In spite of the relevance of Classical Cepheids as primary distance indicators, a general consensus on the dependence of the Period-Luminosity (PL) relation on the Cepheid chemical composition has not been achieved yet. From the theoretical point of view, our previous investigations were able to reproduce some empirical tests for suitable assumptions on the helium to metal relative enrichment, but those results relied on specific assumptions concerning the Mass-Luminosity relation and the efficiency of the convective transfer in the pulsating envelopes. In this paper, we investigate the effects of the assumed value of the mixing length parameter l/Hp on the pulsation properties and we release the assumption of a fixed Mass-Luminosity relation. As a whole, we show that our pulsation relations appear fully consistent with the observed properties of Galactic and Magellanic Cloud Cepheids, supporting the predicted steepening and brightening of the PL relations when moving from metal-rich to metal-poor variables. Moreover, we show that the distances inferred by the predicted PW relations agree with recently measured trigonometric parallaxes, whereas they suggest a correction to the values based on the Infrared Surface Brightness technique, as already found from an independent method. Finally, also the pulsation metal contents suggested by the predicted PW relations appear in statistical agreement with spectroscopic [Fe/H] measurements.
There is strong evidence that the period-luminosity (PL) relation for the Large Magellanic Cloud (LMC) Cepheids shows a break at a period around 10 days. Since the LMC PL relation is extensively used in distance scale studies, the non-linearity of the LMC PL relation may affect the results based on this LMC calibrated relation. In this paper we show that this problem can be remedied by using the Wesenheit function in obtaining Cepheid distances. This is because the Wesenheit function is linear although recent data suggests that the PL and the period-colour (PC) relations that make up the Wesen…
We perform a detailed analysis of Cepheids in NGC 4258, Magellanic Clouds and Milky Way in order to verify the reliability of the theoretical scenario based on a large set of nonlinear convective pulsation models. We derive Wesenheit functions from the synthetic BVI magnitudes of the pulsators and we show that the sign and the extent of the metallicity effect on the predicted Period-Wesenheit (P-W) relations change according to the adopted passbands. These P-W relations are applied to measured BVI magnitudes of NGC 4258, Magellanic and Galactic Cepheids available in the literature. We find tha…
We have performed systematic frequency analysis of the LMC Cepheids observed by OGLE project. Several new types of pulsation behaviour are identified, including triple-mode and amplitude-modulated double-mode pulsations. In ~10% of the first overtone Cepheids we find low amplitude secondary periodicities corresponding to nonradial modes. This is the first evidence for excitation of nonradial oscillations in Classical Cepheid variables.
A number of recent works have suggested that the period-luminosity (PL) relation for the Large Magellanic Cloud (LMC) Cepheids exhibits a controversial nonlinear feature with a break period at 10 days. Therefore, the aim of this Research Note is to test the linearity/nonlinearity of the PL relations for the LMC Cepheids in BVIcJHKs band, as well as in the Wesenheit functions. We show that simply comparing the long and short period slopes, together with their associate d standard deviations, leads to a strictly larger error rate than applying rigorous statistical tests such as the F-test. We ap…
I investigate the discrepancy between the evolution and pulsation masses for Cepheid variables. A number of recent works have proposed that non-canonical mass-loss can account for the mass discrepancy. This mass-loss would be such that a 5Mo star loses approximately 20% of its mass by arriving at the Cepheid instability strip; a 14Mo star, none. Such findings would pose a serious challenge to our understanding of mass-loss. I revisit these results in light of the Padova stellar evolutionary models and find evolutionary masses are ($17\pm5$)% greater than pulsation masses for Cepheids between 5…
Within the current uncertainties in the treatment of the coupling between pulsation and convection, limiting amplitude, nonlinear, convective models appear the only viable approach for providing theoretical predictions about the intrinsic properties of radial pulsators. In this paper we present the results of a comprehensive set of Cepheid models computed within such theoretical framework for selected assumptions on their original chemical composition.
