Short-period Stars Research Articles

Photometric Study of ZZ Mic Variable Star

Abstract We report on the periodicity determination and O-C study of ZZ Mic, a short-period variable star. We used photometric data obtained on October 16, 2015 at Bosscha Observatory using the STEVia telescope (diameter 11 inches) and CCD ST8-XME. We also used 51 light curve data archives from the American Association of Variable Star Observers (AAVSO) database and 33 time of maxima records from various publications. The whole data set ranges from 1960 to 2015. We determine that ZZ Mic has a pulsation period of 0.067179 days with first overtone period of 0.053678 days. We fit the O-C diagram with a polynomial function we determine that the pulsation period of ZZ Mic has been decreasing at a rate of 8.28014 × 10−9 days per year. Using a known period-radius relation, we obtain the radius of ZZ Mic is 1.705 solar radii.

Frequency Analysis of High Amplitude δ Scuti Star V593 Lyr by American Association of Variable Stars Observers International Database

The new results of light variability and frequency analysis of a high amplitude δ Scuti star (HADS) V593 Lyr were reported. δ Scuti-type variable stars are fascinating short-period stars with luminosity classes from III to V and spectral types from A0 to F6. Their location on the Hertzsprung-Russell (H-R) diagram is in the classical cepheid instability strip. The photometric data in the V filter and clear filter of V593 Lyr were collected from the American Association of Variable Stars Observers (AAVSO) international database. Discrete Fourier transformation and pre-whitening methods were used in the Period04 package to examine time-series light curve data. During 15 observational nights, we obtained 28 times of new light maxima magnitude variability and significant pulsation frequencies for the star. The study found that the fundamental frequency of V593 Lyr is 9.7894180 ± 0.0000002 d−1, corresponding to the main pulsation period of about 0.1021511 d and detected its second to fifth harmonics with a signal-to-noise ratio (S/N) of more than four. A study of the pulsation frequencies and period change of HADS stars can estimate their pulsation modes and evolution. Moreover, HADS stars are suitable targets for understanding asteroseismology, which is the subject that applies a star's pulsations to investigate its internal structure, and the high amplitude stars provide reference points for calculating cosmic distances in the Universe.

Unscreening of f(R) gravity near the galactic center black hole: Testability through pericenter shift below S0-2’s orbit

General Relativity (GR) has been tested extensively in the solar system and is being tested in the new environment of the Galactic Center (GC) black hole where the dimensionless gravitational potential ([Formula: see text]) is 100 times stronger than the one encountered in solar system. Therefore, the neighborhood of the GC black hole is a naive opportunity to test modified theories of gravity. In this work, effect of [Formula: see text] gravity near the black hole is studied. The difference of pericenter shift between GR and [Formula: see text] gravity is studied for compact orbits having semi-major axis equal to and below [Formula: see text] au (S0-2 like orbits). In a model-dependent approach, we choose [Formula: see text] (power law gravity) model which is cosmologically motivated and study the deviation in orbital pericenter shift for both zero spin and non-zero spin of the black hole. It is found that effect of [Formula: see text] gravity becomes prominent for compact orbits. In model-independent approach to [Formula: see text] gravity with the generic scalaron fields ([Formula: see text]), we extract the parameters of [Formula: see text] gravity from the current bounds on Parametrized Post-Newtonian (PPN) parameters ([Formula: see text]) near the GC black hole. The screening of [Formula: see text] gravity is also investigated for these bounds on PPN parameters. It has been found that sufficiently massive scalarons ([Formula: see text][Formula: see text]eV) are completely screened but light and intermediate mass scalarons ([Formula: see text][Formula: see text]eV and [Formula: see text][Formula: see text]eV) are unscreened towards S0-2 like orbits as well as in the orbit of the newly discovered short period star S4716 ([Formula: see text][Formula: see text]au). The possibility of detection of the [Formula: see text] gravity effects due to these unscreened scalarons is forecasted with existing and upcoming astrometric capabilities of Extremely Large Telescopes (ELTs).

