text:
# Study of changes in the pulsation period of 148 Galactic Cepheid variables

G. Csörnyei,1,2,3 L. Szabados,1,4 L. Molnár,1,4,5 B. Cseh,1 N. Egei,1 Cs.
Kalup,1,6 V. Kecskeméthy,1,6 R. Könyves-Tóth,1,7 K. Sárneczky,1 and R.
Szakáts1
1Konkoly Observatory, ELKH Research Centre for Astronomy and Earth Sciences,
Konkoly Thege Miklós út 15-17, 1121 Budapest, Hungary
2Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741
Garching, Germany
3Physics Department, Technische Universität München, James-Franck-Str. 1.,
85741 Garching, Germany
4CSFK Lendület Near-Field Cosmology Research Group, Konkoly Thege Miklós út
15-17, 1121 Budapest, Hungary
5ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány
1/A, 1117 Budapest, Hungary
6ELTE Eötvös Loránd University, Department of Astronomy, Pázmány Péter sétány
1/A, 1117 Budapest, Hungary
7Department of Optics & Quantum Electronics, University of Szeged, Dóm tér 9,
6720 Szeged, Hungary E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Investigating period changes of classical Cepheids through the framework of
O-C diagrams provides a unique insight to the evolution and nature of these
variable stars. In this work, the new or extended $O-C$ diagrams for 148
Galactic classical Cepheids are presented. By correlating the calculated
period change rates with the Gaia EDR3 colours, we obtain observational
indications for the non-negligible dependence of the period change rate on the
horizontal position within the instability strip. We find period fluctuations
in 59 Cepheids with a confidence level of 99%, which are distributed uniformly
over the inspected period range. Correlating the fluctuation amplitude with
the pulsation period yields a clear dependence, similar to the one valid for
longer period pulsating variable stars. The non-negligible amount of Cepheids
showing changes in their $O-C$ diagrams that are not or not only of
evolutionary origin points toward the need for further studies for the
complete understanding of these effects. One such peculiar behaviour is the
large amplitude period fluctuation in short period Cepheids, which occurs in a
significant fraction of the investigated stars. The period dependence of the
fluctuation strength and its minimum at the bump Cepheid region suggests a
stability enhancing mechanism for this period range, which agrees with current
pulsation models.

###### keywords:

Stars: variables: Cepheids – binaries: general – stars: oscillations – stars:
evolution

††pubyear: 2021††pagerange: Study of changes in the pulsation period of 148
Galactic Cepheid variables–A.2

##  1 Introduction

The classical Cepheid variable stars (hereafter Cepheids) are objects of
fundamental importance for both stellar astrophysics and extragalactic
distance determination. These variables represent a brief phase in the post-
main-sequence evolution of intermediate-mass (4-12 M⊙) central helium burning
stars, which populate the classical instability strip in the
Hertzsprung–Russell (H–R) diagram. Their importance lies in the periodic
variability of these objects: the characteristic feature of their pulsation,
the period, allows for the calibration of the cosmic distance ladder and an
independent estimation of the Hubble constant via the famous period-luminosity
(PL) relation (Freedman et al., 2001; Riess et al., 2019) or its reddening
free formulation (Wesenheit function). Measuring the exact value of the period
and the rate of its change can also give implications on the physical
parameters of these stars and their evolution using pulsation (Bono et al.,
1999; Marconi et al., 2010) and evolutionary models that take into account
various recently discovered effects, e.g. realistic core overshooting and
meridional mixing from rapid rotation (Maeder & Meynet, 2001; Anderson et al.,
2014). It has been shown that, apart from the pulsation period, which is
obviously the most crucial property for these variables, light curve
parameters (amplitude, skewness and acuteness) can play an important role in
the estimation of physical properties (Bellinger et al., 2020), however, the
role of period changes on the precision of these estimations has not been
investigated. Since Cepheids in different locations within the instability
strip exhibit different period change rates, these values can be used to infer
the probable value of physical properties of the stars, as well (Turner et
al., 2006).

The pulsation period of Cepheids can change for multiple reasons. In most
cases, the physical process behind these changes is stellar evolution, i.e.
the movement of the Cepheid on the H–R diagram. This trajectory through the
instability strip is associated with changes in the physical properties, which
in turn change the period of pulsation: if the star evolves towards the cool
edge of the instability strip, the period increases, while it decreases when
evolving towards the hot edge. Superimposed on the curve corresponding to
stellar evolution, low-frequency quasi-cyclic period variations can also
appear, e.g., in the case of S Vul (Mahmoud & Szabados, 1980), several long-
period Cepheids in the LMC (Rodríguez-Segovia et al., 2021) or for many
Cepheids mentioned in this article. These variations are due to period and
phase fluctuations accumulating in time and building up a phase-lag that will
appear as a change in the pulsation period (Balázs-Detre & Detre, 1965). The
timescale of these fluctuations varies within a wide range, from decades to
even centuries. Due to these variations, period determination using $O-C$
diagrams can become uncertain, thus other methods might be necessary for this
purpose when the baseline of the $O-C$ diagrams is not long enough to reliably
separate the signals originating from the evolution and the fluctuations
(Lombard & Koen, 1993). In most of the cases, long term photometry allows for
the reliable separation of the stellar evolution from the fluctuations on the
$O-C$ diagram. However, in some cases, as for BY Cas and DX Gem, the time span
covered by observations and the timescale of fluctuations are comparable,
which can even lead to the misidentification of the Cepheid’s crossing number,
as it was shown by Berdnikov (2019a). Although now there are photometric data
from long enough timescales for the effects of evolution and fluctuations to
be separated reliably for bright Cepheids, fluctuations can still prove a
problem for the period change determination of fainter Cepheids that are
observed less regularly. Physical processes behind such fluctuations could be
either short-term variations in the atmospheric structure (Deasy & Wayman,
1985), or minor changes in helium abundance gradients in upper layers of
Cepheids (Cox, 1998). The century long temporal coverage of light curves not
only allows the investigation of period changes through the simple methodology
of $O-C$ diagrams (Sterken, 2005), but it can give us an unprecedented insight
into the nature of these fluctuations as well, which facilitates the
refinement of current pulsation models.

Apart from the evolutionary changes and the random fluctuations of the
pulsation period, binarity of the Cepheid can also cause observable effects in
the $O-C$ diagram. One of these is the light-time effect (LiTE), which arises
from the orbital motion of the variable star. This effect can be easily
distinguished from the evolutionary effect, since it results in a periodic
feature in the $O-C$ graph. The orbital modulation can be detected in the
$O-C$ diagram of binary Cepheids when the orbital period is sufficiently long,
because in the case of short orbital periods the amplitude of this cyclic
pattern would be too small to recognize. A very clear indication that a
superimposed wavelike signal is caused by the LiTE instead of fluctuations is
the stability of the signal; since the fluctuation waves are caused by
accumulating random processes, the shape of the resulting quasi-cyclic signal
will get distorted, however, as the LiTE is caused by a deterministic process,
its shape will remain stable over time. The importance of light time effect
lies in the fact that it can be used for a rough estimation of the orbital
parameters. So far only a few Cepheids have been found to show LiTE (AW Per
and RX Aur, Szabados 1992 and Vinkó 1993), with a few more awaiting
spectroscopic confirmation.

Another effect characteristic of binary Cepheids is a phase jump or phase slip
seen in the $O-C$ diagram. A phase jump in the $O-C$ diagram has a stepwise,
while the phase slip has a sawtooth-like shape; in the first case the period
remains the same while the phase of the pulsation suddenly changes, while in
the second case, the period of pulsation changes rapidly, then returns to the
previous value after a given amount of time (period jump-rejump). Such phase
jumps have been found in the $O-C$ diagrams of several Cepheids (e.g. Szabados
1989). One of the most prominent examples of Cepheids exhibiting such features
in their $O-C$ diagrams is Polaris (Turner et al., 2005), which displayed a
parabolic $O-C$ trend corresponding to a continuously increasing period that
was interrupted by a phase jump after which the period continued to increase
with the previous rate, with a significant phase difference.

The idea that phase jumps appear only in the $O-C$ diagrams of binary Cepheids
is based solely on empirical data. Each Cepheid showing such an effect is
either a member of a known spectroscopic binary or suspected to have a
companion based on independent evidence. Moreover, no phase jump was ever
detected for any well-studied single Cepheid. The size of a phase jump is
usually in the order of several hundredths of the pulsation period.
Unfortunately, there is no theoretical explanation for the occurrence of such
phase jumps or slips. For Polaris, Turner et al. (2005) suggested the brief
interruption of the Cepheid’s regular evolution by a short-lived blueward
evolution associated with a small change in the average radius of the star, or
alternatively, by a sudden increase in the mass of the Cepheid. However, these
effects appear to be too spontaneous and peculiar to explain the phase jumps
generally, especially when the phase jump occurs multiple times for a given
star, which was observed to occur among Cepheids. Szabados (1992) suspects
that the occurrence of phase jumps is governed by the orbital motion. In this
case, a possible explanation for the phase jumps could be the perturbing
effect of the companion star that exerts increased influence on the upper
layers of the atmosphere of the Cepheid (where the pulsation takes place) near
periastron passage. The stepwise structure of the $O-C$ diagrams in the case
of phase jumps contradicts current evolutionary models, which predict
continuous change of pulsation period otherwise.

Even though only a few Cepheid variables were observed in the programs of the
_Coriolis_ , _MOST_ , _Kepler_ and _CoRoT_ space missions, the advent of high
precision space photometry brought improvements in the period change studies
of these variables as well. Based on the observations of the Solar Mass
Ejection Imager (SMEI) instrument onboard the _Coriolis_ satellite, Spreckley
& Stevens (2008) connected slow changes in the light curve of the Polaris to a
quite fast evolutionary phase and by supplementing the existing $O-C$ diagram
suggested that another phase jump occurred. Berdnikov & Stevens (2010) studied
the period changes of short period Cepheids based on SMEI photometry and
detected random fluctuations in four of the observed stars. By analysing the
_MOST_ observations of SZ Tau and RT Aur, Evans et al. (2015) revealed cycle-
to-cycle variations that are a function of the pulsation phase. The stability
of their pulsation was also investigated and, besides finding temporal
variations on a scale of decades, it was pointed out that the pulsation of
overtone Cepheids is more erratic than that of fundamental mode variables. In
the case of the _Kepler_ mission, there was only one Cepheid in the observed
field, V1154 Cyg. Derekas et al. (2012, 2017) provided a detailed analysis of
the data and found cycle-to-cycle fluctuations on the $O-C$ diagram, which
were attributed to instabilities in the light-curve shape caused by convection
and hot spots based on the suggestion of Neilson & Ignace (2014). Poretti et
al. (2015) performed the fluctuation tests similar to the work by Evans et al.
(2015) on the _CoRoT_ data for seven Cepheids, and found small cycle-to-cycle
variations. Since these space missions covered only a few Cepheids, our
understanding on these small fluctuations is limited, which is expected to
change through future works based on the data from the currently ongoing
_TESS_ project (Plachy et al., 2021).

