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We review the phenomenology of classical Cepheids (CCs), Anomalous Cepheids (ACs) and type II Cepheids (TIICs) in the Milky Way (MW) and in the Magellanic Clouds (MCs). We also examine the Hertzsprung progression in different stellar systems by using the shape of I-band light curves (Fourier parameters) and observables based on the difference in magnitude and in phase between the bump and the minimum in luminosity. The distribution of Cepheids in optical and in optical–near infrared (NIR) color–magnitude diagrams is investigated to constrain the topology of the instability strip. The use of Cepheids as tracers of young (CCs), intermediate (ACs) and old (TIICs) stellar populations are brought forward by the comparison between observations (MCs) and cluster isochrones covering a broad range in stellar ages and in chemical compositions. The different diagnostics adopted to estimate individual distances (period–luminosity, period–Wesenheit, period–luminosity–color relations) are reviewed together with pros and cons in the use of fundamental and overtones, optical and NIR photometric bands, and reddening free pseudo magnitudes (Wesenheit). We also discuss the use of CCs as stellar tracers and the radial gradients among the different groups of elements (iron, α, neutron-capture) together with their age-dependence. Finally, we briefly outline the role that near-future space and ground-based facilities will play in the astrophysical and cosmological use of Cepheids.

Distance measurements beyond geometrical and semi-geometrical methods, rely mainly on standard candles. As the name suggests, these objects have known luminosities by virtue of their intrinsic properties and play a major role in our understanding of modern cosmology. The main caveats associated with standard candles are their absolute calibration, contamination of the sample from other sources and systematic uncertainties. The absolute calibration mainly depends on their chemical composition and age. To understand the impact of these effects on the distance scale, it is essential to develop methods based on different sample of standard candles. Here we review the fundamental properties of young and intermediate-age distance indicators such as Cepheids, Mira variables and Red Clump stars and the recent developments in their application as distance indicators.

In the context of a project aimed to provide an updated theoretical scenario for various classes of radially pulsating stars, we present the first results obtained for anomalous Cepheids. By adopting reliable and updated evolutionary prescriptions concerning the luminosity levels for various core He-burning stellar models with masses suitable for entering the instability strip, we have computed nonlinear convective pulsation models for both fundamental and first-overtone mode anomalous Cepheids by exploring the impact of varying the metal abundance as well as the efficiency of super-adiabatic convection. These numerical simulations have allowed us to retrieve the boundaries of the instability strip and all relevant pulsation properties, namely period, amplitude, bolometric light and radial velocity curves. This theoretical scenario has been transformed into the Gaia photometric system to derive the first theoretical Period–Luminosity–Colour and Period–Wesenheit relations in the Gaia bands.

The analysis of fifty empirical period-radius relations and forty-three empirical period-luminosity relations is performed for the Cepheids. It is found that most of these relations have significant systematic errors. A new metrological method is suggested to exclude these systematic errors using the new empirical metrological relations and the empirical temperature scale of the various samples of the Cepheids. In this regard, the reliable relations between the mass, radius, effective surface temperature, luminosity, absolute magnitude on the one hand, and the pulsation period on the other hand, as well as the reliable dependence of the radius on the mass are determined for the Cepheids of types δ Cephei and δ Scuti from the Galaxy. These reliable relations permit us to accurately determine the empirical value of the pulsation constant for the Cepheids of both types for the first time. It is found that the pulsation constant very weakly depends on the pulsation period of the Cepheid, contrary to the known theoretical calculation. Hence, the Cepheids pulsate almost as a unified whole and homogeneous spherical body in wide ranges of a star’s mass and evolutionary state with an extremely inhomogeneous distribution of stellar substance over its volume. Therefore, it is first suggested that the pulsation of the Cepheid is, first of all, the pulsation of the almost unified whole and homogenous shell of its gravitational mass. This pulsation is triggered by well-known effects; for example, the local optical opacity of the stellar substance and overshooting, using the usual pulsation of the stellar substance.

A revision of the orbital parameters of the spectroscopic binary Cepheid TX Del, often classified as a BL Her subtype of T2 variables, has been carried out. The pulsation radius is determined from two series of photometric observations, including Gaia DR3: for the () bands and for the () bands, respectively. The amplitude of the radius variations is about . The new radius estimates are significantly smaller than the previous estimates, which were in the range of 37 to 60 . Using the Bayesian distance, pc, the absolute values of TX Del are estimated to be mag and mag, respectively. The absolute magnitude and mean radius of TX Del are in excellent agreement with the predictions of the ‘‘period–luminosity’’ and ‘‘period–radius’’ relations for T2 Cepheids with a period of days; TX Del is approximately fainter than classical Cepheids with the same period. It is shown that for orbital plane inclination angles the satellite can be more massive than the main component, which leads to an evolutionary scenario with mass exchange. The current Roche sphere filling degree is . The orbit of the binary system TX Del calculated based on astrometric data from Gaia DR3 and radial velocity is limited by a kpc torus with the inner and outer radii equal to 7.2 and 8.3 kpc, respectively, while the vertical velocity is km s; the period of galactovertical oscillations is about 77 Myrs. The tangential velocity is 220–255 km s. The high metallicity and kinematic characteristics allow us to refer TX Del to the population of the old disk of the Milky Way with a vertical scale kpc (or to a thick disk with kpc). At the same time, in terms of pulsation characteristics (period, luminosity, and position in the instability strip), TX Del is close to Population II Cepheids with (including globular cluster Cepheids) that are at the evolutionary stage after the horizontal branch. To finally clarify the evolutionary status of such variables, additional evolutionary and pulsation calculations are required over a wider range of periods and luminosities.