Short-period pulsating hot-subdwarf stars observed by TESS

We present results of a Transiting Exoplanet Survey Satellite (TESS) search for short-period pulsations in compact stellar objects observed in years 1 and 3 of the TESS mission, during which the southern ecliptic hemisphere was targeted. We describe the TESS data used and the details of the search method. For many of the targets, we use unpublished spectroscopic observations to classify the objects. From the TESS photometry, we clearly identify 43 short-period hot-subdwarf pulsators, including 32 sdB stars, eight sdOB stars, two sdO stars, and, significantly, one He-sdOB star, which is the first of this kind to show short-period pulsations. Eight stars show signals at both low and high frequencies, and are therefore “hybrid” pulsators. We report the list of prewhitened frequencies and we show the amplitude spectra calculated from the TESS data. We make an attempt to identify possible multiplets caused by stellar rotation, and we select four candidates with rotation periods between 1 and 12.9 days. The most interesting targets discovered in this survey should be observed throughout the remainder of the TESS mission and from the ground. Asteroseismic investigations of these data sets will be invaluable in revealing the interior structure of these stars and will boost our understanding of their evolutionary history. We find three additional new variable stars but their spectral and variability types remain to be constrained.

Extreme Mass Loss in Low-mass Type Ib/c Supernova Progenitors

Many core-collapse supernovae (SNe) with hydrogen-poor and low-mass ejecta, such as ultra-stripped SNe and type Ibn SNe, are observed to interact with dense circumstellar material (CSM). These events likely arise from the core collapse of helium stars that have been heavily stripped by a binary companion and have ejected significant mass during the last weeks to years of their lives. In helium star models run to days before core collapse we identify a range of helium core masses ≈2.5–3 M ⊙ whose envelopes expand substantially due to the helium shell burning while the core undergoes neon and oxygen burning. When modeled in binary systems, the rapid expansion of these helium stars induces extremely high rates of late-stage mass transfer () beginning weeks to decades before core collapse. We consider two scenarios for producing CSM in these systems: either mass transfer remains stable and mass loss is driven from the system in the vicinity of the accreting companion, or mass transfer becomes unstable and causes a common envelope event (CEE) through which the helium envelope is unbound. The ensuing CSM properties are consistent with the CSM masses (∼10−2–1 M ⊙) and radii (∼1013–1016 cm) inferred for ultra-stripped SNe and several type Ibn SNe. Furthermore, systems that undergo a CEE could produce short-period neutron star binaries that merge in less than 100 Myr.

RRab variables with identical light-curve shapes at different pulsation periods

ABSTRACT In this paper, we report on the detection of RRab stars with quasi-identical-shape light curves but period differences as large as 0.05–0.21 d using the Galactic bulge data of the OGLE-IV survey. We have examined stars with shorter periods than the Oosterhoff I ridge of the bulge. These stars generally have smaller amplitudes and larger Fourier phase-differences than the typical bulge RRab stars have at the same period. Many of these ‘anomalous’ stars have good-quality light curves without any sign of the Blazhko modulation. Examining their Fourier parameters revealed that several of these stars show very similar light curve to the typical bulge RR Lyrae. We found hundreds of quasi-identical-shape light-curve pairs with different periods between the ‘anomalous’- and the ‘normal’-position RRab stars based on the OGLE I-band data. The OGLE V-band, and the archive VVV and MACHO surveys Ks-, b- and r-band data of these stars were also checked for light-curve-shape similarity. Finally, 149 pairs with identical-shape light curves in each available photometric band were identified. Calculating the physical properties of the variables using empirical formulae, on average, −0.5 dex, −0.13 mag, 0.67, and 165 K differences between the [Fe/H], MV, R/R⊙, and Teff values of the members of the pairs are derived, being the short-period stars less metal-poor, fainter, smaller, and hotter than the long-period variables. To explain the existence of variables with different physical properties and pulsation periods but with identical-shape light curves is a challenging task for modelling.

