Investigating the period-luminosity relations of δ Scuti stars: A pathway to distance and 3D dust map inference

The Cepheid period-luminosity (PL) relation is important in distance scale studies and stellar pulsation studies. This relation has been thought to be linear for a long time. However, recent work has strongly suggested that the PL relation for Large Magellanic Cloud (LMC) Cepheids is nonlinear: there are two relations, with a break at 10 days. In addition, the LMC periodcolor (PC) relation is also shown to be nonlinear. The main motivation for this dissertation is to investigate the nonlinearity of the LMC PL and PC relations and its implications for distance scale studies and for stellar structure, pulsation, and evolution. Due to the intrinsic dispersion of the PL relation and the relatively small number (∼100) of LMC Cepheids used in previous studies, the nonlinear nature of the LMC PL relation was not discovered until large numbers (∼1000) and high-quality LMC Cepheids became available from the OGLE (Optical Gravitational Lensing Experiment) and MACHO (Massive Compact Halo Objects) surveys. To test the existence of a nonlinear LMC PL relation, we apply a rigorous statistical test, the F-test, to the OGLE and MACHO data. After applying proper extinction corrections to both samples, the F-test results strongly suggested that the LMC PL relation is not linear in the VI bands (from OGLE data) and the VR bands (from MACHO data), with more than 99.5% confidence. A similar result from two totally independent samples suggested that the nonlinear LMC PL relation is real and is not due to artifacts of photometric reductions, extinction, or sample selections. The nonlinear nature of the LMC PL relation is further extended to the JH bands, but not the K band, with Two Micron All Sky Survey data. The fundamental reason that the LMC PL relation is nonlinear is that the LMC PC relation is also nonlinear. F-test results again strongly suggest that the LMC PC relation is nonlinear but that the Galactic PC relation is linear. This is because both the PL and PC relations follow the period-luminosity-color (PLC) relations for Cepheid variables; hence, understanding the nonlinear nature of the PC relation helps in understanding the nonlinear PL relation. Note that the PC relation here is actually the PC relation at mean light, which is the properties averaged over a pulsation cycle. Any “anomaly” of the color at certain phases of the pulsation (e.g., at maximum or minimum light) for some phase or period range would affect the mean-light properties of the PC relation and hence produce the observed break in the PC and PL relations. Consider that the relation , where V V ∝ log T log T T min max max min max and are the temperature at maximum and minimum light, Tmin respectively, which is directly related to the color and V min , is just the amplitude of the Cepheid. If versus period V log T max max has a flat slope for a given period range, then there is a relation between the amplitude and , hence the amplitude-color log Tmin relation, and vice versa. This flatness of the PC relation at maximum and/or minimum light could in principle influence the PC relation at mean light and produce the observed nonlinear PC (mean) relation. One mechanism that can produce a flat PC relation at maximum (and/or minimum) light is the interaction of the hydrogen ionization front (HIF) with the photosphere (defined at an optical depth of ), which halts the temperature of the photosphere 3 at some characteristic values that are independent of the global properties. Using stellar pulsation codes that include a recipe for turbulent convection, we constructed the Galactic and LMC Cepheid models to investigate this interaction. Comparing the “distance” as a function of period between the HIF and photosphere (in mass distribution) for these models, we found evidence that (1) the HIF-photosphere interaction for long-period LMC models occurs at maximum light, which is similar to the Galactic models, and (2) the same interaction for short-period LMC models occurs at minimum light, which is the opposite of Galactic models and behaves more like RR Lyrae stars. We believe that these different behaviors of the HIF-photosphere interaction between the longand short-period LMC models lead to the observed nonlinear PC (mean) and PL (mean) relations. In addition, we also constructed empirical multiphase PC relations from the LMC and Galactic Cepheids. The results show that the empirical LMC PC relations are nonlinear for most of the pulsation phases (especially near phase ∼0.8, where phase zero corresponds to maximum light), in contrast to the empirical Galactic multiphase PC relations. Cepheid distances were usually derived through a Wesenheit function (WF), a linear combination of the PL and PC relations. Although the LMC PL and PC relations are nonlinear, the WF for the LMC Cepheids is found to be linear, as the nonlinearities for both PL and PC relations cancel out when the WF is constructed. Comparisons of Cepheid distances obtained with the (correct) nonlinear PL and PC relations using the WF will only introduce a !0.07 mag (systematic) error in the distance modulus, or ∼4% in distance. However, a full understanding of the nonlinear nature of the Cepheid PL and PC relations is important in terms of stellar pulsation and evolutionary studies.