In this work, our primary aim is to expand the collection of existing $O-C$
diagrams of known binary Cepheids by using century-long photometric
observational data of these variables. To get a more complete set of diagrams,
the initial set of variables was extended with stars that, to our current
knowledge, do not belong to binary systems.

The structure of this paper is as follows. In Section 2 we describe the
available data and the differences of our method from the standard $O-C$
method. In Section 3 we discuss the results obtained for Cepheids that
exhibited some peculiarities on their $O-C$ diagrams individually. Then, in
Section 4 we analyse our sample collectively, first by placing the period
change rates into broader context by comparing the results with previous
works, then by discussing the period fluctuations in general. Finally, in
Section 5 we summarize our results and conclude the paper.

##  2 Data and the method of the analysis

###  2.1 Data and observations

To achieve the near century long coverage of the temporal behaviour of the
pulsation period, we attempted to acquire all available processed photometric
data for every star. The first half of the 20th century was covered by the
Harvard College Observatory Plate database (DASCH project, Grindlay et al.
2012, for stars on the northern galactic hemisphere) and by the observations
of the American Association of Variable Star Observers (AAVSO, Kafka 2021, for
suitably bright Cepheids). The later part of the century was covered by
various articles from the literature, and by the earlier works of Szabados
(1989), Szabados (1991) and Berdnikov et al. (1997). The complete set of
references for articles used for the individual Cepheids can be found in Table
A.1. To complete this dataset with more recent observations, we used the
photometric data obtained by the Hipparcos (Perryman et al., 1997), the All
Sky Automated Survey (ASAS, Pojmanski 2001), INTEGRAL-OMC (Winkler et al.,
2003), KELT111https://exoplanetarchive.ipac.caltech.edu/, ASAS-SN (Shappee et
al., 2014; Kochanek et al., 2017), the Optical Gravitational Lensing
Experiment (OGLE, Udalski et al. 2015), the Solar Mass Ejection Imager (SMEI,
Eyles et al. 2003), as well as the Kamogata Sky Survey (Morokuma et al., 2014)
projects. To supplement these data, we have monitored some Cepheids with the
60/90 cm Schmidt telescope at Piszkéstető Observatory. The photometric data
acquired through this telescope can be found in Table 1.

Cepheid | JD | V
---|---|---
YZ Aur | 2458806.402 | 9.8815
YZ Aur | 2458807.618 | 9.8637
YZ Aur | 2458823.473 | 10.392
YZ Aur | 2458824.418 | 10.129
YZ Aur | 2458828.623 | 10.052
YZ Aur | 2458855.319 | 10.683
… | … | …
Table 1: List of observations taken at the Piszkéstető Observatory. The
brightness values are differential magnitudes except for X Lac, for which we
used previously taken unpublished measurements which are not transformed to
the international system. The full list of observations is available in the
online supplement of the article.

###  2.2 Construction of the $O-C$ diagrams

We followed the changes in the pulsation period of individual Cepheids using
the method of $O-C$ diagrams (Sterken, 2005). To analyse the photometric
datasets of each Cepheid, we applied Discrete Fourier Transformation (DFT,
Deeming 1975) on the measurements, for which we used the Period04 software
(Lenz & Breger, 2005). We analysed the data from the various sources
separately, to account for differences in circumstances valid for individual
observational series, by keeping the frequencies and the relative phases of
the harmonics fixed to a pre-calculated parameter set, which were based on the
dataset with the widest coverage and most data points, while allowing the
amplitudes to change during the fitting procedure.

As a first step of the analysis, every set of observations were split into
smaller subsets. For each survey, depending on the temporal coverage of the
data, 300-450 day long temporal bins were defined depending on the length of
the pulsation period, in which each data point was moved to a new subset. The
folded light curves of each previously created subset of measurements were
calculated, which then were used to determine the observed ($O$) epoch values
of the average (zero point) brightness on the brightening edge of the light
curve. To determine the $O$ values, we fitted the phase folded light curves
corresponding to the different epochs with the previously calculated Fourier
components representative of the data source at hand, by only allowing the
mean brightness and phase offset of the model light curve to change. The $O$
values were then simply calculated from the phase offset of the best fit
model. The use of the median brightness on the ascending branch was motivated
based on the Fourier fit qualities: for longer and longer pulsation periods,
the complexity of the light curve increases, which in turn means that more and
more Fourier components will need to be taken into account. For older, less
accurate datasets the fitting of the high order components could become
uncertain, which in turn could introduce a bias to the inferred $O-C$ values,
when they are based on the epoch of maximum brightness. However, it was found
that the rising branch itself, and the epoch of average brightness on it can
be fitted with suitable accuracy even in such cases, thus providing a viable
alternative for datasets with lower amount of observations. The advantage of
using the epoch of average brightness over that of the maximum becomes even
more important for Cepheids with pulsation periods between 8-10 days, due to
the secondary maximum present on the light curve. It was also found by Derekas
et al. (2012), that the most accurate $O-C$ values can be obtained through
measuring the epoch of average brightness on the rising branch. The average
brightness value was determined as the baseline for the sinusoidal components
at the Fourier fitting of the separate datasets. By this definition the
brightness value at the zero point is slightly different for various datasets
as it will account for the differences between the different surveys and
measurement systems, thus the resulting $O-C$ points will be comparable.
Although the phase folding method inevitably decreases the resolution of the
resulting $O-C$ curve, the precision of the results increases, since the error
of the phase calculation will decrease significantly.

Figure 1: Sample Cepheids from our set showing no features apart from the
evolutionary signal. The size of the datapoints indicate the inverse of the
uncertainty with a non-linear scaling.

To complete the construction of the $O-C$ diagram, the errors of the
individual $O-C$ differences had to be calculated. The most important factor
that governs the precision of the individual $O-C$ values is the number of
photometric data points used for their derivation. Since we only fit the
average brightness and the phase shift of a given light curve shape, even a
low number of data points is enough for the $O-C$ calculation, however to make
sure that the results we obtain are meaningful, we set the lower limit on this
number to 8. Another factor that needs to be taken into account when
estimating the accuracy of the $O-C$ points is the phase distribution of
photometric data points in each of the folded light curves. Obviously, if the
phase curves created with the method explained above were well-covered, i.e.
the calculated phases covered the possible range in an approximately uniform
manner, then the standard least-squares method could be used to fit the known
light-curve shape to the phase curve, which would allow for the
straightforward calculation of the covariance matrix. However, if the phase-
curves were not well covered, which occurred frequently in the case of long
period variables observed in a series of short-cadence observation runs (e.g.
with IOMC), then the above method could significantly underestimate the error
of the $O-C$ differences. To avoid this problem, the errors have been
calculated by applying bootstrapping throughout the fitting procedure; at each
step 75% of the data points have been randomly selected, and fitted with the
precomputed light-curve shape. This step was carried out a hundred times for
each phase curve. The final uncertainty of the individual $O-C$ residuals was
calculated as the square root of the sum of overall phase shift and light-
curve zero point variance.

The $O-C$ derivation method inherently contains a minor bias through the use
of Fourier decomposition for the fitting, as it can obviously just approximate
the actual shape of the light curve, which could cause biases in cases when a
dozen or more Fourier components are required for the description. Although
these biases can be minimized properly by measuring the moment of the average
brightness on the rising branch instead of that of the maximum, a more precise
and general method could be achieved by using a non-parametric light curve
fitting instead of the Fourier decomposition. Nevertheless, since we used the
same amount of Fourier components for every dataset of the individual stars,
these biases affect each of the reductions similarly, thus they did not lead
to noticeable artefacts between the different data sources.

##  3 Remarks on individual Cepheids

The $O-C$ curves corresponding to evolutionary changes are usually represented
as parabolic trends, which correspond to constant rate of period change (for a
sample of $O-C$ diagrams showing solely the effects of stellar evolution, see
Fig.1). There have been numerous studies from the last decades showing such
behaviour (e.g. Parenago 1956, Szabados 1983, Berdnikov et al. 1997). The rate
of period change depends on the position of the Cepheid within the instability
strip and the crossing number, and is usually modelled by a linear function of
time, which rates are well approximated by current pulsation and evolutionary
models (Anderson et al., 2016).

However, there is no physical model that restricts the period change of a star
to be linear, since the luminosity and the temperature themselves change in a
nonlinear manner. Hence, as it has been argued by Fernie (1990a), in some
cases higher order polynomial fits are necessary for certain $O-C$ diagrams,
which indicate not only the change of period itself, but variations in the
rate of period change as well. Such behaviour is common among Cepheids,
significant deviations can be observed from parabolic fits to the $O-C$ plot
in numerous cases. Determining the period change rate for a large set of
Cepheids allows for the statistical study of this phenomenon, as well as
giving way for the refinement of existing period-based empirical relations.
However, there are quite a few classical Cepheid variables that exhibit
peculiar $O-C$ diagrams, for which the statistical description is not yet
available due to the small number of such objects, thus these stars have to be
treated separately.

In this section we present and discuss the $O-C$ diagrams exhibiting features
which could not be explained by evolutionary changes with superimposed
relatively small amplitude fluctuations, such as phase jumps or slips, or
large amplitude waves that partially, if not completely, obscure the effect of
evolution on the diagram. In the large sample of analysed Cepheids, we found
16 of such $O-C$ diagrams, which are presented below.

