Petersen Diagram Research Articles

Multimode δ Sct stars from the Zwicky Transient Facility Survey

We obtain the largest catalog of multimode δ Sct stars in the northern sky to date using the Zwicky Transient Facility (ZTF) Data Release 20. The catalog includes 2254 objects, of which 2181 are new to our study. Among these multimode δ Sct stars, 2142 objects are double-mode δ Sct, while 109 objects are triple-mode δ Sct and three are quadruple-mode δ Sct. By analyzing the light curves in the r and g bands of the ZTF, we determine the basic parameters of multimode δ Sct stars, including the periods and amplitudes. Periods are checked by comparison with the Optical Gravitational Lensing Experiment catalog of double-mode δ Sct stars. On the Petersen diagram, multimode δ Sct stars form six sequences. We find that in Galactic coordinates, the periods of 1O/F double-mode δ Sct stars at high latitudes are shorter than those of 1O/F double-mode δ Sct stars in the disk, due to metallicity variations. In the future, our catalog can be used to establish the period–luminosity relation and the period–metallicity relation of double-mode δ Sct stars, and to study the Galactic structure.

The VELOCE modulation zoo I. Spectroscopic detection of non-radial modes in the first-overtone Cepheids BG Crucis, QZ Normae, V0391 Normae, and V0411 Lacertae

The photometric observations from the recent decade revolutionized our view on classical pulsators. Low-amplitude signals have been detected photometrically in addition to the dominant high-amplitude radial mode pulsations in many RR Lyrae stars and classical Cepheids. First-overtone pulsators with an additional low-amplitude signal at a period ratio of around 0.61 with the main mode, the so-called 0.61 stars, form the most populous group among these stars. The nature of this signal has been attributed to non-radial pulsations. Another mysterious group are stars in which the additional signal forms a period ratio of around 0.68. These are the 0.68 stars. The origin of the signal remains unknown. Here, we search for similar phenomena in spectroscopic observations of first-overtone classical Cepheids collected as part of the project. We performed a frequency analysis of several parameters derived from cross-correlation functions (CCFs), including radial velocity, the full width at half maximum, the bisector inverse span, and the CCF depth (contrast). Using standard pre-whitening, we searched for additional low-amplitude signals. We identified the location of these stars in various sequences of the Petersen diagram. We detect additional signals in four first-overtone classical Cepheids: BG Cru, QZ Nor, V0391 Nor, and V0411 Lac. We classified BG Cru, QZ Nor, and V0391 Nor as 0.61 stars based on their period ratios. V0411 Lac, however, exhibits a ratio of 0.68 between the two modes, and the additional signal has a longer period. This type of multi-periodicity remains unexplained. CCFs yield the first spectroscopic detections of non-radial pulsation modes in classical Cepheids. This opens an asteroseismic window for pursuing a more detailed understanding of these important stars. While the 0.61 signal of BG Cru, QZ Nor, and V0391 Nor is understood to originate from non-radial modes of moderate degrees, the 0.68 signal of V0411 Lac still lacks a physical explanation.

Double-mode RR Lyrae stars observed by K2: analysis of high-precision Kepler photometry

ABSTRACT The results of a Fourier analysis of high-precision Kepler photometry of 75 double-mode RR Lyrae (RRd) stars observed during NASA’s K2 Mission (2014-18) are presented. Seventy-two of the stars are ‘classical’ RRd (cRRd) stars lying along a well-defined curve in the Petersen diagram and showing no evidence of Blazhko modulations. The remaining three stars are ‘anomalous’ RRd (aRRd) stars that lie well below the cRRd curve in the Petersen diagram. These stars have larger fundamental-mode amplitudes than first-overtone amplitudes and exhibit Blazhko variations. Period-amplitude relations for the individual pulsation components of the cRRd stars are examined, as well as correlations involving Fourier phase-difference and amplitude-ratio parameters that characterize the light curves for the two radial modes. A simple statistical model relating the fundamental (P0) and first-overtone (P1) periods to [Fe/H] provides insight into the functional form of the Petersen diagram. A calibration equation for estimating [Fe/H]phot abundances of ‘classical’ RRd stars is derived by inverting the model and using 211 field and 57 globular cluster cRRd stars with spectroscopic metallicities to estimate the model coefficients. The equation is used to obtain [Fe/H]phot for the full sample of 72 K2 cRRd stars and for 2130 cRRd stars observed by the ESA Gaia Mission. Of the 49 K2 cRRd stars that are in the Gaia DR3 catalogue only five were found to be correctly classified, the remainder having been misclassified ‘RRc’ or ‘RRab’.