ABSTRACTIn order to determine color excess in the uvbyfl color system for high amplitude – Scuti stars,reddening free [ m 1 ], [ c 1 ], and fl indices data were obtained from the existing literature for 21 stars.Then, the three intrinsic relations of ( biy ) 0 i [ m 1 ], ( biy ) 0 i [ c 1 ], and ( biy ) 0 ifl were investigated.Among these, it was shown that the ( biy ) 0 i [ c 1 ] relation is the most useful. By establishing intrinsic( biy ) 0 i [ c 1 ] relations for six reddening calibration stars, color excesses of other stars were determined. Key words : variable star– high amplitude – Scuti star– color excess– uvbyfl systemI. INTRODUCTIONUnlike classical cepheids, the investigation of HighAmplitude – ScutiStars(HADS)hasbeenconcentratedon the study of pulsation characteristics and stellar evo-lution. It is true that HADS are not as important asthe classical cepheids. Although the PL relation wasderived for HADS already, their role as a distant indi-cator has been limited due to their low luminosity andlow amplitude. There is, however, a possibility thatthey could be used to determine the distance of extra-galaxies if big telescopes can be used to observe HADSin the future. To be used as a distant indicator, it isabsolutely necessary to determine their reliable colorexcesses.If color excess cannot be determined accurately, wecannot reliably determine distances even though thePL relation is well established. Therefore, a reliableestimation of color excess is as important as the de-termination of a reliable PL relation. This is the rea-son why many different methods have been developedto estimate color excess using various photometric sys-tems for classical cepheids. Also, various spectroscopicmethods were devised too (see Kim 1987, 2008; refer-ences therein) for a long time. However, color excessesderived from these different methods show significantdiscrepancies. In addition, with poorly determinedcolor excess, reliable atmospheric parameters such aseffective temperature, surface gravity, and metallicitycannot be estimated accurately.Unfortunately, color excess of HADS has not beenwidely investigated. For HADS, in most cases, intrin-sic (

We present a theoretical study of the radially pulsating δ Scuti and SX Phoenicis variables, concentrating on the blue straggler SX Phoenicis variables found in globular clusters. We have evolved a grid of stellar models with the metal abundance of the globular cluster M55, including models with alpha-enhanced metal abundances, and tested these models for radial pulsations observed in the high-amplitude δ Scuti and SX Phoenicis stars. Our grid includes models with globally enriched helium content to simulate the effects of stellar collisions and global mixing possible in blue stragglers. We find that global enrichment of helium strongly affects the temperature and luminosity of a given star, but the location of the instability strip blue edge and the slope of the period-luminosity (PL) relation are unchanged. This suggests that the PL relation is not affected by blue straggler formation if blue stragglers are fully mixed stellar mergers. Our blue edges and PL relations are in agreement with other theoretical determinations and also with the observational PL relation of M55, but they are not in agreement with the PL relation previously derived for high-amplitude δ Scuti stars in the field. Analysis of the double-mode variable, V41, suggests either that the star may not be pulsating in the first and second overtones as claimed or that normal stellar models may not be accurate models of blue straggler stars.

We present results from a well-studied δ Scuti star discovered in the Large Magellanic Cloud (LMC). The absolute magnitude of the variable was determined from the period-luminosity (P-L) relation for Galactic δ Scuti stars and from theoretical modeling of the observed B,V,I light curves with nonlinear pulsation models. The two methods give distance moduli for the LMC of 18.46 ± 0.19 and 18.48 ± 0.15, respectively, for a consistent value of the stellar reddening of E(B - V) = 0.08 ± 0.02. We have also analyzed 24 δ Scuti candidates discovered in the OGLE II survey of the LMC, and seven variables identified in the open cluster LW 55 and in the galaxy disk by Kaluzny and coworkers . We find that the LMC δ Scuti stars define a P-L relation whose slope is very similar to that defined by the Galactic δ Scuti variables, and yield a distance modulus for the LMC of 18.50 ± 0.22 mag. We compare the results obtained from the δ Scuti variables with those derived from the LMC RR Lyrae stars and Cepheids. The corresponding distance moduli are as follows: δ Scuti stars, 18.48 ± 0.02 mag (standard deviation of the weighted average of the three above solutions); RR Lyrae stars, 18.49 ± 0.06 mag; and Cepheids, 18.53 ± 0.02 mag. We have assumed an average color excess of E(B - V) = 0.08 ± 0.02 mag for both δ Scuti stars and Cepheids. Within the observational uncertainties, the three groups of pulsating stars yield very similar distance moduli. These moduli are all consistent with the long astronomical distance scale for the Large Magellanic Cloud.

