  0.05662 |  – |  0.03668
04796 |  0.05560 |  0.04323 |  – |  0.02886 |  08748 |  0.07421 |  0.05741 |  – |  0.03777
04811 |  0.05351 |  0.04146 |  0.03255 |  – |  08878 |  – |  0.07977 |  0.06379 |  0.05302
04850 |  0.06728 |  0.05212 |  – |  0.03514 |  08879 |  0.07907 |  0.06126 |  – |  0.04135
04886 |  0.07451 |  0.05897 |  – |  0.04091 |  08920 |  0.05238 |  0.04056 |  – |  0.02532
04913 |  – |  0.17671 |  0.14014 |  0.11836 |  08936 |  0.06523 |  0.05044 |  0.04070 |  –
04936 |  0.11774 |  0.09209 |  0.07463 |  – |  08958 |  0.07793 |  0.06037 |  – |  0.04056
04947 |  0.08381 |  0.06529 |  – |  0.04294 |  08984 |  0.05811 |  0.04542 |  0.03696 |  –
04955 |  0.10526 |  0.07978 |  0.06663 |  – |  08989 |  0.06728 |  0.05342 |  0.04373 |  –
04970 |  0.08629 |  0.06664 |  0.05358 |  – |  09015 |  – |  0.07507 |  0.05908 |  0.04964
04971 |  – |  0.18468 |  0.14630 |  0.12069 |  09031 |  0.10386 |  0.08051 |  – |  0.05473
04976 |  0.07827 |  0.06056 |  0.04733 |  – |  09041 |  – |  0.08383 |  0.06622 |  0.05644
04982 |  0.05676 |  0.04388 |  0.03446 |  – |  09055 |  0.05471 |  0.04204 |  0.03383 |  –
05003 |  – |  0.15232 |  0.12108 |  0.10001 |  09058 |  0.08041 |  0.06205 |  – |  0.04085
05037 |  0.07117 |  0.05505 |  – |  0.03589 |  09071 |  0.06233 |  0.04816 |  0.03902 |  –
05049 |  0.05262 |  0.04065 |  – |  0.02797 |  09074 |  0.06019 |  0.04675 |  – |  0.03048
05056 |  0.13231 |  0.10188 |  0.07904 |  – |  09080 |  0.06657 |  0.05169 |  – |  0.03329
05068 |  – |  0.18621 |  0.14830 |  0.12330 |  09109 |  – |  0.12151 |  0.09671 |  0.07981
05082 |  0.04567 |  0.03529 |  – |  0.02281 |  09122 |  – |  0.08508 |  0.06823 |  0.05703
05115 |  0.09895 |  0.07693 |  0.06344 |  – |  09153 |  0.10064 |  0.07793 |  – |  0.05163
05117 |  0.08683 |  0.06793 |  0.05714 |  – |  09170 |  0.08871 |  0.06905 |  – |  0.04725
05136 |  0.05944 |  0.04592 |  – |  0.02994 |  09177 |  – |  0.14894 |  0.11885 |  0.10295
05190 |  0.07345 |  0.05918 |  – |  0.03813 |  09199 |  0.07233 |  0.05604 |  – |  0.03785
05207 |  0.06937 |  0.05383 |  – |  0.03662 |  09226 |  – |  0.08207 |  0.06362 |  0.05628
05245 |  0.07664 |  0.05917 |  – |  0.03973 |  09241 |  0.08137 |  0.06295 |  0.05289 |  –
05260 |  0.05724 |  0.04452 |  – |  0.03046 |  09260 |  0.08861 |  0.06853 |  – |  0.04606
05303 |  0.06756 |  0.05161 |  0.03996 |  – |  09335 |  0.09732 |  0.07646 |  – |  0.05043
05333 |  – |  0.13050 |  0.10385 |  0.08604 |  09364 |  0.06525 |  0.05067 |  0.03961 |  –
05356 |  0.06610 |  0.05112 |  – |  0.03632 |  09371 |  0.09370 |  0.07247 |  – |  0.05022
05361 |  0.07466 |  0.05904 |  – |  0.03881 |  09472 |  0.07293 |  0.05649 |  – |  0.03785
05371 |  0.06851 |  0.05310 |  0.04292 |  – |  09493 |  0.07490 |  0.05786 |  0.04705 |  –
05401 |  0.09677 |  0.07477 |  – |  0.04990 |  09501 |  0.06633 |  0.05162 |  0.04244 |  –
05413 |  0.08262 |  0.06400 |  – |  0.04309 |  09517 |  0.08307 |  0.06437 |  – |  0.04175
05421 |  0.06303 |  0.04863 |  – |  0.03380 |  09524 |  0.05173 |  0.04011 |  – |  0.02655
05428 |  0.07761 |  0.05994 |  – |  0.04025 |  09566 |  0.07385 |  0.05729 |  – |  0.03785
05468 |  0.10423 |  0.08080 |  – |  0.05396 |  09581 |  0.05767 |  0.04487 |  – |  0.02984
05477 |  0.07137 |  0.05517 |  0.04436 |  – |  09637 |  0.05263 |  0.04085 |  – |  0.02723
05480 |  0.08081 |  0.06257 |  – |  0.04199 |  09640 |  – |  0.16176 |  0.12848 |  0.10364
05501 |  0.08191 |  0.06337 |  0.05085 |  – |  09666 |  0.27319 |  0.21228 |  0.16877 |  –
05514 |  0.10416 |  0.08141 |  – |  0.05698 |  09728 |  – |  0.10558 |  0.08550 |  0.07145
05548 |  – |  0.11320 |  0.09035 |  0.07428 |  09746 |  0.09799 |  0.07745 |  0.06154 |  –
05572 |  0.09416 |  0.07267 |  0.05755 |  – |  09778 |  0.04818 |  0.03722 |  – |  0.02609
05580 |  0.06215 |  0.04805 |  – |  0.03112 |  09840 |  0.12257 |  0.09468 |  0.07604 |  –
05589 |  0.19935 |  0.15262 |  0.12808 |  – |  09888 |  0.08871 |  0.06851 |  0.05747 |  –
05617 |  0.09154 |  0.07221 |  – |  0.04987 |  09903 |  0.06119 |  0.04721 |  0.03683 |  –
05622 |  0.11379 |  0.08995 |  – |  0.05747 |  09905 |  0.07090 |  0.05483 |  0.04270 |  –
05638 |  0.05860 |  0.04545 |  – |  0.02760 |  09913 |  0.05354 |  0.04134 |  – |  0.02751
05692 |  0.11939 |  0.09206 |  – |  0.06120 |  10048 |  0.06033 |  0.04658 |  – |  0.02987
05695 |  0.05903 |  0.04595 |  – |  0.03229 |  10050 |  0.04966 |  0.03868 |  – |  0.02690
05720 |  0.12480 |  0.09612 |  0.07621 |  – |  10100 |  0.09560 |  0.07382 |  0.05925 |  –