We present a machine learning method to estimate the physical parameters of classical pulsating stars such as RR Lyrae and Cepheid variables based on an automated comparison of their theoretical and observed light curve parameters at multiple wavelengths. We train artificial neural networks (ANNs) on theoretical pulsation models to predict the fundamental parameters (mass, radius, luminosity, and effective temperature) of Cepheid and RR Lyrae stars based on their period and light-curve parameters. The fundamental parameters of these stars can be estimated up to 60 percent more accurately when the light-curve parameters are taken into consideration. This method was applied to the observations of hundreds of Cepheids and thousands of RR Lyrae in the Magellanic Clouds to produce catalogs of estimated masses, radii, luminosities, and other parameters of these stars.

We discuss the impact of Gaia, the cornerstone mission of the European Space Agency (ESA), on the calibration of the period–luminosity and luminosity–metallicity relations of Cepheids and RR Lyrae stars, with specific reference to data published as part of the most recent Gaia releases: Early Data Release 3 (EDR3), on 19 December 2020, and Data Release 3 (DR3) on 13 June 2022. We provide future perspectives for the Gaia mission, including extensions approved by ESA and a tentative schedule of the data releases that will take place in the next few years. We briefly present plans for cross-Coordination Unit processing of Gaia data of Cepheids and RR Lyrae stars for DR4 and conclude by outlining the expected improvement in astrometry at the end of the extended Gaia mission, which will help to further strengthen the calibration of the first rung of the cosmic distance ladder.

The instability strip (IS) of classical Cepheids has been extensively studied theoretically. Comparison of the theoretical IS edges with those obtained empirically, using the most recent Cepheids catalogs available, can provide us with insights into the physical processes that determine the position of the IS boundaries. We investigate the empirical positions of the IS of the classical Cepheids in the Large Magellanic Cloud (LMC) using data of classical fundamental-mode and first-overtone LMC Cepheids from the OGLE-IV variable star catalog, together with a recent high-resolution reddening map from the literature. We studied their position on the Hertzsprung-Russell diagram and determined the IS borders by tracing the edges of the color distribution along the strip. We obtain the blue and red edges of the IS in V- and I-photometric bands, in addition to Teff and log L⊙. The results obtained show a break located at the Cepheids’ period of about 3 days, which was not reported before. This phenomenon is most likely explained by the depopulation of second and third crossing classical Cepheids in the faint part of the IS, since blue loops of evolutionary tracks in this mass range do not extend blueward enough to cross the IS at the LMC metallicity. Furthermore, our empirical borders show good agreement with theoretical ones published in the literature. This proves that our empirical IS is a useful tool to put constraints on theoretical models.

We present new theoretical period–luminosity (PL) and period–radius (PR) relations at multiple wavelengths (Johnson–Cousins–Glass and Gaia passbands) for a fine grid of BL Herculis models computed using mesa-rsp. The non-linear models were computed for periods typical of BL Her stars, i.e. 1 ≤ P(days) ≤ 4, covering a wide range of input parameters: metallicity (−2.0 dex ≤ [Fe/H] ≤ 0.0 dex), stellar mass (0.5–0.8 ), luminosity (50–300 ) and effective temperature (full extent of the instability strip; in steps of 50K). We investigate the impact of four sets of convection parameters on multi-wavelength properties. Most empirical relations match well with theoretical relations from the BL Her models computed using the four sets of convection parameters. No significant metallicity effects are seen in the PR relations. Another important result from our grid of BL Her models is that it supports combining PL relations of RR Lyrae and Type II Cepheids together as an alternative to classical Cepheids for the extragalactic distance scale calibration.

We have collected 2330 Cepheids to establish an intuitive 3D map of the Milky Way’s disk. As regards the warp amplitude, the Cepheid disk agrees well with the gas disk for radii up to 15 kpc. However, the mean line of nodes (LON) of the Cepheid disk deviates from the Galactic Center–Sun direction by 17.5±1.0°. This is a new and different result compared with previous results. The LON is not stable at any given radius, but it twists. The twisted pattern suggests that the formation of the Milky Way’s warp is dominated by the massive inner disk. The kinematic warp defined by the Cepheids is also in concordance with the spatial warp. In the 2020 era, the anticipated increasing number of new Cepheids will provide a key opportunity to view our Milky Way’s disk as a whole, and we expect that our knowledge of the disk’s main structural features will be much improved.