11institutetext: Nicolaus Copernicus Astronomical Centre, Polish Academy of
Sciences, Bartycka 18, 00-716 Warszawa, Poland
email<EMAIL_ADDRESS>22institutetext: Sternberg Astronomical Institute,
Moscow State University, 119234, Moscow, Russia 33institutetext: Institute of
Astronomy, Russian Academy of Sciences, 119017, Moscow, Russia

# Precise Fourier parameters of Cepheid Radial Velocity Curves††thanks: Full
Tables • ‣ A, LABEL:Tab:data_funda_quality1, LABEL:Tab:data_funda_quality1a,
LABEL:Tab:data_fourier_quality2, LABEL:Tab:data_fourier_quality1_overtone and
LABEL:Tab:unknown are only available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/cat/J/A+A/

V. Hocdé 11 0000-0002-3643-0366 P. Moskalik 11 0000-0003-3142-0350 N. A.
Gorynya 2233 R. Smolec 11 0000-0001-7217-4884 R. Singh Rathour 11
0000-0002-7448-4285 O. Ziółkowska 11 0000-0002-0696-2839

(Received … ; accepted …)

###### Abstract

Context. Radial velocity (RV) curves of Classical Cepheids are indispensable
in studying resonances between their pulsation modes. They allow precise
determination of the resonant periods, which in turn helps to constrain
fundamental parameters of these stars. The RV curves of Cepheids are also
needed to identify their pulsation modes, which is essential for accurate
calibration of the Cepheid period-luminosity (PL) relation. Finally, RV curves
are necessary for the distance determination with the parallax-of-pulsation
method.

Aims. The primary goal of this paper is to derive precise Fourier parameters
of the radial velocity (RV) curves for fundamental and first-overtone Galactic
Cepheids. The secondary objectives are then to analyze progression of the
Fourier parameters up to the 7th harmonic as well as to propose identification
of the pulsation modes of the stars.

Methods. For each star, we carefully selected RV measurements available in the
literature which yield the highest precision of Fourier parameters according
to the procedure that follows. We performed a Fourier decomposition of the RV
curves using unweighted least-square method. We corrected for zero-point
differences between data sets. We subtracted RV modulations caused by binary
motion and have removed other residual trends. Uncertainty of Fourier
parameters are derived using the covariance matrix. Finally, we have displayed
and analyzed qualitatively the progressions found for low- and for high-order
Fourier parameters. We applied a standard identification of their pulsation
mode based on their Fourier phase difference $\phi_{21}$.

Results. We more than doubled the number of Cepheids with published Fourier
parameters of their RV curves, determined together with their uncertainties.
Our sample includes 178 fundamental-mode and 33 first overtone pulsators, as
well as 7 additional Cepheids whose pulsation mode is uncertain or
undetermined according to our criteria. For the fundamental-mode Cepheids, the
precision of the obtained low-order Fourier phases is about 7 times better and
of the low-order Fourier amplitude ratios about 25% better as compared to
precision achieved in previously published Fourier parameter surveys. The
precision of the first-overtone Fourier parameters is improved by about 50%.
With highly accurate RV Fourier phases $\phi_{21}$, we are able to firmly
identify V495 Cyg as a new first-overtone Cepheid. We confirm first-overtone
nature of several other stars, tentatively identified as such with the shape
of their lightcurves. We also show that $\alpha$ UMi should be definitely
classified as a first-overtone pulsator. In 3 objects (VY Per, AQ Pup and QZ
Nor) we found significant $\gamma$-velocity variations that we attribute to
spectroscopic binarity after careful analysis RV measurements of these stars.
Finally, the analysis of the fundamental mode Fourier parameters up to 7th
order reveals very tight progression of Fourier phases for all pulsation
periods. They are also tightly correlated with each other. For $P<10\,$day we
find a very well defined upper limit for the Fourier amplitude $A_{1}$.

Conclusions. We have provided new precise Fourier parameters of Cepheid
velocity curves, determined from RV measurements available in the literature
together with unpublished data. The pulsation period coverage and the
precision obtained, in particular for Fourier phase $\phi_{21}$, will be
useful for studying the dynamics of Cepheid pulsations with the help of
hydrodynamical models. Further radial velocity measurements from modern high-
resolution spectroscopic instruments will be important to improve these
results.

###### Key Words.:

Techniques : Observational– Radial velocities - stars: variables: Cepheids

##  1 Introduction

The shape of the radial velocity (RV) curve of Cepheids provides information
on their stellar structure and dynamic. In the case of fundamental Cepheid
light curves, Ludendorff (1919); Hertzsprung (1926) noticed that their shape
varies systematically with the pulsation period. A secondary bump superimposed
on the light curve appears first on its descending branch at a period of about
$P\simeq 6\,$day. With increasing period, it moves toward maximum light,
reaches the top at $P\simeq 10\,$day and then moves down the ascending branch
at periods $P>10\,$day. This systematic behavior is called the Hertzsprung
progression or bump progression, and was later verified among Cepheids in the
Magellanic Clouds and Andromeda Galaxy (Parenago & Kukarkin 1936; Shapley &
McKibben 1940; Payne-Gaposchkin 1947). Similar bump progression is observed
for the RV curves (Joy 1937; Ledoux & Walraven 1958; Simon & Teays 1983;
Kovács et al. 1990; Gorynya 1998; Moskalik et al. 2000).

The bump progression was first assumed to be caused by a pressure wave which
propagates inward and reflects on the stellar core, then travels back to reach
the surface and produces the observed bump (also known as the echo model,
Whitney 1956; Christy 1968, 1975). Following hydrodynamical models of Stobie
(1969), Simon & Schmidt (1976) conjectured that the bump progression is due to
the $P_{2}$/$P_{0}=0.5$ resonance between the fundamental and the second
overtone pulsation modes characterized by a pulsation period of $P_{0}$ and
$P_{2}$ respectively. Subsequently, the crucial role of the 2:1 resonance was
confirmed with the extensive set of hydrodynamical models of Buchler et al.
(1990); Moskalik et al. (1992), and understood analytically with the amplitude
equations formalism (Buchler & Kovács 1986; Kovács & Buchler 1989). Since the
resonance is a dynamical phenomenon, RV curves are better suited for
comparison with models than the light curves. Moreover, modelling light curves
is subject to larger uncertainties because it is sensitive to the radiative
transfer treatment in the outer layers of the model. For example, RV curves of
first-overtone Cepheids display a prominent resonance around 4.6 day (Kienzle
et al. 1999), in agreement with hydrodynamical models which predict a
$P_{4}/P_{1}=0.5$ resonance between the fourth and the first overtone (Kienzle
et al. 1999; Feuchtinger et al. 2000). In contrast, the light curves present a
sharp feature at 3.2 day which is not centered at the $P_{4}/P_{1}=0.5$
resonance (Kienzle et al. 1999). This demonstrates the importance of RV curves
for studying dynamical effects.

Simon & Lee (1981); Simon & Teays (1983) showed that the Fourier parameters
can describe efficiently the shape of the light and RV curves. Therefore, they
can be used to quantitatively compare the models and the observations. In
particular, the Fourier phase $\phi_{21}$ of RV curve of the fundamental mode
depends almost exclusively on the resonant period ratio $P_{2}$/$P_{0}$ and is
numerically robust (Buchler et al. 1990). The same is true for first-overtone
Cepheids, where the Fourier phase $\phi_{21}$ depends on $P_{4}$/$P_{1}$
(Kienzle et al. 1999). Thus, using observed RV curves it is possible to derive
the period ratios for individual Cepheids, which in turn can constraint their
internal structure as well as their Mass-Luminosity relation (Moskalik & Krzyt
1998).

RV curves are also indispensable in the parallax-of-pulsation technique for
individual distance determination of Cepheids (also known as Baade-Wesselink
(BW) method, Lindemann 1918; Baade 1926; Becker 1940; Wesselink 1946).
Although some authors emphasized the importance of homogeneous and consistent
RV measurements for applying BW method (see, for example Anderson 2018b;
Borgniet et al. 2019), in practice the community has to deal with multiple
data set obtained from different instruments and cross-correlation measurement
methods. Hence, a consistent method for fitting RV measurements of several
data sets altogether would be valuable.

Therefore, obtaining precise Fourier parameters, and for a large sample of
short and long-period Cepheids, is essential to reconstruct the Hertzsprung
progression and explore it up to higher-order. Of paramount importance is the
precision of observational $\phi_{21}$ from RV curves that not only impacts
directly the error on the period ratio $P_{2}/P_{1}$ and $P_{4}/P_{1}$
(Moskalik & Krzyt 1998), but is also a very useful tool to discriminate
fundamental and first-overtone mode Cepheids, especially for pulsation period
above 5 days (Antonello et al. 1990).

Owing to continuous effort of observers, the number of stars available to
reconstruct the bump progression and the precision of Fourier parameters
increased over time. Using a compilation of 9 different sources from the
literature, Simon & Teays (1983) analyzed for the first time RV curves for 11
classical fundamental-mode and 1O-mode Cepheids. A compilation of Fourier
parameter of 57 RV curves, with median uncertainties of
$\sigma(\phi_{21})=0.18$, was later provided by Kovács et al. (1990) who
compiled RV data from 41 sources. Moskalik et al. (2000) also collected
measurements for 131 Cepheids and reported a median uncertainty of
$\sigma(\phi_{21})=0.08$. Storm et al. (2011) provided Fourier parameters
together with their uncertainty for 76 Galactic Cepheids using a compilation
of 60 sources from the literature, providing an average uncertainty
$\sigma(\phi_{21})=0.18$. In the case of first-overtone Cepheids, Kienzle et
al. (1999) published Fourier parameters for 16 stars with a median uncertainty
$\sigma(\phi_{21})=0.09$.

In recent years, many new high-quality RV observations have been collected
(see, e.g. Anderson et al. 2016a; Borgniet et al. 2019; Gallenne et al. 2019;
Nardetto et al. 2023). In this work, we take advantage of these recent
measurements, in combination of an extensive dataset from the literature, to
enlarge the number of Cepheids with known Fourier parameters and improve their
precision. The paper is dedicated to the Galactic Classical (Population I)
Cepheids, both pulsating either in the fundamental mode and in the first-
overtone mode. We present our data set and our method for Fourier
decomposition in Sect. 2. We present our results on precision of Fourier
parameters in Sect. 3.1, identification of first-overtone in Sect. 3.2 and
binary Cepheids in Sect. 3.3. We discuss the progression of the low- and high-
order Fourier parameters with pulsation period as well as intrarelation of
these parameters in Sect. 4. We conclude in Sect. 5.

##  2 Fourier decomposition of radial velocity curves

###  2.1 Data set of radial velocity measurements

For each star we first collected the RV measurements available in the
literature. All RV measurements used in this study are derived from high-
resolution spectroscopy observations of metallic lines. Indeed, metallic lines
represent a robust and long standing verified approach to obtain reliable RV
measurement from the Cepheid photosphere. However, the use of Gaia Radial
Velocity Spectrometer (RVS) (Gaia Collaboration et al. 2016, 2023)
measurements based on Calcium infrared triplet lines (Munari 1999; Katz et al.
2023) is beyond the scope of this paper. Although Ripepi et al. (2023)
reported the reliability of the Gaia RV measurements for Cepheids, dynamical
effects observed in the Calcium triplet of several Galactic Cepheids
(Wallerstein et al. 2015, 2019; Hocdé et al. 2020) require more investigation
to evaluate their impact on RV measurements compared to optical metallic
spectral lines. Last, most of the RV measurements we collected were obtained
from the cross-correlation technique (optical and numerical, Griffin 1967;
Baranne et al. 1979, 1996) which is an important requirement for the
application of the Baade-Wesselink method (Borgniet et al. 2019).

Then, we included in our final sample only the Cepheids for which we derived
the most precise Fourier parameters, as describes in the following sections.
This sample consists of 178 fundamental and 33 first overtone Cepheids and 7
Cepheids of uncertain or undetermined pulsation mode. The entire data set used
in the analysis is composed of 82 different sources of radial velocity
measurements which are listed in Table • ‣ A. For several Cepheids, the
Fourier decomposition based on highest quality RV data has already been
published by Kienzle et al. (1999), Pont et al. (2001) and Baranowski et al.
(2009). For the sake of uniformity, we decided to reanalyze all these stars,
supplementing the old data with newer measurements, if available (5 stars).
The resulting Fourier parameters were found within the errors to be the same
as in the original papers. All these stars are included in Table • ‣ A and in
following tables. In the next sections, we detail the steps of the Fourier
decomposition, the different corrections applied and the quality
classification of the fits.

###  2.2 Fourier decomposition

Precision on Fourier parameters depends primarily on the uncertainties of the
RV observations, the number of data points, and the phase coverage. Collecting
in the literature datasets from various observations has the advantage of
maximizing the number of data points and completing the phase coverage for
many Cepheids. However, we have to deal with inhomogeneous uncertainty and
datasets that can be far from each other in time. Indeed, there has been a
significant improvement of the precision of RV measurements over time, from
$\approx 1\,$km/s in the 80’s down to $\approx 10\,$m/s for the most precise
high-resolution spectrograph. Moreover, individual error determinations are
rather uncertain before the 90’s and are most often not provided but estimated
from the dispersion of the measurements. On the other hand, definition of
uncertainties varies widely in the literature and given formal error are often
underestimated as compared to long-term uncertainties. Indeed, uncertainties
of RV measurements depends on multiple factors including deviations in the
adopted radial velocity of standard stars, misalignment of stars within the
spectrograph slit, and potential discrepancies in the velocity scale between
Cepheids and standard stars (Barnes et al. 2005). The precision is also
strongly dependent on the choice of metallic lines, correlation template and
the computation method used (Borgniet et al. 2019). In order to avoid case-by-
case assumption on the adopted uncertainties, we shall proceed with a
unweighted least-square method which is also a standard approach in the field.
We note that while heteroskedasticity of the dataset bias the precision of the
least-square estimator, the accuracy of the fitted parameter is still
preserved according to the Gauss-Markov theorem. Nevertheless, we show that
the precision obtained are also robust in Sect. 3.1, and, thus, that the
assumption of homoskedasticity is also excellent.

We decomposed the RV curves into Fourier sine series by least-square fitting
of the data (unweighted) with the following formula:

$V_{r}(t)=A_{0}+\sum_{k=1}^{n}A_{k}\mathrm{sin}[k\omega t+\phi_{k}]$ (1)

where $V_{r}(t)$ is the radial velocity measured at the time $t$, $A_{k}$ and
$\phi_{k}$ are the amplitude and the phase of the $k$ Fourier component. Each
harmonic is a multiple of the fundamental frequency $\omega=2\pi/P$. The
pulsation period, $P$, is also optimized in the fitting process. We used the
dimensionless Fourier parameters introduced by Simon & Lee (1981):

$\displaystyle R_{i1}=\frac{A_{i}}{A_{1}}$ (2)
$\displaystyle\phi_{i1}=\phi_{i}-i\phi_{1}$ (3)

where $\phi_{i1}$ are then adjusted to lie between 0 and 2$\pi$. Choosing the
appropriate order of the fit $n$ is important because a too low order may lead
to higher uncertainties on the Fourier parameters, while the noise might be
fitted if an order is too high. Many studies simply fix the order of the fit
depending on the pulsation period of the star. We adopt a more adaptative
approach where the order of the fit $n$ is iterated until
$A_{n}/\sigma_{A_{n}}$¿4. In some cases, we included one or two higher
harmonics in order to stabilize the fit. This method is purely empirical and
is based on practical results.

The standard deviation of the fit is given by

$\sigma^{2}=\frac{1}{N-2n-2}\sum^{N}_{i=1}(V_{r}(t_{i})-\hat{V}_{i})^{2}$ (4)

where $N$ is the number of data points, $n$ the order of the fit,
$V_{r}(t_{i})$ and $\hat{V}_{i}$ are the radial velocity data and the Fourier
model respectively. The derived standard deviation of the fit is then used to
apply an iterative 3-$\sigma$ data clipping. In this step we remove data
points deviating from the fit by more than 3$\sigma$. This is a standard
outlier rejection technique, commonly used in the literature.

###  2.3 Errors of Fourier parameters

The standard deviation of the fit presented above is a good indicator for the
goodness of fit although it does not reflect the uncertainty of the Fourier
parameters. In fact, it can be misleading to interpret this quantity as an
accuracy of Fourier parameters. For example, for low amplitude variations we
may obtain a fit with very low dispersion, but uncertainties of the Fourier
phases $\phi_{i1}$ might still be large. This may be so because harmonic
components of the fit may be very low and difficult to measure (see e.g.
Polaris). On the contrary, in case of high amplitude RV curves the dispersion
of the fit may be higher, yet the precision of the Fourier parameters may be
very good. In some cases, missing observational data in a specific phase range
can lead to a small fit dispersion but rather uncertain Fourier parameters.
Finally, we should note that precision of the Fourier parameters depends also
on the number of measurements. Adding more and more datapoints of the same
quality does not change the dispersion of the fit, but the errors of all
Fourier parameters become progressively smaller.

The analytical formulae of the Fourier parameters obtained from the least-
square method were derived by Petersen (1986). This method was applied in
comparable RV studies to derive Fourier parameter uncertainties (Kovács et al.
1990; Kienzle et al. 1999; Moskalik et al. 2000; Baranowski et al. 2009; Storm
et al. 2011) but also in the case of light-curves (see, e.g. Deb & Singh
2010). In this approach, the errors of the Fourier amplitude ratios
$\sigma(R_{k1})$ and phase differences $\sigma(\phi_{k1})$ at the order $k$
are obtained from propagation of the variance-covariance matrix of the fitted
Fourier parameters (see Eqs. 9 to 12 in Petersen 1986).

In addition to this basic Fourier fitting method, we applied several
corrections as described in the following.

###  2.4 Zero-points correction

As discussed in Sect. 2.1, we used data sets compiled from many different
sources from the literature. Thus, zero-point offsets between data sets of the
same star might be present, because of different instruments used, different
choice of metallic lines and different methods of extracting RV from the
spectra (see, e.g., Evans et al. 2015; Borgniet et al. 2019; Nardetto et al.
2023).In order to correct for these effects, we treated zero-point differences
between datasets as free parameters and determined them iteratively from the
least-square fit. We note that zero-point corrections have been applied by
other authors as well (see, e.g., Herbig & Moore 1952; Lloyd Evans 1982; Kiss
& Vinkó 2000; Pont et al. 2001; Anderson et al. 2015; Breitfelder et al. 2016;
Torres 2023). Similar corrections have also been used when combining
photometric data (see, e.g., Coulson & Caldwell 1985; Antonello et al. 1990).
We note that applying zero-point correction is not suitable to detect time-
variability of average velocities that would be caused by orbital motion.
However, the systematic detection of binary Cepheids is out of the scope this
paper, and we only focus on removing orbital motion contribution as presented
in the following section.