###  3.1 Cepheids showing wavelike $O-C$ structure

Out of the 16 Cepheids that exhibit peculiar $O-C$ diagrams, we found 9 stars
that show a remarkable wavelike modulation. Such wavelike signals can arise
either due to the presence of a companion star through LiTE, or due to the
fluctuations present in the pulsation of the Cepheid, in which case the
modulation can even obscure the effect of evolution on the $O-C$ diagram. To
model these wavelike signals we simultaneously fitted the underlying parabolic
trend corresponding to stellar evolution and the superimposed modulation
(except for UY Mon, where we could not calculate such a fit reliably, see Sec.
3.1.7). By this procedure, we fitted a single sinusoidal waveform for each
Cepheid, except for IR Cep, where involvement of one additional sinusoidal
component is motivated based on the systematic deviations present in the
residuals calculated with respect to the parabolic plus single sinusoidal
model fit. In the case of the other Cepheids, we either did not find any
additional signals in the residual data after fitting the parabola and the
modulation, or due to the randomness and possible fluctuation origin of the
modulation (as in the case of VZ CMa and DX Gem, see Sec. 3.1.2 and 3.1.6) we
decided that adding further components would not change the outcome of the
analysis significantly.

For these Cepheids, similarly to the other Cepheids showing evolutionary
changes, we then statistically analysed the residuals calculated by various
models (for the constant period, the linear period-change, and the linear
period change plus sinusoidal signal cases) using F-test, in order to
determine the significance of the wavelike modulation in the modelling. The
final F-statistics and the associated p-values can be found in the summary
table in the Appendix (Table B.1). In a later section (Sec. 4.2) we then used
these solutions along with those of Cepheids showing evolutionary changes for
the collective analysis.

####  3.1.1 RX Aurigae

RX Aur is a bright intermediate period Cepheid, whose binarity was long
suspected based on the slope of its colour-colour loop and the LiTE-like
nature of its $O-C$ diagram (Szabados, 1988). The total mass of the system
inferred based on this diagram did not rule out the presence of a companion
(Szabados, 1988), and it is further supported by the RV measurements of
Gorynya et al. (1996), who marked the star as a possible binary.

By extending the $O-C$ diagram with more recent photometric measurements we
confirm the LiTE explanation of the $O-C$ diagram (Fig. 2). The $O-C$ diagram
was calculated using the following elements:

$C_{\textrm{med}}=2443830.177+11.6241\cdot E$

Figure 2: $O-C$ diagram of RX Aurigae, exhibiting a parabolic change
corresponding to evolution of the Cepheid and a wavelike signature
superimposed (top panel), the residual $O-C$ values with the corresponding fit
(middle panel) and the available mean RV values. Figure 3: Phase diagram of
the residual $O-C$ variation of RX Aur. The elements used for the calculation
of the phase diagram were $T_{0}=2442312.825$ and $P=37928.243$ d.

Fitting the evolutionary trend and the superimposed modulation (Fig. 3)
yielded an orbital period of $P_{\rm orb}=37928\pm 7678$ d $=103.84\pm 21.02$
yr and an amplitude of $A=0.255\pm 0.107$ d, which are approximately 1.92 and
1.15 times larger, respectively, than the previously calculated values
(Szabados, 1988). Since, as seen in Fig. 2, the maximum of the variation in
the residual terms is only covered once by our dataset, a more detailed and
reliable orbital parameter determination is not yet possible, as the few data
points near that extremum do not provide sufficient constraint for the fitting
procedure. However, the knowledge of the orbital period and semi-major axis
(amplitude) allows for the determination of the total mass of the system
through the

$\left(\frac{A}{\sin i}\right)^{3}P_{\textrm{orb}}^{-2}=M_{1}+M_{2}$ (1)

formula (since $A=a\sin i$, with $a$ being the semi-major axis of the orbit),
where $A$ is measured in astronomical units, $P_{\textrm{orb}}$ in years. The
resulting parameters are as follows:

$\begin{split}a\sin i=(44.152\pm 18.515)\textrm{ AU }\\\ =(6.605\pm
2.769)\cdot 10^{9}\textrm{ km }\\\ (M_{1}+M_{2})\sin^{3}i=(7.999\pm
6.239)M_{\odot}.\end{split}$

The resulting mass is consistent with the values inferred by Kervella et al.
(2019a) with the assumption that the orbital plane is oriented along or near
the line of sight ($i\gtrsim 60^{\circ}$), and the obtained orbital period
also falls into the range proposed by them. However, no significant signal can
be extracted from the available RV data (from the measurements of Schmidt
1974, Barnes et al. 1987, Gorynya et al. 1996, Imbert 1999 and Borgniet et al.
2019) displayed in Fig. 2, which show a low scatter ($\sigma_{\textrm{RV}}\sim
1.155$ km/s) and their distribution is not aligned in phase with the suspected
LiTE either, hence it is not possible to reach a solid conclusion on the
origin of the observed $O-C$ signal yet.

####  3.1.2 VZ Canis Maioris

VZ CMa is a short period Cepheid whose pulsation mode was investigated in
various studies, yielding different results: Kienzle et al. (1999) classified
this star as an overtone pulsator, while according to Groenewegen & Oudmaijer
(2000) this Cepheid is a fundamental mode pulsator. Its multiplicity was also
studied in various articles: it was first suspected by Stobie & Balona (1979)
and Szabados (1993) based on its color indices and amplitude ratios
respectively, then was finally confirmed spectroscopically by Szabados (1996).
The $O-C$ diagram of the object was studied by Berdnikov & Pastukhova (1994),
who found it to be parabolic.

By extending the previously available data with the archival measurements of
Hacke & Richert (1990), and with that of modern surveys, we found that the
$O-C$ diagram of VZ CMa exhibits a large amplitude wave (Fig. 4), which was
interpreted as a parabolic pattern before, due to the lack of sufficient data.
The $O-C$ diagram was calculated with the following elements:

$C_{\textrm{med}}=2452663.377+3.126151\cdot E$

Figure 4: $O-C$ diagram of VZ CMa exhibiting a larger amplitude quasi-periodic
signal, which could either be attributed to LiTE (which however, would not
explain the systematic deviations of data points at HJD 2450000 on the lower
panel) or to period fluctuations.

Fitting the parabolic trend and the wavelike signal simultaneously, we
estimated the period of the modulation to be 24475 d (67.01 yr) and its peak-
to-peak amplitude to be approximately 0.3 d. Although the values inferred here
could indicate a LiTE origin for the signal, the unusual, non-sinusoidal shape
points towards the presence of fluctuations in the pulsation. This ambiguity
is not lifted by the RV datasets either, as the combined observations of
Stobie & Balona (1979) and Kienzle et al. (1999) cover a much shorter timespan
than the period of the signal, henceforth we cannot conclude whether this wave
originates from binarity or period fluctuations.

####  3.1.3 BY Cassiopeiae

BY Cas is short period Cepheid with a long history of cluster membership and
binarity studies. Membership in NGC 663 was first suggested by Malik (1965),
which was contradicted by the later study of Usenko & Klochkova (2015) citing
large color excess, parallax and RV differences, concluding that BY Cas is a
foreground star, in agreement with the result of Anderson et al. (2013) too.
However, according to the latest analysis of Glushkova et al. (2015), this
Cepheid shows connection not only to its nearest cluster NGC 663, but also to
the clusters NGC 654, NGC 659, and the association Cas 8, too. BY Cas was also
revealed to have a companion with an orbital period of $\sim$560 d (Gorynya et
al., 1994).

The period changes of this variable were investigated by Szabados (1991) and
Berdnikov & Pastukhova (1994), and more recently by Berdnikov (2019b). The
former analysis described the variations as a sequence of sudden period
changes, while the latter work proposed evolutionary changes with a wavelike
pattern superimposed. Our independent analysis showed that the $O-C$ diagram
can indeed be explained by evolutionary changes with a wavelike signal on top
of it (Fig. 5). The presented $O-C$ diagram was calculated with the following
elements:

$C_{\textrm{med}}=2448755.506+3.222019\cdot E$

Figure 5: $O-C$ diagram of BY Cas, showing an evolutionary change with a large
amplitude wavelike signal superimposed.

Using the parabolic fit we estimated the period change rate to be 0.338
s$\cdot$yr-1, which is two orders of magnitude smaller than the one estimated
by Berdnikov & Pastukhova (1994), and places this object in the group of third
crossing Cepheids, in agreement with the findings of Berdnikov (2019b). The
best explanation for the superimposed modulation is the period fluctuation,
which is relatively common for these Cepheids (as shown in the later
sections).

####  3.1.4 IR Cephei

IR Cep is a short period Cepheid which was first studied in detail by Szabados
(1977), revealing strong period change. This Cepheid was found to exhibit a
peculiar light curve: according to Kun & Szabados (1988), the light curve of
this Cepheid rather resembles that of larger amplitude Cepheids. Thus it was
suggested that IR Cep might be a classical Cepheid, but with a bright
companion whose constant contribution to the emitted light reduces the
observable amplitude of the variable, however, no close binary component was
found by the RV analysis of Marschall et al. (1993). Kun & Szabados (1988)
also suggested the possible association of this Cepheid to Cep OB2.

Figure 6: $O-C$ diagram of IR Cep (top panel). In the bottom panel the
parabolic term corresponding to the evolution of the Cepheid had been
subtracted. A simple Fourier fit of two harmonics can be seen plotted on the
residual data points. The current amount of data points did not allow the
addition of more harmonics terms, but this fit illustrates well that the shape
of the residual signal remains relatively unchanged during the time interval
covered by observations.

IR Cep turned out to be peculiar in its pulsation as well: this Cepheid was
classified as a first overtone pulsator by Groenewegen & Oudmaijer (2000), but
Szabó et al. (2007) showed that a first overtone classification would lead to
excessive phase lag discrepancy for this star. They note however, that
according to its position on the phase lag diagram it would quite naturally
fit in the class of fundamental mode Cepheids.

Based on the acquired photometric measurements the corresponding $O-C$ diagram
has been drawn, which covers more than a century (Fig. 6). The presented
diagram was calculated using the following elements:

$C_{\textrm{med}}=2453453.824+2.114088\cdot E$

The rate of period change is 0.042 s$\cdot$yr-1, while the approximate period
and peak-to-peak amplitude of the seemingly periodic signal are 15150 d (41.47
yr) and 0.29 d. The origin of this wavelike signal is uncertain, as the
amplitude could allow for LiTE, which is also favoured by the more stable form
of the signal, however due to the short temporal coverage of the RV data
obtained by Marschall et al. (1993) and Gorynya et al. (1998) this could not
be confirmed spectroscopically, hence the fluctuation origin cannot be
excluded either.