Frequency analysis of the first-overtone RR Lyrae stars based on Extended Aperture Photometry from K2 data

Context. Additional low-amplitude signals have been observed in many RR Lyrae stars separate from pulsations in radial modes. The most common of these are short-period signals forming a period ratio of around 0.60–0.65 with the first overtone and long-period signals forming a period ratio of around 0.68. The RR Lyrae stars may also exhibit quasi-periodic modulation in their light curves, the so-called Blazhko effect. Aims. We used the extensive sample of the first-overtone RR Lyrae stars observed by the Kepler telescope during the K2 mission to search for and characterize additional low-amplitude signals. The K2 data provides space-based photometry for a statistically significant sample. Hence, this data is excellent for studying the pulsation properties of RR Lyrae stars in detail. Methods. We used K2 space-based photometry for RR Lyrae candidates from Campaigns 0–19. We selected RR Lyrae stars pulsating in the first overtone and performed a frequency analysis for each star to characterize their frequency contents. Results. We classified 452 stars as first-overtone RR Lyrae. From that sample, we selected 281 RR0.61 stars, 67 RR0.68 stars, and 68 Blazhko stars. We found particularly interesting stars that show all of the above phenomena simultaneously. We detected signals in RR0.61 stars that form period ratios lower than those observed for the majority of stars of this type. These signals likely form a new sequence in the Petersen diagram, around a period ratio of 0.60. In 32 stars, we detected additional signals that form a period ratio close to that expected in RRd stars, but the classification of these stars as RRd is uncertain. We also report a discovery of additional signals in eight stars that form a new group in the Petersen diagram around the period ratio of 0.465–0.490. The nature of this periodicity remains unknown.

On the use of field RR Lyrae as galactic probes – VI. Mixed mode RR Lyrae variables in Fornax and in nearby dwarf galaxies

ABSTRACT We investigate the properties of the mixed-mode (RRd) RR Lyrae (RRL) variables in the Fornax dwarf spheroidal (dSph) galaxy by using B- and V-band time series collected over 24 yr. We compare the properties of the RRds in Fornax with those in the Magellanic Clouds and in nearby dSphs, with special focus on Sculptor. We found that the ratio of RRds over the total number of RRLs decreases with metallicity. Typically, dSphs have very few RRds with 0.49≲ P0 ≲0.53 d, but Fornax fills this period gap in the Petersen diagram (ratio between first overtone over fundamental period versus fundamental period). We also found that the distribution in the Petersen diagram of Fornax RRds is similar to Small Magellanic Cloud (SMC) RRds, thus suggesting that their old stars have a similar metallicity distribution. We introduce the Period–Amplitude RatioS diagram, a new pulsation diagnostics independent of distance and reddening. We found that Large Magellanic Cloud (LMC) RRds in this plane are distributed along a short- and a long-period sequence that we identified as the metal-rich and the metal-poor component. These two groups are also clearly separated in the Petersen and Bailey (luminosity amplitude versus logarithmic period) diagrams. This circumstantial evidence indicates that the two groups have different evolutionary properties. All the pulsation diagnostics adopted in this investigation suggest that old stellar populations in Fornax and Sculptor dSphs underwent different chemical enrichment histories. Fornax RRds are similar to SMC RRds, while Sculptor RRds are more similar to the metal-rich component of the LMC RRds.

Double-mode Cepheids from the Zwicky Transient Facility Survey

Multimode Cepheids pulsate simultaneously in more than one mode of oscillation. They provide an independent means to test stellar models and pulsation theories. They can also be used to derive metallicities. In recent years, the number of known multimode Cepheids has increased dramatically with the discovery of a large number of Galactic double-mode Cepheids. To date, 209 double-mode Cepheids have been detected in the Galactic bulge and disk, mostly based on the Optical Gravitational Lensing Experiment’s (OGLE) catalog. In this paper, we conduct a comprehensive search for double-mode Cepheids in the northern sky based on Zwicky Transient Facility Data Release 5. We found 72 such objects in the Milky Way. The periods of the 30 sample objects already included in the OGLE catalog show excellent agreement with the OGLE periods. The period ratios of our new Cepheids are consistent with those of known double-mode Cepheids, as evidenced by their loci in the so-called “Petersen diagram.” Compared with OGLE, the completeness of our double-mode Cepheid sample is around 71%. The much improved temporal sampling of the Zwicky Transient Facility offers significant scope to find more double-mode Cepheids, especially at the distribution’s short-period end.