We have used NASA’s TESS mission to study catalogued δ Scuti stars. We examined TESS light curves for 434 stars, including many for which few previous observations exist. We found that 62 are not δ Scuti pulsators, with most instead showing variability from binarity. For the 372 δ Scuti stars, we provide a catalogue of the period and amplitude of the dominant pulsation mode. Using Gaia DR3 parallaxes, we place the stars in the period–luminosity (P–L) diagram and confirm previous findings that most stars lie on a ridge that corresponds to pulsation in the fundamental radial mode, and that many others fall on a second ridge that is a factor two shorter in period. This second ridge is seen more clearly than before, thanks to the revised periods and distances. We demonstrate the value of the P–L diagram in distinguishing δ Scuti stars from short-period RR Lyrae stars, and we find several new examples of high-frequency δ Scuti stars with regular sequences of overtone modes, including XX Pyx and 29 Cyg. Finally, we revisit the sample of δ Scuti stars observed by Kepler and show that they follow a tight period–density relation, with a pulsation constant for the fundamental mode of Q = 0.0315 d.

Context. Over the last few years, δ Scuti stars have been at the center of the attention of the asteroseismology community thanks to the derivation of seismic indices connected to stellar parameters. The statistical analysis of the wealth of data offered by a large space survey such as the Transiting Exoplanet Survey Satellite (TESS), the identification of new δ Scuti stars, and the correlation between asteroseismic indices and stellar parameters in the resulting sample are therefore of utmost interest. Aims. The goal of our study is to analyze the statistical properties of stellar parameters and characterize the asteroseismic indices of δ Scuti stars observed in TESS cycle 4. Methods. We used TESS 2 min cadence photometric data and the corresponding Fourier transform to identify δ Scuti stars. The asteroseismic indices for these stars were determined using an empirical relation and a 2D autocorrelation method. Results. We discovered 765 δ Scuti stars from the data obtained by the TESS mission, from Sectors 40–55, corresponding to cycle 4 and observed with a 2 min cadence. Of these stars, 179 δ Scuti stars have low-resolution spectral parameters from LAMOST. We first analyzed the relation between pulsation and stellar parameters from TESS observations and the distribution of δ Scuti stars with two different stellar parameters, TESS Input Catalog (TIC) and Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST), within the classical instability strip. Most of the stars lie within the instability strip and follow the period-luminosity relation of δ Scuti stars. Additionally, the majority of the stars exhibit pulsation properties consistent with those expected for δ Scuti stars, including periods falling within the typical range, amplitudes at the millimagnitude level, and fundamental parameters such as spectral type, effective temperature, log g, and luminosity that match the characteristics of δ Scuti stars. This confirms the reliability of the δ Scuti stars we have identified. We subsequently obtained the large frequency separation (∆ν), vmax, and ν(Amax) for 179 δ Scuti stars with LAMOST parameters by using an empirical relation and a 2D autocorrelation method, and obtained the relations between these asteroseismic indices. These stars will provide significant support for a deeper study of the internal structure and evolution of stars.

δ Scuti variables are found at the intersection of the classical instability strip and the main sequence on the Hertzsprung–Russell diagram. With space-based photometry providing millions of light curves of A-F type stars, we can now probe the occurrence rate of δ Scuti pulsations in detail. Using the 30 minutes cadence light curves from NASA's Transiting Exoplanet Survey Satellite's first 26 sectors, we identify variability in 103,810 stars within 5–24 cycles per day down to a magnitude of T = 11.25. We fit the period–luminosity relation of the fundamental radial mode for δ Scuti stars in the Gaia G band, allowing us to distinguish classical pulsators from contaminants for a subset of 39,367 stars. Out of this subset, over 15,918 are found on or above the expected period–luminosity relation. We derive an empirical red edge to the classical instability strip using Gaia photometry. The center where the pulsator fraction peaks at 50%–70%, combined with the red edge, agrees well with previous work in the Kepler field. While many variable sources are found below the period–luminosity relation, over 85% of sources inside of the classical instability strip derived in this work are consistent with being δ Scuti stars. The remaining 15% of variables within the instability strip are likely hybrid or γ Doradus pulsators. Finally, we discover strong evidence for a correlation between pulsator fraction and spectral line broadening from the Radial Velocity Spectrometer on board the Gaia spacecraft, confirming that rotation has a role in driving pulsations in δ Scuti stars.