###  2.5 Orbital motion correction

Since a significant fraction of Cepheids are binary or multiple systems,
(Evans et al. 2015; Szabados 2003; Kervella et al. 2019), some of the RV
curves can be strongly affected by orbital motion. In this paper, we stress
that we do not aim to resolve the Keplerian elements of the orbit, but simply
to subtract the trend induced by the orbital motion. To this end, the orbital
motion is modeled by an additional Fourier series, with the orbital period
$P_{\mathrm{orb}}$. Thus, the fitting formula for a binary Cepheid consists of
the two Fourier sums: one for pulsation variations (see Eq.1), and one
additional for orbital motion. This procedure was used for example by Kovács
et al. (1990). In this method, both pulsation and orbital motion are fitted
simultaneously. As mentioned above, a Fourier sine series cannot determine the
Keplerian elements of the orbit, but is naturally suitable for accurately
determine the orbital period. Although this method is convenient for
algorithmic purposes we note that we introduce additional degree of freedoms
(DOFs) as compared to a Keplerian model. However, it must be kept in mind that
the number of data points is significantly larger than the DOF for the
Cepheids of our study, and, thus, this additional cost is largely mitigated.
For example, in the case of SU Cyg we have 23 parameters, instead of 16 using
Keplerian orbit, as compared to 331 datapoints, this yield an increase of the
fit dispersion by only 0.8%. For only three binaries this increase is slightly
above 1%, namely RX Cam (4.1%), S Sge (1.6%) and V350 Sgr (1.1%). In exchange,
we get a much simpler fitting formulae (linear in Fourier parameters). The
orbital period, $P_{\mathrm{orb}}$, is either taken from the literature or
determined from the fit. If the orbital period is consistent with the period
given in the literature (which is usually the case), we choose the one with
smaller formal error. We summarized the orbital period adopted in Table 4. In
Table LABEL:Tab:binary_comp we compare our new orbital periods with the
periods previously publish in the literature.

###  2.6 Trend correction

In some cases, after subtracting the pulsation variation from the RV curve, we
observe a slow linear or parabolic trend in the residuals. Such a trend can be
of instrumental origin or it could result from a slow orbital motion (see,
e.g., Anderson et al. 2015). In order to model both linear and parabolic
trends in the data, we used a single Fourier component with a constant long
period of $P_{\mathrm{trend}}=50000\,$day. Alternatively, the trend can also
be modeled by a polynomial function, but we chose the Fourier model for
practical algorithmic purposes. Hence, Fourier sum for both pulsation
variations and the trend are fitted simultaneously. For each star, we
indicated if an additional trend is fitted (see Tables
LABEL:Tab:data_funda_quality1, LABEL:Tab:data_funda_quality1a,
LABEL:Tab:data_fourier_quality2, LABEL:Tab:data_fourier_quality1_overtone and
LABEL:Tab:unknown).

###  2.7 ($O-C$) correction for long-period Cepheids

The four Cepheids with the longest pulsation periods ($P>40\,$day), namely SV
Vul, GY Sge, V1496 Aql and S Vul, all display very fast, irregular $O-C$
variations, or equivalently, period variations (Berdnikov et al. 2004, 2007;
Turner et al. 2009). Phasing their radial velocity data with constant period
will not yield satisfactory results. Therefore, we proceeded differently: we
corrected for the $O-C$ variations by applying phase offsets (shifts),
calculated for each season separately (see, e.g., Berdnikov et al. 2004;
Anderson et al. 2015; Kervella et al. 2017). A usual method is to use an $O-C$
diagram determined from photometric observations in order to calculate the
required phase shift for each epoch (see, e.g., Kervella et al. 2017). Instead
of this approach, we measured the phase shifts directly from the RV data for
each individual dataset, similarly to Anderson et al. (2015) in case of
$\delta$ Cep. To this end, we first divided the data into seasons and chose
the best sampled season as a reference. The pulsation period $P$ of the star
was determined from this reference season. Then we applied individual phase
shifts to all the remaining seasons. These phase shifts were determined for
all datasets simultaneously by the least square method in which we minimized
the dispersion of the overall Fourier fit. Our procedure of measuring the
required $O-C$ shifts is in its idea equivalent to a Hertzsprung method
(Hertzsprung 1919). We note, that this approach is valid only if each seasonal
chunk of data covers both the ascending branch and the descending branch of
the RV curve. We applied this method only to the four long-period Cepheids
mentioned above and to a single short period Cepheid V Cen, for which we had
to combine data collected over many years (see individual fitting treatment in
Sect. C.1).

###  2.8 Quality classification of the fits

It is difficult to define a consistent criteria to quantify the quality of the
fit for every star. Indeed, quality of the different fits depends on the
number of measurements, coverage of the pulsation cycle, standard deviation of
the fit and presence of instabilities. Thus, we rather visually inspected each
Fourier fit and we classified them into two main categories (see Table 1).
Fourier parameters of highest quality radial velocity curves labeled ”1” are
most accurate, as resulting from fits free of any significant instabilities
and with an excellent phase coverage. By ”instability” we mean a small
artificial undulation of the RV curve, produced by an ill-defined fit and
having no physical meaning. In some cases the fits display slight
instabilities, hence we assigned to them the quality flag ”1a”. Fourier fits
classified as quality ”2” have useful Fourier parameters but they can be
affected by stronger instabilities, scatter or poorer coverage of the
pulsation cycle. For each star of quality ”2” we provided a flag which
describes the type of deficiencies we identified (see Notes to Tables
LABEL:Tab:data_fourier_quality2 and LABEL:Tab:data_fourier_quality1_overtone).
We emphasize that this classification has to be considered only as an
indication to guide the reader.

##  3 Results

###  3.1 New precise Fourier parameters

The plots of entire collection of Cepheid radial velocity curves are available
as an on-line material. We displayed selected velocity curves in Fig. 1, which
nicely exhibits the bump progression in the fundamental-mode Cepheids. We also
present the amplitude of the first Fourier term $A_{1}$ (Fig. 2), the low-
order Fourier parameters (Fig. 3) and high-order Fourier parameters (Fig. 4).
Fourier parameters for all the star are provided in Tables
LABEL:Tab:data_funda_quality1, LABEL:Tab:data_funda_quality1a,
LABEL:Tab:data_fourier_quality2, LABEL:Tab:data_fourier_quality1_overtone and
LABEL:Tab:unknown. The different trends of Fourier parameters are
qualitatively discussed in Sect. 4.

Our sample consists of 218 variables. This is the largest collection of
Cepheids for which Fourier parameters of their RV curves are determined. In
Fig. 5 we present the precision of the Fourier fits and of the derived Fourier
parameters. The median number of data points per star and the median
dispersion of the fit, together with quartiles, are (see Figs. 5(a) and 5(b)):

$\displaystyle\mathrm{Ndat}=74_{47}^{110},\quad\sigma_{\mathrm{fit}}=0.72_{0.38}^{1.15}\,\mathrm{km/s}.$
(5)

For the best defined RV curves (quality flags 1 and 1a) we obtain

$\displaystyle\mathrm{Ndat}=88_{61}^{130},\quad\sigma_{\mathrm{fit}}=0.56_{0.26}^{0.90}\,\mathrm{km/s}.$
(6)

Although for many Cepheids we obtained fits of excellent quality (see RV
curves plots online), the dispersion of the fit alone is a poor indicator of
the Fourier parameters precision. Thus, we also display the derived
uncertainties of low-order Fourier parameters (up to the third harmonics) in
Figs 5(c), 5(e), 5(d) and 5(f). For Cepheids of our sample we find a median
uncertainty of

$\displaystyle\sigma(\phi_{21})=0.04_{0.02}^{0.07},\quad\sigma(R_{21})=0.009_{0.005}^{0.018},$
(7)
$\displaystyle\sigma(\phi_{31})=0.08_{0.04}^{0.16},\quad\sigma(R_{31})=0.009_{0.005}^{0.016}.$
(8)

It is also possible to verify the obtained precision using the trends of the
Fourier phases of low and high-orders (Figs. 3 and 4). When plotted vs.
pulsation period, the Fourier phases $\phi_{n1}$ display remarkably tight
progressions, which is in qualitative agreement with theoretical predictions
(Buchler et al. 1990). More importantly, the dispersion of each Fourier phase
progression is in excellent agreement with the derived uncertainties, which
demonstrates that they the are realistic. Intrarelations of Fourier phases
discussed in Sect. 4.6 and displayed in Fig. 7 provide further evidences of
internal consistency on the obtained precisions, and, therefore, of the
robustness of our fitting method.

We compare the precision achieved in our work with the precision of the
similar Fourier parameter surveys published in the literature. First, we
cross-matched our results with those of Storm et al. (2011). We display
histograms of their errors in Figs 5(c), 5(e), 5(d) and 5(f). Comparing the
median of each sample we conclude that our median precision for the same set
of stars is 8 times better in case of $\sigma(\phi_{21})$, 6 times better in
case of $\sigma(\phi_{31})$, 18% in case of $\sigma(R_{21})$ and 22% better in
case of $\sigma(R_{31})$. For the first overtone Cepheids, we obtain a median
uncertainty of

$\displaystyle\sigma(\phi_{21})=0.06_{0.04}^{0.09},\quad\sigma(R_{21})=0.011_{0.006}^{0.017},$
(9)

as compared to previous results from Kienzle et al. (1999)

$\displaystyle\sigma(\phi_{21})=0.09_{0.06}^{0.16},\quad\sigma(R_{21})=0.0155_{0.013}^{0.018},$
(10)

which yield an improved precision of about 33 and 29% respectively.

As a conclusion, we not only obtained the largest sample of available Fourier
parameters of RV curves (218 stars, determined from a same method), but we
also significantly improved the precision of Fourier parameters. This improved
precision is valuable for fine-tuning hydrodynamical pulsation models (see,
e.g. Paxton et al. 2019; Kovács et al. 2023), a problem that will be addressed
in forthcoming papers in preparation in our group. As an immediate by-product,
however, we are able to identify first-overtone mode Cepheids based on the
Fourier phase $\phi_{21}$ in Sect. 3.2 and to identify some binary Cepheids
with their orbital period in Sect. 3.3.

Table 1: Quality classification of the Cepheids sample.

Quality | F | 1O | U
---|---|---|---
1 | 85 | 30 | 4
1a | 27 | 1 |
2 | 66 | 2 | 3
Total | 178 | 33 | 7

  * •

Notes : Consecutive rows list numbers of stars with quality classification 1,
1a and 2, as defined in Sect. 2.8, for fundamental (F), first-overtone (1O)
and undetermined mode (U) Cepheids.

(a)

(b)

Figure 1: Radial velocity curves and Fourier fits for a set of fundamental-
mode (a) and first-overtone Cepheids (b). The scale is indicated in the left
bottom corner of the plots.  Figure 2: First Fourier term, $A_{1}$, for
fundamental-mode and first-overtone Cepheids. Fundamental-mode variables of
Quality 1 and 1a are plotted with black points, while gray points are used for
variables of Quality 2. A vertical dashed line indicates a period of 10 day
and it is plotted to guide the eye in the resonance vicinity.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Low-order Fourier parameters for RV curves of fundamental-mode and
first-overtone Cepheids. For the fundamental mode black points and grey points
are variables of Quality (1+1a) and 2 respectively (see Table 1). Vertical
dashed lined indicate a pulsation period of 4.68 and 10 day to guide the eye
in the resonance vicinity of first-overtone and fundamental mode Cepheids
respectively.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Fourier parameters of 5th, 6th and 7th terms for RV curves of the
fundamental-mode Cepheids. Black points and Grey points are variables of
Quality (1+1a) and 2 respectively (see Table 1).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Histograms of (a) the standard deviation of the fits
$\sigma_{\mathrm{fit}}$ and (b) the number of data points Ndat for our Cepheid
sample. Histogram of uncertainties of low-order Fourier parameters are
displayed for (c) $\sigma(\phi_{21})$, (d) $\sigma(R_{21})$, (e)
$\sigma(\phi_{31})$ and (f) $\sigma(R_{31})$ for Cepheids of this paper and,
for comparison, for Cepheids from Storm et al. (2011). In each plot, the
median of each sample is represented by a vertical dashed line. See Sect. 3.

###  3.2 Identification of First-overtone Cepheids

Many known overtone Cepheids in the literature have been identified with the
help of the Fourier phases of their light curves (Antonello & Poretti 1986;
Antonello et al. 1990; Mantegazza & Poretti 1992; Poretti 1994).
Unfortunately, light-curve Fourier parameters cannot differentiate the
pulsation modes at longer periods ($P>5.0-5.5\,$day) because $\phi_{21}$ and
$\phi_{31}$ sequences of fundamental and first-overtone merge with each other
(Antonello et al. 1990; Soszyński et al. 2008). We stress that inability to
identify the pulsation mode from the light-curve in this period range is not a
matter of precision of the observation; it is rather an intrinsic physical
property of classical Cepheids.

Alternatively, other photometric survey of Galactic Cepheids such as OGLE
(Soszyński et al. 2008, 2010) or Gaia used a criterion based on the
$R_{21}$-$P$ plane (Clementini et al. 2016), which distinguishes the first-
overtone and the fundamental-mode pulsators. This criterion is robust for a
large samples of Cepheids, but can fail in individual cases. Indeed, we still
encounter the same problem as in the case of $\phi_{21}$, namely, that light-
curve $R_{21}$ sequences of the fundamental mode and the first-overtone merge
for periods $P>5.5\,$day. Moreover, the light-curve $R_{21}$ of the
fundamental-mode Cepheids depends on the metallicity (Klagyivik & Szabados
2007; Szabados & Klagyivik 2012; Majaess et al. 2013; Hocdé et al. 2023) and
the location of the star within the instability strip (Sandage et al. 2004,
2009). In particular, fundamental-mode Cepheids located close to the edge of
the instability strip might have a low amplitude and, consequently, low
$R_{21}$, in the range of those observed in the overtone pulsators. The
fundamental-mode Cepheid Y Oph is a good example of such a behaviour (this
paper; Hocdé et al. 2023b, submitted). In the framework of specific processing
and validation of all sky Cepheids from Gaia DR3, Ripepi et al. (2022, 2023)
performed a re-classification of Gaia DR2 and DR3 pulsation modes obtained
from The Specific Objects Study (SOS) Cep&RRL pipeline, after in-depth
analysis of light-curves, period-luminosity relation and results from
literature. For Cepheids of pulsation period above 5 days, most of their re-
classification change fundamental into first-overtone mode as shown in Table
2.

In the light of these difficulties, the mode identification based on
$\phi_{21}$ of the RV curves is a more robust, and thus preferable, approach.
As shown already by Kienzle et al. (1999), the progression of Fourier phases
for radial velocity curves is different than for the light curves and allows
to discriminate between the overtone and the fundamental-mode at all periods
(except a narrow range around $P=5\,$day, where the two progressions cross).
In fact, for Cepheids with $P>5.5$ day analysis of the RV curves is the only
unambiguous way to identify the pulsation mode. Using this property, we
identify certainty V495 Cyg as a new overtone Cepheid. This star has been
classified by Gaia as fundamental-mode pulsators, but this cannot be
considered reliable, because of its pulsation period above $5.5\,$day. We
confirm overtone nature of several other Cepheids, classified as such by Gaia:
VZ CMa, V351 Cep, DT Cyg, V532 Cyg, V411 Lac and EV Sct. We also confirm
overtone nature of GH Car, originally classified by Kienzle et al. (1999).
This last star has later been re-classified by Gaia as a fundamental-mode
pulsator. However, for $P=5.7\,$day Gaia’s classification based on the light
curve is not reliable. Finally, we have examined the pulsation mode of
Polaris, which was debated in the last decade (see, e.g. Turner et al. 2013;
Neilson 2014; Anderson 2018a). Although a consensus emerged for Polaris to be
a first-overtone pulsator based on different level of evidences (see Torres
(2023) and references therein), we are able to firmly conclude, on the basis
of Fourier parameters of its radial velocity curve, that Polaris is an
overtone pulsator.

(a)

(b)

Figure 6: (a) Velocity Fourier phases $\phi_{21}$ for first-overtone and
short-period fundamental-mode Cepheids. Orange labels correspond to overtone
Cepheids for which our results disagree either by Gaia SOS Cep&RRL pipeline or
reclassification from (Ripepi et al. 2023) (see Table 2). Purple symbols
correspond to stars with pulsation mode uncertain or unknown according to our
criteria based on $\phi_{21}$. V1726 Cyg and GH Lup cannot be plotted since
the 2nd Fourier term has not been detected in these stars, and V636 Cas has
very high $\phi_{21}\simeq 4.5$ rad. All three variables are presented in the
bottom plot. (b) Astrometric Based Luminosities in the Wesenheit G-band.
Relations for the first-overtone and for the fundamental-mode Cepheids are
also plotted as red and black curves, respectively.

####  3.2.1 Comparison with mode identification of Gaia survey

As discussed above, with the mode identification method used by the SOS
Cep&RRL pipeline from Gaia, some Cepheids might be misclassified. For example,
the two well established first-overtones Cepheids MY Pup (Kienzle et al. 1999)
and V440 Per (Kienzle et al. 1999; Baranowski et al. 2009) are classified as
Fundamental mode from Gaia DR3 catalogue. In order to demonstrate the
robustness of our criteria based on velocity $\phi_{21}$, we compared
systematically our classification with that of Gaia DR3 catalogue. In total,
we found 5 first-overtones Cepheids that are classified in Gaia DR3 as
fundamental-mode pulsators : X Lac, MY Pup, GH Car, V495 Cyg and V440 Per. We
display these stars in Fig. 6(a) and mark them with orange labels. As
expected, we find disagreements only for pulsation periods above 5.4 days,
since criteria based on the light-curves are ambiguous in this period range.
Four of these stars have very recently been reclassified by Ripepi et al.
(2023) as first-overtone variables in agreement with our results. In Fig. 6(a)
we also display several Cepheids, pulsation mode of which is uncertain or
unknown according to our criteria, but can be identified according to Gaia DR3
(CK Cam, BD Cas, CE Cas A, V924 Cyg). These variables are plotted as purple
points. Finally, we also mark in the Figure the Cepheid FF Aql, that is the
only Cepheid of our sample below 5 days for which the results from Gaia DR3
SOS Cep&RRL pipeline and of Ripepi et al. (2023) disagree. We summarize the
mode identifications for all these stars in Table 2.

In Fig. 6(b) we verify our mode identifications with the help of the
astrometry-based luminosity (ABL) (Feast & Catchpole 1997; Arenou & Luri
1999), established from reddening-free Wesenheit magnitudes, $W_{G}$ as
defined by Ripepi et al. (2019), i.e. $W_{G}=G-1.90(G_{Bp}-G_{Rp})$. The ABL
vs. period relations displayed in Fig. 6(b) for first-overtone and for
fundamental-mode variables are calibrated with a clean sample of Cepheids
(RUWE ¡ 1.41, $\delta\omega/\omega<0.2$) (Ripepi et al. 2023). The relations
are plotted together with their dispersion adopted from Ripepi et al. (2023).
We have also derived observed ABL for our sample of stars (Table 2), using the
apparent Wesenheit magnitudes and Gaia DR3 parallaxes (corrected from zero-
point, Lindegren et al. 2021). We have also applied residual parallax offset
(Riess et al. 2021). As we can see from Fig. 6(b), all the stars that we have
identified as long period first-overtone Cepheids (red symbols) are in
excellent agreement with the ABL vs. period relation for the first overtone.
Only for one star, GH Car, the ABL plot yields ambiguous conclusion since the
star falls between the ABL relations of the two modes, deviating from either
of them by more than 3$\sigma$. Still, it is twice closer to the first
overtone line. Among the stars with pulsation mode uncertain/unknown according
to our criteria, BD Cas, V1726 Cyg and V924 Cyg agree with the first-overtone
ABL relation while CK Cam, CE Cas A and GH Lup are in better agreement with
the fundamental-mode relation.