####  3.1.5 V532 Cygni

V532 Cyg is a short period Cepheid known to be a binary system member. The
presence of a companion star was first suspected by Madore (1977) based on the
shape of its colour-colour loop, then a similar conclusion was reached by
Usenko (1990) and Szabados (1991) later on. This was later confirmed by the RV
observations of Gorynya et al. (1996), finding a 388 d orbital period.

The period changes of this Cepheid were studied several times in the
literature: Szabados (1977, 1991) first described the $O-C$ diagram as a
sequence of period jumps, however, it was later shown that it can be described
by evolutionary changes and a large wavelike signal superimposed (Berdnikov,
2019c). The latter behaviour of the $O-C$ diagram is also confirmed by our
independent study, as shown in Fig. 7. The presented $O-C$ diagram was
calculated using the following elements:

$C_{\textrm{med}}=2441705.834+3.283536\cdot E$

Figure 7: $O-C$ diagram of V532 Cyg exhibiting an evolutionary trend with
large amplitude wavelike modulation superimposed.

As it is shown in the $O-C$ diagram, the period of the Cepheid is indeed
decreasing due to its evolution with a rate of $-$0.104 s$\cdot$yr-1, which
places it among the second crossing Cepheids. The large amplitude ($A=0.612\pm
0.013$ d) of the superimposed signal suggests that the fluctuation of the
period is a suitable explanation.

####  3.1.6 DX Geminorum

DX Gem is a short period Cepheid which was for long considered one of the few
first crossing Cepheids due to its large period change rate. The first period
analysis on this Cepheid was done by Szabados (1977, 1991), in which a series
of phase jumps that appeared to be in accord with the binarity (Burki, 1985)
of the object were suggested as interpretation. However, Berdnikov (2019a)
showed that the $O-C$ diagram can be described by an evolutionary change with
a wavelike signal superimposed.

Figure 8: $O-C$ diagram of DX Gem (top panel). In the bottom panel the
parabolic term corresponding to the evolution of the Cepheid had been
subtracted. The remaining residual term exhibits a wavelike pattern, which is
unstable and too large in amplitude to be connected to LiTE.

According to our independent analysis, the $O-C$ diagram of DX Gem indeed
cannot be explained by evolutionary changes alone (see Fig. 8). Our $O-C$
diagram was calculated using the following elements:

$C_{\textrm{med}}=2441866.183+3.136779\cdot E$

From the obtained fit it is apparent that the Cepheid exhibits a shortening
period with a rate of $-$0.186 s$\cdot$yr-1, hence it belongs to the class of
second crossing Cepheids. The residual large amplitude modulation ($A=0.917\pm
0.027$ d) cannot be explained by LiTE, thus the most reasonable explanation
for it is the presence of fluctuations in the period.

####  3.1.7 UY Monocerotis

UY Mon is a short period Cepheid that was rather neglected in the past: it was
first correctly classified by Imbert (1981), as it was considered as an
eclipsing binary before, and it has not been studied in terms of RV data and
cluster membership yet. The period changes of this Cepheid were investigated
before by Berdnikov et al. (1997), who found an increasing period.

In our analysis, we extended the $O-C$ diagram compiled by Berdnikov et al.
(1997) in both directions using the Harvard Observatory plate archives and
more recent observations. The $O-C$ diagram was calculated using the following
elements:

$C_{\textrm{med}}=2432500.253+2.398185\cdot E$

According to our results, the $O-C$ diagram of UY Mon looks like a slanted
sinusoidal signal, with a possible period longer than 45000d ($\sim$120 yr).
As the amplitude of this oscillation is small compared to the length of the
period, the possibility of LiTE cannot be excluded. However, the time interval
covered by the compiled data does not allow us to determine whether this
signal is indeed periodic, thus further photometric and long-term RV
observations are necessary for a firm conclusion.

Figure 9: $O-C$ diagram of UY Mon exhibiting a long-term low amplitude
wavelike signal, which could be a result of LiTE caused by a formerly unknown
binary component in the vicinity of the Cepheid.

####  3.1.8 Y Ophiuchi

Y Oph is an intermediate period Cepheid that was frequently analysed in the
past. It is a known binary with an orbital period of $\sim$1222.5 d (Szabados,
1989): the presence of the companion was first suspected by Pel (1978) based
on photometry, as Y Oph appeared too blue for its pulsation period. However,
no such companion was detected from the IUE spectra (Evans, 1992) nor through
NACO lucky imaging Gallenne et al. (2014). On this basis, its true nature
remains elusive.

By collecting not only the more recent measurements, but the results from
previous works on this Cepheid as well (Parenago, 1956; Szabados, 1989;
Fernie, 1990a) we compiled an $O-C$ diagram spanning almost a century (Fig.
10). To calculate the $O-C$ diagram, we assumed the following elements:

$C_{\textrm{med}}=2439848.803+17.127827\cdot E.$

Figure 10: $O-C$ diagram of Y Oph exhibiting a wavelike trend corresponding to
the LiTE superimposed on the secular period change of the Cepheid (top panel),
the residual $O-C$ diagram obtained after subtracting the parabola (middle
panel) and the available RV data points whitened for both the pulsation and
the orbital motion of the Cepheid (bottom panel). Figure 11: Phase diagram of
the residual $O-C$ data points of Y Oph and the obtained fit. The elements
used for calculating the phase diagram were $T_{0}=2451548.800$ and
$P=27948.469$ days.

The obtained $O-C$ diagram can be described by a combination of the parabolic
trend corresponding to the evolution of the Cepheid and a wavelike signal
superimposed, with the period of the modulation being $P=27948\pm 1164$ d
$=76.52\pm 3.19$ yr and the amplitude $A=0.461\pm 0.024$ d (Fig. 11). If we
assume that this periodic variation originated from LiTE, then the obtained
amplitude corresponds to a semi-major axis of $a\sin i=79.82\pm 4.19$ AU
$=(1.194\pm 0.063)\cdot 10^{10}$ km, which agrees well with the long period of
this wave. Although the duration of the covered time interval does not allow
for a detailed fit of the residual curve, the accuracy of the simple
sinusoidal fit suggests that the assumed orbit has a low inclination, which
also agrees well with this period–semi-major axis pair. Since the period is
much longer than the one obtained for the previously suspected companion, the
light-time interpretation of the wavelike signal would suggest that a third
component is present in the system of Y Oph, but no signal with such a period
could be extracted from the presently available RV data (bottom panel of Fig.
10). Henceforth, we cannot yet confirm the presence of the third component,
however, as we will show it in Sec. 4, the amplitude of the modulation we
found is larger and can be considered an outlier compared to the fluctuation
amplitudes of similar period Cepheids, which also favours the new companion
and the LiTE explanation.

####  3.1.9 AW Persei

AW Per is a short period Cepheid in a binary system. The binary nature of the
star was discovered by Miller & Preston (1964) based on Ca II and K line
measurements, with the orbit being investigated later on by Welch & Evans
(1989), Vinkó (1993), and then by Evans et al. (2000). This object was also
found to exhibit LiTE that could be matched to RV observations (Szabados,
1991).

Figure 12: $O-C$ diagram of AW Per exhibiting a wavelike trend corresponding
to the LiTE superimposed on the secular period change of the Cepheid (top
panel) and the residual $O-C$ diagram obtained after subtracting the parabola
(bottom panel). The fitting of this wavelike trend can be used for the
determination of orbital elements as described in the text.

To further investigate the signal created by the LiTE we extended the $O-C$
dataset of Szabados (1991) with more modern observations from various sources.
The resulting $O-C$ diagram (Fig. 12) was calculated with the following
elements:

$C_{\textrm{med}}=2442708.076+6.463688\cdot E.$

Similarly to RV data, the LiTE also allows for the detailed estimation of
orbital parameters. We must note however, that we did not aim to refine the
set of parameters obtained by Evans et al. (2000), we only investigated this
possibility for confirmation purposes. The $O-C$ variation due to the orbital
motion can be described as

$O-C=\frac{a\sin
i}{c}(1-e^{2})\left[\frac{\sin(\nu+\omega)}{1+e\cos\nu}-\frac{\sin(\nu_{0}+\omega)}{1+e\cos\nu_{0}}\right],$
(2)

where $a$, $e$, $i$, and $\omega$ refer to the semi-major axis, eccentricity,
inclination and longitude of periastron of the orbit of the pulsating star,
respectively. In the equation, $\nu$ represents the true anomaly of the
pulsator at a given time, while $\nu_{0}$ denotes the true anomaly at the
epoch chosen for the $O-C$ calculation. Based on Eq. 3 the semi-amplitude can
also be defined:

$K=\frac{a\sin i}{c}(1-e^{2})$ (3)

To fit this profile to the obtained set of $O-C$ residuals, we utilized the
RadVel RV fitting toolkit developed by Fulton et al. (2018), which we
appropriately modified using Eq. 3 for the fitting of $O-C$ diagrams. Using
the Monte Carlo based fitting procedure of the code we obtained a fit plotted
in Fig. 13 and for which the fitting parameters are listed in Table 2. As it
is clearly visible from the obtained results, the precision of the $O-C$
diagram based orbital parameters is far worse than the ones obtained by Evans
et al. (2000) based on higher precision RV measurements, but our results are
consistent with the values obtained previously. It is worth mentioning
however, that there is a significant deviation between the orbital period
values, as our result for this parameter differs from the one obtained by
Evans et al. (2000) ($P=14594\pm 324$ d) by more than $7\sigma$. Based on the
fact that our photometric dataset covers a longer timeframe than the one
covered by RV observations, our result suggests that the orbital period of the
binary is indeed significantly shorter than previously thought. To reach a
firm conclusion, however, further photometric and spectroscopic observations
are required.