Context. Pulsating extremely low-mass pre-white dwarf stars (pre-ELMV), with masses between ~0.15 M⊙ and ~0.30 M⊙, constitute a new class of variable stars showing g- and possibly p-mode pulsations with periods between 320 and 6000 s (frequencies between 14.4 and 270 c/d), driven by the κ mechanism operating in the second He ionization zone. On the other hand, main sequence δ Scuti stars, with masses between 1.2 and 2.5 M⊙, pulsate in low-order g and p modes with periods in the range [700–28 800] s (frequencies in the range [3–123] c/d), driven by the κ mechanism operating in the He II ionization zone and the turbulent pressure acting in the HI ionization layer. Interestingly enough, the instability strips of pre-ELM white dwarf and δ Scuti stars nearly overlap in the Teff vs. log g diagram, leading to a degeneracy when spectroscopy is the only tool to classify the stars and pulsation periods only are considered. Aims. Pre-ELM white dwarf and δ Scuti stars are in very different stages of evolution and therefore their internal structure is very distinct. This is mirrored in their pulsational behavior, thus employing asteroseismology should allow us to distinguish between these groups of stars despite their similar atmospheric parameters. Methods. We have employed adiabatic and non-adiabatic pulsation spectra for models of pre-ELM white dwarfs and δ Scuti stars, and compare their pulsation periods, period spacings, and rates of period change. Results. Unsurprisingly, we found substantial differences in the period spacing of δ Scuti and pre-ELM white dwarf models. Even when the same period range is observed in both classes of pulsating stars, the modes have distinctive signature in the period spacing and period difference values. For instance, the mean period difference of p-modes of consecutive radial orders for δ Scuti model are at least four times longer than the mean period spacing for the pre-ELM white dwarf model in the period range [2000–4600] s (frequency range [18.78–43.6] c/d). In addition, the rate of period change is two orders of magnitudes larger for the pre-ELM white dwarfs compared to δ Scuti stars. In addition, we also report the discovery of a new variable star, SDSSJ075738.94+144827.50, located in the region of the Teff versus log g diagram where these two kind of stars coexist. Conclusions.The characteristic spacing between modes of consecutive radial orders (p as well as g modes) and the large differences found in the rates of period change for δ Scuti and pre-ELM white dwarf stars suggest that asteroseismology can be employed to discriminate between these two groups of variable stars. Furthermore, we found that SDSSJ075738.94+144827.50 exhibits a period difference between p modes characteristic of a δ Sct star, assuming consecutive radial order for the observed periods.

Context. In the era of the Hubble tension, it is crucial to obtain a precise calibration of the period-luminosity (PL) relations of classical pulsators. Type II Cepheids (T2Cs; often exhibiting negligible or weak metallicity dependence on PL relations) used in combination with RR Lyraes and the tip of the red giant branch may prove useful as an alternative to classical Cepheids for the determination of extragalactic distances. Aims. We present new theoretical PL and period-Wesenheit (PW) relations for a fine grid of convective BL Her (the shortest period T2Cs) models computed using MESA-RSP in the Gaia passbands and we compare our results with the empirical relations from Gaia DR3. Our goal is to study the effect of metallicity and convection parameters on the theoretical PL and PW relations. Methods. We used the state-of-the-art 1D non-linear radial stellar pulsation tool MESA-RSP to compute models of BL Her stars over a wide range of input parameters: metallicity (−2.0 dex ≤ [Fe/H] ≤ 0.0 dex), stellar mass (0.5 M⊙ − 0.8 M⊙), stellar luminosity (50 L⊙ − 300 L⊙), and effective temperature (across the full extent of the instability strip; in steps of 50 K). We used the Fourier decomposition technique to analyse the light curves obtained from MESA-RSP and Gaia DR3 and then compared the theoretical and empirical PL and PW relations in the Gaia passbands. Results. The BL Her stars in the All Sky region exhibit statistically different PL slopes compared to the theoretical PL slopes computed using the four sets of convection parameters. We find the empirical PL and PW slopes from BL Her stars in the Magellanic Clouds to be statistically consistent with theoretical relations computed using the different convection parameter sets in the Gaia passbands. There is a negligible effect coming from the metallicity on the PL relations in the individual Gaia passbands. However, there is a small but significant negative coefficient of metallicity in the PWZ relations for the BL Her models using the four sets of convection parameters. This could be attributed to the increased sensitivity of bolometric corrections to metallicities at wavelengths shorter than the V band. Our BL Her models also suggest a dependence of the mass-luminosity relation on metallicity. We found the observed Fourier parameter space to be covered well by our models. Higher mass models (> 0.6 M⊙) may be needed to reliably model the observed light curves of BL Her stars in the All-Sky region. We also found the theoretical light curve structures (especially the Fourier amplitude parameters) to be affected by the choice of convection parameters.