####  3.2.2 Comments on individual overtone Cepheids

Below we enclose comments on five individual stars : three identified long-
period overtone Cepheids and on Polaris and FF Aql. Comments on remaining
overtone Cepheids and on Cepheids of uncertain/unknown mode identifications
are deferred to Appendix C.4 and C.5 respectively.

X Lac, V495 Cyg: The radial velocity curves of X Lac and V495 Cyg have already
been discussed by Kienzle et al. (1999), who referred to unpublished results
of Krzyt, Moskalik, Gorynya et al. (1999). The Fourier amplitudes $A_{1}$ for
both variables are above 10 km/s and are rather typical for fundamental-mode
Cepheids. On the other hand, the values of their RV Fourier phases $\phi_{21}$
are well below the fundamental-mode progression (by 5.2$\sigma$ and
8.9$\sigma$, respectively) and are similar to those of long-period overtone
Cepheids. Fourier phase $\phi_{31}$ is available only for V495 Cyg and it
deviates from the fundamental-mode progression as well (by 7.6$\sigma$).
Kienzle et al. (1999) have suggested that X Lac and V495 Cyg might be overtone
pulsators, but they have stopped short of classifying them as such. Following
the approach of Antonello et al. (1990), who recommended using for mode
identification solely $\phi_{21}$ and $\phi_{31}$, we conclude with certainty
that both Cepheids are overtone pulsators. We also provide their RV Fourier
parameters for the first time. We note, that both stars are classified in Gaia
DR3 catalog as fundamental-mode pulsators. Only X Lac is re-classified in
Ripepi et al. (2023) as overtone pulsator. We have to keep in mind that at
pulsation periods of 5.4 day and 6.7 day Gaia’s mode identification, which is
based on the light curves, cannot be reliable (Antonello et al. 1990).

GH Car: Antonello et al. (1990) have pointed out that this Cepheid is located
in the light-curve Fourier plots at the period where the first overtone and
the fundamental-mode progressions merge. Therefore, its mode of pulsation
cannot be established on the basis of the light-curve Fourier phases. With the
radial velocity curve, GH Car has been identified as an overtone Cepheid by
Kienzle et al. (2000), who however have not published its Fourier parameters.
We confirm overtone nature of GH Car and provide its RV Fourier parameters for
the first time. We note that GH Car is classified in Gaia DR3 catalog as
fundamental-mode pulsator, but in Ripepi et al. (2023) as overtone pulsator.
For the same reason as stated in the previous case, using the RV curves is the
only way to unambiguously identify the pulsation mode in this star.

FF Aql: our first-overtone identification aligns well with Gaia DR3 catalog,
but Ripepi et al. (2023) re-classifies it as a fundamental-mode pulsator. The
value of velocity Fourier phase of this Cepheid, $\phi_{21}=3.477\pm 0.012$,
places it firmly among overtone pulsators and 29$\sigma$ away from the
fundamental-mode progression (see Fig. 6(a)). The light-curve Fourier phase
$\phi_{21}$ is measured less accurately, but its value leads to the same mode
identification (Antonello & Poretti 1986). On this basis we can conclude with
certainty that FF Aql is an overtone pulsator.

$\alpha$ UMi (Polaris): pulsation mode of Polaris has been vigorously debated
in the last decade (see, e.g. van Leeuwen 2013; Turner et al. 2013; Neilson
2014; Anderson 2018a; Evans et al. 2018). This variable has the lowest
pulsation amplitude of all Cepheids and, consequently, almost sinusoidal light
and radial velocity curves. Using Fourier phases to establishing its pulsation
mode was so far unsuccessful. The primary objective of this study is to detect
and characterize the first harmonic of the RV curve of Polaris, which is a
notably low amplitude component (below 0.1km/s). This necessitates a large
number of measurements to beat down the errors. A challenging aspect lies in
the fact that Polaris exhibits fast variations in both its amplitude and
period (Neilson et al. 2012; Usenko et al. 2018; Anderson 2019; Usenko et al.
2020). For this reason, phasing together long datasets yields excessive
scatter of the RV curve. To address this problem, we require a short dataset
to ensure that neither amplitude nor phase undergo significant variability
during the observations. At the same time, it is imperative that this short
dataset consists of a large number of measurements. Regrettably, this
criterion is not met when examining any of the subsets within the very precise
dataset of Anderson (2019). However, the conditions are met by a subset of
data from Eaton (2020). The subset is 83-day long, from HJD=2454713 to
HJD=2454796 and consists of 167 data points. This dataset yields Fourier fit
with dispersion of 0.09 km/s and allows to determine the RV Fourier phase of
$\phi_{21}=3.736\pm 0.113$.

The measured value of $\phi_{21}$ is much more accurate, but within error it
is the same as determined previously by Moskalik & Ogłoza (2000). This is so,
despite an over 50% increase of the pulsation amplitude between the two
epochs. The new value of $\phi_{21}$ fits very well the overtone progression
and deviates from the fundamental-mode progression by 5.4$\sigma$. Thus, for
the first time we are able to firmly demonstrate with the RV Fourier
parameters, that Polaris is an overtone Cepheid. Our mode identification
agrees with earlier result of Moskalik & Ogłoza (2000), who reached the same
conclusion with a very different method, namely with measuring the phase lag
between light and the radial velocity curves. We note, that the mode
identifications of our paper and of Moskalik & Ogłoza (2000) depend neither on
distance to Polaris nor on its interstellar reddening.

Table 2: Comparison of mode identification of our work and of Gaia survey.

Star | Period | GDR3 | R23 | This work
---|---|---|---|---
CK Cam | 3.29 | F | - | U/F
BD Cas | 3.65 | 1O | - | U/1O
V1726 Cyg | 4.24 | 1O | - | U/1O
FF Aql | 4.47 | 1O | F | 1O
CE Cas A | 5.14 | F | - | U/F
X Lac | 5.44 | F | 1O | 1O
V924 Cyg | 5.57 | F | 1O | U/1O
MY Pup | 5.69 | F | 1O | 1O
GH Car | 5.73 | F | 1O | 1O
V495 Cyg | 6.72 | F | - | 1O
V440 Per | 7.57 | F | 1O | 1O
V636 Cas | 8.38 | F | - | U/F
GH Lup | 9.28 | F | - | U/F

  * •

Notes : For each star we indicate the fundamental or first-overtone mode
identification from The Specific Objects Study Cep&RRL pipeline (Gaia
Collaboration et al. 2023) (GDR3), re-classification from Ripepi et al. (2023)
(R23) when applicable, and our identification based on radial velocity
$\phi_{21}$. For uncertain or undetermined mode (U), we also indicate the mode
identified from PL-ABL relation (see Fig. 6(b)).

###  3.3 Detected spectroscopic Cepheid binaries

In five Cepheids of our sample we have found statistically very significant
drift of the $\gamma$-velocity. Such $\gamma$-velocity variation might be
caused by orbital motion in a long period binary system. For two of these
variables, BP Cir and LR TrA, the binary nature has been already recognized
(Petterson et al. 2005; Szabados et al. 2013b). To our knowledge the
spectroscopic binarity of the other three Cepheids is identified here for the
first time.

VY Per: we have found a slow $\gamma$-velocity drift, amounting to
$\approx$7.0 km/s in 500 days. The observed change of $\gamma$-velocity
indicates a binary orbital motion, with the period much longer than the
duration of the observations.

AQ Pup: Anderson et al. (2016a) noticed a small fluctuation of the
$\gamma$-velocity in older archival data, but they attributed this value to
problem of phase folding the data because of fast and irregular period
changes. They also reported no significant $\gamma$-velocity variations within
their precise data set. We reached a different conclusion by analyzing in
details the single very precise dataset of Anderson et al. (2016a). Fitting
simultaneously pulsation RV curve and a trend, we have found a slow,
approximately parabolic $\gamma$-velocity drift, amounting to $\simeq 2.0$
km/s in 800 days. Considering an excellent quality of the data, this drift is
highly significant. To refine the analysis, we have divided the measurements
into four separate subsets and determined the $\gamma$-velocity for each
subset independently. This method of analysis is not sensitive to a possible
variations of the pulsation period which might affect the phasing of the RV
curve. We have found that the $\gamma$-velocity of AQ Pup has changed between
JD=2456652 and JD=2457459 by $\Delta V_{\gamma}=1.81\pm 0.045\,$km/s. The
observed change of $\gamma$-velocity indicates a binary orbital motion, with
the period much longer than the duration of the observations.

QZ Nor: the analysis of this star has been performed with a single very
accurate dataset of Anderson (2014). Fitting simultaneously pulsation RV curve
and a trend, we have found a slow, linear $\gamma$-velocity drift, amounting
to $\simeq 0.5$ km/s in 1000 days. Considering an excellent quality of the
data, this drift is highly significant. Proceeding in the same was as in case
of AQ Pup, we have divided the measurements into four separate subsets and
determined the $\gamma$-velocity for each of them independently. We have found
that the $\gamma$-velocity of QZ Nor has changed between JD=2455708 and
JD=2456689 by $\Delta V_{\gamma}=0.48\pm 0.046$ km/s. The observed change of
$\gamma$-velocity also indicates a binary orbital motion, with the period much
longer than the duration of the observations.

##  4 Discussion on Fourier parameters trend

###  4.1 Amplitude of the first Fourier term

In case of the fundamental-mode Cepheids of short periods ($P<10\,$day), we
observe in Fig. 2, a very sharply defined upper envelope limit of the
pulsation amplitude. This limit defines empirically a maximum $A_{1}$ of
Galactic Cepheids to be between 16.5 and 15 km/s, depending on the pulsation
period. Conversely, the amplitudes of Cepheids are distributed below this
upper limit. In case of the light curve, the amplitude of fundamental-mode
Cepheids is highest on the blue edge of the instability strip and decreases
towards the red edge (Sandage et al. 2004, 2009; Hocdé et al. 2023). It has
been suggested in the literature that the radial velocity amplitude might be
correlated with the location within the instability strip as well (Cogan 1980;
Pont et al. 2001).

The light curve amplitude is also sensitive to the metallicity (Klagyivik &
Szabados 2007; Szabados & Klagyivik 2012; Majaess et al. 2013; Hocdé et al.
2023). In order to test if the RV amplitudes of Cepheids with $P<10\,$day
depend on metallicity, we cross-matched our data with metallicity determined
with the HR spectroscopy (Luck 2018). We found no significant correlation
between the velocity amplitude $A_{1}$ and the spectroscopic metallicity of
the Cepheid. This is different from result of (Szabados & Klagyivik 2012), who
found that the RV amplitude decreases with [Fe/H]. Extending the Cepheid
sample to metal-poor objects of the Magellanic Clouds seems necessary to
clarify this issue.

For the fundamental-mode Cepheids with $P>10\,$day, we observe in Fig. 2 a
sharp rise of the pulsation amplitude $A_{1}$ between 10 and 15 day. The upper
limit envelope of $A_{1}$ reaches a broad maximum of about 25 km/s for periods
of 15-21 day. For example, SV Mon ($P=15.23\,$day) and RZ Vel ($P=20.40$day)
have amplitudes of $A_{1}=24.87\pm 0.18\,$km/s and $A_{1}=24.64\pm 0.03\,$km/s
respectively. Some stars pulsate with much lower amplitudes though, e.g. V340
Nor, SZ Car and Y Oph, all with $A_{1}<10\,$km/s. There is only a small number
of Cepheids with pulsation period longer than 21 day. Their amplitude
generally declines with increasing period, but with a few noticeable
exceptions (e.g. AQ Pup).

Finally, in Fig. 2 we see an upper limit envelope of the pulsation amplitude
also for the overtone Cepheids with $P<5\,$day. This limit is not as sharply
defined as for the fundamental-mode. Nevertheless, it clearly defines a
maximum amplitude $A_{1}$ of overtone Cepheids to be between 9.5km/s and
9.0km/s, depending on the pulsation period.

###  4.2 Amplitude ratio $R_{21}$, $R_{31}$ and $R_{41}$

For the fundamental-mode Cepheids, the amplitude ratio of the second over the
first Fourier component $R_{21}$ displays an almost symetric Z-shape centered
on $R_{21}=0.3$ and $P=10\,$day (see Fig. 3(b)). The maximum of $R_{21}$,
which corresponds to the most asymmetrical RV curves, is located at $P\approx
8\,$day. In our sample, the maximum value is reached by W Gem with
$R_{21}=0.61\pm 0.01$ at $P=7.91\,$day. Given the smooth variation of $A_{1}$
for $P<10\,$day, we can show that this sharp rise of $R_{21}$ is caused by the
amplitude of the second Fourier component $A_{2}$. Interestingly, $R_{21}$ of
the light curves display a somewhat different trend, with more asymmetrical
curves (highest $R_{21}$) at $P\approx 2.5\,$day followed by decrease of light
curve asymmetry until $P\approx 10\,$day (see, e.g. Soszyński et al. 2008,
2010; Pietrukowicz et al. 2021). Thus, the amplitude $A_{2}$ of the light-
curves starts decreasing already at $P\simeq 2.5\,$day, while for the RV
curves it continues to increase up to $P\simeq 8\,$day. Moreover, $R_{21}$ of
the $V$-band light curve is known to have a minimum value at about $P\simeq
10\,$day, i.e. close to the resonance period, while in case of the RV curves
the minimum of $R_{21}$ occurs at a longer period of $\sim 12.5\,$day. For
example, we found $R_{21}=0.029\pm 0.002$ for XY Car ($P$=12.44 day). Although
a quantitative comparison is needed, these differences between light and RV
curves seem to be reproduced by hydrodynamical models (Moskalik et al. 1992;
Paxton et al. 2019).

The plots of $R_{31}$ and $R_{41}$ appear more scattered and both parameters
have larger relative uncertainties than $R_{21}$. This is expected since the
third and fourth Fourier component have lower amplitudes and, thus, are less
accurately determined from the data. Nevertheless, our amplitude ratios are
precise enough to observe a first maximum of $R_{31}$ at a period close to 6
day. Among quality 1 Cepheids in our sample, the highest value are reached by
AW Per ($R_{31}=0.25\pm 0.01$) and XX Sgr ($R_{31}=0.24\pm 0.01$) at $P\approx
6.5\,$day. For $P>6.5\,$day a sudden decline of $R_{31}$ is observed, however
it is difficult to precisely locate the minimum, because of a deficiency of
our Cepheids sample with periods between 8 and $10\,$day. This defiency could
be due to the so-called zone of avoidance (van Genderen 1970; Buchler et al.
1997; Szabados & Klagyivik 2012; Udalski et al. 2018). Moskalik et al. (1992)
postulated that this dip which appears around 8 day could be associated with
the $P_{4}/P_{0}=1/3$ resonance between the fundamental and fourth overtone
(Moskalik & Buchler 1989). $R_{31}$ has its second maximum at around
$P=10\,$day, then it declines again, reaching minimum around 17 day, i.e. at a
period slightly longer than in case of $R_{21}$. Beyond 17 day, $R_{31}$
increases again and, similarly to $R_{21}$, it reaches its next maximum at
around $P\simeq 40\,$day. In the case of $R_{41}$ the dispersion of the data
is too large to describe the behaviour of the fourth Fourier term.

###  4.3 Fourier phase $\phi_{21}$

Low-order Fourier parameters of fundamental-mode Cepheids presented in Fig. 3
display a characteristic variation shaped by the $P_{2}/P_{0}=0.5$ resonance
around 10 day. At short-periods below about 8 day, we are able to confirm the
tight progression of Fourier phases with the pulsation period as observed by
Moskalik et al. (2000), although this time with a better precision of the
parameters. This tight progression was also predicted by hydrodynamical models
(Buchler et al. 1990). In that study, they show that the dispersion of
$\phi_{21}$ can be used to derive the width of the instability strip. On the
other hand, we observe a large scatter of $\phi_{21}$ for periods between 9 to
13 day. This can be explained by the strong sensitivity of $\phi_{21}$ to the
value of resonant period ratio $P_{2}/P_{0}$, which at a given period depends
on the fundamental parameters of the star (mainly metallicity, but also on
mass, luminosity and temperature). This sensitivity of $\phi_{21}$ to stellar
parameters is strongest within the resonance region, as predicted by
hydrodynamical calculation (see, e.g., Buchler et al. 1990; Moskalik et al.
1992; Buchler et al. 1997). By comparing the light curves of Cepheids in the
Milky Way with those in metal-poor galaxies, it appears that with decreasing
metallicity the resonance center shifts towards longer periods (Beaulieu 1998;
Bhardwaj et al. 2015). That implies that any dispersion of [Fe/H] within the
Cepheid sample will cause the dispersion $P_{2}/P_{0}$ at a given pulsation
period (Buchler et al. 1990). We note, that for our Cepheids with periods of
9-13 days, the spectroscopic metallicities of Luck (2018) range from
[Fe/H]=$-0.2$ to $+0.3\,$dex. This range of metallicity could be large enough
to explain, at least in part, the large scatter of $\phi_{21}$ observed in the
resonance region.

At longer periods we observe an almost constant value of radial velocity
$\phi_{21}$ from about $P=13\,$day to $P=70\,$day. In the case of light curves
this branch appears in general more scattered, even for a small sample of
Galactic Cepheids (Pont et al. 2001; Simon & Moffett 1985; Hocdé et al. 2023).
This scatter is attributed to differences in metallicity between the stars
(Hocdé et al. 2023). Since the velocity $\phi_{21}$ in long-period Cepheids is
almost constant, the observed metallicity effect on light-curve $\phi_{21}$ is
likely of photospheric origin. Additional RV curves for long-period Cepheids
are necessary to analyze this issue in more detail.

###  4.4 Fourier phases $\phi_{31}$ and $\phi_{41}$

The Fourier phase $\phi_{31}$ undergoes a rather smooth, monotonic increase
across the entire range of pulsation periods. Contrary to $\phi_{21}$, it does
not display a sharp variation in the vicinity of the $P_{2}/P_{0}=0.5$
resonance, but only a somewhat increased scatter in the progression. The
intrarelations of Fourier phases presented in Sec. 4.6 show in the resonance
region a linear correlation between $\phi_{21}$ and $\phi_{31}$. The phase
$\phi_{31}$ increases moderately fast with the pulsation period up to
$P\approx 15\,$day, then for $P=15-20\,$day it undergoes a very steep increase
(possibly a jump) by 2 rad. At even longer periods, we observe a very slow
increase of $\phi_{31}$ from about 5 to 6 rad. Thus, in the long-period
Cepheids, both lowest order Fourier phases display a slow variation with the
pulsation period.

