Pulsation Period Research Articles

The Lack of RR Lyrae Variables in Ultrafaint Dwarf Galaxy Sextans II

Abstract RR Lyrae in ultrafaint dwarf (UFD) galaxies, if found, would be valuable to constrain the distances of these UFD. In this work, we report our search of RR Lyrae in a recently discovered UFD—Sextans II. Based on multiband (griz) time-series archival DECam imaging data, we do not find any RR Lyrae in Sextans II. On the contrary, the sparse multiband DECam light curves allowed recovery of a known foreground RR Lyrae, BB Sex, which happened to fall within the DECam images, and its pulsation period was correctly identified. Therefore, it is possible that Sextans II lacks RR Lyrae, similar to several other fainter UFDs (with M V ≳ −5.0 mag) that do not have RR Lyrae.

High-cadence stellar variability studies of RR Lyrae stars with DECam: New multiband templates

We present the most extensive set to date of high-quality RR Lyrae light curve templates in the griz bands, based on time-series observations of the Dark Energy Camera Plane Survey (DECaPS) East field, located in the Galactic bulge at coordinates (RA, Dec)(J2000) = (18:03:34, -29:32:02), obtained with the Dark Energy Camera (DECam) on the 4-m Blanco telescope at the Cerro Tololo Inter-American Observatory (CTIO). Our templates, which cover both fundamental-mode (RRab) and first-overtone (RRc) pulsators, can be especially useful when there is insufficient data for accurately calculating the average magnitudes and colors, hence distances, as well as to inform multiband light curve classifiers, as will be required in the case of the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST). In this paper, we describe in detail the procedures that were adopted in producing these templates, including a novel approach to account for the presence of outliers in photometry. Our final sample comprises 136 RRab and 144 RRc templates, all of which are publicly available. Lastly, in this paper we study the inferred Fourier parameters and other light curve descriptors, including rise time, skewness, and kurtosis, as well as their correlations with the pulsation mode, period, and effective wavelength.

Recovering Pulsar Periodicity from Time-of-arrival Data by Finding the Shortest Vector in a Lattice

The strict periodicity of pulsars is one of the primary ways through which their nature and environment can be studied, and it has also enabled precision tests of general relativity and studies of nanohertz gravitational waves using pulsar timing arrays (PTAs). Identifying such a periodicity from a discrete set of arrival times is a difficult algorithmic problem, In particular when the pulsar is in a binary system. This challenge is especially acute in γ-ray pulsar astronomy, as there are hundreds of unassociated Fermi-LAT sources that may be produced by γ-ray emission from unknown pulsars. Recovering their timing solutions will help reveal their properties and may allow them to be added to PTAs. The same issue arises when attempting to recover a strict periodicity for repeating fast radio bursts (FRBs). Such a detection would be a major breakthrough, providing us with the FRB source’s age, magnetic field, and binary orbit. The problem of recovering a timing solution from sparse time-of-arrival data is currently unsolvable for pulsars in unknown binary systems, and incredibly hard even for isolated pulsars. In this paper, we frame the timing recovery problem as the problem of finding a short vector in a lattice and obtain the solution using off-the-shelf lattice reduction and sieving techniques. As a proof of concept, we solve PSR J0318+0253, a millisecond γ-ray pulsar discovered by FAST in a γ-ray-directed search, in a few CPU minutes. We discuss the assumptions of the standard lattice techniques and quantify their performance and limitations.

CHEOPS observations confirm nodal precession in the WASP-33 system

We aim to observe the transits and occultations of which orbits a rapidly rotating delta Scuti pulsator, with the goal of measuring the orbital obliquity via the gravity-darkening effect, and constraining the geometric albedo via the occultation depth. We observed four transits and four occultations with CHEOPS, and employ a variety of techniques to remove the effects of the stellar pulsations from the light curves, as well as the usual CHEOPS systematic effects. We also performed a comprehensive analysis of low-resolution spectral and Gaia data to re-determine the stellar properties of system. We measure an orbital obliquity degrees, which is consistent with previous measurements made via Doppler tomography. We also measure the planetary impact parameter, and confirm that this parameter is undergoing rapid secular evolution as a result of nodal precession of the planetary orbit. This precession allows us to determine the second-order fluid Love number of the star, which we find agrees well with the predictions of theoretical stellar models. We are unable to robustly measure a unique value of the occultation depth, and emphasise the need for long-baseline observations to better measure the pulsation periods.