Mira variables in the Milky Way’s nuclear stellar disc: discovery and classification

ABSTRACT The properties of the Milky Way’s nuclear stellar disc give crucial information on the epoch of bar formation. Mira variables are promising bright candidates to study the nuclear stellar disc, and through their period–age relation dissect its star formation history. We report on a sample of 1782 Mira variable candidates across the central $3\times 3\, \mathrm{deg}^2$ of the Galaxy using the multi-epoch infrared VISTA Variables in Via Lactea (VVV) survey. We describe the algorithms employed to select candidate variable stars and then model their light curves using periodogram and Gaussian process methods. By combining with WISE, 2MASS, and other archival photometry, we model the multiband light curves to refine the periods and inspect the amplitude variation between different photometric bands. The infrared brightness of the Mira variables means many are too bright and missed by VVV. However, our sample follows a well-defined selection function as expected from artificial star tests. The multiband photometry is modelled using stellar models with circumstellar dust that characterize the mass-loss rates. We demonstrate how ≳90 per cent of our sample is consistent with O-rich chemistry. Comparison to period–luminosity relations demonstrates that the bulk of the short period stars are situated at the Galactic Centre distance. Many of the longer period variables are very dusty, falling significantly under the O-rich Magellanic Cloud and solar neighbourhood period–luminosity relations and exhibit high mass-loss rates of $\sim 2.5\times 10^{-5}M_\odot \, \mathrm{yr}^{-1}$. The period distribution appears consistent with the nuclear stellar disc forming $\gtrsim 8\, \mathrm{Gyr}$ ago, although it is not possible to disentangle the relative contributions of the nuclear stellar disc and the contaminating bulge.

A search for period changes of eight short-period Type II Cepheids

ABSTRACT In this study, we investigate the period changes of eight short-period Type II Cepheids of the BL Her subtype, i.e. with periods in the 1–4 d range. The O − C diagrams for these stars are constructed using all suitable observational data from ground and space surveys. This spans a time interval of over one century and includes digitized photographic plates as well as photometry from the literature. The O − C diagrams show parabolic evolutionary trends, which indicate the presence of both increasing and decreasing periods for these eight short-period stars. These period changes are in good agreement with the recent theoretical evolutionary framework and stellar evolution models for BL Her stars. The pulsation stability test proposed by Lombard and Koen also suggests that the changes in the periods are real.

Revisiting bright δ Scuti stars and their period–luminosity relation with TESS and Gaia DR3

ABSTRACT 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.

Pulsating B stars in the Scorpius–Centaurus Association with TESS

ABSTRACT We study 119 B stars located in the Scorpius–Centaurus Association using data from NASA’s TESS Mission. We see pulsations in 81 stars (68 per cent) across the full range of effective temperatures. In particular, we confirm previous reports of low-frequency pulsations in stars whose temperatures fall between the instability strips of SPB stars (slowly pulsating B stars) and δ Scuti stars. By taking the stellar densities into account, we conclude that these cannot be p modes and confirm previous suggestions that these are probably rapidly rotating SPB stars. We also confirm that they follow two period–luminosity relations that are consistent with prograde sectoral g modes that are dipole (l = m = 1) and quadrupole (l = m = 2), respectively. One of the stars (ξ2 Cen) is a hybrid pulsator that shows regular spacings in both g and p modes. We confirm that α Cru has low-amplitude p-mode pulsations, making it one of the brightest β Cephei stars in the sky. We also find several interesting binaries, including a very short-period heartbeat star (HD 132094), a previously unknown eclipsing binary (π Lup), and an eclipsing binary with high-amplitude tidally driven pulsations (HR 5846). The results clearly demonstrate the power of TESS for studying variability in stellar associations.