We present a detailed comparison between predicted and empirical PL_{I,K} relations and Wesenheit function for Galactic and Magellanic Clouds (MCs) First Overtone (FO) Cepheids. We find that zero-points predicted by Galactic Cepheid models based on a noncanonical (mild overshooting) Mass-Luminosity (ML) relation are in very good agreement with empirical zero-points based on HIPPARCOS parallaxes, while those based on canonical (no overshooting) ML relation are about 0.2-0.3 mag brighter. We also find that predicted and empirical PL_K relation and Wesenheit function give, according to optical (V…
We report the discovery of 20 1H/2H and 7 F/1H beat Cepheids in the SMC by the MACHO Project. We utilize the 20 1H/2H stars to determine lightcurve shape for the SMC second-overtone (2H) mode of Cepheid pulsation. We predict, similar to the findings of Alcock et al. (1997, ApJ, submitted), that 2H Cepheids will have nearly or purely sinusoidal light variations; that the P-L relation for 2H Cepheids will not be distinguishable from the P-L relation for 1H Cepheids within photometric accuracy; and that 2H stars may be discernable from F and 1H stars using the amplitude-period diagram and Fourier…
We have obtained mosaic images of NGC 3109 in the V and I bands on 74 nights, spanning approximately one year. From these data, we have conducted an extensive search for Cepheid variables over the entire field of the galaxy, resulting in the discovery of 113 variables with periods ranging from 3.4 to 31.4 days. In this sample, 76 Cepheids, including many long-period variables, were not known before. For the previously known 45 Cepheids in this galaxy, our data proved that reported periods were wrong for 14 objects; for nearly all other previously known Cepheid variables we were able to signifi…
In a previous paper, we reported on the discovery of more than a hundred new Cepheid variables in the Sculptor Group spiral NGC 300 from wide-field images taken in the B and V photometric bands at ESO/La Silla. In this paper, we present additional VI data, derive improved periods and mean magnitudes for the variables, and construct period-luminosity relations in the V, I and the reddening-independent (V-I) Wesenheit bands using 58 Cepheid variables with periods between 11 and 90 days. We obtain tightly defined relations, and by fitting the slopes determined for the LMC Cepheids by the OGLE II …
MACHO Project photometry of 45 LMC FO/SO beat Cepheids which pulsate in the first and second overtone (FO and SOo, respectively) has been analysed to determine the lightcurve characteristics for the SO mode of Cepheid pulsation. We predict that singly-periodic SO Cepheids will have nearly sinusoidal lightcurves; that we will only be able to discern SO Cepheids from fundamental (F) and (FO) Cepheids for P <= 1.4 days; and that the SO distribution will overlap the short-period edge of the LMC FO Cepheid period-luminosity relation (when both are plotted as a function of photometric period). We al…
A theoretical investigation of the pulsation behavior of so-named ``anomalous'' Cepheids is presented. The study is based on nonlinear convective pulsation models with $Z$=0.0001 and 0.0004, mass in the range 1.3-2.2 Mo and various luminosity levels. Based on these computations, we derive period, bolometric light curves and the edges of the instability strip, showing that a variation of the metal abundance from Z=0.0001 to 0.0004 has quite small effects on these quantities. Then, using bolometric corrections and color-temperature transformations, we are able to provide the predicted relations …
We investigate a selected sample of Galactic classical Cepheids with available distance and reddening estimates in the framework of the theoretical scenario provided by pulsation models, computed with metal abundance Z=0.02, helium content in the range of Y=0.25 to 0.31, and various choices of the stellar mass and luminosity. After transforming the bolometric light curve of the fundamental models into BVRIJK magnitudes, we derived analytical relations connecting the pulsation period with the stellar mass, the mean (intensity-averaged) absolute magnitude, and the color of the pulsators. These r…
We present a differential study of 500 Magellanic Cepheids with 3 million measurements obtained as a by-product of the EROS microlensing survey. The data-set is unbiased and provides an excellent basis for a differential analysis between LMC and SMC. We investigate the pulsational properties of the Cepheids as a function of metallicity and the constraints they give to stellar pulsation, stellar evolution theory and opacities. Moreover, our empirical analysis shows a small metallicity effect on the period-luminosity relation which has a significant implication to recent distance estimates based…