Metallicity of RR Lyrae stars from the Gaia Data Release 3 catalogue computed with Machine Learning algorithms

Abstract We present new P − φ31 − [Fe/H] and P − φ31 − A2 − [Fe/H] relations for fundamental-mode (RRab) and first-overtone mode (RRc) RR Lyrae stars (RRLs), respectively. The relations were calibrated based on pulsation periods and Fourier parameters of the RRL light curves in the Gaia G-band published in the Gaia Data Release 3 (DR3), and accurate spectroscopically measured metallicities available in the literature. We apply the feature selection algorithm to identify the most relevant parameters for the determination of metallicity. To fit the relations, we used the Bayesian approach, which allowed us to carefully take into account uncertainties in various parameters and the intrinsic scatter of the relations. The root mean squared errors of the predicted metallicity values in the training samples are 0.28 dex and 0.21 dex for RRab and RRc stars, respectively, comparable with the typical uncertainty of low/intermediate resolution spectroscopic metallicity measurements. We applied the new relations to measure individual metallicities and distances to ∼ 134,000 RRLs from the Gaia DR3 catalogue, as well as mean metallicities and distances to 38 Milky Way globular clusters. We also estimate the mean metallicity and distance to the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC): [Fe/H]LMC = −1.63 ± 0.36 and μLMC = 18.55 ± 0.18 mag, [Fe/H]SMC = −1.86 ± 0.36 dex and μSMC = 19.01 ± 0.17 mag, respectively, in excellent agreement with previous measurements.

Gas layer charge influence law study of underwater explosion bubble pulsation

Abstract To discuss the influence of the gas layer on underwater explosion load, the influence of gas layer thickness on bubble pulsation period maximum radius, pressure peak value, and specific impulse during underwater explosion was first discussed. A simulation model of underwater explosion with gas layer charge is established by numerical simulation method, and the influence law of gas layer on the pulsation of underwater explosion bubble is calculated and analyzed. The results show that the pulsation period and maximum radius of the underwater explosion bubble increase with the increase of gas layer thickness. The peak value of bubble pulsation pressure decreases, while the specific impulse increases first and then decreases. The research conclusion shows that in a certain gas layer, the damage risk of protective structure may be increased due to the increase of pulsating pressure ratio impulse or the resonance between pulsating frequency and structure.

Research on speed fluctuation suppression of PMSM based on SADRC and QRC cascade control

The objective of this research is to reduce the speed fluctuation of Permanent Magnet Synchronous Motor (PMSM) due to periodic torque fluctuation. To achieve this, the authors designed a cascade control system based on Linear/Nonlinear Switching Active Disturbance Rejection Control (SADRC) and Quasi Resonant Controller (QRC). Periodic torque pulsations, mainly the first, second, and fourth harmonics, are analyzed first. The architecture of the SADRC and QRC cascade control systems is then established. Subsequently, the parameters of SADRC and QRC are calibrated by the bandwidth method, and a closed-loop system model of cascade control is developed. Ultimately, periodic load fluctuations with step load fluctuations are applied at the load side to ascertain the efficacy of the SADRC and QRC cascade control systems. The SADRC and QRC cascade control reduce the first, second, and fourth harmonics of speed fluctuations by 74%, 33%, and 47.5%, respectively, through the use of QRC current compensation in the presence of peak 2 N m periodic torque fluctuations. The SADRC and QRC cascade control systems demonstrate excellent performance in the context of cyclic and step load fluctuations.