Figure 13: Best fit to the residual $O-C$ data of AW Per calculated by the RadVel software. The grey diamonds represent phase folded $O-C$ residual data, the blue curve the calculated model fit, while the red points show the binned average values of the data points. For the calculation of phases the orbital period of 12078 d has been used. Parameter | Credible Interval | Max. L. | Units
---|---|---|---
$P$ | $12078^{+340}_{-280}$ | $12052$ | d
| $33.07^{+0.93}_{-0.77}$ | $32.99$ | yr
$T_{\rm{conj}}$ | $2422894^{+770}_{-760}$ | $2422915$ |
$e$ | $0.45^{+0.21}_{-0.2}$ | $0.49$ |
$\omega$ | $2.38^{+0.47}_{-5.1}$ | $2.6$ | rad
$K$ | $0.051^{+0.01}_{-0.012}$ | $0.053$ | d
$a\sin i$ | $1.66^{+1.14}_{-0.58}\cdot 10^{9}$ | $1.81\cdot 10^{9}$ | km
| $11.08^{+7.62}_{-3.88}$ | 12.10 | AU
Table 2: Orbital elements obtained by fitting the residual $O-C$ diagram of AW
Per using the Monte Carlo based code. The Max. L. columns shows the maximum
likelihood estimates of the given parameters.

###  3.2 Cepheids showing phase jumps/slips

Of the $O-C$ diagrams we found to be peculiar, 5 could be explained through
phase jumps or slips. In case of a phase jump, the phase of the pulsation
changes rapidly, while the period remains relatively the same, leading to a
stepwise $O-C$ graph. In the case of a phase slip however, the period of the
pulsation seems to change rapidly, just to return to the previous value after
a given amount of time (period jump-rejump), resulting in a sawtooth-shaped
$O-C$ diagram. Such behaviours, which cannot be explained through stellar
evolution, have been empirically associated with the Cepheids being binaries,
as only some of those were found to exhibit it. However, the complete
explanation for this phenomenon is still elusive.

####  3.2.1 XZ Carinae

XZ Car is a long period Cepheid which was found to be possibly linked to
various open clusters in the past: Glushkova et al. (2015) claimed a possible
relation between the Cepheid and the open cluster Ruprecht 93, while Chen et
al. (2015) found the membership in ASCC 64 to be more probable, based on
proper motion, age, and instability-strip selection criteria, although none of
these associations were marked likely by Anderson et al. (2013) earlier.
Anderson et al. (2016) found the Cepheid to exhibit time dependent
$\gamma$-velocity changes, indicating the binary nature of the star.

The period changes of this object were not investigated before, despite being
a relatively bright star. According to our analysis, the $O-C$ diagram of XZ
Car can only be explained through a phase jump, as the data points available
from before HJD 2440000 show a systematic offset from the rest of the data,
and no evolutionary signal was visible (Fig. 14). The $O-C$ diagram was
calculated using the following elements:

$C_{\textrm{med}}=2452624.168+16.652232\cdot E$

The phase jump occurred between HJD 2438000 and 2442000, and altered the phase
of the pulsation by 0.67 d (i.e. 0.040 phase), assuming that the period was
unchanged before and after the jump (Fig. 14). Apart from the phase jump, no
other signal could be extracted that would indicate period change in the
Cepheid. We note that due to the low amount of available data before the
assumed occurrence of the phase jump this interpretation remains uncertain,
however, if true, it could yield an additional empirical link between the
presence of a companion star and the phase jump nature of the $O-C$ diagram.

Figure 14: $O-C$ diagram of XZ Car exhibiting a phase jump between HJD 2438000
and 2442000, which altered the phase of the pulsation by 0.77 d.

####  3.2.2 UX Carinae

UX Car is a short period Cepheid, possibly linked to the open cluster IC 2581
based on its photometric and kinematic properties (Glushkova et al., 2015).
However, the latest Gaia measurements suggest that the Cepheid is outside the
cluster, with the Gaia EDR3 distance of the star being 1.52 kpc (Gaia
Collaboration, 2016; Bailer-Jones et al., 2021), while the distance of IC 2581
being 2.45 kpc (Kaltcheva & Golev, 2012). This association was also marked
unlikely by Anderson et al. (2013) previously.

This Cepheid was not investigated in terms of period changes before. Based on
our studies, this variable exhibits rapid period jumps, as it can be seen in
Fig. 15. The $O-C$ diagram was calculated using the following elements:

$C_{\textrm{med}}=2453453.824+3.682171\cdot E$

As seen in the figure, the $O-C$ diagram shows a series of period jumps and
rejumps, which is best explained by a series of constant period segments:
before HJD 2425000: $P=3.6822968\pm 0.0000129$ d
HJD 2425000$-$2429000: $P=3.6821750\pm 0.0000285$ d
HJD 2429000$-$2451000: $P=3.6822724\pm 0.0000030$ d
after HJD 2451000: $P=3.6821861\pm 0.0000052$ d

Figure 15: $O-C$ diagram of UX Car, interpreted as a series of period jumps.
Three distinct jumps were covered by the observational data, along with the
possibility of a fourth one: the first occurred at approximately HJD 2424900,
while the second one at HJD 2428900, and the third at HJD 2451000. A fourth
period jump could occur at HJD 2457700, but to confirm this, additional
photometric observations would be necessary.

Since UX Car was found to be a possible binary by Kervella et al. (2019b)
based on proper motion studies (although with low probability, due to the
large separation), binarity could explain the shape of the $O-C$ diagram,
which cannot be interpreted by evolution alone.

####  3.2.3 R Crucis

R Cru is a short period Cepheid that has been frequently studied in the past
owing to its vicinity to and possible membership in the open clusters NGC 4349
(Majaess et al., 2012) or Loden 624 (Anderson et al., 2013). The membership in
the former cluster was rejected by Chen et al. (2015), based on the distance
difference between the cluster and the Cepheid and the fact the R Cru is
brighter than the instability strip of the open cluster.

The binary nature of R Cru was first suspected by Lloyd Evans (1982) based on
its $\gamma$-velocity drift, then the presence of the companion was confirmed
by HST imaging (Evans et al., 2016b) and a possible X-ray emission was also
identified at the position of the resolved companion (Evans et al., 2016a).
This binary companion was also confirmed by Kervella et al. (2019b, a) through
finding a slight offset in separation between the _HST_ and _Gaia_ observation
epochs. They also determined the spectral type of R Cru B, which was found to
be compatible with the X-ray radiation.

Figure 16: $O-C$ diagram of R Cru exhibiting a period jump at HJD 2433000,
which modified the period of the Cepheid until HJD 2446000. The jump–rejump
nature of the $O-C$ diagram provides another indication for the existence of
the binary component.

Despite being relatively bright, no $O-C$ diagram has been constructed yet for
this Cepheid. According to our analysis, R Crucis underwent a period jump
approximately at HJD 2433000 and pulsated with a modified period until
approximately HJD 2446000 (Fig. 16):
before HJD 2432900: $P=5.8257576\pm 0.0000141$ d
HJD 2432900$-$2446100: $P=5.8255291\pm 0.0000109$ d
after HJD 2446100: $P=5.8257880\pm 0.0000061$ d

The presented $O-C$ diagram was calculated using the following elements:

$C_{\textrm{med}}=2448157.512+5.825771\cdot E$

After the phase jump the period of the Cepheid returned to a value slightly
longer than before, which suggests that the period of the Cepheid is
increasing, however this could only be determined by observing another period
jump. Nevertheless, the jump–rejump structure of the $O-C$ diagram could be
linked to the presence of a companion, as evolutionary changes alone cannot
explain this behaviour.

####  3.2.4 DT Cygni

DT Cyg is a short period Cepheid which was frequently observed in the past
decades. The $O-C$ diagram of the Cepheid constructed by Szabados (1991)
showed peculiar period jump–rejump behaviour. This Cepheid was also suggested
to be in a binary system, first by Leonard & Turner (1986), then by Szabados
(1991) based on RV measurements, however, the orbital motion is yet to be
confirmed.

By analysing the obtained $O-C$ diagram of DT Cyg we confirm the previous
statement by Szabados (1991), that a second rapid period change occurred near
HJD 2446000 (Fig. 17). Our $O-C$ diagram was calculated using the following
elements:

$C_{\textrm{med}}=2452688.108+2.499063\cdot E$

The pulsation of DT Cyg can be characterized with the following periods:
before HJD 2441300: $P=2.4992250\pm 0.0000093$ d
HJD 2441300$-$2445900: $P=2.4990380\pm 0.0000116$ d
after HJD 2445900: $P=2.4992363\pm 0.0000025$ d

Figure 17: $O-C$ diagram of DT Cyg, exhibiting a clear case of period jump.

It is not possible to determine how the period changed before HJD 2428600
based on the current diagram. The peculiar structure of the $O-C$ diagram
could be linked to the presence of a companion, since evolutionary changes
alone cannot explain it. Moreover, as it can be seen from the values above,
the average pulsation period of DT Cyg became shorter after the rejump, which
suggests that the star belongs to the second crossing Cepheids.

####  3.2.5 BN Puppis

BN Pup in an intermediate period Cepheid which was not investigated in terms
of period changes before. The Cepheid was suspected to be connected to NGC
2533, but this turned out to be unlikely based on the large distance and age
differences (Havlen, 1976; Anderson et al., 2013). The binary nature of the
object was not investigated before, with only limited amount of RV data being
available (Coulson & Caldwell, 1985; Pont et al., 1994; Storm et al., 2011).

Figure 18: $O-C$ diagram of BN Pup, with a phase jump of 0.2 d visible at
approximately HJD 2444500 which indicates binarity of the Cepheid.

Based on the compiled $O-C$ diagram plotted in Fig. 18, we have some
indication that the Cepheid might have a companion. The phase jump in the case
of this Cepheid is less ambiguous than in other cases since it is detected in
data from a single source (DASCH photometry) and it is also supported by the
precise photometric measurements of Pel (1976). The presented diagram was
calculated using the following elements:

$C_{\textrm{max}}=2452782.845+13.672693\cdot E$

The phase jump occurred at approximately HJD 2444500 and altered the phase of
the pulsation with $\sim$0.2 d. The values of the pulsation period before and
after the phase jump are in agreement within errors. To determine whether a
long period signal can be extracted from the RV data, we reanalysed the
measurements of Coulson & Caldwell (1985), Pont et al. (1994), and Storm et
al. (2011). According to the Fourier decomposition of the signal, a period of
$\sim$4300 d can be extracted, but the sparseness and the uncertainty of the
RV data does not allow us to determine whether it is truly connected to the
binary nature of the Cepheid. The phase jump exhibited by the $O-C$ diagram
could suggest that BN Pup is possibly a member in a binary system, if one
assumes an empirical correspondence between the two, although further RV
measurements are necessary to validate this assumption.