A Spectroscopic Angle on Central Engine Size Scales in Accreting Neutron Stars

Abstract Analyses of absorption from disk winds and atmospheres in accreting compact objects typically treat the central emitting regions in these systems as point sources relative to the absorber. This assumption breaks down if the absorbing gas is located within a few × 1000 GM/c 2, in which case a small component of the absorber’s Keplerian motion contributes to the velocity width of absorption lines. Here, we demonstrate how this velocity-broadening effect can be used to constrain the sizes of central engines in accreting compact objects via a simple geometric relationship, and develop a method for modeling this effect. We apply this method to the Chandra/HETG spectra of three ultracompact and short-period neutron star X-ray binaries in which evidence of gravitationally redshifted absorption, owing to an inner-disk atmosphere, has recently been reported. The significance of the redshift is above 5σ for XTE J1710−281 (this work) and 4U 1916−053, and is inconsistent with various estimates of the relative radial velocity of each binary. For our most sensitive spectrum (XTE J1710−281), we obtain a 1σ upper bound of 310 km s−1 on the magnitude of this geometric effect and a central engine of size R CE < 60 GM/c 2 (or < 90 GM/c 2 at the 3σ level). These initial constraints compare favorably to those obtained via microlensing in quasars and approach the sensitivity of constraints via relativistic reflection in neutron stars. This sensitivity will increase with further exposures, as well as the launch of future microcalorimeter and grating missions.

MG1-688432: A Peculiar Variable System

Abstract The short-period variable star MG1-688432 has been discovered to exhibit occasional extremely high energy optical outbursts as high as 1038 erg. Outbursts are typically several hours in duration. These events are often highly structured, resembling sequential associated releases of energy. Twenty years of time sequence photometry are presented, indicating a basic sinusoidal light curve of mean period 6.65 days, with some phase shifting and long-term temporal trends in amplitude and mean brightness. Spectroscopy reveals a peculiar star, best resembling a K3 subgiant that has evolved off the main sequence moderately redward of the giant branch. Spectroscopic and radial velocity analyses indicate a binary system orbiting its barycenter with an unseen companion to the K3 IV primary. This is not an eclipsing system, with the inclination of the orbit precluding eclipse by the secondary. The system is at a distance of 1.5 kpc, and analysis of Gaia observations leads to the conclusion that the H-R diagram position of MG1-688432 is established by an intrinsic feature of the system, most likely either the stellar evolutionary state of the observed star or the presence of small (nongray) dust within the system. Two mechanisms (or combinations thereof) that might give rise to characteristics of the system are (1) magnetically induced chromospheric activity and (2) impacts with tidally disrupted planetary debris.

Four ‘Peculiar’ RRd stars observed by K2

ABSTRACT Four stars pulsating simultaneously with a dominant period PD ∈ (0.28, 0.39) d and an additional period PA ∈ (0.20, 0.27) d have been identified from among the more than 3000 RR Lyrae stars observed by the Kepler space telescope during NASA’s K2 Mission. All four stars are located in the direction of the Galactic Bulge and have period ratios, PA/PD, significantly smaller than those of most double-mode RR Lyrae (RRd) stars: PA/PD ∈ (0.694, 0.710) versus P1/P0 ∈ (0.726, 0.748). Three of the stars are faint (〈V〉 = 18–20 mag) and distant and are among the ‘peculiar’ RRd (pRRd) stars discovered by Prudil et al. (2017); the fourth star, EPIC 216764000 (= V1125 Sgr), is a newly discovered pRRd star several magnitudes brighter than the other three stars. In this paper, the high-precision long-cadence K2 photometry is analysed in detail and used to study the cycle-to-cycle light variations. The pulsational characteristics of pRRd stars are compared with those of ‘classical’ and ‘anomalous’ RRd (cRRd, aRRd) stars. The conclusion by Prudil et al. that pRRd stars form a separate group of double-mode pulsators and are not simply very short-period cRRd stars is confirmed. V1127 Aql and AH Cam are identified as other probable members of the class of pRRd stars.

Calcium-rich Transient SN 2019ehk in a Star-forming Environment: Yet Another Candidate for a Precursor of a Double Neutron-star Binary

Abstract We present optical and near-infrared observations of SN 2019ehk, which was initially reported as a Type Ib supernova (SN). We show that it evolved to a Ca-rich transient according to its spectral properties and evolution in late phases. However, it shows a few properties distinct from those of the canonical Ca-rich transients: a short-duration first peak in the light curve, high peak luminosity, and association with a star-forming environment. Indeed, some of these features are shared with iPTF14gqr and iPTF16hgs, which are candidates for a special class of core-collapse SNe: the so-called ultra-stripped envelope SNe, i.e., a relatively low-mass He (or C+O) star explosion in a binary as a precursor of short-period double neutron star (NS) binaries. The estimated ejecta mass (0.4M ⊙) and explosion energy (1.7 × 1050 erg) are consistent with this scenario. The analysis of the first peak suggests the existence of dense circumstellar material in the vicinity of the progenitor, implying a CCSN origin. Based on this analysis, we suggest SN 2019ehk is another candidate for a low-mass He star explosion. It might create a double NS binary, but with a wide separation. These candidates for low-mass stripped envelope SNe, including ultra-stripped envelope SN candidates, seem to form a subpopulation among Ca-rich transients, associated with young population. We propose that the key to distinguishing this population is the early first peak in their light curves.