Research on Helmholtz self-excited oscillating jets enhanced heat transfer in horizontal tubes

The self-excited oscillating jets have several advantages, including simple structures, environmental friendliness, and high efficiency. It has become a prominent research focus in the heat transfer field. This study focuses on analyzing the influence of the self-excited oscillation chamber structural parameters on the interaction between coherent structures and heat transfer. Combining the heat transfer enhancement of the pulsating backflow vortex with boundary layer theory, a self-excited oscillation horizontal tube model is constructed. The Large Eddy Simulation (LES) method simulates the self-excited oscillation in horizontal tubes. The heat transfer and flow resistance mechanisms in the tube are investigated. The results indicate that backflow vortices in the downstream pipe alternate near the shear layer. As the flow approaches the oscillator outlet, discrete vortices decrease, the pulsation effect weakens, the temperature gradient diminishes in the tube, and the heat transfer boundary layer thickens. The pressure drops and resistance coefficient in the horizontal tubes exhibit periodic pulsations. The optimal structural parameters of the Helmholtz horizontal tubes are obtained through comprehensive analysis: diameter ratio of the chamber to the upstream pipe DT/d1 = 9.5, ratio of the chamber length to the upstream pipe diameter LT/d1 = 4.3, collision wall angle α = 100°. The relationship between the structural parameters of the oscillator and the heat transfer factors has been established, guiding the application of the Helmholtz oscillating jets in the field of enhanced heat transfer.

Effect of electrical field on pulsation characteristics of laser-induced cavitation bubble

An experimental platform for laser-induced cavitation under the action of an electric field was built, and the bubble pulsation under two different conditions of infinite domain and solid wall was experimentally studied. The effects of voltage, laser energy, and liquid viscosity on bubble pulsation and the difference in the effect of voltage on bubble induced by different laser energies and bubble pulsation in different viscous liquids were explored. It is found that the size, velocity and period of bubble pulsation in infinite domain increase with the increase of laser energy, and the law is similar at the solid wall. When the voltage is applied, the size and period of bubble pulsation in infinite domain decrease with the increase of voltage, the bubble expansion speed decreases with the increase of voltage, but the bubble collapse speed increases with the increase of voltage. The diameter and velocity of bubble pulsation in infinite domain decrease with the increase of liquid viscosity, while the whole pulsation period of bubble increases with the increase of viscosity. Due to the damping effect of liquid, the effect of electric field on the bubble pulsation with higher viscosity is low.

X-Ray Hardening Preceding the Onset of SGR 1935+2154's Radio Pulsar Phase

Magnetars are neutron stars with extremely strong magnetic fields, frequently powering high-energy activity in X-rays. Pulsed radio emission following some X-ray outbursts has been detected, albeit its physical origin is unclear. It has long been speculated that the origin of magnetars’ radio signals is different from those from canonical pulsars, although convincing evidence is still lacking. Five months after magnetar SGR 1935+2154's X-ray outburst and its associated fast radio burst 20200428, a radio pulsar phase was discovered. Here we report the discovery of X-ray spectral hardening associated with the emergence of periodic radio pulsations from SGR 1935+2154 and a detailed analysis of the properties of the radio pulses. The observations suggest that radio emission originates from the outer magnetosphere of the magnetar, and the surface heating due to the bombardment of inward-going particles from the radio emission region is responsible for the observed X-ray spectral hardening.

Stellar Pulsation and Evolution: A Combined Theoretical Renewal and Updated Models (SPECTRUM). I. Updating Radiative Opacities for Pulsation Models of Classical Cepheid and RR-Lyrae

Pulsating stars are universally recognized as precise distance indicators and tracers of stellar populations. Their variability, combined with well-defined relationships between pulsation properties and intrinsic evolutionary parameters such as luminosity, mass, and age, makes them essential for understanding galactic evolution and retrieving star formation histories. Therefore, accurate modeling of pulsating stars is crucial for using them as standard candles and stellar population tracers. This is the first paper in the “Stellar Pulsation and Evolution: a Combined Theoretical Renewal and Updated Models” project, which aims to present an update of Stellingwerf’s hydrodynamical pulsation code, by adopting the latest radiative opacity tables commonly used in stellar evolution community. We assess the impact of this update on pulsation properties, such as periods, instability strip topology, and light-curve shapes, as well as on period–Wesenheit and period–luminosity relations for Classical Cepheids and RR Lyrae stars, comparing the results with those derived using older opacity data. Our results indicate that the opacity update introduces only minor changes: instability strip boundary locations shift by no more than 100 K in effective temperature, and pulsation periods vary within 1σ compared to previous evaluations. Light curves exhibit slight differences in shape and amplitude. Consequently, the theoretical calibration of the Cepheid- or RRL-based extragalactic distance scale remains largely unaffected by the opacity changes. However, achieving consistency in opacity tables between stellar evolution and pulsation codes is a significant step toward a homogeneous and self-consistent stellar evolution and pulsation framework.