The behavior of $\phi_{41}$ is somewhat different. This Fourier phase
continues its steep, almost linear increase up to the longest period of 70
day. Contrary to $\phi_{31}$, it increases with nearly constant rate also at
periods between 15 and 70 day. Despite the amplitude ratio $R_{41}$ versus
period diagram being very scattered, the Fourier phase $\phi_{41}$ shows a
very tight progression with the pulsation period. We also confirm that there
is no break of $\phi_{41}$ around 7 day, contrary to what is observed for the
light-curves (Simon & Moffett 1985). The difference between $\phi_{41}$ of RV
curves and of light-curves was first noted by Kovács et al. (1990) who
attributed it to unknown photospheric phenomena.

###  4.5 High-order Fourier parameters

The presence of resonance features affecting higher order Fourier components,
due to $P_{0}/P_{1}=3/2$ at 24 day and to $P_{0}/P_{3}=3$ at 27 day, was
suggested by Antonello & Morelli (1996); Antonello et al. (2009). Theoretical
high-order Fourier parameters were also computed with hydrodynamical model
(Aikawa & Antonello 2000) to investigate these effects. In Fig. 4 we displayed
high-order Fourier parameters (5th, 6th and 7th Fourier terms) for the RV
curves of all Cepheids of our sample. Fourier phases $\phi_{51}$ and
$\phi_{61}$ display a linear trend over the entire range of pulsation periods
similarly to $\phi_{41}$. At a close inspection, we notice that additional
structures seem to be present in the plots. In case of $\phi_{51}$, we observe
an excessive scatter at periods close to 10 day, i.e. close to the resonance
center. In this period range, several Cepheids deviate significantly upwards
from the general $\phi_{51}$ progression. On the other hand, the progression
of $\phi_{61}$ vs. pulsation period seems to split into two parallel
sequences, that coexist for periods between 10 and 20 day. We have to note,
however, that this observation is based on a rather limited number of long-
period Cepheids. The existence of the two $\phi_{61}$ sequences has to be
confirmed when more RV curves becomes available.

The amplitude ratios $R_{51}$ and $R_{61}$ appear to be more scattered,
therefore it is difficult to distinguish any trends (see Figs. 4(b) and 4(d)).
However, a weak minimum around the resonance period at $P\sim$10 day is
visible, especially in case of $R_{61}$ (see in Fig. 4(d)). From these data,
we are not able to identify any resonance features suggested by Antonello &
Morelli (1996); Antonello et al. (2009). Interestingly, the amplitude ratios
of low and high order reach a secondary maximum at long periods, before a
sharp decrease that follows when periods becomes even longer. For $R_{21}$,
$R_{31}$ and $R_{41}$ the maximum occurs at $P=45\,$day, and for $R_{51}$ and
$R_{61}$ at $P=27\,$day. As we can see from Hertzsprung progression displayed
in Fig. 1(a), the latter value roughly coincides with a period at which a
secondary bump reaches the top of the radial velocity curve (see T Mon).

###  4.6 Intrarelation of Fourier parameters

We inspected intrarelation between Fourier parameters, in particular between
the Fourier phases, similarly to Kovács et al. (1990) and Simon & Moffett
(1985) in the case of RV curves and light curves respectively. In the latter
case, the authors found a discontinuity between the short-period ($P<10\,$day)
and the long-period Cepheids and speculated that it might be caused by a
different attractor or saturation mechanism, as originally proposed by Klapp
et al. (1985). From Fig. 7 we see however that there is no apparent break at
any phase-phase diagram. In particular, we observe a continuous variation of
$\phi_{41}$ vs. $\phi_{31}$ which is not the case for the light curves (see
Fig. 17 in Simon & Moffett 1985), in agreement with our comments in Sect. 4.4.
Moreover, $\phi_{31}$ and $\phi_{41}$ seem to be tightly correlated outside
the resonance region. As expected, the intrarelations displayed in Fig. 7 are
less scattered than the ones presented by Kovács et al. (1990) (see their
Figure 5). We can notice for the first time a qualitative agreement between
the shape of $\phi_{21}$ vs. $\phi_{31}$ relation determined observationally
and that computed with hydrodynamical pulsation models of Buchler et al.
(1990) presented in Kovács et al. (1990). Therefore, it is now possible to use
Fourier parameters of RV curves to constrain hydrodynamical pulsation models.

For Cepheids in the resonance region between 9 and 12.4 day, we observe a
linear relation between $\phi_{21}$ and $\phi_{31}$ (blue dots in Fig. 7).
This suggests that in this period range the variations of these two Fourier
phases are likely sensitive to the same stellar parameters. However, when
$\phi_{41}$ is plotted vs. $\phi_{21}$ or vs. $\phi_{31}$ for the same stars,
we see no trend but rather an unstructured cloud of points (Figs. 7(c) and
7(d)). This suggests that the variation of $\phi_{41}$ in the resonance
vicinity depends on different parameters than in case of $\phi_{21}$ or
$\phi_{31}$. Last, the variation of $R_{21}$ with $\phi_{21}$ (Fig. 7(b))
displays an interesting loop, where $\phi_{21}$ variation lags behind the
$R_{21}$ variation. In particular, Cepheids in the resonance region between 9
and 12.4 day are characterized by a decrease of $R_{21}$ while $\phi_{21}$
continues to increase. In this period range, the relation between the two
seems to be linear although with a large dispersion.

(a)

(b)

(c)

(d)

Figure 7: Intrarelation between low-order Fourier phases, and intrarelation
between Fourier phase $\phi_{21}$ and amplitude ratio $R_{21}$. The meaning of
the different symbols are indicated in the figure (a).

##  5 Conclusions

In this paper, we took the advantage of the observational effort of the last
decades which allows precise determination of the Fourier parameters of
Cepheid velocity curves. These parameters are extremely valuable for
constraining the dynamics of hydrodynamical pulsation models and for studying
resonances between modes. The collected radial velocity curves will also be
very useful for distance determination with parallax-of-pulsation method. In
this work we presented a careful analysis of each Cepheid independently. First
of all, we selected for each star the highest quality RV measurements
available in the literature. Since the datasets were often obtained with
different instruments, we corrected for zero-point offsets between the
datasets. We removed the contribution from orbital motion and residual trends.
For four longest period variables in our sample, we also corrected for the
rapid and irregular period change. We then used these data to perform a
Fourier fit of the radial velocity curve of each Cepheid. The resulting
Fourier fits were classified into two categories depending on their quality,
taking into account the stability of the fit, the phase coverage and the
dispersion of fit residuals.

Our final sample consists of 178 fundamental and 33 first overtone Cepheids
with new precise radial velocity Fourier parameters. We also report 7
additional Cepheids whose pulsation mode is uncertain or undetermined. We
displayed Fourier parameters up to order six, we analyzed these parameters
qualitatively and we compared some characteristic features to the literature.
In 3 Cepheids, namely VY Per, AQ Pup and QZ Nor we have found previously
unreported significant $\gamma$-velocity variations, indicative of a binary
(or multiplicity) motion with long orbital period. We have also confirmed
$\gamma$-velocity variations in LR TrA and BP Cir. We identified V495 Cyg as a
new first-overtone Cepheids and we confirmed the first-overtone nature of
several others including Polaris. Obtaining more RV measurements for Cepheids
located near the resonance center and for metal-poor Cepheids of the
Magellanic Clouds, or of the outskirt of the Galaxy, will be essential to
fully investigate the internal physics and pulsation dynamics of these
variables. The pulsation period coverage and the precision obtained, in
particular for Fourier phase $\phi_{21}$, will be useful for studying the
dynamics of Cepheid pulsations with the help of hydrodynamical models. Further
radial velocity measurements from modern high-resolution spectroscopic
instruments will be valuable to improve these results.

###### Acknowledgements.

We are grateful to astronomers who dedicated years of work to the measurement
of radial velocity of Cepheids used in this paper. We thank Louis Balona,
Jurek Krzesiński, Simon O’Toole and Frédéric Pont for making available to us
their unpublished RV data. Natalia Gorynya thanks the Simeiz Observatory for
generously granting the observing time on 1m telescope during 33 years of her
Cepheid radial velocity program. VH, RS, OZ and RSR are supported by the
National Science Center, Poland, Sonata BIS project 2018/30/E/ST9/00598. This
research made use of the SIMBAD and VIZIER databases at CDS, Strasbourg
(France) and the electronic bibliography maintained by the NASA/ADS system.
This research also made use of Astropy, a community-developed core Python
package for Astronomy (Astropy Collaboration et al. 2018, 2022).