Pulsational instability of pre-main-sequence models from accreting protostars

Context. The physics of early stellar evolution (e.g. accretion processes) is often not properly included in the calculations of pre-main-sequence models, leading to insufficient model grids and hence systematic errors in the results. Aims. We aim to investigate current and improved approaches for the asteroseismic modelling of pre-main-sequence δ Scuti stars. Methods. We calculated an extensive grid of pre-main-sequence models including the early accretion phase and used the resulting equilibrium models as input to calculate theoretical frequency spectra. These spectra were used to investigate different approaches in modelling echelle diagrams to find the most reliable methods. By applying Petersen diagrams, we present a simple algorithm to extract echelle diagrams from observed pulsation frequencies. Results. We show that model grids with insufficient input physics and imperfect modelling approaches lead to underestimated uncertainties and systematic errors in the extracted stellar parameters. Our re-discussion of HD 139614 leads to different stellar parameters than the ones derived by Murphy et al. (2021, MNRAS, 502, 1633). We performed a model comparison between this previous investigation and our results by applying the Akaike and Bayesian information criteria. While the results with regard to our ten-dimensional model are inconclusive, they show (very) strong evidence of a six-dimensional model with fixed accretion parameters (leading to almost identical stellar parameters to those of the ten-dimensional model) to be preferred over the model applied by Murphy et al. (2021, MNRAS, 502, 1633). In general, our modelling approach can provide narrow constraints on the stellar parameters (i.e. ΔR ∼ 0.05 R⊙, Δlog g ≲ 0.01, and ΔM⋆ ∼ 0.1 M⊙). Conclusions. The extensively tested modelling approaches and automatic extraction of echelle diagrams should allow us to study many more pre-main-sequence δ Scuti stars in the future and lead to reliable stellar parameters.

OGLE-GAL-ACEP-091: The First Known Multi-mode Anomalous Cepheid

Abstract Anomalous Cepheids (ACs) are metal-deficient, core-helium-burning pulsating stars with masses in the range 1.2–2.2 . Until recently, all known ACs were pure single-mode pulsators. The first candidate for an AC pulsating in more than one radial mode—OGLE-GAL-ACEP-091—was recently identified in the Milky Way based on the photometric database of the Optical Gravitational Lensing Experiment (OGLE) survey. We analyze this object showing that it is actually a triple-mode pulsator. Its position in the Petersen diagram, the light-curve morphology quantified by Fourier coefficients, and absolute magnitudes derived from the Gaia parallax are consistent with the assumption that OGLE-GAL-ACEP-091 is an AC. Our grid of linear pulsation models indicates that OGLE-GAL-ACEP-091 is a 1.8 star with a metallicity of about [Fe/H] = −0.5 dex.

The pulsation properties of λ bootis stars I. the southern TESS sample

ABSTRACT We analyse TESS light curves for 70 southern λ Boo stars to identify binaries and to determine which of them pulsate as δ Scuti stars. We find two heartbeat stars and two eclipsing binaries among the sample. We calculate that 81 per cent of λ Boo stars pulsate as δ Sct variables, which is about twice that of normal stars over the same parameter space. We determine the temperatures and luminosities of the λ Boo stars from photometry and Gaia DR2 parallaxes. A subset of 40 λ Boo stars have 2-min TESS data, reliable temperatures and luminosities, and δ Sct pulsation. We use Petersen diagrams (period ratios), échelle diagrams, and the period–luminosity relation to identify the fundamental mode in 20 of those 40 stars and conclude that a further 8 stars are not pulsating in this mode. For the remaining 12, the fundamental mode cannot be unambiguously identified. Further mode identification is possible for 12 of the fundamental mode pulsators that have regular sequences of pulsation overtones in their échelle diagrams. We use stellar evolution models to determine statistically that the λ Boo stars are only superficially metal weak. Simple pulsation models also better fit the observations at a metallicity of Z = 0.01 than at Z = 0.001. The TESS observations reveal the great potential of asteroseismology on λ Boo stars, for determining precise stellar ages and shedding light on the origin(s) of the λ Boo phenomenon.