Mass-loss Rate of Highly Evolved Stars in the Magellanic Clouds

Asymptotic giant branch stars (AGBs) and red supergiant stars (RSGs) exhibit significant mass-loss phenomena and are considered important sources of interstellar dust. In this work, we employed a uniform method of spectral energy distribution fitting to analyze a large, and hence statistically significant, sample of approximately 40,000 RSGs and AGBs in the Magellanic Clouds (MCs), providing a new catalog of evolved stars that includes stellar parameters and dust properties. Our results reveal that the total dust-production rate (DPR) of the Large Magellanic Cloud is approximately 9.69 × 10−6 M ☉ yr−1, while it is around 1.75 × 10−6 M ☉ yr−1 for the Small Magellanic Cloud, with a few stars significantly contributing to the total DPR. No significant differences were observed in the contributions to DPR from carbon-rich and oxygen-rich (O-rich) evolved stars in the MCs. We explored the relations between stellar parameters (luminosity, infrared color, period, amplitude) and mass-loss rate (MLR) for evolved stars. A prominent turning point at log(L/L☉)≈4.4 appears in the luminosity–MLR diagram of RSGs, potentially related to their mass-loss mechanism. The luminosity–MLR relation of AGBs is highly scattered. The DPR of AGBs shows a clear change with pulsation period and amplitude, with DPR exhibiting a drastic increase at pulsation periods of approximately 300 days and I-band amplitudes greater than 0.5 mag. Metallicity has some impact on the DPR of O-rich stars, with lower metallicity seeming to result in lower mean DPR and a higher proportion of optically thin stars.

Probing the Extended Atmospheres of AGB Stars. I. Synthetic Imaging of 1D Hydrodynamical Models at Radio and (Sub-)millimeter Wavelengths

We investigate the observable characteristics of the extended atmospheres of asymptotic giant branch (AGB) stars across a wide range of radio and (sub-)millimeter wavelengths using state-of-the-art 1D dynamical atmosphere and wind models over one pulsation period. We also study the relationships between the observable features and model properties. We further study practical distance ranges for observable sources assuming the capabilities of current and upcoming observatories. We present time-variable, frequency-dependent profiles of pulsating AGB stars’ atmospheres, illustrating observable features in resolved and unresolved observations, including disk brightness temperature, photosphere radius, and resolved and unresolved spectral indices. Notably, temporal variations in disk brightness temperature closely mirror the temperature variability of the stellar atmosphere. We find that while the photospheric radius decreases due to gas dilution in the layers between consecutive shocks, the increase in the observed stellar radius reflects shock propagation through the atmosphere during the expansion phase, providing a direct measurement method for the shock velocity. Furthermore, our models indicate that enhanced gas temperatures after the passage of a strong shock might be observable in the high-frequency Atacama Large Millimeter/submillimeter Array (ALMA) bands as a decrease in the brightness temperature with increasing frequency. We demonstrate that synthetic observations based on state-of-the-art dynamical atmosphere and wind models are necessary for proper interpretations of current (ALMA and Very Large Array (VLA)) and future (Square Kilometre Array and next-generation VLA) observations and that multiwavelength observations of AGB stars are crucial for empirical studies of their extended atmospheres.

Instability in Supernova Fallback Disks and Its Effect on the Formation of Ultralong Period Pulsars

Several pulsars with unusually long periods were discovered recently, comprising a potential population of ultralong period pulsars (ULPPs). The origin of their long periodicity is not well understood, but may be related to magnetars spun down by surrounding fallback disks. While there are few systematic investigations on the fallback-disk-assisted evolution of magnetars, instability in the disk has received little attention, which determines the lifetime of the disk. In this work we simulate the evolution of the magnetic field, spin period, and magnetic inclination angle of magnetars with a supernova fallback disk. We find that a thermal viscous instability in the disk could significantly affect the formation of ULPPs. Our simulation results also reveal that a large fraction of ULPPs seem to be nearly aligned and orthogonal rotators. This might help place ULPPs above the death line in the pulse period–period derivative plane. However, some extra mechanisms seem to be required to account for the radio emission of ULPPs.