###  3.3 Overtone Cepheids

The remaining two Cepheids that were found to exhibit peculiar $O-C$ diagrams
show erratic or semi-erratic period changes, which, to our current best
understanding, can only be explained through a set of linear segments
(consecutive intervals of random, but otherwise constant-period pulsations)
and is a characteristic of overtone pulsators. During the observed timeframe,
none of these Cepheids showed recurrence in the change of their pulsation
period.

####  3.3.1 X Lacertae

X Lac is a short period Cepheid which is possibly connected to the open
cluster FSR 384 (Glushkova et al., 2015) (although this association was marked
as unlikely by Anderson et al. (2013) before). The period changes of this
Cepheid were extensively studied by Evans et al. (2015), finding that it shows
an erratic period change. Apart from the erratic period change, the low
pulsation amplitude also points towards the overtone nature of the pulsation
(Storm et al., 2011), which makes this object interesting, as only a few
overtone pulsators are known with such a long period.

Figure 19: $O-C$ diagram of X Lac exhibiting an erratic behaviour which can
best be described with a series of rapid period changes.

Our analysis of the available photometric data yielded very similar results to
those presented by Evans et al. (2015): the resulting $O-C$ diagram is best
fitted with a series of linear segments as shown in Fig. 19. The presented
$O-C$ diagram was calculated with the following elements:

$C_{\textrm{med}}=2442737.707+5.444322\cdot E$

According to our analysis, the $O-C$ diagram of X Lac can be best described
with the following periods:
before HJD 2428000: $P=5.4442119\pm 0.0000659$ d
HJD 2435500$-$2437000: $P=5.4445175\pm 0.0001767$ d
HJD 2438500$-$2441500: $P=5.4456094\pm 0.0002935$ d
HJD 2442000$-$2445000: $P=5.4443820\pm 0.0000715$ d
HJD 2447500$-$2449500: $P=5.4444422\pm 0.0001904$ d
HJD 2452500$-$2455000: $P=5.4451025\pm 0.0003423$ d
after HJD 2455000: $P=5.4448548\pm 0.0000447$ d

Our analysis shows that the period change sequence of X Lac is indeed erratic
throughout the observed timeframe, which supports the assumption that this
Cepheid is an overtone pulsator.

####  3.3.2 EU Tauri

EU Tau is a short period Cepheid and an overtone pulsator (Gieren et al.,
1990). The binary nature of the object was first suspected based on RV data
(Gorynya et al., 1996), however a later study by Evans et al. (2015) found
this unlikely. The period changes of EU Tau were investigated by Berdnikov et
al. (1997) and Evans et al. (2015), finding first a linear, then erratic
period change, respectively.

The obtained $O-C$ diagram (Fig. 20) was calculated using the following
elements:

$C_{\textrm{med}}=2432499.902+2.10241\cdot E.$

According to our analysis, which matches with the finding of Evans et al.
(2015), the $O-C$ diagram can be best described as three constant period
segments:

before HJD 2435000: $P=2.1023336\pm 0.0000028$ d
HJD 2437000$-$2451000: $P=2.1025187\pm 0.0000017$ d
after HJD 2452500: $P=2.1022962\pm 0.0000047$ d

Since the transitions between these segments were not covered by observations,
their nature cannot be established. To determine whether the pulsation periods
are truly erratic, or if there is any connection between them, further
observations are required.

Figure 20: $O-C$ diagram of EU Tau, showing two rapid period changes, between
which the pulsation of the star can be described with constant period values.

##  4 Collective analysis on period change rates and fluctuations

###  4.1 Colour-magnitude diagram and period change rates

Apart from the Cepheids detailed in the previous sections, we have obtained
$O-C$ diagrams for 132 additional Cepheids (see Table A.1 for the complete
list), which all show a parabolic shape, thus dominated by the evolution of
the variable star. As a result, our Cepheid sample contains 141 stars that
exhibit clear evolutionary changes in their $O-C$ diagrams, including those
that were termed peculiar based on additional features in their $O-C$ diagrams
in Sect 3.1.

The recent advancement in astrometry owing to the measurements of the Gaia
space probe allows for the construction of the colour-magnitude diagram (CMD)
for the current set of Cepheids with an unprecedented accuracy. Figure 22
shows the CMD based on the Gaia DR2 and Gaia EDR3 data (Gaia Collaboration,
2016, 2018, 2021; Lindegren et al., 2021; Riello et al., 2021) as a
comparison. Since neither of the data releases contained measurements of the
Galactic extinction for all the Cepheids, we applied the attenuation
correction on the individual photometric measurements based on the colour-
excess values listed in the DDO database of Galactic Classical
Cepheids222http://www.astro.utoronto.ca/DDO/research/cepheids/cepheids.html.
For the Cepheids which were not listed on the website, or scattered far from
the rest of the variables on the CMD due to the false value of the colour
excess (KL Aql, V898 Cen) we used the more recent measurements (Schmidt, 2015;
Groenewegen, 2020), but in the case of V898 Cen the resulting position was
still slightly different from the rest of the dataset. For the additional
Cepheids that have no published colour-excess values at all (EV Cir, V5738
Sgr, GP Mus, V2744 Oph, CE Pup, DU Pyx and V520 Vel) we assumed the extinction
values inferred by Anders et al. (2019). The colour excess values were then
converted to the Gaia passband system using the scaling given in Casagrande &
VandenBerg (2018).

As seen in Fig. 22, the data from EDR3 show a much smaller scatter and exhibit
a much clearer trend for the pulsation period of the Cepheids, which
demonstrates well the improvement of the Gaia data compared to the previous
data releases. On the right panel of Fig. 22 we attempted to draw the limiting
lines of the strip populated by the Cepheids; for the approximate alignment of
these lines we omitted all Cepheids with uncertain or non-published extinction
mentioned above. The resulting lines are notably not parallel, which indicates
that for longer period Cepheids the possible range of colour index values is
broader, which means that the instability _strip_ of Cepheids could in fact be
an instability _wedge_ , as proposed earlier by Fernie (1990b). This also
matches the results previously obtained from the observations of IUE for a
smaller sample of Cepheids (Evans, 1992). To examine this possibility even
further, more accurate extinction values would be required, which are planned
to be published with the complete third data release of Gaia in 2022.

Figure 21: Colour-magnitude diagrams of the investigated Cepheids calculated
from the Gaia DR2 (left) and Gaia EDR3 (right) datasets. The dashed lines show
the assumed edges of the instability strip. The error bars show the
uncertainties arising due to the parallax and the reddening errors. Figure 22:
Period-luminosity relation of the investigated Cepheids, calculated from the
Gaia DR2 (left) and Gaia EDR3 (right) datasets. The individual points are
coloured by the dereddened Gaia $BP-RP$ values. The right panel demonstrates
well the improvement of quality in the Cepheid distances between the two data
releases.

By fitting the $O-C$ diagrams of the investigated Cepheids with second order
polynomials, along with a sinusoidal modulation superimposed when
statistically motivated, and then applying the method described by Sterken
(2005) we could determine the rates of period change for each of the variable
stars. Our results and conclusions on period change rates essentially match
those presented by Turner et al. (2006): the values inferred by us are well
explained by the evolutionary models discussed in that work and the ratio of
Cepheids showing positive and negative period changes is also identical, if
one considers the possible selection effects (70% of our sample Cepheids show
positive period changes, while this value was 67% in Turner et al. (2006)).
The deviations from evolutionary models we find in our sample are also very
similar to the previous results: several short period Cepheids exhibit a
period change rate that fall between the value ranges expected for first and
third crossing Cepheids, while several intermediate period Cepheids show a
period change rate that is smaller than predicted by evolutionary models.

As it was discussed by Neilson et al. (2012), the inclusion of convective core
overshooting and enhanced mass loss can explain these deviations. Moreover,
they showed that the difference between the ratio of stars showing positive
and negative period changes for observations and predictions (where the
unmodified evolutionary models estimate significantly larger fraction of
positive period change Cepheids, namely 85%) can also be explained through
mass loss, although they note that there is currently no explanation for large
mass loss rate they found to be required. Complementary to this work, Miller
et al. (2020) investigated how rotation and convective core overshooting alter
the distribution of period change rates, finding that they do not assert
significant influence on then, despite being important parameters for the
evolution of Cepheids.

As it was noted by Turner et al. (2006), the period change rate could be an
important probe for the estimation of stellar properties. To test this
observationally, we correlated the period change rates of Cepheids to their
colours for a moderate range of pulsation periods (between 2 and 5 days) as
seen in Fig. 23. The period range was chosen based on the CMD, as the relevant
effect was visible primarily for short period Cepheids, while for longer
periods the scatter on the CMD was too large for detecting a meaningful
effect. As seen in the plot, bluer Cepheids exhibit slightly larger period
change rates, which decrease toward the red edge of the instability strip. The
individual points are coloured by the pulsation period length, which show no
significant correlation with either the period change rate or the colour
within the considered period range. This agrees well and observationally
validates the statement in Turner & Berdnikov (2004), in which they note that
at a given pulsation period, Cepheids at the high-temperature edge of the
instability strip should exhibit larger rate of period change than those at
the low-temperature edge, due to the former having larger masses.

This demonstrates that the period change rate can indeed provide a valuable
observational probe for the determination of physical parameters. In addition
to the pulsation period, which can be used to estimate both the “vertical” and
the “horizontal” position of the Cepheid within the instability strip, and the
corresponding physical parameters owing to the period-luminosity and period-
colour relations (assuming an underlying mass-luminosity relation and a
pulsation model, see, e.g., Beaulieu et al. 2001), the period change rate can
independently probe the “horizontal” direction. Since the period and period
change rate values are not tightly correlated due to the various crossing
modes, mass loss, and convective core overshooting rates, this correlation is
not bound to the pulsation period, which is also well shown in Fig. 23.
Henceforth, the period change rate could be used to enhance the precision of
colour and temperature estimation of Cepheids when coupled with the period
values. Further investigation of this correlation is beyond the scope of this
work, however we note that it would be worthwhile to analyse to what extent
can it be used for the inference of physical properties. Combined with the new
data from Gaia it could provide a simple way to constrain the effective
temperature of the variables compared to the conventional spectroscopic
method, yet it could prove to be more precise than the current, solely colour
based empirical relations.

Figure 23: Correlation plot of the colours and period changes of short period
Cepheids (pulsation period range of 2-5 days). The red line shows the best
linear fit to the whole dataset, while the grey shaded area shows the
$1\sigma$ quantile of the fit. Period changes decrease within errors towards
the red edge of the instability strip.