It has been observed that Cepheids in the Magellanic Clouds have lower masses for the same luminosity than those in the Milky Way. The model, from Neilson & Lester (2008), of pulsation-driven mass loss for Cepheids is applied to theoretical models of Cepheids with metallicity consistent with the Milky Way and Large and Small Magellanic Clouds. The mass-loss model is analyzed using the metallicity correction of the Period-Luminosity relation to compare the ratio of mass loss of Cepheids with lower metallicity to that of Cepheids with solar metallicity. It is determined that mass loss may be lar…
An analytical derivation is presented for computing mass-loss rates of Cepheids by using the method of Castor, Abbott, & Klein (1975) modified to include a term for momentum input from pulsation and shocks generated in the atmosphere. Using this derivation, mass-loss rates of Cepheids are determined as a function of stellar parameters. When applied to a set of known Cepheids, the calculated mass-loss rates range from 10^{-10} to 10^{-7}M_{Sun}/yr, larger than if the winds were driven by radiation alone. Infrared excesses based on the predicted mass-loss rates are compared to observations from …
We present nonlinear theoretical results concerning the predicted pulsation behaviour of stellar models suitable for the sample of Cepheids belonging to the Large Magellanic Cloud (LMC) star cluster NGC 1866. The blue and red edges of the instability strip, transformed into the observative plane (V, B-V) by adopting current values for the distance modulus (DM=18.57 mag., Welch et al. 1991) and for the reddening (E(B-V)=0.06 mag., Arp 1967; Walker 1974; Walker 1987), are in agreement with the observed distribution of Cepheids in the quoted cluster. Moreover, the distribution of observational da…
The longstanding problem of modeling double-mode behaviour of classical pulsators was solved with the incorporation of turbulent convection into pulsation hydrocodes. However, the reasons for the computed double-mode behaviour were never clearly identified. In our recent papers (Smolec & Moskalik 2008a,b) we showed that the double-mode behaviour results from the neglect of negative buoyancy effects in some of the hydrocodes. If these effects are taken into account, no stable non-resonant double-mode behaviour can be found. In these proceedings we focus our attention on the role of negative buo…
We have selected a sample of Galactic Cepheids for which accurate estimates of radii, distances, and photometric parameters are available. The comparison between their pulsation masses, based on new Period-Mass-Radius (PMR) relations, and their evolutionary masses, based on both optical and NIR Color-Magnitude (CM) diagrams, suggests that pulsation masses are on average of the order of 10% smaller than the evolutionary masses. Current pulsation masses show, at fixed radius, a strongly reduced dispersion when compared with values published in literature.The increased precision in the pulsation …
We present new theoretical Period-Radius (PR) relations for first overtone Galactic Cepheids. Current predictions are based on several sequences of nonlinear, convective pulsation models at solar chemical composition (Y=0.28, Z=0.02) and stellar masses ranging from 3.0 to 5.5 Mo. The comparison between predicted and empirical radii of four short-period Galactic Cepheids suggests that QZ Nor and EV Sct are pulsating in the fundamental mode, whereas Polaris and SZ Tau pulsate in the first overtone. This finding supports the mode identifications that rely on the comparison between direct and Peri…
We have investigated line profiles in a large sample of Cepheid spectra, and found four stars that show unusual (for Cepheids) line profile structure (bumps or/and asymmetries). The profiles can be phase dependent but the behavior persists over many cycles. The asymmetries are unlikely to be due to the spectroscopic binarity of these stars or the specific velocity field in their atmospheres caused by shock waves. As a preliminary hypothesis, we suggest that the observed features on the line profiles in the spectra of X Sgr, V1334 Cyg, EV Sct and BG Cru can be caused by the non-radial oscillati…
We report on the discovery of a Cepheid population in the Sculptor Group spiral galaxy NGC 247 for the first time. On the basis of wide-field images collected in photometric surveys in V and I bands which were conducted with three different telescopes and cameras, 23 Cepheid variables were discovered with periods ranging from 17 to 131 days. We have constructed the period-luminosity relations from these data and obtain distance moduli to NGC 247 of 28.20 $\pm$ 0.05 mag (internal error) in V, 28.04 $\pm$ 0.06 mag in I, and 27.80 $\pm$ 0.09 mag in the reddening-independent Wesenheit index. From …
We have obtained mosaic images of NGC 6822 in V and I bands on 77 nights. From these data, we have conducted an extensive search for Cepheid variables over the entire field of the galaxy, and we have found 116 such variables with periods ranging from 1.7 to 124 days. We used the long-period ($>$ 5.6 days) Cepheids to establish the period-luminosity relations in V, I and in the reddening-independent Wesenheit index, which are all very tightly defined. Fitting the OGLE LMC slopes in the various bands to our data, we have derived distance values for NGC 6822 in V, I and ${\rm W}_{\rm I}$ which ag…