A “Wonderful” Reference Dataset of Mira Variables

The conditions in Mira variable atmospheres make them wonderful laboratories to study a variety of stellar physics such as molecule–grain formation, dust production, shock chemistry, stellar winds, mass loss, opacity-driven pulsation, and shocks. We were awarded an NSF grant to analyze over a decade of synoptic observations from the Palomar Testbed Interferometer (PTI) of 106 Miras to curate a Mira Reference Dataset. The Miras included in this dataset include M-types, S-types, and C-types, and span a wide range of pulsation periods. PTI measured K-band angular sizes that when combined with a distance allow us to directly determine fundamental stellar parameters such as effective temperature, radial size, and bolometric flux. Supplementing observations with interferometric measurements of the stars opens the Mira laboratory to a wealth of different experiments. We provide two case studies to serve as examples of the power of the Mira Reference Dataset. The first case study describes combining PTI measurements with Spitzer IRS spectra of M-type Miras, which allowed us to fully characterize CO2 gas in their atmospheres. The second case study examines how PTI narrow-band data can be used to study phase-dependent pulsation effects on the stellar atmosphere. We provide a list of all the Miras (with coordinates) included in the set for anyone who would like to add them to their observing programs. All the data we produce and collate for this Mira Reference Dataset will be hosted and curated on a website open to the public so that other researchers and citizen scientists can participate in expanding the utility and body of knowledge on this set of “wonderful” stars.

Separating the causes of O–C variations in an RR Lyrae star with the Blazhko effect

Context. To date, puzzlingly few bona fide RR Lyrae stars have been identified in binaries. Binarity in pulsating stars can be revealed through well-timed photometric data over sufficiently long time bases because of the light travel time effect (LTTE) on the pulsations, which manifests as variation in the timings of maximum light in the O−C (observed minus calculated) diagram. However, O−C variations can also have other causes, such as the Blazhko effect or a sudden or gradual change in the main pulsation period. Aims. We approach this challenge by disentangling the Blazhko effect and period changes from the potential LTTE on V1109 Cas, an RR Lyrae star suspected to be part of a binary system based on its O−C data in the GEOS database. In doing so, we aim to uncover the subtler signals indicative of a companion (LTTE). Methods. We analysed nine years of ground-based photometric data, using Fourier analysis to model the pulsation modulated by the Blazhko effect. From the fit to the observed light curves, we constructed a refined O−C diagram without the scatter caused by Blazhko modulation. Subsequently, we considered different possible scenarios, because distinguishing intrinsic period changes or breaks from the LTTE is challenging. Results. If the remaining O−C variation is due to a period break, refining the O−C diagram can almost entirely remove the trends. If we interpret the variation as an LTTE, we can consider possible configurations. While we currently favour a period break as the explanation for the residual O−C variations, more data on the star in the forthcoming years will help reveal whether the O−C variations are due to the LTTE or intrinsic period changes. Conclusions. Our study, based on a detailed light-curve fitting of V1109 Cas, offers insights into the discernment between Blazhko modulation, intrinsic period breaks, and the LTTE in binary systems. Our results highlight the complexity of determining binarity from O−C data traditionally used for detecting binary motion, calling for caution in this process.