Owing to the long temporal coverage of the newly calculated $O-C$ diagrams one
can not only calculate the period change rate, but its acceleration as well:
in such cases the resulting $O-C$ diagrams systematically deviate from the
parabolic fit, since their complete description would require higher-order
polynomials, as it was also noted in previous works (Szabados, 1983; Fernie,
1984; Turner et al., 2006). In our sample, most of the long period Cepheids
exhibit such $O-C$ diagrams (most notably, EV Aql, S Vul and SV Vul).
Naturally, all Cepheids are expected to show such high order terms in their
$O-C$ diagrams, but due to the short observed timeframes these terms are
usually negligible for Cepheids that exhibit smaller period change rates.

To visualize this expectation we attempted to recover the higher order period
change from theoretical models: by adopting the evolutionary trajectory models
and instability strip edges from Georgy et al. (2013) and Anderson et al.
(2016) respectively, and the period-mass-luminosity-temperature relations of
De Somma et al. (2020), we calculated how the pulsation period would change
for a single star due to stellar evolution. An example of the results is shown
in Fig. 24: the theoretical models also yield non-linearly changing period,
which validates the expectation for higher order terms in the $O-C$ values.
Currently, the interpretation of these higher order terms is not yet complete,
but they will also yield additional valuable information about the pulsation
of these stars similarly to the period change rates.

Figure 24: Period values estimated for the second crossing of a 7M⊙ star at
different times (right panel), showing a curved evolution indicating a non-
linearly changing period, with the left panel showing the corresponding
evolutionary trajectory (with the important segment highlighted).

###  4.2 Period fluctuations

Apart from examining the overall rate of period change for the Cepheids in our
sample, the $O-C$ diagrams also made it possible to investigate random
fluctuations present in the pulsation. An example for such fluctuation signal
can be seen in the $O-C$ diagram of SV Vul (Fig. 25): it shows a long-term
trend corresponding to the evolution of the Cepheid and a quasi-periodic
wavelike signal superimposed, with varying shape. Such wavelike signals cannot
be explained by LiTE, as their amplitudes are too large compared to their
period, and the shape of the waves also changes in time. The physical
explanation for such modulations is that random fluctuations in the pulsation
period accumulate and cause wavelike signals of varying shapes. For long, this
was the only classical Cepheid for which fluctuations were detected (Turner et
al., 2009). Recently, Rodríguez-Segovia et al. (2021) also found that five
additional long period Cepheids in the LMC reliably show such large amplitude
fluctuations, highlighting their ubiquity. It was expected that fluctuations
are present in most Cepheid variables as well, but detection would require
continuous space-based photometry, where consecutive epochs can be observed
Turner et al. (2010). Multiple studies attempted to detect and quantify
fluctuations this way, as we discussed in Sect. 1.

Figure 25: $O-C$ diagram of SV Vul (top panel), showing a long-term change
corresponding to stellar evolution, with superimposed period fluctuations, and
the residual $O-C$ diagram (bottom panel) after subtracting the evolutionary
changes.

As we will show below, such period fluctuations are more common among Cepheids
than previously thought. It has been shown that longer period variables, like
RV Tauri or Mira stars, can exhibit long term quasi-periodic waves in their
$O-C$ diagrams due to the period fluctuations, which becomes more prominent
for longer pulsation periods (Percy et al., 1997; Percy & Colivas, 1999;
Molnár et al., 2019), however, no such conclusions were drawn for classical
Cepheids. In the sample of stars we investigated, we found that 51 Cepheid
showed wavelike signals in the residual $O-C$ diagrams that most probably
originated from fluctuations, apart from the Cepheids that were termed
‘peculiar’ in the previous sections. To select these Cepheids from our sample,
we performed F-tests on their $O-C$ diagram fits to analyse the residuals of
the models and to check whether they indicate that the use of more complex
models (i.e. which also fit for an assumed wavelike modulation) are validated.
We compared three different models for the test: the constant period model
(linear $O-C$ diagram), the linear period change model (parabolic $O-C$
diagram) and the linear period change plus wavelike modulation (superposition
of a parabolic and a sinusoidal change), in which case we simultaneously fit
for both the evolutionary changes and the modulation. Table B.1 of the
Appendix show the results obtained for the different model pairs. We included
Cepheids in the fluctuating sample when the F-test conducted for the
comparison of the parabolic and parabolic plus wave fits yielded an
F-statistics larger than 10, which means that the more complex model was
favoured at a significance level of 0.99 (with the difference between the
degrees of freedom of the compared models being $DF=3$). To remove possible
contaminations, or Cepheids for which the fitting code yielded erroneous
results, we rejected every star where the period of the sinusoidal fit was
either shorter than 3000 days (8-10 times the folding time used for the
construction of the $O-C$ diagrams, to avoid sampling effects) or longer than
five quarters of the observed time frame (in which case the sinusoidal
waveform overfitted the parabolic change).

In Fig. 26 we plotted the distribution of the investigated Cepheids according
to their pulsation periods along with the information about what fraction of
Cepheids show fluctuations in their $O-C$ diagram. As seen on the plot, all
long period ($\log_{10}P>1.25$) Cepheids show fluctuations (with the sole
exception being EZ Vel, however the temporal coverage of its photometry did
not allow for a detailed fit for the $O-C$ diagram), which was expected based
on the behaviour RV Tauri stars with similar periods. The ratio of stars
exhibiting such modulations starts to drop for intermediate period stars and
falls to a value of $\sim$25–40%, which remains approximately the same for
shorter period Cepheids too. We suspect that at least a fraction of the stars
that were not included in the fluctuating sample also show wavelike
modulations of such nature as well, however their overall amplitude was lower
than the scatter of the residual $O-C$ diagram, hence they remained below
detectability.

Figure 26: Distribution of Cepheids according to their pulsation periods. The
blue histogram shows all investigated Cepheids, while the yellow ones show
only those that show a wavelike signal in their $O-C$ diagram superimposed the
trend corresponding to the evolution. In addition, the red histogram shows the
fraction corresponding to the Cepheids termed ‘peculiar’ in the previous
sections, while showing wavelike signals in the $O-C$ diagrams.

It is important to note that the apparently general fact, that all longer
period ($\log_{10}P>1.25$) Cepheids exhibit period fluctuations to some
extent, has not been known before, thus it can have important implications for
the calibration of the distance ladder via PL relation of Cepheids.
Extragalactic Cepheids are generally difficult to observe, as their light
curve spans only a few hundred or a thousand days at most, with sparse
sampling. Consequently, it is impossible to determine their fluctuation-free,
evolution based “true period”, and the resulting period value can be affected
by the cumulative effect of the period fluctuations. Whereas the shorter
period ($1.25<\log_{10}P<1.48$) population of these Cepheids exhibit only
smaller fluctuations and the possible error on the inferred period values
remains below a few hundredths of a day, for longer period Cepheids (e.g., for
one of the longest period Cepheids in our sample, for S Vul) such fluctuations
could lead to a cumulative effect that alters the inferred period by more than
0.5 day, which would then add a detectable term to the final uncertainty.
While such effects might not affect the PL relationship significantly due to
the use of a larger sample of stars or the inclusion of shorter period
variables (for which the period determination is more accurate) as was done in
Riess et al. (2019), it could severely increase the uncertainty of the results
based entirely on long or ultra-long period Cepheids (e.g., Bird et al. 2009).

We also measured how the strength and size of these accumulated period
fluctuations change for different pulsations periods. For this purpose, we
applied the Eddington–Plakidis method (E–P method) (Eddington & Plakidis,
1929) on each Cepheid in our sample that showed signs of fluctuations.
According to this method, one must whiten the calculated $O-C$ values for the
effect of evolution, then take the absolute values of all delays
$u(x)=|a(r+x)-a(r)|$ for each $r$th maximum, $a(r)$, for every possible $x$
cycle separation in the residual dataset. According to Eddington & Plakidis
(1929), if the irregular signal is present due to the fluctuation of the phase
and period at the same time, then the $\langle u(x)\rangle$ means of all
accumulated delays should be related to the random period fluctuation
$\epsilon$ by Eq. 4:

$\langle u(x)\rangle^{2}=2\alpha^{2}+x\epsilon^{2}$ (4)

where $\alpha$ characterizes the errors in the measured times used for the
$O-C$ data points.

We produced Fig. 28 by comparing the resulting $\epsilon$ period fluctuation
parameter (obtained by fitting the E–P diagrams for $x<100$ cycle separations)
with the corresponding pulsation period, which shows that there is a
significant connection between the size of fluctuations and the pulsation
period.

However, the E–P method has two drawbacks which are inherent to its
definition, one of which is that one has to manually set what maximal cycle
separation is considered for the linear fit described above. There is no
statistical background on what one should choose for the maximal cycle
separation, and since Cepheids show a very large variation of such E–P
diagrams, one has to set this value manually. This makes the method infeasible
for large datasets, since one either has to tune this value for each star, or
pick a general value, which might not be optimal for every sample member. Due
to this, the resulting $\epsilon$ fluctuation parameter will also be more
uncertain, and its value will be based on a modelling choice. An example for
this is also shown in Fig. 27: as seen in the figure, multiple ranges could be
considered for the fitting of the diagram, which sets a range for the final
fluctuation parameter values as well. Additionally, the E-P method also has
the disadvantage of not being able to provide an upper limit for the
fluctuation parameter, therefore if the $O-C$ diagram is dominated by
uncorrelated noise, the resulting E–P diagram will show no structures, unlike
in Fig. 27. Hence, no linear fitting can be carried out in those scenarios.

To circumvent this issue, we calculated a correlation diagram similar to Fig.
28 using the result of the $O-C$ diagram fits instead. For each of the
Cepheids where fluctuations were found by the F-test, we simply correlated the
measured amplitude of the wavelike modulation with the pulsation period of the
star. Unlike in the case of the Eddington-Plakidis method, here we were able
to determine a detectability limit even when the amplitude of the modulation
could not be measured, i.e., when no fluctuations were found: in this case the
detectability limit could still be determined as the maximal value of the
Fourier spectrum calculated for the residuals after fitting and subtracting
the evolutionary changes.