We present new Period-Age (PA) and Period-Age-Color (PAC) relations for fundamental and first overtone classical Cepheids. Current predictions rely on homogeneous sets of evolutionary and pulsation models covering a broad range of stellar masses and chemical compositions. We found that PA and PAC relations present a mild dependence upon metal content. Moreover, the use of different PA and PAC relation for fundamental and first overtone Cepheids improves the accuracy of age estimates in the short-period (log P < 1) range (old Cepheids), because they present smaller intrinsic dispersions. At the…
We present the results of an extensive theoretical investigation on the pulsation behavior of Bump Cepheids. We constructed several sequences of full amplitude, nonlinear, convective models by adopting a chemical composition typical of Large Magellanic Cloud (LMC) Cepheids (Y=0.25, Z=0.008) and stellar masses ranging from 6.55 to 7.45 solar masses. We find that theoretical light and velocity curves reproduce the HP, and indeed close to the blue edge the bump is located along the descending branch, toward longer periods it crosses at first the luminosity/velocity maximum and then it appears alo…
We analyse a sample of 236 Cepheids from the Hipparcos catalog, using the method of ``reduced parallaxes'' in V, I, K and the reddening-free ``Wesenheit-index''. We compare our sample to those considered by Feast & Catchpole (1997) and Lanoix et al. (1999). We stress the importance of two systematic effects which influence the distance to the LMC: the slopes of the Galactic PL-relations and metallicity corrections. In order to assess the influence of these various effects, we present 27 distance moduli (DM) to the LMC, based on three different colours (V,I,K), three different slopes, and three…
We report the discovery of eleven beat Cepheids in the Small Magellanic Cloud, using data obtained by the EROS microlensing survey. Four stars are beating in the fundamental and first overtone mode (F/1OT), seven are beating in the first and second overtone (1OT/2OT). The SMC F/1OT ratio is systematically higher than the LMC F/1OT, while the 1OT/2OT period ratio in the SMC Cepheids is the same as the LMC one.
In this paper we present an extended set of nonlinear convective pulsation models at varying the metallicity and DY/DZ ratio. The predicted instability strip and bolometric light curves are discussed by comparing the new models with our previous ones. In particular, the dependence on both metal and helium abundances is investigated. By transforming the bolometric light curves into the observational bands we are able to derive both Period-Color-Luminosity and Wesenheit relations for each selected chemical composition. Synthetic Period-Luminosity relations are obtained by populating the instabil…
The Galactic Cepheid period histogram has a strong dip between 8 and 10 days that has defied an explanation based on evolutionary and linear pulsation studies. We show here that this deficiency is caused by the instability of the nonlinear fundamental pulsation cycle in this period range. The strong metallicity dependence of this instability is consistent with the absence of a corresponding minimum in the Magellanic Cloud data. Our results also suggest that the Galactic Cepheids must have a large spread in metallicity.
XZ Ceti is the only known anomalous Cepheid in the Galactic field. Being the nearest and brightest such variable star, a detailed study of XZ Ceti may shed light on the behaviour of anomalous Cepheids whose representatives have been mostly detected in external galaxies. CCD photometric and radial velocity observations have been obtained. The actual period and amplitude of pulsation were determined by Fourier analysis. The long time scale behaviour of the pulsation period was studied by the method of the O-C diagram using the archival Harvard photographic plates and published photometric data. …
We investigate the theoretical scenario concerning the large sample of variables recently discovered in the dwarf, metal-poor irregular galaxy Leo A, focusing the attention on the "Anomalous" Cepheid phenomenon and its correlation with RR Lyrae stars, Classical and Population II Cepheids. To this purpose, we make use of suitable stellar and pulsation models to depict the pulsational history of evolutionary structures with metallicity Z=0.0004, finding that He-burning pulsators are expected only outside the mass interval ~ 0.8-1.7 Mo. The dependence on metal content of this scenario has been in…
A short analysis is presented of the effects on the cepheid light curve shape, i.e. on the Fourier parameters usually adopted for its description, of the blending of the stellar image with other close stars. The conclusion is that, within reasonable error, the effects are in general small and the Fourier decomposition is confirmed to be a useful tool for pulsation mode discrimination. A large effect has been found on the phase differences in a narrow period range corresponding to the known resonance centers between pulsation modes.
