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For both periods of the double-mode Cepheid V371 Per and for the Cepheid OGLE-LMC-CEP-2132 we have constructed diagrams spanning a time interval of 126 and 119 years, respectively. The diagrams have the shape of parabolas, which has allowed us for the first time to determine the quadratic light elements and to calculate the rates of evolutionary changes in their periods: s yr and s yr for the fundamental mode and the first overtone of V371 Per, respectively, and s yr for OGLE-LMC-CEP-2132, in agreement with the results of theoretical calculations for the first crossing of the instability strip. The pulsation stability test proposed by Lombard and Koen has confirmed that the increase in the periods is real.

Old-aged stellar distance indicators are present in all Galactic structures (halo, bulge, disk) and in galaxies of all Hubble types and, thus, are immensely powerful tools for understanding our Universe. Here we present a comprehensive review for three primary standard candles from Population II: (i) RR Lyrae type variables (RRL), (ii) type II Cepheid variables (T2C), and (iii) the tip of the red giant branch (TRGB). The discovery and use of these distance indicators is placed in historical context before describing their theoretical foundations and demonstrating their observational applications across multiple wavelengths. The methods used to establish the absolute scale for each standard candle is described with a discussion of the observational systematics. We conclude by looking forward to the suite of new observational facilities anticipated over the next decade; these have both a broader wavelength coverage and larger apertures than current facilities. We anticipate future advancements in our theoretical understanding and observational application of these stellar populations as they apply to the Galactic and extragalactic distance scale.

Cepheid variable mass Cepheid variable stars have been important in the development of modern astrophysics through their use in establishing cosmic distances, but despite extensive research they retain some of their mysteries. One is the mass discrepancy problem, the fact that the masses of classical Cepheid supergiants calculated from pulsation theory (it is pulsation that causes their luminosity to vary) are smaller than the masses calculated from stellar evolution models. The ideal system in which to make an accurate mass determination would be a well-detached double-lined eclipsing binary in which one of the components was a classical Cepheid. Pietrzynski et al . report the discovery of just such a system in the Large Magellanic Cloud. The resultant mass determination, to a precision of one per cent, is in agreement with the mass as predicted by pulsation theory. Masses of pulsating classical Cepheid supergiants derived from stellar pulsation theory are smaller than the masses derived from stellar evolution theory. An independent determination for a classical Cepheid in a binary system is needed to determine which is correct. These authors report the discovery of a classical Cepheid in the Large Magellanic Cloud. They determine the mass to a precision of one per cent and show that it agrees with its pulsation mass. Stellar pulsation theory provides a means of determining the masses of pulsating classical Cepheid supergiants—it is the pulsation that causes their luminosity to vary. Such pulsational masses are found to be smaller than the masses derived from stellar evolution theory: this is the Cepheid mass discrepancy problem 1 , 2 , for which a solution is missing 3 , 4 , 5 . An independent, accurate dynamical mass determination for a classical Cepheid variable star (as opposed to type-II Cepheids, low-mass stars with a very different evolutionary history) in a binary system is needed in order to determine which is correct. The accuracy of previous efforts to establish a dynamical Cepheid mass from Galactic single-lined non-eclipsing binaries was typically about 15–30% (refs 6 , 7 ), which is not good enough to resolve the mass discrepancy problem. In spite of many observational efforts 8 , 9 , no firm detection of a classical Cepheid in an eclipsing double-lined binary has hitherto been reported. Here we report the discovery of a classical Cepheid in a well detached, double-lined eclipsing binary in the Large Magellanic Cloud. We determine the mass to a precision of 1% and show that it agrees with its pulsation mass, providing strong evidence that pulsation theory correctly and precisely predicts the masses of classical Cepheids.