The spectral Petersen diagram as a new tool to map pulsation modes in variable stars

Additional pulsation modes have been discovered in many Cepheids, RR Lyrae, and other variable stars. Fourier transforms are used to find, fit and subtract the main pulsation period and its harmonics to reveal additional modes. Commonly, for every star, the strongest of these modes is presented in a "Petersen diagram", where the shorter-to-longer period ratio is plotted against the longer period. This diagram discards the information about temporal variations, multiple pulsation modes, and signals which are below some chosen signal to noise threshold. I here present a new tool to map pulsation modes in variable stars, dubbed the "spectral Petersen diagram". It shows all signals, irrespective of their multiplicity or significance. Many (thousands) of light curves can be stacked to improve the signal to noise ratio and reveal unprecedented detail about the pulsation mode space. This tool is useful to constraint the parameters of variable star models.

TYC 3637-1152-1 – A high amplitude δ Scuti star with peculiar pulsational properties

In some δ Scuti stars, only one or two radial modes are excited (usually the fundamental mode and/or first overtone mode) and the observed peak-to-peak amplitudes exceed 0.3 mag (V). These stars are known as High Amplitude Delta Scuti (HADS) variables.We here present a detailed photometric and spectroscopic analysis of the HADS star TYC 3637-1152-1. We have derived a metallicity close to solar, a spectral type of F4 V and an age of logt=9.1. Employing archival time series data from different sources, two frequencies f0 = 10.034 c/d and f1 = 12.681 c/d and their harmonics and linear combinations were identified. The period ratio of f0/f1=0.791 puts this star into a peculiar position in the Petersen diagram, from which we conclude that TYC 3637-1152-1 is a unique object with peculiar pulsational properties that indicate a transitional state between HADS stars pulsating in the fundamental and first overtone modes and stars pulsating in higher overtones.

The additional-mode garden of RR Lyrae stars

Space-based photometric missions revealed a surprising abundance of millimagnitude-level additional modes in RR Lyrae stars. The modes that appear in the modulated fundamental-mode (RRab) stars can be ordered into four major categories. Here we present the distribution of these groups in the Petersen diagram, and discuss their characteristics and connections to additional modes observed in other RR Lyrae stars.

Petersen diagram revolution

Over the recent years, the Petersen diagram for classical pulsators, Cepheids and RR Lyr stars, populated with a few hundreds of new multiperiodic variables. We review our analyses of the OGLE data, which resulted in the significant extension of the known, and in the discovery of a few new and distinct forms of multiperiodic pulsation. The showcase includes not only radial mode pulsators, but also radial-non-radial pulsators and stars with significant modulation observed on top of the beat pulsation. First theoretical models explaining the new forms of stellar variability are briefly discussed.

Peculiar double-periodic pulsation in RR Lyrae stars of the OGLE collection – II. Short-period stars with a dominant radial fundamental mode

We report the discovery of a new group of double-periodic stars in the OGLE Galactic bulge photometry. In 38 stars identified as fundamental-mode RR Lyrae and four classified as first-overtone RR Lyrae, we detected an additional shorter periodicity. The periods of the dominant variability in the newly discovered group are 0.28 < PD < 0.41 d. Period ratios (0.68–0.72) are smaller than the period ratios of the Galactic bulge RRd stars. The typical amplitude ratio (of the additional to the dominant periodicity) is 20 per cent for the stars identified as fundamental-mode RR Lyrae and 50 per cent for stars classified as first-overtone RR Lyrae. 10 stars from our sample exhibit equidistant peaks in the frequency spectrum, which suggests the Blazhko-type modulation of the main pulsation frequency and/or additional periodicity. The Fourier coefficients R21 and R31 are some of the lowest among fundamental-mode RR Lyrae stars, but among the highest for the first-overtone pulsators. For the phase Fourier coefficients φ21 and φ31, our stars lie between RRab and RRc stars. The stars discussed were compared with radial linear pulsation models. Their position in the Petersen diagram cannot be reproduced by assuming that two radial modes are excited and their physical parameters are like those characteristic of RR Lyrae stars. The non-radial-mode scenario also faces difficulties. We conclude that the dominant variability is most likely due to pulsation in the radial fundamental mode, which applies to stars classified as first-overtone mode pulsators. At this point, we cannot explain the nature of the additional periodicity. Even more, the classification of the stars as RR Lyrae should be treated as tentative.