Observations of the southern Cepheid l Car to yield the mean angular diameter and angular pulsation amplitude have been made with the Sydney University Stellar Interferometer (SUSI) at a wavelength of 696 nm. The resulting mean limb-darkened angular diameter is 2.990+-0.017 mas (i.e. +-0.6 per cent) with a maximum-to-minimum amplitude of 0.560+-0.018 mas corresponding to 18.7+-0.6 per cent in the mean stellar diameter. Careful attention has been paid to uncertainties, including those in measurements, in the adopted calibrator angular diameters, in the projected values of visibility squared at …
We utilize nonlinear pulsation models to reproduce the observed light and color curves for two samples of bump Cepheid variables, 19 from the Large Magellanic Cloud and 9 from the Small Magellanic Cloud. This analysis determines the fundamental parameters mass, luminosity, effective temperature, metallicity, distance and reddening for the sample of stars. The use of light curve shape alone to determine metallicity is a new modelling technique introduced here. The metallicity, distance and reddening distributions for the two samples are in agreement with those of similar stellar populations in …
We present the discovery and CCD observations of the first eclipsing binary with a Type II Cepheid component in our Galaxy. The pulsation and orbital periods are found to be 4.1523 and 51.38 days, respectively, i.e. this variable is a system with the shortest orbital period among known Cepheid binaries. Pulsations dominate the brightness variations. The eclipses are assumed to be partial. The EB-subtype eclipsing light curve permits to believe that the binary's components are non-spherical.
The theoretical period-luminosity relation for Cepheids was deduced from stellar evolution and pulsation computations. Questions concerning the metallicity dependence of the relation as well as the constancy of the pulsation `constant' Q in the instablility strip were addressed. We found no significant metallicity dependence of either the slope or of the zero-point of the period-luminosity relation. The Q-value, however, varies across the instability strip, in effective temperature as well as in luminosity; its metallicity dependence, on the other hand, is marginal.
The discovery of a large number of beat Cepheids in the Large Magellanic Cloud in the MACHO survey, provides an opportunity to compare the characteristics of such Cepheids over a range of metallicities. We produced a large grid of linear nonadiabatic pulsation models using the OPAL opacity tables and with compositions corresponding to those of the Milky Way, and the Large and Small Magellanic Clouds. Using the relationship between the period ratio and the main pulsation period, we are able to define a range of models which correspond to the observed beat Cepheids, and thereby constrain the phy…
We have computed stellar evolutionary models for stars in a mass range characteristic of Cepheid variables ($3<m/\Msol<12$) for different metallicities representative of the Galaxy and the Magellanic Clouds populations. The stellar evolution calculations are coupled to a linear non adiabatic stability analysis to get self-consistent mass-period-luminosity relations. The period - luminosity relation as a function of metallicity is analysed and compared to the recent EROS observations in the Magellanic Clouds. The models reproduce the observed width of the instability strips for the SMC and LMC.…
For many years modeling of double-mode pulsation of classical pulsators was a challenging problem. Inclusion of turbulent convection into pulsation hydrocodes finally led to stable double-mode models. However, it was never analysed, which factor of turbulent convection is crucial. We show that the double-mode behaviour displayed in the computed models results from incorrect assumptions adopted in some of the pulsation hydrocodes, namely from the neglect of buoyant forces in convectively stable layers. This leads to significant turbulent energies and consequently to strong eddy-viscous damping …
We examine the properties of 17 long period (80-180 days) and very luminous (median absolute magnitude of M_I= -7.93 and M_V= -7.03) Cepheids to see if they can serve as an useful distance indicator. We find that these Ultra Long Period (ULP) Cepheids have a relatively shallow Period-Luminosity (PL) relation, so in fact they are more "standard candle"-like than classical Cepheids. In the reddening-free Wesenheit index, the slope of the ULP PL relation is ~10 times less steep than the standard PL relation for the SMC Cepheids. The scatter of our sample about the W_I PL relation is 0.22 mag, app…
We constructed several sequences of classical Cepheid envelope models at solar chemical composition ($Y=0.28, Z=0.02$) to investigate the dependence of the pulsation properties predicted by linear and nonlinear hydrodynamical models on input physics. To study the dependence on the equation of state (EOS) we performed several numerical experiments by using the simplified analytical EOS originally developed by Stellingwerf and the recent analytical EOS developed by Irwin. Current findings suggest that the pulsation amplitudes as well as the topology of the instability strip marginally depend on …