## References

  * Abt (1959) Abt, H. A. 1959, ApJ, 130, 769
  * Aikawa & Antonello (2000) Aikawa, T. & Antonello, E. 2000, A&A, 363, 593
  * Albrow & Cottrell (1996) Albrow, M. D. & Cottrell, P. L. 1996, MNRAS, 280, 917
  * Anderson (2014) Anderson, R. I. 2014, A&A, 566, L10
  * Anderson (2016) Anderson, R. I. 2016, MNRAS, 463, 1707
  * Anderson (2018a) Anderson, R. I. 2018a, A&A, 611, L7
  * Anderson (2018b) Anderson, R. I. 2018b, in Proceedings of the Polish Astronomical Society, Vol. 6, The RR Lyrae 2017 Conference. Revival of the Classical Pulsators: from Galactic Structure to Stellar Interior Diagnostics, ed. R. Smolec, K. Kinemuchi, & R. I. Anderson, 193–200
  * Anderson (2019) Anderson, R. I. 2019, A&A, 623, A146
  * Anderson et al. (2016a) Anderson, R. I., Casertano, S., Riess, A. G., et al. 2016a, ApJS, 226, 18
  * Anderson et al. (2016b) Anderson, R. I., Mérand, A., Kervella, P., et al. 2016b, MNRAS, 455, 4231
  * Anderson et al. (2015) Anderson, R. I., Sahlmann, J., Holl, B., et al. 2015, ApJ, 804, 144
  * Antonello & Morelli (1996) Antonello, E. & Morelli, P. L. 1996, A&A, 314, 541
  * Antonello & Poretti (1986) Antonello, E. & Poretti, E. 1986, A&A, 169, 149
  * Antonello et al. (1990) Antonello, E., Poretti, E., & Reduzzi, L. 1990, A&A, 236, 138
  * Antonello et al. (2009) Antonello, E., Speroni, L., Kanbur, S., Richter, I., & Szabó, R. 2009, Communications in Asteroseismology, 158, 274
  * Arellano Ferro (1984) Arellano Ferro, A. 1984, MNRAS, 209, 481
  * Arellano Ferro & Madore (1985) Arellano Ferro, A. & Madore, B. F. 1985, The Observatory, 105, 207
  * Arenou & Luri (1999) Arenou, F. & Luri, X. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 167, Harmonizing Cosmic Distance Scales in a Post-HIPPARCOS Era, ed. D. Egret & A. Heck, 13–32
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Baade (1926) Baade, W. 1926, Astronomische Nachrichten, 228, 359
  * Babel et al. (1989) Babel, J., Burki, G., Mayor, M., Waelkens, C., & Chmielewski, Y. 1989, A&A, 216, 125
  * Balog et al. (1997) Balog, Z., Vinkó, J., & Kaszas, G. 1997, AJ, 113, 1833
  * Baranne et al. (1979) Baranne, A., Mayor, M., & Poncet, J. L. 1979, Vistas in Astronomy, 23, 279
  * Baranne et al. (1996) Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373
  * Baranowski et al. (2009) Baranowski, R., Smolec, R., Dimitrov, W., et al. 2009, MNRAS, 396, 2194
  * Barnes et al. (2005) Barnes, Thomas G., I., Jeffery, E. J., Montemayor, T. J., & Skillen, I. 2005, ApJS, 156, 227
  * Barnes et al. (1988) Barnes, Thomas G., I., Moffett, T. J., & Slovak, M. H. 1988, ApJS, 66, 43
  * Beaulieu (1998) Beaulieu, J. P. 1998, Mem. Soc. Astron. Italiana, 69, 21
  * Becker (1940) Becker, W. 1940, ZAp, 19, 289
  * Benedict et al. (2007) Benedict, G. F., McArthur, B. E., Feast, M. W., et al. 2007, AJ, 133, 1810
  * Berdnikov et al. (2010) Berdnikov, L. N., Kniazev, A. Y., Usenko, I. A., Kovtyukh, V. V., & Kravtsov, V. V. 2010, Astronomy Letters, 36, 490
  * Berdnikov et al. (2007) Berdnikov, L. N., Pastukhova, E. N., Gorynya, N. A., Zharova, A. V., & Turner, D. G. 2007, PASP, 119, 82
  * Berdnikov et al. (2004) Berdnikov, L. N., Samus, N. N., Antipin, S. V., et al. 2004, PASP, 116, 536
  * Bersier (2002) Bersier, D. 2002, ApJS, 140, 465
  * Bersier et al. (1994) Bersier, D., Burki, G., Mayor, M., & Duquennoy, A. 1994, A&AS, 108, 25
  * Bhardwaj et al. (2015) Bhardwaj, A., Kanbur, S. M., Singh, H. P., Macri, L. M., & Ngeow, C.-C. 2015, MNRAS, 447, 3342
  * Böhm-Vitense et al. (1998) Böhm-Vitense, E., Evans, N. R., Carpenter, K., et al. 1998, ApJ, 505, 903
  * Borgniet et al. (2019) Borgniet, S., Kervella, P., Nardetto, N., et al. 2019, A&A, 631, A37
  * Breitfelder et al. (2016) Breitfelder, J., Mérand, A., Kervella, P., et al. 2016, A&A, 587, A117
  * Breitfellner & Gillet (1993a) Breitfellner, M. G. & Gillet, D. 1993a, A&A, 277, 553
  * Breitfellner & Gillet (1993b) Breitfellner, M. G. & Gillet, D. 1993b, A&A, 277, 541
  * Buchler et al. (1997) Buchler, J. R., Goupil, M. J., & Piciullo, R. 1997, ApJ, 491, L99
  * Buchler & Kovács (1986) Buchler, J. R. & Kovács, G. 1986, ApJ, 303, 749
  * Buchler et al. (1990) Buchler, J. R., Moskalik, P., & Kovács, G. 1990, ApJ, 351, 617
  * Burki & Benz (1982) Burki, G. & Benz, W. 1982, A&A, 115, 30
  * Butler (1993) Butler, R. P. 1993, ApJ, 415, 323
  * Christy (1968) Christy, R. F. 1968, QJRAS, 9, 13
  * Christy (1975) Christy, R. F. 1975, in NASA Special Publication, Vol. 383, 85–98
  * Clementini et al. (2016) Clementini, G., Ripepi, V., Leccia, S., et al. 2016, A&A, 595, A133
  * Cogan (1980) Cogan, B. C. 1980, ApJ, 239, 941
  * Coker et al. (1989) Coker, R., Madore, B., Mould, J., et al. 1989, PASP, 101, 594
  * Coulson (1983) Coulson, I. M. 1983, MNRAS, 205, 1135
  * Coulson & Caldwell (1985) Coulson, I. M. & Caldwell, J. A. R. 1985, South African Astronomical Observatory Circular, 9, 5
  * Coulson et al. (1985) Coulson, I. M., Caldwell, J. A. R., & Gieren, W. P. 1985, ApJS, 57, 595
  * Coulson et al. (1986) Coulson, I. M., Caldwell, J. A. R., & Gieren, W. P. 1986, ApJ, 303, 273
  * Cseh et al. (2023) Cseh, B., Csörnyei, G., Szabados, L., et al. 2023, arXiv e-prints, arXiv:2310.10218
  * Deb & Singh (2010) Deb, S. & Singh, H. P. 2010, MNRAS, 402, 691
  * Derekas et al. (2017) Derekas, A., Plachy, E., Molnár, L., et al. 2017, MNRAS, 464, 1553
  * Eaton (2020) Eaton, J. A. 2020, JAAVSO, 48, 91
  * Evans (1976) Evans, N. R. 1976, ApJS, 32, 399
  * Evans (1988) Evans, N. R. 1988, ApJS, 66, 343
  * Evans et al. (1992) Evans, N. R., Arellano Ferro, A., & Udalska, J. 1992, AJ, 103, 1638
  * Evans et al. (2011) Evans, N. R., Berdnikov, L., Gorynya, N., Rastorguev, A., & Eaton, J. 2011, AJ, 142, 87
  * Evans et al. (2015) Evans, N. R., Berdnikov, L., Lauer, J., et al. 2015, AJ, 150, 13
  * Evans et al. (2018) Evans, N. R., Karovska, M., Bond, H. E., et al. 2018, ApJ, 863, 187
  * Evans & Lyons (1986) Evans, N. R. & Lyons, R. 1986, AJ, 92, 436
  * Evans & Sugars (1997) Evans, N. R. & Sugars, B. J. A. 1997, AJ, 113, 792
  * Evans et al. (2000) Evans, N. R., Vinkó, J., & Wahlgren, G. M. 2000, AJ, 120, 407
  * Evans et al. (1990) Evans, N. R., Welch, D. L., Scarfe, C. D., & Teays, T. J. 1990, AJ, 99, 1598
  * Evans et al. (1993) Evans, N. R., Welch, D. L., Slovak, M. H., Barnes, Thomas G., I., & Moffett, T. J. 1993, AJ, 106, 1599
  * Feast (1967) Feast, M. W. 1967, MNRAS, 136, 141
  * Feast & Catchpole (1997) Feast, M. W. & Catchpole, R. M. 1997, MNRAS, 286, L1
  * Feuchtinger et al. (2000) Feuchtinger, M., Buchler, J. R., & Kolláth, Z. 2000, ApJ, 544, 1056
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Gaia Collaboration et al. (2023) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1
  * Gallenne et al. (2019) Gallenne, A., Kervella, P., Borgniet, S., et al. 2019, A&A, 622, A164
  * Gallenne et al. (2018) Gallenne, A., Kervella, P., Evans, N. R., et al. 2018, ApJ, 867, 121
  * Gieren (1981) Gieren, W. 1981, ApJS, 46, 287
  * Gieren (1985) Gieren, W. P. 1985, A&A, 148, 138
  * Gieren & Brieva (1992) Gieren, W. P. & Brieva, E. 1992, A&A, 253, 126
  * Gieren et al. (1989) Gieren, W. P., Matthews, J. M., Moffett, T. J., et al. 1989, AJ, 98, 1672
  * Gieren et al. (1994) Gieren, W. P., Welch, D. L., Mermilliod, J.-C., Matthews, J. M., & Hertling, G. 1994, AJ, 107, 2093
  * Gorynya (1998) Gorynya, N. A. 1998, Information Bulletin on Variable Stars, 4636, 1
  * Gorynya et al. (1992) Gorynya, N. A., Irsmambetova, T. R., Rastorgouev, A. S., & Samus, N. N. 1992, Soviet Astronomy Letters, 18, 316
  * Gorynya et al. (1996a) Gorynya, N. A., Rastorguev, A. S., & Samus, N. N. 1996a, Astronomy Letters, 22, 33
  * Gorynya et al. (1996b) Gorynya, N. A., Samus, N. N., Rastorguev, A. S., & Sachkov, M. E. 1996b, Astronomy Letters, 22, 175
  * Gorynya et al. (2000) Gorynya, N. A., Samus, N. N., Sachkov, M. E., Antipin, S. V., & Rastorgouev, A. S. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 203, IAU Colloq. 176: The Impact of Large-Scale Surveys on Pulsating Star Research, ed. L. Szabados & D. Kurtz, 242–243
  * Gorynya et al. (1998) Gorynya, N. A., Samus, N. N., Sachkov, M. E., et al. 1998, Astronomy Letters, 24, 815
  * Griffin (1967) Griffin, R. F. 1967, ApJ, 148, 465
  * Griffin (2016) Griffin, R. F. 2016, The Observatory, 136, 209
  * Groenewegen (2008) Groenewegen, M. A. T. 2008, A&A, 488, 25
  * Groenewegen (2013) Groenewegen, M. A. T. 2013, A&A, 550, A70
  * Groenewegen & Jurkovic (2017) Groenewegen, M. A. T. & Jurkovic, M. I. 2017, A&A, 604, A29
  * Harris & Wallerstein (1984) Harris, H. C. & Wallerstein, G. 1984, AJ, 89, 379
  * Harris et al. (1987) Harris, H. C., Welch, D. L., Kraft, R. P., & Schmidt, E. G. 1987, AJ, 94, 403
  * Herbig & Moore (1952) Herbig, G. H. & Moore, J. H. 1952, ApJ, 116, 348
  * Hertzsprung (1919) Hertzsprung, E. 1919, Astronomische Nachrichten, 210, 17
  * Hertzsprung (1926) Hertzsprung, E. 1926, Bull. Astron. Inst. Netherlands, 3, 115
  * Hocdé et al. (2020) Hocdé, V., Nardetto, N., Borgniet, S., et al. 2020, A&A, 641, A74
  * Hocdé et al. (2023) Hocdé, V., Smolec, R., Moskalik, P., Ziółkowska, O., & Singh Rathour, R. 2023, A&A, 671, A157
  * Imbert (1981) Imbert, M. 1981, A&AS, 44, 319
  * Imbert (1983) Imbert, M. 1983, A&AS, 53, 85
  * Imbert (1984) Imbert, M. 1984, A&AS, 58, 529
  * Imbert (1996) Imbert, M. 1996, A&AS, 116, 497
  * Imbert (1999) Imbert, M. 1999, A&AS, 140, 79
  * Joy (1937) Joy, A. H. 1937, ApJ, 86, 363
  * Katz et al. (2023) Katz, D., Sartoretti, P., Guerrier, A., et al. 2023, A&A, 674, A5
  * Kervella et al. (2019) Kervella, P., Gallenne, A., Evans, N. R., et al. 2019, A&A, 623, A116
  * Kervella et al. (2017) Kervella, P., Trahin, B., Bond, H. E., et al. 2017, A&A, 600, A127
  * Kienzle et al. (1999) Kienzle, F., Moskalik, P., Bersier, D., & Pont, F. 1999, A&A, 341, 818
  * Kienzle et al. (2000) Kienzle, F., Pont, F., Moskalik, P., & Bersier, D. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 203, IAU Colloq. 176: The Impact of Large-Scale Surveys on Pulsating Star Research, ed. L. Szabados & D. Kurtz, 239
  * Kiss & Vinkó (2000) Kiss, L. L. & Vinkó, J. 2000, MNRAS, 314, 420
  * Klagyivik & Szabados (2007) Klagyivik, P. & Szabados, L. 2007, Astronomische Nachrichten, 328, 825
  * Klapp et al. (1985) Klapp, J., Goupil, M. J., & Buchler, J. R. 1985, ApJ, 296, 514
  * Kovács & Buchler (1989) Kovács, G. & Buchler, J. R. 1989, ApJ, 346, 898
  * Kovács et al. (1990) Kovács, G., Kisvarsányi, E. G., & Buchler, J. R. 1990, ApJ, 351, 606
  * Kovács et al. (2023) Kovács, G. B., Nuspl, J., & Szabó, R. 2023, MNRAS, 521, 4878
  * Ledoux & Walraven (1958) Ledoux, P. & Walraven, T. 1958, Handbuch der Physik, 51, 353
  * Lindegren et al. (2021) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, A&A, 649, A2
  * Lindemann (1918) Lindemann, F. A. 1918, MNRAS, 78, 639
  * Lloyd Evans (1968) Lloyd Evans, T. 1968, MNRAS, 141, 109
  * Lloyd Evans (1980) Lloyd Evans, T. 1980, South African Astronomical Observatory Circular, 1, 257
  * Lloyd Evans (1982) Lloyd Evans, T. 1982, MNRAS, 199, 925
  * Luck (2018) Luck, R. E. 2018, AJ, 156, 171
  * Ludendorff (1919) Ludendorff, H. 1919, Astronomische Nachrichten, 209, 217
  * Majaess et al. (2013) Majaess, D., Turner, D., Gieren, W., Berdnikov, L., & Lane, D. 2013, Ap&SS, 344, 381
  * Mantegazza & Poretti (1992) Mantegazza, L. & Poretti, E. 1992, A&A, 261, 137
  * Marschall et al. (1993) Marschall, L. A., Seeds, M. A., & Davis, R. J. 1993, AJ, 106, 1123
  * Matthews et al. (1995) Matthews, J. M., Gieren, W. P., Mermilliod, J.-C., & Welch, D. L. 1995, AJ, 110, 2280
  * Mermilliod et al. (1987) Mermilliod, J. C., Mayor, M., & Burki, G. 1987, A&AS, 70, 389
  * Mermilliod et al. (2008) Mermilliod, J. C., Mayor, M., & Udry, S. 2008, A&A, 485, 303
  * Metzger et al. (1991) Metzger, M. R., Caldwell, J. A. R., McCarthy, J. K., & Schechter, P. J. 1991, ApJS, 76, 803
  * Metzger et al. (1992) Metzger, M. R., Caldwell, J. A. R., & Schechter, P. L. 1992, AJ, 103, 529
  * Moskalik & Buchler (1989) Moskalik, P. & Buchler, J. R. 1989, ApJ, 341, 997
  * Moskalik et al. (1992) Moskalik, P., Buchler, J. R., & Marom, A. 1992, ApJ, 385, 685
  * Moskalik & Krzyt (1998) Moskalik, P. & Krzyt, T. 1998, in Astronomical Society of the Pacific Conference Series, Vol. 135, A Half Century of Stellar Pulsation Interpretation, ed. P. A. Bradley & J. A. Guzik, 29
  * Moskalik et al. (2000) Moskalik, P., Krzyt, T., Gorynya, N. A., & Samus, N. N. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 203, IAU Colloq. 176: The Impact of Large-Scale Surveys on Pulsating Star Research, ed. L. Szabados & D. Kurtz, 233–234
  * Moskalik & Ogłoza (2000) Moskalik, P. & Ogłoza, W. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 203, IAU Colloq. 176: The Impact of Large-Scale Surveys on Pulsating Star Research, ed. L. Szabados & D. Kurtz, 237–238
  * Munari (1999) Munari, U. 1999, Baltic Astronomy, 8, 73
  * Nardetto et al. (2023) Nardetto, N., Gieren, W., Storm, J., et al. 2023, A&A, 671, A14
  * Neilson (2014) Neilson, H. R. 2014, A&A, 563, A48
  * Neilson et al. (2012) Neilson, H. R., Engle, S. G., Guinan, E., et al. 2012, ApJ, 745, L32
  * Parenago & Kukarkin (1936) Parenago, P. P. & Kukarkin, B. W. 1936, ZAp, 11, 337
  * Paxton et al. (2019) Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10
  * Payne-Gaposchkin (1947) Payne-Gaposchkin, C. 1947, AJ, 52, 218
  * Pedicelli et al. (2010) Pedicelli, S., Lemasle, B., Groenewegen, M., et al. 2010, A&A, 518, A11
  * Petersen (1986) Petersen, J. O. 1986, A&A, 170, 59
  * Petterson et al. (1998) Petterson, O. K. L., Albrow, M. D., & Cottrell, P. L. 1998, Southern Stars, 38, 9
  * Petterson et al. (2004) Petterson, O. K. L., Cottrell, P. L., & Albrow, M. D. 2004, MNRAS, 350, 95
  * Petterson et al. (2005) Petterson, O. K. L., Cottrell, P. L., Albrow, M. D., & Fokin, A. 2005, MNRAS, 362, 1167
  * Pietrukowicz et al. (2021) Pietrukowicz, P., Soszyński, I., & Udalski, A. 2021, Acta Astron., 71, 205
  * Pont et al. (1994) Pont, F., Burki, G., & Mayor, M. 1994, A&AS, 105, 165
  * Pont et al. (2001) Pont, F., Kienzle, F., Gieren, W., & Fouqué, P. 2001, A&A, 376, 892
  * Pont et al. (1997) Pont, F., Queloz, D., Bratschi, P., & Mayor, M. 1997, A&A, 318, 416
  * Poretti (1994) Poretti, E. 1994, A&A, 285, 524
  * Reyes & Anderson (2023) Reyes, M. C. & Anderson, R. I. 2023, A&A, 672, A85
  * Riess et al. (2021) Riess, A. G., Casertano, S., Yuan, W., et al. 2021, ApJ, 908, L6
  * Ripepi et al. (2021) Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2021, MNRAS, 508, 4047
  * Ripepi et al. (2022) Ripepi, V., Clementini, G., Molinaro, R., et al. 2022, VizieR Online Data Catalog, J/A+A/674/A17
  * Ripepi et al. (2023) Ripepi, V., Clementini, G., Molinaro, R., et al. 2023, A&A, 674, A17
  * Ripepi et al. (2019) Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14
  * Sandage et al. (2004) Sandage, A., Tammann, G. A., & Reindl, B. 2004, A&A, 424, 43
  * Sandage et al. (2009) Sandage, A., Tammann, G. A., & Reindl, B. 2009, A&A, 493, 471
  * Schmidt (1974) Schmidt, E. G. 1974, MNRAS, 167, 613
  * Shapley & McKibben (1940) Shapley, H. & McKibben, V. 1940, Proceedings of the National Academy of Science, 26, 105
  * Simon & Lee (1981) Simon, N. R. & Lee, A. S. 1981, ApJ, 248, 291
  * Simon & Moffett (1985) Simon, N. R. & Moffett, T. J. 1985, PASP, 97, 1078
  * Simon & Schmidt (1976) Simon, N. R. & Schmidt, E. G. 1976, ApJ, 205, 162
  * Simon & Teays (1983) Simon, N. R. & Teays, T. J. 1983, ApJ, 265, 996
  * Slovak & Barnes (1987) Slovak, M. H. & Barnes, T. G. 1987, in Bulletin of the American Astronomical Society, Vol. 19, 753
  * Soszyński et al. (2008) Soszyński, I., Poleski, R., Udalski, A., et al. 2008, Acta Astron., 58, 163
  * Soszyński et al. (2010) Soszyński, I., Poleski, R., Udalski, A., et al. 2010, Acta Astron., 60, 17
  * Stibbs (1955) Stibbs, D. W. N. 1955, MNRAS, 115, 363
  * Stobie (1969) Stobie, R. S. 1969, MNRAS, 144, 485
  * Stobie & Balona (1979) Stobie, R. S. & Balona, L. A. 1979, MNRAS, 189, 641
  * Storm et al. (2004) Storm, J., Carney, B. W., Gieren, W. P., et al. 2004, A&A, 415, 531
  * Storm et al. (2011) Storm, J., Gieren, W., Fouqué, P., et al. 2011, A&A, 534, A94
  * Sugars & Evans (1996) Sugars, B. J. A. & Evans, N. R. 1996, AJ, 112, 1670
  * Szabados (1990) Szabados, L. 1990, MNRAS, 242, 285
  * Szabados (2003) Szabados, L. 2003, Information Bulletin on Variable Stars, 5394, 1
  * Szabados et al. (2013a) Szabados, L., Anderson, R. I., Derekas, A., et al. 2013a, MNRAS, 434, 870
  * Szabados et al. (2014) Szabados, L., Cseh, B., Kovács, J., et al. 2014, MNRAS, 442, 3155
  * Szabados et al. (2013b) Szabados, L., Derekas, A., Kiss, L. L., et al. 2013b, MNRAS, 430, 2018
  * Szabados & Klagyivik (2012) Szabados, L. & Klagyivik, P. 2012, A&A, 537, A81
  * Szabados & Pont (1998) Szabados, L. & Pont, F. 1998, A&AS, 133, 51
  * Szabó et al. (2011) Szabó, R., Szabados, L., Ngeow, C. C., et al. 2011, MNRAS, 413, 2709
  * Torres (2023) Torres, G. 2023, MNRAS, 526, 2510
  * Turner et al. (2013) Turner, D. G., Kovtyukh, V. V., Usenko, I. A., & Gorlova, N. I. 2013, ApJ, 762, L8
  * Turner et al. (2009) Turner, D. G., Majaess, D. J., Lane, D. J., et al. 2009, in American Institute of Physics Conference Series, Vol. 1170, Stellar Pulsation: Challenges for Theory and Observation, ed. J. A. Guzik & P. A. Bradley, 108–110
  * Udalski et al. (2018) Udalski, A., Soszyński, I., Pietrukowicz, P., et al. 2018, Acta Astron., 68, 315
  * Usenko et al. (2014) Usenko, I. A., Kniazev, A. Y., Berdnikov, L. N., Fokin, A. B., & Kravtsov, V. V. 2014, Astronomy Letters, 40, 435
  * Usenko et al. (2018) Usenko, I. A., Kovtyukh, V. V., Miroshnichenko, A. S., Danford, S., & Prendergast, P. 2018, MNRAS, 481, L115
  * Usenko et al. (2020) Usenko, I. A., Miroshnichenko, A. S., Danford, S., Kovtyukh, V. V., & Turner, D. G. 2020, Odessa Astronomical Publications, 33, 65
  * van Genderen (1970) van Genderen, A. M. 1970, A&A, 7, 244
  * van Leeuwen (2013) van Leeuwen, F. 2013, A&A, 550, L3
  * Vinkó et al. (1998) Vinkó, J., Evans, N. R., Kiss, L. L., & Szabados, L. 1998, MNRAS, 296, 824
  * Wallerstein et al. (2015) Wallerstein, G., Albright, M. B., & Ritchey, A. M. 2015, PASP, 127, 503
  * Wallerstein et al. (2019) Wallerstein, G., Anderson, R. I., Farrell, E. M., et al. 2019, PASP, 131, 094203
  * Wallerstein et al. (1992) Wallerstein, G., Jacobsen, T. S., Cottrell, P. L., Clark, M., & Albrow, M. 1992, MNRAS, 259, 474
  * Welch & Evans (1989) Welch, D. L. & Evans, N. R. 1989, AJ, 97, 1153
  * Welch et al. (1987) Welch, D. L., Evans, N. R., Lyons, R. W., et al. 1987, PASP, 99, 610
  * Wesselink (1946) Wesselink, A. J. 1946, Bull. Astron. Inst. Netherlands, 10, 91
  * Whitney (1956) Whitney, C. 1956, AJ, 61, 192