DETERMINATION OF PULSATION PERIODS AND OTHER PARAMETERS OF 2875 STARS CLASSIFIED AS MIRA IN THE ALL SKY AUTOMATED SURVEY (ASAS)

ABSTRACT We have developed an interactive PYTHON code and derived crucial ephemeris data of 99.4% of all stars classified as “Mira” in the ASAS database, referring to pulsation periods, mean maximum magnitudes, and whenever possible, the amplitudes among others. We present a statistical comparison between our results and those given by the International Variable Star Index (VSX) of the American Association of Variable Star Observers, as well as those determined with the machine learning automatic procedure of Richards et al. Our periods are in good agreement with those of the VSX in more than 95% of the stars. However, when comparing our periods with those of Richards et al., the coincidence rate is only 76% and most of the remaining cases refer to aliases. We conclude that automatic codes still require more refinements in order to provide reliable period values. Period distributions of the target stars show three local maxima around 215, 275, and 330 days, apparently of universal validity; their relative strength seems to depend on galactic longitude. Our visual amplitude distribution turns out to be bimodal, however, 1/3 of the targets have rather small amplitudes (A &lt; 2.5 m ) and could refer to semiregular variables (SR). We estimate that about 20% of our targets belong to the SR class. We also provide a list of 63 candidates for period variations and a sample of 35 multiperiodic stars that seem to confirm the universal validity of typical sequences in the double period and in the Petersen diagrams.

Non-radial pulsation in first overtone Cepheids of the Small Magellanic Cloud

We analyse photometry for 138 first overtone Cepheids from the Small Magellanic Cloud, in which Optical Gravitational Lensing Experiment (OGLE) team discovered additional variability with period shorter than first overtone period, and period ratios in the (0.60, 0.65) range. In the Petersen diagram these stars form three well separated sequences. The additional variability cannot correspond to other radial mode. This form of pulsation is still puzzling. We find that amplitude of the additional variability is small, typically 2-4 per cent of the first overtone amplitude, which corresponds to 2-5 mmag. In some stars we find simultaneously two close periodicities corresponding to two sequences in the Petersen diagram. The most important finding is the detection of power excess at half the frequency of the additional variability (at subharmonic) in 35 per cent of the analysed stars. Interestingly, power excess at subharmonic frequency is detected mostly for stars of the middle sequence in the Petersen diagram (74 per cent), incidence rate is much lower for stars of the top sequence (31 per cent), and phenomenon is not detected for stars of the bottom sequence. The amplitude and/or phase of the additional periodicities strongly vary in time. Similar form of pulsation is observed in first overtone RR Lyrae stars. Our results indicate that the nature and cause of this form of pulsation is the same in the two groups of classical pulsators; consequently, a common model explaining this form of pulsation should be searched for. Our results favour the theory of the excitation of non-radial modes of angular degrees 7, 8 and 9, proposed recently by Dziembowski.

CzeV293 and CzeV581—Two new high-amplitude double-mode delta Scuti stars

We report on the discovery of two high-amplitude double-mode delta Scuti stars in constellations of Hercules and Auriga. The stars were observed photometrically in five and two seasons, respectively. Frequency analysis revealed that both stars show complex pulsation behaviour with two independent modes and several combination peaks. Placing the stars into the Petersen diagram allowed us to identify the pulsation modes as the fundamental and the first overtone. Both stars follow the general trend for F/1O pulsators in the short-period part of the Petersen diagram and turned out to be classical members of HADS group of variables. Using empirical formulae we roughly estimate visual absolute magnitude, intrinsic (B−V)0 colour index and temperature of the target stars.