Infrared surface brightness technique applied to RR Lyrae stars from the solar neighborhood

Context. The Baade-Wesselink (BW) method, also known as the pulsation parallax method, allows us to estimate distances to individual pulsating stars. Accurate geometric parallaxes obtained by the Gaia mission serve us in the calibration of the method and in the determination of its precision. This method also provides a way of determining mean radii of pulsating stars. Aims. The main aim of this work is to determine the scatter and possible dependence of p-factors of RR Lyrae stars on their pulsation periods. The secondary objective is to determine the mean radius-period relations for these stars. Methods. Our calibrations for RR Lyrae stars are based on photometric data gathered at the Cerro Murphy Observatory and parallaxes from the Data Release 3 of the Gaia space mission. We obtained spectroscopic data specifically for this project using high-resolution spectrographs. We used the infrared surface brightness (IRSB) version of the method that relies on a surface brightness-color relation that is dependent on the (V − K) color. It allows us to estimate stellar angular diameters, while tracing variations of the stellar radius using measurements of the stellar radial velocity obtained from spectroscopy. We present results based on four different empirical surface brightness-color relations, with three relations for dwarfs and subgiants and one for classical Cepheids. Results. We present our calibration of projection factors and determination of the mean radii for nine Galactic RR Lyrae stars. We obtained a spread of p-factors of around 0.07–0.08 for our sample of RR Lyrae stars from the solar neighborhood. However, depending on a given SBCR, we also found relations between the p-factor and the pulsation period for RRab stars with a root mean square (rms) scatter around the relation of around 0.05, but with relatively large uncertainty on the relation parameters. We also present relations between the mean radius and period for RR Lyrae pulsating in the fundamental mode with an rms scatter around the relation of 0.012 R⊙. We observe a clear offset between p-factors obtained using the IRSB technique (with a mean p value between 1.39 and 1.45) and values inferred using the SPIPS tool. This confirms that different implementations of the BW method are sensitive to various components of the p-factor. On the other hand, we obtain a similar scatter for p, as observed in a previous study based on the SPIPS tool. Our period-radius relations are in a good agreement with both the inference based on SPIPS and theoretical predictions.

Finding The Relationship between Pulsation Motion and Hα Emission Lines on γ Cas using Photometric and Spectroscopic Data

Abstract Pulsating stars show variability in brightness due to changes in their volume. There are two types of pulsation/oscillation: radial and non-radial. Radial oscillation keeps the symmetrical shape of the star, while non-radial oscillation does not. We study a pulsating star: γ Cas, classified as a B0.5 IVe variable star because of its pulsations and also an emission star due to the emission lines in its spectra. When a star pulsates, its atmosphere expands and contracts, potentially leading to the formation of shock waves. Observational features such as discontinuities in the radial velocity curve and emission lines support this phenomenon. In this study, we analyse the relationship between pulsation motion and the H α emission lines of γ Cas using photometric and spectroscopic data. Photometric data is obtained from the BRIght Target Explorer (BRITE) catalog, while spectroscopic data comes from the Be Star Spectra (BeSS) catalog. Both datasets cover the same observational dates: August 28 to October 26, 2015. Confirming the pulsation period of γ Cas at 1.21 days, we also discover an added pulsation period of 10 days from photometric data. To confirm this, we examine spectroscopic data from three specific dates: August 28, September 7, and 27, 2015. Notably, the relative flux of the H α emission lines continue to increase, leading us to conclude that these lines are indeed caused by the pulsation of γ Cas.

AstroSat and Insight-HXMT Observations of the Long-period X-Ray Pulsar 4U 2206+54

We report the timing and spectral studies of the accreting X-ray pulsar 4U 2206+54 using AstroSat and Insight-HXMT observations taken in 2016 and 2020 respectively. X-ray pulsations from the system are detected by both missions. The AstroSat discovered a significant periodic signal at ∼5619 s in 2016, and Insight-HXMT found a pulsation period at ∼5291 s in 2020. A comparison of its spin-period evolution with the present spin-period estimates shows that the neutron star in 4U 2206+54 now has recently been undergoing a spin-up episode after attaining to its slow pulsations of 5750 s around 2015 from its prolonged spin-down phase. The present average spin-up rate of the pulsar is found to be at ∼1.2 × 10−13 Hz s−1. The phase-averaged spectra of the pulsar in 1–60 keV could be explained with a high-energy cutoff power-law continuum model; no evident line features are found with AstroSat. The application of Comptonization models such as comptt and compmag to the phase-averaged spectra of 4U 2206+54 reveals a hotter source photon region near the pulsar with an emission size extending to ∼2–2.8 km. Using the quasi-spherical settling accretion theory, we explain the present spin-up and the possibility of the strong magnetic field of the pulsar.

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.