##  Appendix A Table of references

Table 3: Sources of RV data used for Fourier decomposition in Sect. 2.  Star | Period (day) | Reference | Star | Period (day) | Reference
---|---|---|---|---|---
$\eta$ Aql | $7.176\,483$ | 59 | DL Cas | $8.000\,616$ | 12,14,32,41,55,57
U Aql | $7.024\,149\,4$ | 11,14,22 | FM Cas | $5.809\,314$ | 12,34,35
SZ Aql | $17.139\,56$ | 9,11 | VW Cas | $5.993\,868$ | 36
TT Aql | $13.754\,690$ | 9,11,14,36,46,63,73 | V379 Cas | $4.306\,696$ | 34,35,36,47
FF Aql | $4.471\,041\,3$ | 14 | V636 Cas | $8.375\,898$ | 5,12,14,16,33,34,35
FN Aql | $9.481\,502$ | 9,33,34,35,78 | V Cen | $5.494\,326$ | 11,28,49
FM Aql | $6.114\,113$ | 9,33,34,35,36 | XX Cen | $10.952\,98$ | 11,49
KL Aql | $6.107\,969$ | 34,35,36,81 | AZ Cen | $3.211\,75$ | 47
V336 Aql | $7.304\,228$ | 36 | KN Cen | $34.021\,26$ | 4
V496 Aql | $6.807\,073$ | 14,35,36,39,49,73 | V339 Cen | $9.466\,516$ | 19,70
V600 Aql | $7.238\,803$ | 33,34,35,57,65 |  $\delta$ Cep | $5.366\,311\,4$ | 3
V733 Aql | $6.178\,632$ | 33,34,35,40,81 | CP Cep | $17.863\,06$ | 33,34,35,57,65
V916 Aql | $13.443\,415$ | 33,34,35,36 | CR Cep | $6.233\,28$ | 12,57
V1162 Aql | $5.376\,202$ | 33,34,35,36,39 | IR Cep | $2.114\,242\,7$ | 33,34,35,36,53
V1344 Aql | $7.476\,736$ | 49,78 | V351 Cep | $2.805\,441$ | 33,34,35,36
V1496 Aql | $65.924$ | 36 | AV Cir | $3.065\,248\,3$ | 14,47
V340 Ara | $20.809\,97$ | 61,65 | AX Cir | $5.273\,513$ | 49,62
RT Aur | $3.728\,308\,5$ | 23,33,34,35,36,48,64 | BP Cir | $2.398\,103\,1$ | 14,27,62
RX Aur | $11.624\,021$ | 46,64,69 | R Cru | $5.825\,907$ | 11,49,80
SY Aur | $10.145\,52$ | 4 | S Cru | $4.689\,719$ | 11,14
YZ Aur | $18.194\,42$ | 33,36,66,75 | T Cru | $6.733\,36$ | 11
AN Aur | $10.289\,049$ | 33,36 | AG Cru | $3.837\,327$ | 28,51,70
BK Aur | $8.002\,618$ | 33,36 | BG Cru | $3.342\,530$ | 14
RW Cam | $16.413\,83$ | 14,36 | X Cyg | $16.385\,983$ | 9,12,14,15,17,33,34,35,36,48,72
RX Cam | $7.912\,200$ | 45 | SU Cyg | $3.845\,547\,6$ | 14,35,36,44,52
CK Cam | $3.294\,826$ | 35,36,48 | SZ Cyg | $15.110\,73$ | 4
RY CMa | $4.678\,352$ | 8,33,34,35,36 | TX Cyg | $14.711\,62$ | 36,64
SS CMa | $12.352\,72$ | 4 | VX Cyg | $20.135\,65$ | 33,34,35,36
VZ CMa | $3.126\,33$ | 47 | VY Cyg | $7.857\,234$ | 36
$\ell$ Car | $35.561\,03$ | 2 | VZ Cyg | $4.864\,330\,8$ | 9,12,14,73
U Car | $38.838\,67$ | 11,14,82 | BZ Cyg | $10.142\,06$ | 33,34,35,36
V Car | $6.696\,69$ | 50,51,70 | CD Cyg | $17.074\,744$ | 4,14,46
UX Car | $3.682\,284$ | 50,51,70,71 | DT Cyg | $2.499\,318\,8$ | 35,36,48
VY Car | $18.882\,747$ | 4 | GH Cyg | $7.817\,964$ | 36
WZ Car | $23.015\,52$ | 11,19 | MW Cyg | $5.954\,885$ | 33,34,35,36,45
XX Car | $15.711\,13$ | 20,26 | V383 Cyg | $4.612\,19$ | 34,35,36,81
XY Car | $12.436\,283$ | 4,11 | V386 Cyg | $5.257\,624$ | 33,34,35,36
XZ Car | $16.652\,244$ | 4 | V402 Cyg | $4.364\,926\,0$ | 33,34,35,36,46,57
YZ Car | $18.167\,617$ | 4,11,14,27,49,62 | V438 Cyg | $11.210\,99$ | 36,57
AQ Car | $9.769\,513$ | 4 | V459 Cyg | $7.251\,457$ | 34,35,36,57,65
FZ Car | $3.578\,39$ | 47 | V495 Cyg | $6.718\,163$ | 34,35,36,57
GH Car | $5.725\,23$ | 76 | V520 Cyg | $4.049\,071$ | 33,34,35,36
GI Car | $4.430\,75$ | 47 | V532 Cyg | $3.283\,841$ | 33,34,35,36,57
HW Car | $9.199\,134$ | 4 | V538 Cyg | $6.119\,064$ | 34,35,36
IT Car | $7.5396$ | 47 | V924 Cyg | $5.571\,274$ | 34,35,36
RS Cas | $6.295\,991$ | 33,34,35,36 | V1154 Cyg | $4.925\,468\,2$ | 21,33,34,35,36,46,79
RW Cas | $14.789\,173$ | 33,34,35,36 | V1334 Cyg | $3.332\,401$ | 14
RY Cas | $12.139\,66$ | 34,35,36 | V1726 Cyg | $4.237\,45$ | 34,35,36
SU Cas | $1.949\,336\,59$ | 12,14,33,34,35,36,48,72 | V2340 Cyg | $7.966\,28$ | 36
SW Cas | $5.440\,971$ | 35,36 |  $\beta$ Dor | $9.842\,661\,3$ | 14,63
SY Cas | $4.071\,161\,4$ | 12,33,34,35,36,46 |  $\zeta$ Gem | $10.149\,357$ | 14,22,73
SZ Cas | $13.6401$ | 18 | W Gem | $7.913\,356$ | 33,34,35,36,46,64
XY Cas | $4.501\,70$ | 42 | AA Gem | $11.303\,215$ | 46,75
BD Cas | $3.650\,692$ | 33,34,35,36,47,81 | AD Gem | $3.787\,78$ | 43
BY Cas | $3.221\,953$ | 33,34,35,36 | DX Gem | $3.138\,357$ | 11,12
CD Cas | $7.800\,801$ | 36,65 | BB Her | $7.507\,939$ | 33,34,35,46,81
CE Cas A | $5.141\,061$ | 8,33,35 | V Lac | $4.982\,979$ | 33,34,35,36
CF Cas | $4.875\,114$ | 12,33,35,54,57,65 | X Lac | $5.444\,560$ | 9,12
DD Cas | $9.812\,037\,5$ | 4,12 | Y Lac | $4.323\,775\,1$ | 9,14,46
Z Lac | $10.885\,804$ | 9,33,34,35,36,45,74 | V335 Pup | $4.861\,300$ | 1
RR Lac | $6.416\,294$ | 12,46 | U Sgr | $6.745\,309\,6$ | 14,33,34,35,56,72
BG Lac | $5.331\,941$ | 9,46 | W Sgr | $7.595\,015\,4$ | 14,22
V411 Lac | $2.908\,265$ | 36 | X Sgr | $7.012\,724$ | 14,60,73
GH Lup | $9.2802$ | 19 | Y Sgr | $5.773\,380\,2$ | 11,14,73
T Mon | $27.035\,31$ | 22 | WZ Sgr | $21.850\,40$ | 4
SV Mon | $15.234\,20$ | 46 | XX Sgr | $6.424\,252$ | 36,39,73
TW Mon | $7.097\,067$ | 66,67 | YZ Sgr | $9.554\,00$ | 36,73
TZ Mon | $7.428\,134$ | 67 | AP Sgr | $5.058\,064$ | 36
UY Mon | $2.398\,252\,8$ | 46,47 | AV Sgr | $15.411\,10$ | 58,61,65
XX Mon | $5.456\,542$ | 67 | BB Sgr | $6.637\,116$ | 34,35,73
BE Mon | $2.7037$ | 12 | V350 Sgr | $5.154\,239\,3$ | 14,22,39,62
CU Mon | $4.707\,564$ | 66,67 | S Sge | $8.382\,118\,1$ | 9,14,22,25,48
CV Mon | $5.378\,662$ | 56,58,72 | GY Sge | $51.767$ | 36
FG Mon | $4.496\,620\,3$ | 67 | RV Sco | $6.061\,09$ | 6,28,51
FI Mon | $3.287\,823\,7$ | 67 | RY Sco | $20.325\,08$ | 4
FT Mon | $3.421\,772$ | 66,67 | V482 Sco | $4.527\,826$ | 10,28,51
V508 Mon | $4.1349$ | 12 | V500 Sco | $9.316\,60$ | 36,49
V510 Mon | $7.457\,495$ | 66,67 | V636 Sco | $6.796\,958\,8$ | 13,14,62
R Mus | $7.510\,302$ | 14 | V950 Sco | $3.380\,119$ | 47
S Mus | $9.659\,889$ | 14,27,62,82 | X Sct | $4.198\,056$ | 36,39,57,58,68
RT Mus | $3.086\,126$ | 70,71 | Y Sct | $10.341\,39$ | 11,36
S Nor | $9.754\,435\,0$ | 14,27,56,82 | Z Sct | $12.902\,07$ | 4
U Nor | $12.644\,38$ | 11,65 | RU Sct | $19.705\,10$ | 14,34,35,36,56
SY Nor | $12.645\,439$ | 11,12,49,58 | SS Sct | $3.671\,338$ | 36,38,39
QZ Nor | $3.786\,834$ | 1 | TY Sct | $11.054\,18$ | 36,57,58
V340 Nor | $11.288\,642$ | 39,56,58 | UZ Sct | $14.747\,749$ | 57,61,65
Y Oph | $17.125\,75$ | 14,22,63 | CK Sct | $7.416\,227$ | 36,57,58
BF Oph | $4.067\,537$ | 36 | CM Sct | $3.916\,994$ | 49,57,58,65
RS Ori | $7.566\,967$ | 46 | EV Sct | $3.091\,102$ | 56,57,72
GQ Ori | $8.616\,400$ | 46 | CR Ser | $5.301\,266$ | 33,34,35,36
SV Per | $11.129\,454$ | 30,35,36,64,66 | ST Tau | $4.034\,238\,6$ | 12,35,36,46
UX Per | $4.565\,42$ | 75 | SZ Tau | $3.148\,814\,8$ | 12,29,33,34,35,36,48
UY Per | $5.365\,00$ | 18,57 | EU Tau | $2.102\,488\,1$ | 12,31,33,34,35,36
VX Per | $10.882\,532$ | 4 | R TrA | $3.389\,248$ | 14,28
VY Per | $5.531\,85$ | 18 | S TrA | $6.323\,21$ | 6,28,50,51
AW Per | $6.463\,634\,4$ | 37 | LR TrA | $2.428\,405$ | 77
BM Per | $22.961\,94$ | 33,36,64,66 |  $\alpha$ UMi | $3.974\,63$ | 22
V440 Per | $7.572\,14$ | 7 | RY Vel | $28.1201$ | 11,19,65
X Pup | $25.959\,09$ | 4 | RZ Vel | $20.402\,91$ | 77
RS Pup | $41.5849$ | 1 | SW Vel | $23.427\,444$ | 11,63,65
VZ Pup | $23.175\,18$ | 14,73 | BG Vel | $6.923\,89$ | 77
AP Pup | $5.084\,534$ | 47 | AH Vel | $4.227\,412$ | 14
AQ Pup | $30.190\,16$ | 4 | CS Vel | $5.904\,798$ | 56,58
BC Pup | $3.544\,217$ | 66,67 | DR Vel | $11.1954$ | 19,65
BN Pup | $13.672\,484$ | 65,73 | S Vul | $69.43$ | 4,14
HW Pup | $13.457\,11$ | 67 | T Vul | $4.435\,408\,4$ | 9,12,14,24,48
LS Pup | $14.1432$ | 39,73 | U Vul | $7.990\,700\,9$ | 9,12,14,33,34,35,36,45
MY Pup | $5.694\,712$ | 14,47 | X Vul | $6.319\,652$ | 12,33,34,35
|  |  | SV Vul | $44.995$ | 12,14,72

  * •

Notes : We list the pulsation period that we have derived for each star and
which is used in the Fourier fit, see Sect. 2 for details.

  * •

References : 1: Anderson (2014), 2: Anderson (2016), 3: Anderson et al.
(2015), 4: Anderson et al. (2016a), 5: Arellano Ferro (1984), 6: Balona, L. A.
(private communication), 7: Baranowski et al. (2009), 8: Barnes et al. (1988),
9: Barnes et al. (2005), 10: Berdnikov et al. (2010), 11: Bersier (2002), 12:
Bersier et al. (1994), 13: Böhm-Vitense et al. (1998), 14: Borgniet et al.
(2019), 15: Breitfellner & Gillet (1993a), 16: Burki & Benz (1982), 17: Butler
(1993), 18: Coker et al. (1989), 19: Coulson & Caldwell (1985), 20: Coulson et
al. (1985), 21: Derekas et al. (2017), 22: Eaton (2020), 23: Evans (1976), 24:
Evans & Lyons (1986), 25: Evans et al. (1993), 26: Feast (1967), 27: Gallenne
et al. (2019), 28: Gieren (1981), 29: Gieren (1985), 30: Gieren & Brieva
(1992), 31: Gieren et al. (1989), 32: Gieren et al. (1994), 33: Gorynya et al.
(1992), 34: Gorynya et al. (1996b), 35: Gorynya et al. (1998), 36: Gorynya, N.
A. et al. (in preparation), 37: Griffin (2016), 38: Groenewegen (2008), 39:
Groenewegen (2013), 40: Harris & Wallerstein (1984), 41: Harris et al. (1987),
42: Imbert (1981), 43: Imbert (1983), 44: Imbert (1984), 45: Imbert (1996),
46: Imbert (1999), 47: Kienzle et al. (1999), 48: Kiss & Vinkó (2000), 49:
Krzesiński, J. (private communication), 50: Lloyd Evans (1968), 51: Lloyd
Evans (1980), 52: Madore, B. F., from Evans (1988), 53: Marschall et al.
(1993), 54: Matthews et al. (1995), 55: Mermilliod et al. (1987), 56:
Mermilliod et al. (2008), 57: Metzger et al. (1991), 58: Metzger et al.
(1992), 59: Nardetto et al. (2023), 60: O’Toole, S. (private communication),
61: Pedicelli et al. (2010), 62: Petterson et al. (2004), 63: Petterson et al.
(2005), 64: Pont, F. (private communication), 65: Pont et al. (1994), 66: Pont
et al. (1997), 67: Pont et al. (2001), 68: Ripepi et al. (2021), 69: Schmidt
(1974), 70: Stibbs (1955), 71: Stobie & Balona (1979), 72: Storm et al.
(2004), 73: Storm et al. (2011), 74: Sugars & Evans (1996), 75: Szabados &
Pont (1998), 76: Szabados et al. (2013b),77: Szabados et al. (2013a), 78:
Szabados et al. (2014), 79: Szabó et al. (2011), 80: Usenko et al. (2014), 81:
Vinkó et al. (1998), 82: Wallerstein et al. (1992).

##  Appendix B Binary Cepheids

Table 4: Orbital periods of binary Cepheids adopted for the Fourier fit.

Star |  $P_{\mathrm{orb}}$ (day) | Source
---|---|---
U Aql | 1862.8 $\pm$1.1 | This paper
FF Aql | 1428.5 $\pm$1.6 | This paper
V496 Aql | 1351.31 $\pm$0.60 | This paper
V916 Aql | 2434 $\pm$26 | Gorynya et al. (in prep.)
V1344 Aql | 6294 $\pm$30 | Gorynya et al. (in prep.)
RX Cam | 1113.8 $\pm$0.5 | Groenewegen (2013)
YZ Car | 829.68 $\pm$0.18 | This paper
BY Cas | 4584 $\pm$33 | Gorynya et al. (in prep.)
DL Cas | 684.27 $\pm$0.16 | Groenewegen (2013)
XX Cen | 711.7 $\pm$3.5 | This paper
AX Cir | 6532 $\pm$25 | Petterson et al. (2004)
SU Cyg | 549.085$\pm$0.026 | This paper
VZ Cyg | 2183 $\pm$10 | Groenewegen (2013)
MW Cyg | 439.61 $\pm$0.18 | Groenewegen (2013)
V1334 Cyg | 1932.8 $\pm$1.8 | Gallenne et al. (2018)
DX Gem | 1043.1 $\pm$4.8 : | This paper
Z Lac | 382.63 $\pm$0.10 | Groenewegen (2008)
S Mus | 505.335$\pm$0.037 | This paper
SY Nor | 551.86 $\pm$0.27 | This paper
AW Per | 13590 $\pm$120 | This paper
S Sge | 675.389$\pm$0.096 | This paper
W Sgr | 1630.4 $\pm$3.3 | This paper
V350 Sgr | 1466.99 $\pm$0.78 | This paper
V636 Sco | 1320.6 $\pm$1.3 | Gallenne et al. (2019)
U Vul | 2510 $\pm$2.8 | Groenewegen (2008)

  * •

Notes : The orbital period $P_{\mathrm{orb}}$ is a period of additional
Fourier sum representing orbital motion, see Sect. 2.5 for details.

###  B.1 Comments on binary Cepheids

V1344 Aql: the recent study of Cseh et al. (2023) derived for this star
orbital period of about $P_{orb}\simeq 12000$ day, which is about twice longer
than the period derived by Gorynya et al. (in prep). We scrutinized both
datasets to understand the origin of this discrepancy. The dataset of Gorynya
et al. spans more than 23 years, with nearly 200 measurements uniformly
distributed in time. This dataset yields unambiguously the orbital period
close to 6300 days and is not compatible with the period of $\sim 12000$ days.
Thus, the orbital period found by Cseh et al. (2023) can be excluded.

XX Cen: Szabados (1990) determined orbital period of this binary Cepheid to be
$P_{\mathrm{orb}}=909.4\,$day. Later, Groenewegen (2008) has refined this
value to $P_{\mathrm{orb}}=924.1\,$day. Using only the most recent high
quality RV data and assuming circular orbit (as done by Groenewegen 2008), we
find somewhat shorter period of $P_{\mathrm{orb}}=711.7\,$day. The resulting
fit has a dispersion of only 0.34 km/s, which is more than 4 times lower than
for the fit in which $P_{\mathrm{orb}}$ from Groenewegen (2008) is assumed.

DX Gem: this overtone Cepheid is a spectroscopic binary (Bersier 2002). The
total range of orbital velocity variations is $\sim 20\,$km/s, but the
currently available data are not sufficient to uniquely determine the orbital
period. Among several possibilities, we have chosen
$P_{\mathrm{orb}}=1043.1\,$day, which yields the lowest dispersion of the fit.
The Fourier parameters of the pulsation RV curve are almost insensitive to the
choice of $P_{\mathrm{orb}}$.

SY Nor: this is another spectroscopic binary (Bersier 2002). The total range
of orbital velocity variations is $\sim 30\,$km/s. To the best of our
knowledge, the full orbital solution was never published, but the unique
determination of orbital period is already possible. Our value of
$P_{\mathrm{orb}}=551.86\,$day is almost the same as 551.7 days, the value
found by Bersier (2002).

BP Cir: the presence of hot companion of BP Cir was detected for the first
time by International Ultraviolet Explorer satellite (Arellano Ferro & Madore
1985; Evans et al. 1992) and later revealed by long-baseline interferometric
observations (Gallenne et al. 2019). From data of Petterson et al. (2004)
alone, we have found a slow $\gamma$-velocity drift, amounting to $\approx$7.0
km/s in 2300 days. Variability of $\gamma$-velocity of BP Cir has already been
noticed by Petterson et al., who wrote that ”observations indicate continuing
decrease in $\gamma$-velocity”. The observed drift indicates a binary orbital
motion, with the period much longer than the duration of the observations.

LR TrA: a slow $\gamma$-velocity drift has been noticed in this Cepheid
already by Szabados et al. (2013a). We have confirmed this finding, we have
found a $\gamma$-velocity change of 1.5 km/s in 450 days. Considering an
excellent quality of analyzed data, a drift of this size is highly
significant. The observed change of $\gamma$-velocity indicates a binary
orbital motion, with the period much longer than the duration of the
observations.