A new perspective of pulsation in stars within the δ Scuti instability strip has recently emerged as a result of Kepler observations. The majority of stars within the instability strip do not pulsate and practically all δ Scuti stars contain low frequencies. Because γ Doradus stars co-exist with δ Sct stars in the same region of the instability strip, it follows that γ Dor stars are driven by the same mechanism as δ Sct stars. The difference must be due to different mode selection processes. The search for an unknown damping factor which is missing from the models will be essential for further progress. Maia variables and hot γ Dor stars are briefly discussed. Luminosities of roAp stars obtained from Gaia DR2 parallaxes and spectroscopic effective temperatures show that the roAp stars are slightly evolved with temperatures in the range 6300–8300 K, considerably cooler than predicted by the models. The roAp stars and stars with solar-like oscillations share the same mass - temperature - luminosity relation, but with frequencies which are about 50 percent higher. This suggests that roAp frequencies are determined by the critical acoustic frequency, but this frequency is larger than in standard models, perhaps as a result of a temperature inversion in the atmosphere.

Context. Anomalous Cepheids (ACs) are pulsating variable stars, and are less studied compared to the well-known Classical Cepheids (CCs) and RR Lyrae stars. The ACs are metal poor ([Fe/H] < 1.5) and follow distinct period-luminosity (PL) and period-Wesenheit (PW) relations that can be used for distance measurements, and they can pulsate in the fundamental (F) and first overtone (1O) modes. Aims. Our goal is to evaluate the precision and accuracy of distances obtained via PL and PW relations of ACs and thus to assess if they could be used to establish a cosmic distance scale independent from CCs. To this aim, we derived new, precise PL and PW relations for the F mode, the 1O mode, and, for the first time, the combined F+1O mode ACs in the Magellanic Clouds. We investigated the wavelength dependence of these relations and applied them to calculate the distances of various stellar systems in the Local Group hosting ACs, as well as to confirm the classification of these variable stars. Methods. We analyzed near-infrared (NIR) time series photometry in the Y, J, and Ks bands for about 200 ACs in the Magellanic Clouds acquired during 2009–2018 in the context of the VISTA survey of the Magellanic Clouds system (VMC), a European Southern Observatory public survey. The VMC NIR photometry was complemented with optical data from Gaia DR3 and the Optical Gravitational Lensing Experiment IV survey, which also provided the identification, periods, and pulsation mode for the investigated ACs. Custom templates generated from our best light curves were used to derive precise intensity-averaged mean magnitudes for 118 and 75 ACs in the Large (LMC) and Small Magellanic Clouds (SMC), respectively. Results. Optical and NIR mean magnitudes were used to derive multiband PL and PW relations, which were calibrated with the geometric distance modulus to the LMC based on eclipsing binaries. We investigated the dependence of PL relations on wavelength, finding that slopes increase and dispersion decreases when going from optical to NIR bands. We calculated the LMC distance modulus through calibrated AC PW relations in the Milky Way using Gaia parallaxes, the LMC-SMC relative distance modulus, and we confirmed the AC nature of a few new pulsators in Galactic globular clusters. We derived a distance modulus for the Draco dwarf spheroidal galaxy of 19.425 ± 0.048 mag, which is in agreement with recent literature determinations, but a discrepancy of 0.1 mag with RR Lyrae-based distance hints at possible metallicity effects on the AC PL and PW relations. Future spectroscopic surveys and Gaia DR4 will refine the AC distance scale and assess metallicity effects on PLRs and PWRs.