Figure 27: The Eddington–Plakidis diagram of EV Aql, along with the calculated
fit for cycle separations $x<100$. As the diagram shows, multiple choices are
possible for this maximal value, which can influence the final result for the
period fluctuation parameter.

Through this procedure we obtained Fig. 29, which shows similar trends as the
Eddington–Plakidis method based diagram, except for the shorter period
Cepheids. As expected, the detectability limit drawn out by the non-detections
increases slightly with period as the light curves of the Cepheids become more
and more complex, thus more difficult to fit, but still, a reasonable amount
of fluctuating Cepheids were found throughout the investigated period range,
which defines the underlying trend reliably. As it is shown by the average
curve in Fig. 29, the amplitude of the fluctuations decreases linearly against
the logarithm of the period for longer period Cepheids ($\log_{10}P>1.25$),
only to reach a minimum for intermediate period Cepheids around a period of
$\log_{10}P\approx 0.8$ and increases again for short period stars. We note
that the Cepheids where we found no modulations, and hence are marked as non-
detections on the diagram, can significantly influence the position of this
minimum. However, the fact that the largest detected fluctuations in the
intermediate period range stay well below the maximal ones found for both
shorter and longer period Cepheids supports the notion that there is a minimum
for these modulations in the period range of bump Cepheids, suggesting a
quenching mechanism for these variables. Moreover, Fig. 29 also supports the
LiTE explanation described above for the modulation found for RX Aur and Y
Oph, as both stars are outliers for the trend defined by the fluctuations.

We also investigated how easing the F-test based constraints affects the
results: if we lower the confidence level from 0.99 to 0.95, the F-test based
procedure favours the inclusion of further 23 Cepheids into the fluctuation
sample. Most of these stars fall in the period range of bump Cepheids, with
the amplitudes of the wavelike signals being close or at the noise level of
their $O-C$ diagrams. In case these stars are included, the fraction of
Cepheids showing fluctuations jumps to a level of $\sim$50% instead of 25-40%,
and remains constant thorough the lower end of the pulsation period range,
suggesting an even higher frequency for the presence of these modulations. In
terms of fluctuation strength, since the fluctuation amplitudes are close to
the detection limits assumed for these stars in Fig. 29, which were taken into
account for the fitting, neither the actual position of the found fluctuation
minimum, nor the general trend changes significantly if we ease the
constraints.

Figure 28: Correlation between the Eddington-Plakidis $\epsilon$ parameter
calculated for cycle separations $x<100$ and the pulsation period. The blue
and red dashed curves show the two fits described in the text. The black curve
shows the empirical average, while the blue region shows the empirical
standard deviation around it. For the fitting of the coloured curves, the
peculiar Cepheids were not taken into account. Figure 29: Correlation between
the measured fluctuation amplitudes and the pulsation periods. The legend is
the same as for Fig. 28. The non-detection limits calculated from the Fourier
spectrum for individual Cepheids are marked with grey triangles.

As seen in the figures, multiple interpretations are possible: since there are
two separate point clouds in the resulting plot, we can either assume that the
fluctuation features observed in them are of the same origin, or of different
ones.

By assuming that the fluctuations detected for short and long period Cepheids
are of the same origin, the correlation plot can be fitted by a quadratic
function. This fit shows that the size of fluctuations decreases from the
longer periods towards shorter periods, reaching a minimum at the lower end of
the period range of bump Cepheids, then growing back again for shorter period
stars. This picture agrees with the presence of a quenching mechanism for bump
Cepheids, and it also follows well the average curve.

If one assumes that the two parts of the fluctuation diagram are of different
origins (i.e. the fluctuations are of different nature for long and short
period stars), then the plot can be fitted by a broken linear curve. One wing
of this broken linear curve on the Eddington-Plakidis based diagram (Fig. 28)
shows that for long period Cepheids the fluctuations scale linearly with the
pulsation period, which agrees with the findings of Percy & Colivas (1999) for
longer period Mira stars, indicating that this fluctuation effect is a similar
period fluctuation in both types of variable stars. However, in this case
there is no explanation for the short period wing, as this fit would suggest
that fluctuations detected for shorter period Cepheids are of different origin
than the former one. In the case of the amplitude-based diagram, the broken
linear fit shows a similar linear trend for the longer period Cepheids. Based
on these fits, the linear scaling present between the fluctuations and the
period for long period Cepheids weakens considerably on intermediate and short
period ranges, as the level of fluctuations seems not to change as
significantly over this range.

The datapoints at short periods ($\log_{10}P<0.8$) also show a much larger
scatter around the model fits compared to other period ranges, with a clear
gap in the amplitude distribution (around $\log_{10}A\approx-1.0$, where we
find several Cepheids with both significantly larger and smaller fluctuations
in this period range). This can indicate that there are more than one type of
fluctuations present at short periods, one for normal and one for overtone
Cepheids, presumably (with smaller and larger fluctuations, respectively).
Their comprehensive modelling would require a more complete sample of shorter
period Cepheids, but such an analysis is beyond the scope of this paper.
Nevertheless, the supposed models show that either the change in the nature of
fluctuation signals occurs at, or the overall strength of the fluctuations
have a minimal value close to the period range of the bump Cepheids, both of
which suggests that the pulsation of the bump Cepheids is subject to less such
random fluctuations and provides an additional proof for the increased
stability of the pulsation of these stars.

##  5 Summary

In this work we presented updated or completely new $O-C$ diagrams for a total
of 148 Classical Cepheids. Out of these stars, four Cepheids (RX Aur, UY Mon,
Y Oph, and AW Per) show signs of possible LiTE (only the case of AW Per and RX
Aur has been known before). Further five Cepheids (XZ CMa, UX Car, R Cru, DT
Cyg, and BN Pup) exhibit $O-C$ diagrams which either further confirm or raise
the question of their binary nature. For the interesting cases of BY Cas, V532
Cyg, DX Gem and VZ CMa we found large amplitude fluctuations influencing the
pulsation of the variables, while for IR Cep we found a similarly large
amplitude and complex, yet stable wavelike signal superimposed on the
evolutionary changes of the star, which could either be caused by fluctuations
or binarity. The proper understanding of these peculiar variations would
require further analysis of these stars.

By analysing the obtained set of $O-C$ diagrams showing evolutionary changes,
we measured the rates of period change for the individual stars and compared
these values to those given by stellar evolution models. The ratio of the
number of Cepheids showing positive and negative period changes in our random
sample agrees with the ratio given by models that include enhanced mass loss.
By plotting the CMD of the obtained sample of Cepheids using the recently
published Gaia EDR3 data, we could show that the rates of period change do not
increase as consistently toward the cooler and brighter stars as one would
expect based on the models, but the rates of period change vary systematically
from the blue edge to the red edge of the instability strip. Considering this,
it possibly yields an observational evidence that the rate of period change
carries significant information about the physics of these variables.

After fitting and analysing the obtained $O-C$ diagrams, we found that a
significant fraction of the Cepheids in our sample exhibit detectable period
fluctuations in the form of quasi-periodic waves, similar to those found in
the case of other, longer period pulsating stars. By measuring the strength of
the fluctuations using the Eddington-Plakidis method and a model fitting based
approach, we found that the period fluctuation scales linearly for longer
period Cepheids (in a logarithmic sense), which is similar to the behaviour of
Mira stars, while for shorter period Cepheids this scaling significantly
weakens or disappears. The presence of such fluctuations and their increasing
strength for long period Cepheids can have important implications for PL
relation based distance determinations, as they could add an additional
uncertainty term to the pulsation period, that was previously unaccounted for.
Due to the detection limits set by the scatter in the data, we could only
detect these fluctuations in about a third of the investigated short and
intermediate period Cepheids. We showed that the explanation and analysis of
the dependence of fluctuation strength on the periods of short period Cepheids
depends largely on whether one or two fluctuation processes are assumed: in
the first case the fluctuation strength shows a minimum in the range of bump
Cepheids, then it increases back at short periods, while in the latter case we
find that the fluctuation strength depends on only weakly or is independent of
the period for shorter period stars. Either way, the scatter among the
fluctuation strengths increases for short period Cepheids, which suggests a
non-negligible contribution from overtone Cepheids that exhibit larger
fluctuation signals. Whichever interpretation is correct, the lack of strongly
fluctuating bump Cepheids and the fluctuation minimum found by the joint fit
in their period range suggests the increased stability of the pulsation of
these stars, which is in agreement with the presence of a quenching mechanism.

## Data availability

All source data used for our analysis are openly available online, except for
the measurements carried out at the Piszkéstető Observatory, which are
published as online material accompanying the article. The resulting $O-C$
files, from which the conclusions were drawn, are also published in the online
supplement and will be accessible in an accompanying Vizier table. The data
and the code used for our analysis is available on the GitHub page of the
author: https://github.com/Csogeza/O-C_extract.

## Acknowledgements

We thank the referee for their useful comments that helped to improve this
paper. The research was completed with the extensive use of Python, along with
the numpy (Harris et al., 2020), scipy (Virtanen et al., 2020) and astropy
(Astropy Collaboration et al., 2018) modules, and with the R software (R Core
Team, 2020). Some of the data used in the article were made available to the
community through the Exoplanet Archive on behalf of the KELT project team.
This research made use of NASA’s Astrophysics Data System (ADS). For the data
search and retrieval the authors made extensive use of the Vizier, CDS and
Simbad websites of the Strasbourg astronomical Data Center
(https://cds.u-strasbg.fr/). This work has made use of data from the European
Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. We acknowledge with thanks
the variable star observations from the AAVSO International Database
contributed by observers worldwide and used in this research. The research
leading to these results has been supported by the Hungarian National
Research, Development and Innovation Office (NKFIH) grants K-129249,
GINOP-2.3.2-15-2016-00003 and Élvonal KKP-137523 (‘SeismoLab’), and by the
Lendület LP2018-7/2021 grant of the Hungarian Academy of Sciences. Cs. K.
acknowledges the support provided from the ÚNKP-20-2 and ÚNKP-21-2 New
National Excellence Programs of the Ministry of Innovation and Technology from
the source of the National Research, Development and Innovation Fund.

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authors:
G. Cs\"ornyei, L. Szabados, L. Moln\'ar, B. Cseh, N. Egei, Cs. Kalup,
  V. Kecskem\'ethy, R. K\"onyves-T\'oth, K. S\'arneczky, R. Szak\'ats

submitter: Geza Attila Csornyei