arXiv subjects

Adriano Pietrinferni

Publications and source records attributed to Adriano Pietrinferni.

At least 19 recordsLinked to original sources

arXiv2025

The dwarf irregular galaxy NGC 6822. II. Young, intermediate and old stellar populations: comparison between theory and observations

This paper presents a quantitative analysis of the stellar content in the Local Group dwarf irregular galaxy NGC 6822 by comparing stellar evolution models and observations in color-magnitude diagrams (CMDs) and color-color diagrams (CC-Ds). Our analysis is based on optical ground-based g,r,i photometry, and deep archive HST photometry of two fields in the galaxy disk. We compared young, intermediate-age, and old stellar populations with isochrones from the BaSTI-IAC library and found that NGC 6822 hosts a quite metal-rich ([Fe/H] = -0.7 to -0.4) young component with an age ranging from 20 to 100 Myr. The intermediate-age population experienced a modest chemical enrichment between 4 and 8 Gyr ago while stars older than 11 Gyr have a low metal abundance ([Fe/H] ~ -1.70). We also identified the AGB clump population with a luminosity peak at i ~ 23.35 mag. Our analysis of both the CMD and the optical-NIR-MIR CC-Ds of AGB oxygen- and carbon-rich stars, using the PARSEC+COLIBRI isochrones with and without circumstellar dust, reveal that this stellar component exhibits a spread in age from 1 to 2 Gyr and in metallicity between [Fe/H]=-1.30 and -1.70. The stellar models we used reproduce very well the two distinct color sequences defined by AGB O- and C-rich stars in the various optical-NIR-MIR CC-Ds, suggesting that they are reliable diagnostics to identify and characterise intermediate-age stellar populations. However, we also find that evolutionary prescriptions in the optical i-(r-i) CMDs predict, at fixed color, systematically lower luminosities than observed AGB stars.

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arXiv2025

The elderly among the oldest: new evidence for extremely metal-poor RR Lyrae stars

We performed a detailed spectroscopic analysis of three extremely metal-poor RR Lyrae stars, exploring uncharted territories at these low metallicities for this class of stars. Using high-resolution spectra acquired with HARPS-N at TNG, UVES at VLT, and PEPSI at LBT, and employing Non-Local Thermodynamic Equilibrium (NLTE) spectral synthesis calculations, we provide abundance measurements for Fe, Al, Mg, Ca, Ti, Mn, and Sr. Our findings indicate that the stars have metallicities of [Fe/H] = -3.40 \pm 0.05, -3.28 \pm 0.02, and -2.77 \pm 0.05 for HD 331986, DO Hya, and BPS CS 30317-056, respectively. Additionally, we derived their kinematic and dynamical properties to gain insights into their origins. Interestingly, the kinematics of one star (HD 331986) is consistent with the Galactic disc, while the others exhibit Galactic halo kinematics, albeit with distinct chemical signatures. We compared the [Al/Fe] and [Mg/Mn] ratios of the current targets with recent literature estimates to determine whether these stars were either accreted or formed in situ, finding that the adopted chemical diagnostics are ineffective at low metallicities ([Fe/H] $\lesssim -$1.5). Finally, the established horizontal branch evolutionary models, indicating that these stars arrive at hotter temperatures on the Zero-Age Horizontal Branch (ZAHB) and then transition into RR Lyrae stars as they evolve, fully support the existence of such low-metallicity RR Lyrae stars. As a consequence, we can anticipate detecting more of them when larger samples of spectra become available from upcoming extensive observational campaigns.

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arXiv2024

Classical Cepheid Pulsation properties in the Rubin-LSST filters

Homogeneous multi-wavelength observations of classical Cepheids from the forthcoming Rubin-LSST have the potential to significantly contribute to our understanding of the evolutionary and pulsation properties of these pulsating stars. Updated pulsation models for Classical Cepheid stars have been computed under various assumptions about chemical compositions, including relatively low metallicity ($Z$ = $0.004$ with $Y$ =$0.25$ and $Z$=$0.008$ with $Y$ =$0.25$), solar metallicity ($Z$=$0.02$ with $Y$=$0.28$), and supersolar metallicity environments ($Z$ = $0.03$ with $Y$ = $0.28$). From the predicted periods, intensity-weighted mean magnitudes, and colors, we have derived the first theoretical pulsation relations in the Rubin-LSST filters (ugrizy), including period-luminosity-color, period-Wesenheit, and period-age-color relations. We find that the coefficients of these relations are almost insensitive to the efficiency of superadiabatic convection but are significantly affected by the assumption of the mass-luminosity relation and the adopted chemical composition. Metal-dependent versions of these relations are also derived, representing valuable tools for individual distance determinations and correction for metallicity effects on the cosmic distance scale.