Table 5: New orbital periods of binary Cepheid adopted in this paper compared with the previously published values. Star |  $P_{\mathrm{orb}}$ (day) | Reference | Star |  $P_{\mathrm{orb}}$ (day) | Reference
---|---|---|---|---|---
U Aql | 1862.8 $\pm$1.1 | This paper | S Mus | 505.335$\pm$0.037 | This paper
| 1831.4 $\pm$6.5 | Gallenne et al. (2019) |  | 506.3 $\pm$0.5 | Gallenne et al. (2019)
| 1853.6 $\pm$3.0 | Groenewegen (2013) |  | 505.15 $\pm$0.08 | Groenewegen (2008)
| 1856.4 $\pm$4.3 | Welch et al. (1987) |  | 504.9 $\pm$0.1 | Petterson et al. (2004)
|  |  |  | 504.58 | Petterson et al. (1998)
FF Aql | 1428.5 $\pm$1.6 | This paper |  | 505.44 $\pm$0.35 | Evans et al. (1990)
| 1430.3 $\pm$2.6 | Gallenne et al. (2019) |  | 506.4 $\pm$2 | Lloyd Evans (1982)
| 1432.7 $\pm$0.9 | Groenewegen (2013) |  |  |
| 1432.4 $\pm$1.1 | Groenewegen (2008) | AW Per | 13590 $\pm$120 | This paper
| 1434 $\pm$1 | Benedict et al. (2007) |  | 13954 $\pm$181 | Griffin (2016)
| 1434.2 | Gorynya et al. (2000) |  | 14293.70 $\pm$283 | Groenewegen (2013)
| 1433 | Gorynya et al. (1996a) |  | 14594 $\pm$324 | Evans et al. (2000)
| 1429.7 $\pm$1.1 | Evans et al. (1990) |  | 13100 $\pm$1000 | Welch & Evans (1989)
| 1435.0. | Abt (1959) |  |  |
|  |  | S Sge | 675.389$\pm$0.096 | This paper
V496 Aql | 1351.31 $\pm$0.60 | This paper |  | 675.72 $\pm$0.04 | Groenewegen (2008)
| 1351.5 $\pm$5.8 | Borgniet et al. (2019) |  | 675.75 | Gorynya et al. (2000)
| 1066.2 $\pm$1.9 | Groenewegen (2013) |  | 675.77 | Gorynya et al. (1996a)
| 1331 $\pm$6.5 | Groenewegen (2008) |  | 675.99 $\pm$0.14 | Breitfellner & Gillet (1993b)
| 1447 | Gorynya et al. (2000) |  | 675.79 $\pm$0.18 | Evans et al. (1993)
|  |  |  | 676.2 | Slovak & Barnes (1987)
YZ Car | 829.68 $\pm$0.18 | This paper |  | 682 | Herbig & Moore (1952)
| 831.6 $\pm$0.9 | Gallenne et al. (2019) |  |  |
| 830.22 $\pm$0.34 | Anderson et al. (2016a) | W Sgr | 1630.4 $\pm$3.3 | This paper
| 657.3 $\pm$0.3 | Petterson et al. (2004) |  | 1615.5 $\pm$11.0 | Gallenne et al. (2019)
| 850 $\pm$11 | Coulson (1983) |  | 1651 $\pm$8.8 | Groenewegen (2008)
|  |  |  | 1582 $\pm$3 | Benedict et al. (2007)
XX Cen | 711.7 $\pm$3.5 | This paper |  | 1582 $\pm$9 | Petterson et al. (2004)
| 924.1 $\pm$1.1 | Groenewegen (2008) |  | 1603 | Albrow & Cottrell (1996)
| 909.4 $\pm$29.0 | Szabados (1990) |  | 1740 $\pm$43 | Bersier et al. (1994)
|  800 | Coulson et al. (1986) |  | 1780 $\pm$5 | Babel et al. (1989)
SU Cyg | 549.085$\pm$0.026 | This paper | V350 Sgr | 1466.99 $\pm$0.78 | This paper
| 538.5 $\pm$4.2 | Borgniet et al. (2019) |  | 1465.3 $\pm$0.4 | Gallenne et al. (2019)
| 549.24 $\pm$0.02 | Groenewegen (2008) |  | 1468.9 $\pm$0.9 | Groenewegen (2013)
| 549.25 | Gorynya et al. (2000) |  | 1472.91 $\pm$1.33 | Evans et al. (2011)
| 549.16 $\pm$0.08 | Evans (1988) |  | 1482 $\pm$2.4 | Groenewegen (2008)
| 549.20 $\pm$0.03 | Imbert (1984) |  | 1464.9 $\pm$0.4 | Petterson et al. (2004)
|  |  |  | 1481.8 | Gorynya et al. (2000)
SY Nor | 551.88 $\pm$0.27 | This paper |  | 1477.0 $\pm$3.6 | Evans & Sugars (1997)
| 551.7 | Bersier (2002) |  | 1467 | Gorynya et al. (1996a)
|  |  |  | 1129 $\pm$24 | Szabados (1990)

##  Appendix C Fourier parameters

###  C.1 Comments on individual Cepheids of Tables C.1-C.3

KL Aql, V733 Aql: in older literature these two variables are considered to be
Pop. II Cepheids (see, e.g., Balog et al. 1997; Vinkó et al. 1998). Recently,
they have been both reclassified as Classical Cepheids, i.e. Cepheids of Pop.
I (Groenewegen & Jurkovic 2017). Our analysis of the RV curves confirms this
new classification. All Fourier parameters of KL Aql and V733 Aql firmly place
these two variables among fundamental-mode Classical Cepheids.

TT Aql: individual zero-point corrections are applied separately to each
season of Gorynya in prep. measurements. These offsets change from year to
year in a systematic way, suggesting a binary motion with a very small peak-
to-peak amplitude of $\Delta V_{r}\sim 1.0\,$km/s.

SV Vul, GY Sge, V1496 Aql, S Vul: these four long-period Cepheids all display
very fast, irregular $O-C$ variations. To phase their velocity curves
properly, we have taken into account the $O-C$ shifts, calculated for each
season separately. This procedure is described in detail in Sect. 2.7.

V Cen: New data of Bersier (2002) and Krzesiński (private comm.) do not
provide a good coverage of the RV maximum. Therefore, we have included into
the fit also older data of Gieren (1981). Because of a very long timebase
(6614 d), phasing of all these data with a common period does not work too
well and yields RV curve with excessive scatter. In order to improve the
phasing, we have applied individual $O-C$ shifts to each season separately.
This reduces the scatter of the RV curve by $\sim$10% and improves overall
stability of the fit.

$\ell$ Car: because of amplitude instability, only data collected after
HJD=2457000 are used in the analysis. The lack of amplitude stability might be
linked to cycle-to-cycle spectral line and atmospheric velocity gradient
variability observed in this star (Anderson et al. 2016b; Anderson 2016).

YZ Car: because of lower quality, older data of Petterson et al. (2004)
collected before HJD=2450500 are not used in the analysis.

SV Per: because of lower quality, older data of Gieren & Brieva (1992)
collected before HJD=2445000 are not used in the analysis.

RS Pup: because of period instability, only data collected after HJD=2456200
are used in the analysis.

DR Vel: first harmonic of the radial velocity curve has not been detected in
this Cepheid, perhaps because of low quality of available data. The second
harmonic has been marginally detected. The measured value of $\phi_{31}$ is
consistent with fundamental-mode pulsations. Therefore, we classify DR Vel as
a fundamental-mode Cepheid.

Table 6: Low-order Fourier parameters of fundamental-mode Cepheids, stars of Quality 1.  Star | Period | Ndat | $n$ | $\sigma_{\mathrm{fit}}$ | $A_{1}$ | $\phi_{21}$ | $R_{21}$ | $\phi_{31}$ | $R_{31}$ | $\phi_{41}$ | $R_{41}$
---|---|---|---|---|---|---|---|---|---|---|---
SS Sct | $3.67$ | $67$ | 3 | 0.78 | $14.18$ | 2.917 | 0.268 | 5.933 | 0.095 |  |
|  |  |  |  | $0.14$ | 0.041 | 0.010 | 0.108 | 0.010 |  |
RT Aur | $3.73$ | $123$ | 6 | 0.62 | $15.53$ | 3.065 | 0.356 | 6.141 | 0.166 | 2.918 | 0.080
|  |  |  |  | $0.08$ | 0.018 | 0.006 | 0.037 | 0.005 | 0.073 | 0.005
SU Cyg | $3.85$ | $331$ | 5+5 | 1.13 | $16.49$ | 3.010 | 0.329 | 6.060 | 0.145 | 2.925 | 0.069
|  |  |  |  | $0.09$ | 0.020 | 0.006 | 0.041 | 0.005 | 0.083 | 0.005
SY Cas | $4.07$ | $119$ | 7 | 0.95 | $15.57$ | 3.140 | 0.347 | 6.119 | 0.163 | 2.999 | 0.096
|  |  |  |  | $0.14$ | 0.029 | 0.009 | 0.061 | 0.008 | 0.092 | 0.009
T Vul | $4.44$ | $197$ | 7 | 0.52 | $14.50$ | 2.986 | 0.354 | 6.032 | 0.154 | 2.877 | 0.082
|  |  |  |  | $0.06$ | 0.013 | 0.005 | 0.029 | 0.004 | 0.051 | 0.004
XY Cas | $4.50$ | $47$ | 5+t | 0.70 | $14.78$ | 2.969 | 0.322 | 6.154 | 0.164 | 2.998 | 0.080
|  |  |  |  | $0.16$ | 0.038 | 0.013 | 0.075 | 0.011 | 0.142 | 0.012
S Cru | $4.69$ | $53$ | 7 | 0.39 | $15.19$ | 3.066 | 0.365 | 6.244 | 0.191 | 3.307 | 0.090
|  |  |  |  | $0.09$ | 0.027 | 0.007 | 0.033 | 0.006 | 0.065 | 0.006
VZ Cyg | $4.86$ | $89$ | 6+2 | 0.46 | $15.00$ | 3.075 | 0.365 | 6.226 | 0.169 | 3.242 | 0.083
|  |  |  |  | $0.07$ | 0.016 | 0.005 | 0.030 | 0.005 | 0.061 | 0.005
CF Cas | $4.88$ | $87$ | 4 | 0.94 | $13.73$ | 3.069 | 0.345 | 0.149 | 0.164 | 3.577 | 0.051
|  |  |  |  | $0.16$ | 0.038 | 0.012 | 0.083 | 0.011 | 0.208 | 0.012
V1154 Cyg | $4.93$ | $181$ | 4 | 0.93 | $11.40$ | 3.144 | 0.280 | 0.210 | 0.106 | 3.144 | 0.029
|  |  |  |  | $0.10$ | 0.036 | 0.009 | 0.090 | 0.009 | 0.310 | 0.009
AP Pup | $5.08$ | $45$ | 5 | 0.51 | $14.02$ | 3.115 | 0.369 | 0.101 | 0.183 | 3.325 | 0.090
|  |  |  |  | $0.12$ | 0.030 | 0.009 | 0.054 | 0.009 | 0.087 | 0.009
V350 Sgr | $5.15$ | $162$ | 7+4 | 0.30 | $15.09$ | 3.136 | 0.380 | 0.127 | 0.194 | 3.451 | 0.092
|  |  |  |  | $0.03$ | 0.008 | 0.002 | 0.014 | 0.002 | 0.028 | 0.002
V386 Cyg | $5.26$ | $79$ | 5 | 1.19 | $14.41$ | 3.206 | 0.388 | 0.080 | 0.195 | 3.503 | 0.101
|  |  |  |  | $0.19$ | 0.049 | 0.015 | 0.073 | 0.014 | 0.146 | 0.014
BG Lac | $5.33$ | $61$ | 7 | 0.41 | $14.24$ | 3.163 | 0.372 | 0.194 | 0.178 | 3.597 | 0.075
|  |  |  |  | $0.08$ | 0.019 | 0.006 | 0.033 | 0.006 | 0.077 | 0.006
$\delta$ Cep | $5.37$ | $135$ | 11+t | 0.06 | $15.57$ | 3.137 | 0.418 | 0.100 | 0.227 | 3.349 | 0.120
|  |  |  |  | $0.01$ | 0.002 | 0.000 | 0.002 | 0.000 | 0.004 | 0.000
V1162 Aql | $5.38$ | $109$ | 4+t | 1.04 | $12.76$ | 3.170 | 0.356 | 0.476 | 0.154 | 4.197 | 0.049
|  |  |  |  | $0.15$ | 0.041 | 0.012 | 0.078 | 0.012 | 0.243 | 0.011
FM Cas | $5.81$ | $66$ | 7 | 0.75 | $13.55$ | 3.226 | 0.411 | 0.487 | 0.188 | 3.771 | 0.080
|  |  |  |  | $0.16$ | 0.031 | 0.013 | 0.073 | 0.011 | 0.139 | 0.011
MW Cyg | $5.95$ | $202$ | 6+3 | 1.09 | $15.04$ | 3.260 | 0.445 | 0.463 | 0.224 | 3.714 | 0.088
|  |  |  |  | $0.12$ | 0.021 | 0.008 | 0.042 | 0.008 | 0.088 | 0.007
FM Aql | $6.11$ | $139$ | 6+t | 1.03 | $15.79$ | 3.205 | 0.413 | 0.598 | 0.201 | 4.010 | 0.072
|  |  |  |  | $0.13$ | 0.024 | 0.009 | 0.046 | 0.008 | 0.118 | 0.008
V538 Cyg | $6.12$ | $52$ | 5+t | 1.34 | $12.37$ | 3.192 | 0.393 | 0.585 | 0.147 | 3.955 | 0.056
|  |  |  |  | $0.28$ | 0.092 | 0.026 | 0.185 | 0.023 | 0.412 | 0.025
CR Cep | $6.23$ | $41$ | 5+t | 0.81 | $10.50$ | 3.281 | 0.360 | 0.597 | 0.121 | 3.937 | 0.045
|  |  |  |  | $0.20$ | 0.062 | 0.020 | 0.149 | 0.021 | 0.445 | 0.018
AW Per | $6.46$ | $355$ | 6+5 | 0.92 | $15.36$ | 3.295 | 0.486 | 0.555 | 0.248 | 3.801 | 0.117
|  |  |  |  | $0.08$ | 0.014 | 0.005 | 0.025 | 0.005 | 0.044 | 0.005
BB Sgr | $6.64$ | $88$ | 5 | 0.59 | $12.90$ | 3.430 | 0.446 | 0.941 | 0.167 | 4.355 | 0.067
|  |  |  |  | $0.09$ | 0.022 | 0.008 | 0.047 | 0.007 | 0.114 | 0.007
T Cru | $6.73$ | $39$ | 5 | 0.40 | $11.52$ | 3.406 | 0.394 | 0.941 | 0.099 | 3.911 | 0.036
|  |  |  |  | $0.09$ | 0.027 | 0.009 | 0.089 | 0.008 | 0.219 | 0.009
U Sgr | $6.75$ | $320$ | 7 | 0.72 | $14.88$ | 3.378 | 0.503 | 0.835 | 0.207 | 4.069 | 0.071
|  |  |  |  | $0.06$ | 0.011 | 0.004 | 0.022 | 0.004 | 0.058 | 0.004
V636 Sco | $6.80$ | $133$ | 5+3 | 0.56 | $12.49$ | 3.459 | 0.424 | 1.136 | 0.136 | 4.439 | 0.055
|  |  |  |  | $0.08$ | 0.017 | 0.007 | 0.048 | 0.007 | 0.125 | 0.006
V496 Aql | $6.81$ | $314$ | 3+5 | 0.60 | $8.89$ | 3.519 | 0.333 | 1.080 | 0.062 |  |
|  |  |  |  | $0.05$ | 0.021 | 0.006 | 0.093 | 0.005 |  |
BG Vel | $6.92$ | $33$ | 5 | 0.10 | $11.15$ | 3.485 | 0.408 | 1.115 | 0.090 | 4.136 | 0.038
|  |  |  |  | $0.03$ | 0.008 | 0.003 | 0.029 | 0.002 | 0.062 | 0.003
U Aql | $7.02$ | $106$ | 10+3 | 0.20 | $14.71$ | 3.460 | 0.528 | 0.932 | 0.208 | 3.998 | 0.086
|  |  |  |  | $0.03$ | 0.007 | 0.002 | 0.012 | 0.002 | 0.028 | 0.002
$\eta$ Aql | $7.18$ | $98$ | 10+t | 0.05 | $15.51$ | 3.496 | 0.521 | 0.975 | 0.200 | 3.951 | 0.088
|  |  |  |  | $0.01$ | 0.001 | 0.001 | 0.003 | 0.001 | 0.006 | 0.001
V600 Aql | $7.24$ | $93$ | 5 | 1.18 | $13.58$ | 3.536 | 0.491 | 1.042 | 0.142 | 3.881 | 0.069
|  |  |  |  | $0.18$ | 0.037 | 0.015 | 0.105 | 0.013 | 0.201 | 0.014
V459 Cyg | $7.25$ | $68$ | 5 | 1.20 | $14.20$ | 3.531 | 0.548 | 1.194 | 0.179 | 3.938 | 0.101
|  |  |  |  | $0.21$ | 0.050 | 0.019 | 0.108 | 0.017 | 0.184 | 0.015
TZ Mon | $7.43$ | $36$ | 5 | 0.54 | $14.92$ | 3.619 | 0.552 | 0.938 | 0.157 | 4.158 | 0.114
|  |  |  |  | $0.16$ | 0.032 | 0.013 | 0.079 | 0.011 | 0.105 | 0.012
V1344 Aql | $7.48$ | $48$ | 3 | 0.24 | $6.83$ | 3.661 | 0.300 | 0.623 | 0.039 |  |
|  |  |  |  | $0.05$ | 0.030 | 0.008 | 0.195 | 0.007 |  |
BB Her | $7.51$ | $85$ | 5 | 0.72 | $12.96$ | 3.606 | 0.500 | 1.149 | 0.130 | 3.966 | 0.083
|  |  |  |  | $0.12$ | 0.025 | 0.010 | 0.076 | 0.009 | 0.116 | 0.009
IT Car | $7.54$ | $40$ | 2 | 0.44 | $8.03$ | 3.888 | 0.345 |  |  |  |
|  |  |  |  | $0.10$ | 0.043 | 0.014 |  |  |  |
W Sgr | $7.60$ | $135$ | 9+3 | 0.15 | $14.49$ | 3.645 | 0.562 | 1.015 | 0.181 | 4.020 | 0.118
|  |  |  |  | $0.02$ | 0.004 | 0.002 | 0.008 | 0.001 | 0.012 | 0.001
GH Cyg | $7.82$ | $93$ | 4+t | 1.37 | $13.50$ | 3.682 | 0.567 | 1.062 | 0.142 | 4.118 | 0.070
|  |  |  |  | $0.20$ | 0.044 | 0.018 | 0.118 | 0.016 | 0.261 | 0.014
RX Cam | $7.91$ | $85$ | 6+5 | 0.52 | $14.42$ | 3.786 | 0.566 | 1.007 | 0.142 | 4.331 | 0.111
|  |  |  |  | $0.09$ | 0.017 | 0.007 | 0.051 | 0.006 | 0.064 | 0.006
U Vul | $7.99$ | $440$ | 6+4 | 0.74 | $12.00$ | 3.698 | 0.498 | 0.971 | 0.132 | 3.973 | 0.085
|  |  |  |  | $0.05$ | 0.012 | 0.005 | 0.035 | 0.004 | 0.052 | 0.004
DL Cas | $8.00$ | $191$ | 5+4 | 0.76 | $13.25$ | 3.605 | 0.466 | 1.042 | 0.091 | 4.010 | 0.056
|  |  |  |  | $0.08$ | 0.018 | 0.007 | 0.074 | 0.007 | 0.126 | 0.006
S Sge | $8.38$ | $115$ | 6+4 | 0.37 | $14.15$ | 3.947 | 0.542 | 1.131 | 0.131 | 4.557 | 0.112
|  |  |  |  | $0.05$ | 0.010 | 0.004 | 0.030 | 0.004 | 0.035 | 0.004
GQ Ori | $8.62$ | $29$ | 5 | 0.70 | $14.42$ | 3.846 | 0.564 | 1.036 | 0.156 | 4.376 | 0.110
|  |  |  |  | $0.19$ | 0.039 | 0.016 | 0.090 | 0.014 | 0.142 | 0.014
HW Car | $9.20$ | $66$ | 3 | 0.11 | $9.05$ | 5.057 | 0.222 | 1.742 | 0.078 |  |
|  |  |  |  | $0.02$ | 0.011 | 0.002 | 0.029 | 0.002 |  |
V500 Sco | $9.32$ | $75$ | 5 | 1.33 | $13.20$ | 4.535 | 0.480 | 1.065 | 0.159 | 5.498 | 0.070
|  |  |  |  | $0.23$ | 0.047 | 0.019 | 0.109 | 0.018 | 0.247 | 0.018
FN Aql | $9.48$ | $140$ | 4 | 0.77 | $14.49$ | 5.290 | 0.171 | 2.295 | 0.151 | 4.754 | 0.034
|  |  |  |  | $0.10$ | 0.040 | 0.007 | 0.048 | 0.007 | 0.195 | 0.006
YZ Sgr | $9.55$ | $76$ | 5 | 0.58 | $13.91$ | 4.699 | 0.424 | 1.388 | 0.176 | 0.142 | 0.061
|  |  |  |  | $0.09$ | 0.024 | 0.008 | 0.046 | 0.007 | 0.117 | 0.007
S Mus | $9.66$ | $211$ | 5+3 | 0.48 | $12.28$ | 4.233 | 0.482 | 0.895 | 0.142 | 5.198 | 0.075
|  |  |  |  | $0.05$ | 0.012 | 0.004 | 0.033 | 0.004 | 0.062 | 0.004
AQ Car | $9.77$ | $59$ | 10+t | 0.08 | $13.95$ | 5.077 | 0.304 | 1.762 | 0.168 | 6.270 | 0.004
|  |  |  |  | $0.02$ | 0.004 | 0.001 | 0.008 | 0.001 | 0.300 | 0.001
DD Cas | $9.81$ | $123$ | 7 | 0.21 | $14.49$ | 5.225 | 0.232 | 2.003 | 0.165 | 4.423 | 0.018
|  |  |  |  | $0.03$ | 0.010 | 0.002 | 0.015 | 0.002 | 0.119 | 0.002
$\beta$ Dor | $9.84$ | $139$ | 6 | 0.18 | $13.91$ | 5.013 | 0.312 | 1.759 | 0.177 | — | —
|  |  |  |  | $0.03$ | 0.010 | 0.002 | 0.013 | 0.002 | — | —
SY Aur | $10.15$ | $109$ | 5+t | 0.16 | $10.51$ | 4.319 | 0.380 | 0.708 | 0.106 | 5.670 | 0.053
|  |  |  |  | $0.02$ | 0.008 | 0.003 | 0.022 | 0.002 | 0.040 | 0.003
$\zeta$ Gem | $10.15$ | $312$ | 7 | 0.15 | $11.85$ | 5.209 | 0.243 | 1.924 | 0.127 | 3.855 | 0.006
|  |  |  |  | $0.01$ | 0.005 | 0.001 | 0.009 | 0.001 | 0.185 | 0.001
Y Sct | $10.34$ | $72$ | 4 | 1.45 | $18.09$ | 5.072 | 0.206 | 2.020 | 0.194 | 5.097 | 0.048
|  |  |  |  | $0.25$ | 0.079 | 0.014 | 0.100 | 0.014 | 0.290 | 0.015
VX Per | $10.88$ | $118$ | 9 | 0.17 | $13.49$ | 4.452 | 0.465 | 1.124 | 0.148 | 5.577 | 0.076
|  |  |  |  | $0.02$ | 0.005 | 0.002 | 0.014 | 0.002 | 0.026 | 0.002
Z Lac | $10.89$ | $276$ | 7+1 | 0.81 | $18.35$ | 5.018 | 0.197 | 2.017 | 0.191 | 5.323 | 0.039
|  |  |  |  | $0.07$ | 0.022 | 0.004 | 0.024 | 0.004 | 0.102 | 0.004
XX Cen | $10.95$ | $40$ | 7+1 | 0.34 | $16.42$ | 4.866 | 0.289 | 1.735 | 0.179 | 6.012 | 0.017
|  |  |  |  | $0.09$ | 0.021 | 0.006 | 0.042 | 0.006 | 0.334 | 0.005
TY Sct | $11.05$ | $54$ | 3 | 1.26 | $17.38$ | 5.053 | 0.166 | 2.145 | 0.140 |  |
|  |  |  |  | $0.24$ | 0.097 | 0.014 | 0.106 | 0.015 |  |
SV Per | $11.13$ | $80$ | 5 | 0.93 | $21.01$ | 4.681 | 0.089 | 2.364 | 0.161 | 5.431 | 0.092
|  |  |  |  | $0.16$ | 0.085 | 0.007 | 0.050 | 0.008 | 0.087 | 0.007
V340 Nor | $11.29$ | $50$ | 3 | 0.41 | $8.58$ | 4.900 | 0.262 | 1.295 | 0.067 |  |
|  |  |  |  | $0.09$ | 0.039 | 0.010 | 0.152 | 0.010 |  |
AA Gem | $11.30$ | $52$ | 5+t | 0.53 | $17.53$ | 5.561 | 0.047 | 2.659 | 0.142 | 5.420 | 0.074
|  |  |  |  | $0.11$ | 0.133 | 0.006 | 0.047 | 0.006 | 0.089 | 0.006
RX Aur | $11.62$ | $51$ | 5 | 0.47 | $14.30$ | 4.791 | 0.213 | 1.610 | 0.143 | 4.413 | 0.028
|  |  |  |  | $0.09$ | 0.036 | 0.007 | 0.053 | 0.007 | 0.259 | 0.007
SS CMa | $12.35$ | $80$ | 8 | 0.18 | $16.86$ | 4.798 | 0.237 | 1.630 | 0.167 | 5.441 | 0.020
|  |  |  |  | $0.03$ | 0.009 | 0.002 | 0.012 | 0.002 | 0.118 | 0.002
XY Car | $12.44$ | $74$ | 8 | 0.21 | $20.95$ | 3.736 | 0.029 | 2.751 | 0.133 | 5.675 | 0.097
|  |  |  |  | $0.03$ | 0.059 | 0.002 | 0.013 | 0.002 | 0.023 | 0.002
SY Nor | $12.65$ | $42$ | 5+1 | 0.58 | $22.73$ | 3.179 | 0.074 | 2.783 | 0.116 | 5.719 | 0.103
|  |  |  |  | $0.16$ | 0.079 | 0.006 | 0.057 | 0.006 | 0.066 | 0.006
SZ Cas | $13.64$ | $70$ | 2 | 0.76 | $8.30$ | 5.208 | 0.097 |  |  |  |
|  |  |  |  | $0.14$ | 0.162 | 0.016 |  |  |  |
TT Aql | $13.75$ | $253$ | 11 | 0.53 | $23.20$ | 2.970 | 0.164 | 3.106 | 0.066 | 6.107 | 0.097
|  |  |  |  | $0.05$ | 0.014 | 0.002 | 0.032 | 0.002 | 0.024 | 0.002
LS Pup | $14.14$ | $27$ | 5 | 0.60 | $22.14$ | 3.618 | 0.068 | 2.693 | 0.129 | 5.808 | 0.087
|  |  |  |  | $0.19$ | 0.122 | 0.008 | 0.069 | 0.008 | 0.111 | 0.008
SZ Cyg | $15.11$ | $107$ | 12 | 0.14 | $21.94$ | 2.942 | 0.245 | 4.758 | 0.061 | 0.742 | 0.085
|  |  |  |  | $0.03$ | 0.005 | 0.001 | 0.021 | 0.001 | 0.017 | 0.001
SV Mon | $15.23$ | $40$ | 8 | 0.69 | $24.87$ | 2.932 | 0.262 | 4.775 | 0.056 | 0.447 | 0.089
|  |  |  |  | $0.18$ | 0.028 | 0.008 | 0.121 | 0.007 | 0.077 | 0.007
X Cyg | $16.39$ | $467$ | 10 | 0.90 | $23.45$ | 3.026 | 0.294 | 5.263 | 0.067 | 1.049 | 0.081
|  |  |  |  | $0.06$ | 0.010 | 0.003 | 0.040 | 0.003 | 0.033 | 0.003
XZ Car | $16.65$ | $116$ | 13+t | 0.21 | $23.79$ | 3.015 | 0.293 | 5.263 | 0.071 | 1.006 | 0.082
|  |  |  |  | $0.03$ | 0.005 | 0.001 | 0.017 | 0.001 | 0.019 | 0.001
CD Cyg | $17.07$ | $149$ | 10 | 0.39 | $24.84$ | 3.004 | 0.279 | 5.227 | 0.044 | 0.667 | 0.062
|  |  |  |  | $0.05$ | 0.009 | 0.002 | 0.049 | 0.002 | 0.034 | 0.002
Y Oph | $17.13$ | $94$ | 3 | 0.23 | $8.47$ | 3.141 | 0.028 | 2.143 | 0.034 |  |
|  |  |  |  | $0.03$ | 0.147 | 0.004 | 0.122 | 0.004 |  |
YZ Car | $18.17$ | $92$ | 7+2 | 0.23 | $13.97$ | 3.176 | 0.080 | 2.243 | 0.049 | 4.919 | 0.025
|  |  |  |  | $0.04$ | 0.034 | 0.003 | 0.055 | 0.003 | 0.107 | 0.003
VY Car | $18.88$ | $80$ | 10 | 0.23 | $24.06$ | 3.007 | 0.298 | 5.345 | 0.076 | 1.071 | 0.075
|  |  |  |  | $0.04$ | 0.007 | 0.002 | 0.025 | 0.002 | 0.023 | 0.002
RU Sct | $19.71$ | $116$ | 3 | 1.03 | $21.87$ | 3.004 | 0.243 | 5.887 | 0.059 |  |
|  |  |  |  | $0.15$ | 0.030 | 0.007 | 0.112 | 0.007 |  |
RY Sco | $20.33$ | $52$ | 8+t | 0.15 | $16.26$ | 2.957 | 0.155 | 3.567 | 0.006 | 5.633 | 0.016
|  |  |  |  | $0.03$ | 0.013 | 0.002 | 0.320 | 0.002 | 0.120 | 0.002
RZ Vel | $20.40$ | $65$ | 9 | 0.16 | $24.64$ | 3.028 | 0.297 | 5.634 | 0.083 | 1.484 | 0.052
|  |  |  |  | $0.03$ | 0.005 | 0.001 | 0.018 | 0.001 | 0.026 | 0.001
WZ Sgr | $21.85$ | $81$ | 13 | 0.21 | $23.62$ | 3.053 | 0.313 | 5.560 | 0.104 | 1.619 | 0.085
|  |  |  |  | $0.04$ | 0.006 | 0.002 | 0.019 | 0.002 | 0.020 | 0.002
T Mon | $27.04$ | $590$ | 16+t | 0.35 | $20.77$ | 2.980 | 0.342 | 5.736 | 0.182 | 2.368 | 0.137
|  |  |  |  | $0.02$ | 0.004 | 0.001 | 0.006 | 0.001 | 0.008 | 0.001
AQ Pup | $30.19$ | $98$ | 11+t | 0.18 | $25.58$ | 2.996 | 0.337 | 5.673 | 0.141 | 1.983 | 0.096
|  |  |  |  | $0.04$ | 0.009 | 0.002 | 0.013 | 0.002 | 0.016 | 0.001
$\ell$ Car | $35.56$ | $591$ | 12 | 0.26 | $15.89$ | 2.936 | 0.323 | 5.902 | 0.172 | 2.763 | 0.101
|  |  |  |  | $0.02$ | 0.004 | 0.001 | 0.008 | 0.001 | 0.012 | 0.001
SV Vul | $45.00$ | $124$ | 8 | 0.36 | $19.84$ | 2.905 | 0.369 | 5.902 | 0.191 | 2.666 | 0.110
|  |  |  |  | $0.05$ | 0.008 | 0.003 | 0.015 | 0.002 | 0.024 | 0.003
S Vul | $69.43$ | $88$ | 7 | 0.21 | $12.31$ | 3.164 | 0.334 | 0.250 | 0.120 | 3.641 | 0.055
|  |  |  |  | $0.03$ | 0.023 | 0.003 | 0.026 | 0.004 | 0.064 | 0.004