Black Holes and the Supermassive Compact Object at the Galactic Center: Multi-arts of Thought and Nature

Black Holes and the Supermassive Compact Object at the Galactic Center: Multi-arts of Thought and Nature

S62 and S4711: Indications of a Population of Faint Fast-moving Stars inside the S2 Orbit—S4711 on a 7.6 yr Orbit around Sgr A*

Abstract We present high-pass filtered NACO and SINFONI images of the newly discovered stars S4711–S4715 between 2004 and 2016. Our deep H+K-band (SINFONI) and K-band (NACO) data show the S-cluster star S4711 on a highly eccentric trajectory around Sgr A* with an orbital period of 7.6 yr and a periapse distance of 144 au to the supermassive black hole (SMBH). S4711 is hereby the star with the shortest orbital period and the smallest mean distance to the SMBH during its orbit to date. The used high-pass filtered images are based on coadded data sets to improve the signal to noise. The spectroscopic SINFONI data let us determine detailed stellar properties of S4711 like the mass and the rotational velocity. The faint S-cluster star candidates, S4712–S4715, can be observed in a projected distance to Sgr A* of at least temporarily ≤120 mas. From these stars, S4714 is the most prominent, with an orbital period of 12 yr and an eccentricity of 0.985. The stars S4712–S4715 show similar properties, with magnitudes and stellar masses comparable to those of S4711. The MCMC simulations determine confidently precise uncertainties for the orbital elements of S62 and S4711–S4715. The presence of S4711 in addition to S55, S62, and the also newly found star S4714 implies a population of faint stars that can be found at distances to Sgr A* that are comparable to the size of our solar system. These short orbital time period stars in the dense cluster around the SMBH in the center of our Galaxy are perfect candidates to observe gravitational effects such as the periapse shift.

An Adaptive Scheduling Tool to Optimize Measurements to Reach a Scientific Objective: Methodology and Application to Measurements of Stellar Orbits in the Galactic Center

Abstract In various fields of physics and astronomy, access to experimental facilities or to telescopes is becoming more and more competitive and limited. It therefore becomes important to optimize the type of measurements and their scheduling to reach a given scientific objective and to increase the chances of success of a scientific project. In this communication, extending the work of Ford and of Loredo et al., we present an efficient adaptive scheduling tool aimed at prioritizing measurements in order to reach a scientific goal. The algorithm, based on the Fisher matrix, can be applied to a wide class of measurements. We present this algorithm in detail and discuss some practicalities such as systematic errors or measurement losses due to contingencies (such as weather, experimental failure, ...). As an illustration, we consider measurements of the short-period star S0-2 in our Galactic Center (GC). We show that the radial velocity measurements at the two turning points of the radial velocity curve are more powerful for detecting the gravitational redshift than measurements at the maximal relativistic signal. We also explicitly present the methodology that was used to plan measurements in order to detect the relativistic redshift considering systematics and possible measurement losses. For the future, we identify the astrometric turning points to be highly sensitive to the relativistic advance of the periastron. Finally, we also identify measurements particularly sensitive to the distance to our GC: the radial velocities around periastron and the astrometric measurements just before closest approach and at the maximal R.A. astrometric turning point.