We have performed a detailed systematic search for multiperiodicity in the Population I Cepheids of the Large Magellanic Cloud. In this process we have identified for the first time several new types of Cepheid pulsational behaviour. We have found two triple-mode Cepheids pulsating simultaneously in the first three radial overtones. In 9% of the first overtone Cepheids we have detected weak, but well resolved secondary periodicities. They appear either very close to the primary pulsation frequency or at a much higher frequency with a characteristic period ratio of 0.60-0.64. In either case, th…
We compute the phase lags between the radial velocity curves and the light curves $ΔΦ_1= ϕ^{V_r}_1 - ϕ^{mag}_1$ for classical Cepheid model sequences both in the linear and the nonlinear regimes. The nonlinear phase lags generally fall below the linear ones except for high period models where they lie above, and of course for low pulsation amplitudes where the two merge. The calculated phase lags show good agreement with the available observational data of normal amplitude Galactic Cepheids. The metallicity has but a moderate effect on the phase lag, while the mass-luminosity relation and the …
Period-colour (PC) and amplitude-colour (AC) relations at maximum, mean and minimum light are constructed from a large grid of full amplitude hydrodynamic models of Cepheids with a composition appropriate for the SMC (Small Magellanic Cloud). We compare these theoretical relations with those from observations. The theoretical relations are in general good agreement with their observational counterparts though there exist some discrepancy for short period (log [P] < 1) Cepheids. We outline a physical mechanism which can, in principle, be one factor to explain the observed PC/AC relations for th…
Our analysis of the Deep CFHT M33 variability survey database has uncovered 5 Beat Cepheids (BCs) that are pulsating in the fundamental and first overtone modes. With {\it only} the help of stellar pulsation theory and of mass--luminosity (M-L) relations, derived from evolutionary tracks, we can accurately determine the metallicities Z of these stars. The [O/H] metallicity gradient of -0.16 dex/kpc that is inferred from the M33 galacto-centric distances of these Cepheids and from their 'pulsation' metallicities is in excellent agreement with the standard spectroscopic metallicity gradients tha…
An analysis of the observed characteristics of the Galactic Cepheid variables is carried out in the framework of their period-luminosity relation being used as a standard candle for distance measurement. The variation of the observed number density of Galactic Cepheids as function of their period and amplitude along with stellar pulsation characteristics is used to divide the population into two groups: one with low periods, probably multi-mode or higher mode oscillators, and another of high period variables which should be dominantly fundamental mode radial pulsators. Methods to obtain extinc…
It is well known that the mere location of a Beat Cepheid model in a Period Ratio \vs Period diagram (Petersen diagram) puts constraints on its metallicity Z. But these bounds are sensitive to the mixture of elements that are lumped into the parameter $Z$. In this short paper we update the previous results that were based on the Grevesse-Noels solar mixture to the recent, revised Asplund, Grevesse & Sauval (2005) (AGS) solar mixture. We also examine the effect of the envelope depth on the accuracy of the computed pulsation periods. We find that for low period Cepheids with high Z the customary…
We have used the Advanced Camera for Surveys aboard the Hubble Space Telescope to obtain F435W, F555W and F814W single-epoch images of six fields in the spiral galaxy NGC 300. Taking advantage of the superb spatial resolution of these images, we have tested the effect that blending of the Cepheid variables studied from the ground with close stellar neighbors, unresolved on the ground-based images, has on the distance determination to NGC 300. Out of the 16 Cepheids included in this study, only three are significantly affected by nearby stellar objects. After correcting the ground-based magnitu…
The numerical hydrodynamic modelling of beat Cepheid behavior has been a longstanding quest in which purely radiative models have failed miserably. We find that beat pulsations occur naturally when turbulent convection is accounted for in our hydrodynamics codes. The development of a relaxation code and of a Floquet stability analysis greatly facilitates the search for and analysis of beat Cepheid models. The conditions for the occurrence of beat behavior can be understood easily and at a fundamental level with the help of amplitude equations. Here a discriminant D arises whose sign decides wh…