The multi-color CCD photometric study of 27 δ Scuti stars is presented. By using approximately three years of photometric observations, we obtained the times of maxima and magnitude changes during the observation time interval for each star. The ephemerides of our δ Scuti stars were calculated based on the Markov Chain Monte Carlo (MCMC) method using the observed times of maxima and the period of the stars’ oscillations. We used the Gaia EDR3 parallaxes to calculate the luminosities and also the absolute magnitudes of these δ Scuti stars. The fundamental physical parameters of all the stars in our sample such as masses and radii were estimated. We determined the pulsation modes of the stars based on the pulsation constants. Moreover, the period–luminosity (P–L) relation of δ Scuti stars was investigated and discussed. Then, by using a machine learning classification, new P–L relations for fundamental and overtone modes are presented.

The Gaia DR3 parallax approach was used to estimate the absolute parameters of 2375 δ Scuti stars from the ASAS catalog. The selected stars have a variety of observational characteristics, with a higher than 80% probability of being δ Scuti stars. We have displayed all the stars in the Hertzsprung–Russell diagram along with the δ Scuti instability strip, the Zero Age Main Sequence and the Terminal Age Main Sequence. Then, we determined which fundamental and overtone modes each star belongs to using pulsation constant (Q) calculations. In addition, we evaluated the parameters in the Q calculation equation using three machine learning methods, which showed that surface gravity and temperature have the greatest effect on its calculation. The Period–Luminosity (P-L) relationship of the δ Scuti stars was also revisited. Eventually, using least squares linear regression, we made four linear fits for fundamental and overtone modes and updated their relationships.

We use TESS 10-min full-frame images (Sectors 27–55) to study a sample of 1708 stars within 500 pc of the Sun that lie in a narrow colour range in the centre of the δ Scuti instability strip (0.29 < GBP − GRP < 0.31). Based on the Fourier amplitude spectra, we identify 848 δ Scuti stars, as well as 47 eclipsing or contact binaries. The strongest pulsation modes of some δ Scuti stars fall on the period–luminosity relation of the fundamental radial mode but many correspond to overtones that are approximately a factor of two higher in frequency. Many of the low-luminosity δ Scuti stars show a series of high-frequency modes with very regular spacings. The fraction of stars in our sample that show δ Scuti pulsations is about 70 per cent for the brightest stars (G < 8), consistent with results from Kepler. However, the fraction drops to about 45 per cent for fainter stars and we find that a single sector of TESS data only detects the lowest amplitude δ Scuti pulsations (around 50 ppm) in stars down to about G = 9. Finally, we have found four new high-frequency δ Scuti stars with very regular mode patterns, and have detected pulsations in λ Mus that make it the fourth-brightest δ Scuti in the sky (G = 3.63). Overall, these results confirm the power of TESS and Gaia for studying pulsating stars.

We analyse observations of 1568 δ Scuti stars in the public archive of the Kepler satellite. We compare the location of these stars in the Hertzsprung–Russell (HR) diagram with that calculated by linear, non-adiabatic pulsation models. There is generally a good agreement and it appears that many of the hotter δ Scuti stars are pulsating in overtones as high as radial order n= 8. Models predict unstable modes of intermediate and high spherical harmonic, l, which are trapped in the envelope. The instability strip for these modes extends well beyond the blue edge for l≤ 4 and should be visible in the Kepler data. However, stars with these predicted properties are not seen. Moreover, we find that the observed frequency range does not agree with the models. Another challenge is to understand why more than half of the stars in the δ Sct instability strip are not pulsating. The distribution of amplitudes argues strongly against the notion that the pulsation amplitudes are below the Kepler detection limit. The mode density of δ Scuti stars is surprisingly low and suggests that modes of a high degree are not common. We do not find any stars with mode densities as high as that found in the CoRoT observations of HD 50844. The periodograms of stars in the same part of the HR diagram vary widely. However, we have identified a group of δ Scuti stars characterized by a single dominant frequency in which a period–luminosity law is present. In many cases the dominant frequency is accompanied by a close frequency of lower amplitude.