Improving Orbit Estimates for Incomplete Orbits with a New Approach to Priors: with Applications from Black Holes to Planets

Abstract We propose a new approach to Bayesian prior probability distributions (priors) that can improve orbital solutions for low-phase-coverage orbits, where data cover less than ∼40% of an orbit. In instances of low phase coverage—such as with stellar orbits in the Galactic center or with directly imaged exoplanets—data have low constraining power and thus priors can bias parameter estimates and produce underestimated confidence intervals. Uniform priors, which are commonly assumed in orbit fitting, are notorious for this. We propose a new observable-based prior paradigm that is based on uniformity in observables. We compare performance of this observable-based prior and of commonly assumed uniform priors using Galactic center and directly imaged exoplanet (HR 8799) data. The observable-based prior can reduce biases in model parameters by a factor of two and helps avoid underestimation of confidence intervals for simulations with less than ∼40% phase coverage. Above this threshold, orbital solutions for objects with sufficient phase coverage—such as S0-2, a short-period star at the Galactic center with full phase coverage—are consistent with previously published results. Below this threshold, the observable-based prior limits prior influence in regions of prior dominance and increases data influence. Using the observable-based prior, HR 8799 orbital analyses favor low-eccentricity orbits and provide stronger evidence that the four planets have a consistent inclination of ∼30° to within 1σ. This analysis also allows for the possibility of coplanarity. We present metrics to quantify improvements in orbital estimates with different priors so that observable-based prior frameworks can be tested and implemented for other low-phase-coverage orbits.

RR Lyrae Variables in the Crater II Dwarf Galaxy

Abstract We report the detection of RR Lyrae variable stars in Crater II, a recently discovered large and diffuse satellite dwarf galaxy of the Milky Way (MW). Based on B, V time-series photometry obtained with the Korea Microlensing Telescope Network 1.6 m telescope at Cerro Tololo Inter-American Observatory, we identified 83 ab-type and 13 c-type pulsators by fitting template light curves. The detected RR Lyrae stars are centrally concentrated, which ensures that most of them are members of Crater II. In terms of the distribution of RRab stars in the period-amplitude diagram, Crater II is clearly different from ultra-faint dwarf (UFD) galaxies, but very similar to the two classical MW dwarf spheroidal (dSph) galaxies Draco and Carina with Oosterhoff-intermediate (Oo-int) properties. Combined with the mean period of ab-type variables ( ) and the c-type fraction (∼0.14) in Crater II, this suggests an Oo-int classification for Crater II and implies that its nature is more like a dSph rather than a UFD. We also estimated the mean metallicity, reddening, and distance of Crater II, from the photometric and pulsation properties of the RR Lyrae stars. The stellar population model we have constructed indicates that Crater II is dominated by an old population, but is relatively younger than the oldest globular clusters in the MW. With a lack of high-amplitude short-period RRab stars, Crater II, like most of the other less massive dSphs, is probably not a surviving counterpart of the major building blocks of the MW halo.

Spectroscopic properties of a two-dimensional time-dependent Cepheid model

Standard spectroscopic analyses of Cepheid variables are based on hydrostatic one-dimensional model atmospheres, with convection treated using various formulations of mixing-length theory. This paper aims to carry out an investigation of the validity of the quasi-static approximation in the context of pulsating stars. We check the adequacy of a two-dimensional time-dependent model of a Cepheid-like variable with focus on its spectroscopic properties. With the radiation-hydrodynamics code CO5BOLD, we construct a two-dimensional time-dependent envelope model of a Cepheid with $T_\mathrm{eff}= 5600$ K, $\log g=2.0$, solar metallicity, and a 2.8-day pulsation period. Subsequently, we perform extensive spectral syntheses of a set of artificial iron lines in local thermodynamic equilibrium. The set of lines allows us to systematically study effects of line strength, ionization stage, and excitation potential. We evaluate the microturbulent velocity, line asymmetry, projection factor, and Doppler shifts. The mean Doppler shift is non-zero and negative, -1 km/s, after averaging over several full periods and lines. This residual line-of-sight velocity (related to the "K-term") is primarily caused by horizontal inhomogeneities, and consequently we interpret it as the familiar convective blueshift ubiquitously present in non-pulsating late-type stars. Limited statistics prevent firm conclusions on the line asymmetries. Our two-dimensional model provides a reasonably accurate representation of the spectroscopic properties of a short-period Cepheid-like variable star. Some properties are primarily controlled by convective inhomogeneities rather than by the Cepheid-defining pulsations.