  * •

Notes : In this table we provided only the low-order parameters up to fourth
Fourier order and the standard deviation of the fit. The full table with all
derived Fourier parameters and their uncertainties is only available in
electronic form. This table displays the name of the star, the pulsation
period in days, the number of data points used in the fit Ndat, the order of
the fit $n$ which can be additionally supplemented with an orbital motion
correction or a trend t (see Sect 2.5 and 2.6 respectively).

###  C.2 Low-order Fourier parameters of fundamental-mode Cepheids, stars of
Quality 1a.

Table 7: Low-order Fourier parameters of fundamental-mode Cepheids, stars of Quality 1a.  Star | Period | Ndat | $n$ | $\sigma_{\mathrm{fit}}$ | $A_{1}$ | $\phi_{21}$ | $R_{21}$ | $\phi_{31}$ | $R_{31}$ | $\phi_{41}$ | $R_{41}$
---|---|---|---|---|---|---|---|---|---|---|---
ST Tau | $4.03$ | $89$ | 6 | 0.87 | $15.67$ | 3.068 | 0.352 | 5.953 | 0.167 | 2.950 | 0.094
|  |  |  |  | $0.14$ | 0.032 | 0.009 | 0.058 | 0.009 | 0.102 | 0.009
BF Oph | $4.07$ | $64$ | 8 | 0.58 | $15.17$ | 3.001 | 0.318 | 5.956 | 0.173 | 2.814 | 0.080
|  |  |  |  | $0.13$ | 0.028 | 0.010 | 0.058 | 0.008 | 0.107 | 0.008
Y Lac | $4.32$ | $82$ | 6 | 0.91 | $16.50$ | 3.090 | 0.336 | 6.163 | 0.168 | 3.194 | 0.078
|  |  |  |  | $0.15$ | 0.033 | 0.009 | 0.060 | 0.010 | 0.125 | 0.009
V402 Cyg | $4.36$ | $105$ | 5 | 0.87 | $13.46$ | 3.088 | 0.332 | 0.045 | 0.136 | 3.032 | 0.061
|  |  |  |  | $0.13$ | 0.034 | 0.010 | 0.070 | 0.010 | 0.157 | 0.009
AP Sgr | $5.06$ | $72$ | 6+t | 0.70 | $15.65$ | 3.171 | 0.427 | 0.142 | 0.228 | 3.379 | 0.126
|  |  |  |  | $0.13$ | 0.026 | 0.009 | 0.043 | 0.009 | 0.085 | 0.008
AX Cir | $5.27$ | $100$ | 7+2 | 0.33 | $13.17$ | 3.142 | 0.341 | 0.324 | 0.150 | 3.806 | 0.049
|  |  |  |  | $0.05$ | 0.012 | 0.004 | 0.027 | 0.004 | 0.075 | 0.004
CV Mon | $5.38$ | $106$ | 8 | 0.59 | $15.09$ | 3.106 | 0.429 | 0.080 | 0.222 | 3.354 | 0.114
|  |  |  |  | $0.10$ | 0.020 | 0.006 | 0.033 | 0.006 | 0.054 | 0.006
Y Sgr | $5.77$ | $66$ | 6 | 0.33 | $15.07$ | 3.220 | 0.400 | 0.365 | 0.201 | 3.687 | 0.091
|  |  |  |  | $0.06$ | 0.013 | 0.004 | 0.025 | 0.004 | 0.046 | 0.004
R Cru | $5.83$ | $44$ | 6+t | 0.68 | $15.23$ | 3.217 | 0.449 | 0.358 | 0.243 | 3.696 | 0.124
|  |  |  |  | $0.17$ | 0.035 | 0.013 | 0.062 | 0.009 | 0.107 | 0.011
V733 Aql | $6.18$ | $77$ | 4 | 0.95 | $10.59$ | 3.357 | 0.372 | 0.780 | 0.118 | 4.710 | 0.053
|  |  |  |  | $0.17$ | 0.046 | 0.019 | 0.169 | 0.014 | 0.296 | 0.016
RR Lac | $6.42$ | $56$ | 6 | 0.47 | $14.10$ | 3.200 | 0.420 | 0.366 | 0.203 | 3.648 | 0.090
|  |  |  |  | $0.10$ | 0.022 | 0.007 | 0.041 | 0.007 | 0.084 | 0.007
XX Sgr | $6.42$ | $116$ | 6 | 0.90 | $15.24$ | 3.304 | 0.486 | 0.510 | 0.242 | 3.843 | 0.110
|  |  |  |  | $0.12$ | 0.023 | 0.009 | 0.041 | 0.008 | 0.079 | 0.008
X Sgr | $7.01$ | $77$ | 5 | 0.71 | $12.29$ | 3.247 | 0.412 | 0.332 | 0.159 | 3.284 | 0.078
|  |  |  |  | $0.12$ | 0.033 | 0.012 | 0.075 | 0.011 | 0.128 | 0.012
RS Ori | $7.57$ | $38$ | 6 | 0.72 | $14.90$ | 3.550 | 0.562 | 0.892 | 0.193 | 3.863 | 0.127
|  |  |  |  | $0.19$ | 0.029 | 0.015 | 0.077 | 0.012 | 0.104 | 0.012
VY Cyg | $7.86$ | $84$ | 6 | 1.02 | $14.51$ | 3.711 | 0.587 | 1.001 | 0.177 | 4.234 | 0.169
|  |  |  |  | $0.18$ | 0.035 | 0.014 | 0.082 | 0.012 | 0.093 | 0.013
W Gem | $7.91$ | $96$ | 9 | 0.84 | $14.88$ | 3.679 | 0.608 | 0.976 | 0.188 | 4.166 | 0.141
|  |  |  |  | $0.14$ | 0.024 | 0.010 | 0.062 | 0.009 | 0.071 | 0.010
S Nor | $9.75$ | $142$ | 7 | 0.30 | $14.08$ | 4.935 | 0.352 | 1.687 | 0.184 | 0.419 | 0.022
|  |  |  |  | $0.04$ | 0.010 | 0.003 | 0.020 | 0.003 | 0.132 | 0.003
RY Cas | $12.14$ | $40$ | 3 | 1.18 | $15.89$ | 4.890 | 0.206 | 1.376 | 0.173 |  |
|  |  |  |  | $0.27$ | 0.092 | 0.019 | 0.104 | 0.019 |  |
Z Sct | $12.90$ | $130$ | 12+t | 0.23 | $22.37$ | 4.200 | 0.057 | 2.511 | 0.140 | 5.557 | 0.094
|  |  |  |  | $0.03$ | 0.026 | 0.001 | 0.012 | 0.002 | 0.017 | 0.001
TX Cyg | $14.71$ | $61$ | 5 | 1.21 | $22.22$ | 3.040 | 0.104 | 2.720 | 0.084 | 5.667 | 0.076
|  |  |  |  | $0.24$ | 0.108 | 0.010 | 0.127 | 0.010 | 0.134 | 0.011
SZ Aql | $17.14$ | $64$ | 11 | 0.61 | $25.13$ | 3.017 | 0.255 | 4.672 | 0.024 | 0.246 | 0.066
|  |  |  |  | $0.12$ | 0.022 | 0.005 | 0.201 | 0.005 | 0.076 | 0.005