THE CARINA PROJECT. IX. ON HYDROGEN AND HELIUM BURNING VARIABLES

We present new multi-band (UBVI) time-series data of helium burning variables in the Carina dwarf spheroidal galaxy. The current sample includes 92 RR Lyrae-six of them are new identifications-and 20 Anomalous Cepheids, one of which is new identification. The analysis of the Bailey diagram shows that the luminosity amplitude of the first overtone component in double-mode variables is located along the long-period tail of regular first overtone variables, while the fundamental component is located along the short-period tale of regular fundamental variables. This evidence further supports the transitional nature of these objects. Moreover, the distribution of Carina double-mode variables in the Petersen diagram (P_1/P_0 vs P_0) is similar to metal-poor globulars (M15, M68), to the dwarf spheroidal Draco and to the Galactic Halo. This suggests that the Carina old stellar population is metal-poor and affected by a small spread in metallicity. We use trigonometric parallaxes for five field RR Lyrae stars to provide an independent estimate of the Carina distance using the observed reddening free Period--Wesenheit [PW, (BV)] relation. Theory and observations indicate that this diagnostic is independent of metallicity. We found a true distance modulus of \mu=20.01\pm0.02 (standard error of the mean) \pm0.05 (standard deviation) mag. We also provided independent estimates of the Carina true distance modulus using four predicted PW relations (BV, BI, VI, BVI) and we found: \mu=(20.08\pm0.007\pm0.07) mag, \mu=(20.06\pm0.006\pm0.06) mag, \mu=(20.07\pm0.008\pm0.08) mag and \mu=(20.06\pm0.006\pm0.06) mag. Finally, we identified more than 100 new SX Phoenicis stars that together with those already known in the literature (340) make Carina a fundamental laboratory to constrain the evolutionary and pulsation properties of these transitional variables.

Double-mode radial–non-radial RR Lyrae stars. OGLE-IV photometry of two high cadence fields in the Galactic bulge

We analyse the OGLE-IV photometry of the first overtone and double-mode RR Lyrae stars (RRc/RRd) in the two fields towards the Galactic bulge observed with high cadence. In 27 per cent of RRc stars we find additional non-radial mode, with characteristic period ratio, P x /P 1O \in (0.6, 0.64). It strongly corroborates the conclusion arising from the analysis of space photometry of RRc stars, that this form of pulsation must be common. In the Petersen diagram the stars form three sequences. In 20 stars we find two or three close secondary modes simultaneously. The additional modes are clearly non-stationary. Their amplitude and/or phase vary in time. As a result, the patterns observed in the frequency spectra of these stars may be very complex. In some stars the additional modes split into doublets, triplets or appear as a more complex bands of increased power. Subharmonics of additional modes are detected in 20 per cent of stars. They also display a complex structure. Including our previous study of the OGLE-III Galactic bulge data, we have discovered 260 RRc and 2 RRd stars with the additional non-radial mode, which is the largest sample of these stars so far. The additional mode is also detected in two Blazhko RRc stars, which shows that the modulation and additional non-radial mode are not exclusive.

Blazhko-type modulation in the double-mode RR Lyrae stars of the OGLE Galactic bulge collection

We present the analysis of Blazhko-type modulation in double-mode RR Lyrae (RRd) stars found in the OGLE photometry of the Galactic bulge. Modulation is detected in 15 stars. Most of them have non-typical period ratio of the radial modes. In the Petersen diagram, at a given period of the fundamental mode, they are located significantly below or above the sequence formed by the majority of RRd stars. Multi-periodic modulation is very frequent; two or three modulation periods are detected in 8 stars. Modulation periods vary from ~20 to more than 300 days. Radial mode amplitudes can be modulated by a few to nearly hundred per cent. Both radial modes may be modulated with the same period. More commonly however, dominant modulation for the fundamental mode has different period than dominant modulation for the first overtone. Quite often modulation of only one mode is detected in the data. We find a clear feedback between pulsation amplitude of the dominant mode and mean stellar brightness: lower the pulsation amplitude, brighter the star. At phases of low pulsation amplitude, the mode periods are prone to fast changes. All the stars share the common feature: their pulsation properties are non-stationary. Amplitudes and phases of the radial modes vary irregularly on a long time-scale of a few hundred or thousand days. The short-term modulations are also irregular. One of the stars has switched the pulsation mode recently: from single-mode fundamental mode pulsation to RRd state. In other star the non-radial mode with characteristic ~0.61 period ratio to the first overtone is detected. This non-radial mode is likely modulated with the same period as the radial modes.