Galactic Centre cepheids Cepheid variable stars are used by astronomers as standard candles to establish interstellar distances, thanks to the strong link between their luminosity and pulsation periods. As classical Cepheids have pulsation periods that decrease with increasing age, they can also be used to probe star formation history based on the distribution of their periods. A near-infrared survey of the region around the Galactic Centre has now revealed three classical Cepheids in the nuclear bulge. Their properties suggest that there was a period of elevated star formation around 25 million years ago, a timescale that is comparable with that of the cyclic gas accumulation predicted for the central part of the Milky Way. The nuclear bulge is a region with a radius of about 200 parsecs around the centre of the Milky Way 1 . It contains stars with ages 2 , 3 , 4 ranging from a few million years to over a billion years, yet its star-formation history and the triggering process for star formation remain to be resolved. Recently, episodic star formation, powered by changes in the gas content, has been suggested 5 . Classical Cepheid variable stars have pulsation periods that decrease with increasing age 6 , so it is possible to probe the star-formation history on the basis of the distribution of their periods 7 , 8 . Here we report the presence of three classical Cepheids in the nuclear bulge with pulsation periods of approximately 20 days, within 40 parsecs (projected distance) of the central black hole. No Cepheids with longer or shorter periods were found. We infer that there was a period about 25 million years ago, and possibly lasting until recently, in which star formation increased relative to the period of 30–70 million years ago.

Since the expansion of the universe was first established by Edwin Hubble and Georges Lemaître about a century ago, the Hubble constant H0 which measures its rate has been of great interest to astronomers. Besides being interesting in its own right, few properties of the universe can be deduced without it. In the last decade, a significant gap has emerged between different methods of measuring it, some anchored in the nearby universe, others at cosmological distances. The SH0ES team has found H0=73.2±1.3kms-1Mpc-1 locally, whereas the value found for the early universe by the Planck Collaboration is H0=67.4±0.5kms-1Mpc-1 from measurements of the cosmic microwave background. Is this gap a sign that the well-established ΛCDM cosmological model is somehow incomplete? Or are there unknown systematics? And more practically, how should humble astronomers pick between competing claims if they need to assume a value for a certain purpose? In this article, we review results and what changes to the cosmological model could be needed to accommodate them all. For astronomers in a hurry, we provide a buyer’s guide to the results, and make recommendations.

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.

We review the local determination of the Hubble constant, H0, focusing on recent measurements of a distance ladder constructed from geometry, Cepheid variables and Type Ia supernovae (SNe Ia). We explain in some detail the components of the ladder: (1) geometry from Milky Way parallaxes, masers in NGC 4258 and detached eclipsing binaries in the Large Magellanic Cloud; (2) measurements of Cepheids with the Hubble Space Telescope (HST) in these anchors and in the hosts of 42 SNe Ia; and (3) SNe Ia in the Hubble flow. Great attention to negating systematic uncertainties through the use of differential measurements is reviewed. A wide array of tests are discussed. The measurements provide a strong indication of a discrepancy between the local measure of H0 and its value predicted by Λ Cold Dark Matter theory, calibrated by the cosmic microwave background (Planck), a decade-long challenge known as the ‘Hubble Tension’. We present new measurements with the James Webb Space Telescope of >320 Cepheids on both rungs of the distance ladder, in a SN Ia host and the geometric calibrator NGC 4258, showing good agreement with the same as measured with HST. This provides strong evidence that systematic errors in HST Cepheid photometry do not play a significant role in the present Hubble Tension. Future measurements are expected to refine the local determination of the Hubble constant.

The Milky Way is a barred spiral galaxy, with physical properties inferred from various tracers informed by the extrapolation of structures seen in other galaxies. However, the distances of these tracers are measured indirectly and are model-dependent. We constructed a map of the Milky Way in three dimensions, based on the positions and distances of thousands of classical Cepheid variable stars. This map shows the structure of our Galaxy’s young stellar population and allows us to constrain the warped shape of the Milky Way’s disk. A simple model of star formation in the spiral arms reproduces the observed distribution of Cepheids.

Here we discuss impacts of distance determinations on the Galactic disk traced by relatively young objects. The Galactic disk, ∼ 40 kpc in diameter, is a cross-road of studies on the methods of measuring distances, interstellar extinction, evolution of galaxies, and other subjects of interest in astronomy. A proper treatment of interstellar extinction is, for example, crucial for estimating distances to stars in the disk outside the small range of the solar neighborhood. We’ll review the current status of relevant studies and discuss some new approaches to the extinction law. When the extinction law is reasonably constrained, distance indicators found in today and future surveys are telling us stellar distribution and more throughout the Galactic disk. Among several useful distance indicators, the focus of this review is Cepheids and open clusters (especially contact binaries in clusters). These tracers are particularly useful for addressing the metallicity gradient of the Galactic disk, an important feature for which comparison between observations and theoretical models can reveal the evolution of the disk.