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arXiv2023

The updated BaSTI stellar evolution models and isochrones.IV. alpha-depleted calculations

This is the fourth paper of our new release of the BaSTI (a Bag of Stellar Tracks and Isochrones) stellar model and isochrone library. Following the updated solar-scaled, alpha-enhanced, and white dwarf model libraries, we present here alpha-depleted ([alpha/Fe] = -0.2) evolutionary tracks and isochrones, suitable to study the alpha-depleted stars discovered in Local Group dwarf galaxies and in the Milky Way. These calculations include all improvements and updates of the solar-scaled and alpha-enhanced models, and span a mass range between 0.1 and 15 Msun, 21 metallicities between [Fe/H] = -3.20 and +0.45 with a helium-to-metal enrichment ratio dY/dZ = 1.31, homogeneous with the solar-scaled and alpha-enhanced models. The isochrones -- available in several photometric filters -- cover an age range between 20 Myr and 14.5 Gyr, including the pre-main-sequence phase. We have compared our isochrones with independent calculations of alpha-depleted stellar models, available for the same alpha-element depletion adopted in present investigation. We have also discussed the effect of an alpha-depleted heavy element distribution on the bolometric corrections in different wavelength regimes. Our alpha-depleted evolutionary tracks and isochrones are publicly available at the BaSTI website.

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arXiv2022

On the dwarf irregular galaxy NGC 6822. I. Young, intermediate and old stellar populations

We present accurate and deep multi-band ($g,r,i$) photometry of the Local Group dwarf irregular galaxy NGC 6822. The images were collected with wide field cameras at 2m/4m- (INT,CTIO,CFHT) and 8m-class telescopes (SUBARU) covering a 2 square degrees FoV across the center of the galaxy. We performed PSF photometry of $\approx$7,000 CCD images and the final catalog includes more than 1 million objects. We developed a new approach to identify candidate field and galaxy stars, and performed a new estimate of the galaxy center by using old stellar tracers finding that it differs by 1.15 (RA) and 1.53 (DEC) arcmin from previous estimates. We also found that young (Main Sequence, Red Supergiants), intermediate (Red Clump, Asymptotic Giant Branch [AGB]) and old (Red Giant Branch [RGB]) stars display different radial distributions. Old stellar population is spherically distributed and extends to radial distances larger than previously estimated ($\sim$1 degree). The young population shows a well defined bar and a disk-like distribution, as suggested by radio measurements, that is off-center compared with old population. We discuss pros and cons of the different diagnostics adopted to identify AGB stars and develop new ones based on optical-NIR-MIR color-color diagrams (CCDs) to characterize Oxygen and Carbon (C) rich stars. We found a mean population ratio between Carbon and M-type (C/M) stars of 0.67$\pm$0.08 (optical/NIR/MIR) and we used the observed C/M ratio with empirical C/M-metallicity relations to estimate a mean iron abundance of [Fe/H]$\sim$-1.25 ($σ$=0.04 dex) that agrees quite well with literature estimates.

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arXiv2021

The updated BaSTI stellar evolution models and isochrones. III. White Dwarfs

We present new cooling models for carbon-oxygen white dwarfs with both H- and He-atmospheres, covering the whole relevant mass range, to extend our updated BaSTI (a Bag of Stellar Tracks and Isochrones) stellar evolution archive. They have been computed using core chemical stratifications obtained from new progenitor calculations, adopting a semiempirical initial-final mass relation. The physics inputs have been updated compared to our previous BaSTI calculations: ^{22}Ne diffusion in the core is now included, together with an updated CO phase diagram, and updated electron conduction opacities. We have calculated models with various different neon abundances in the core, suitable to study white dwarfs in populations with metallicities ranging from super-solar to metal poor, and have performed various tests/comparisons of the chemical stratification and cooling times of our models. Two complete sets of calculations are provided, for two different choices of the electron conduction opacities, to reflect the current uncertainty in the evaluation of the electron thermal conductivity in the transition regime between moderate and strong degeneracy, crucial for the H and He envelopes. We have also made a first, preliminary estimate of the effect -- that turns out to be generally small -- of Fe sedimentation on the cooling times of white dwarf models, following recent calculations of the phase diagrams of carbon-oxygen-iron mixtures. We make publicly available the evolutionary tracks from both sets of calculations, including cooling times and magnitudes in the Johnson-Cousins, Sloan, Pan-STARSS, Galex, Gaia-DR2, Gaia-eDR3, HST-ACS, HST-WFC3, and JWST photometric systems.

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arXiv2021

Period-Age-Metallicity and Period-Age-Color-Metallicity relations for Classical Cepheids: an application to the Gaia EDR3 sample

Based on updated pulsation models for Classical Cepheids, computed for various assumptions about the metallicity and helium abundance, roughly representative of pulsators in the Small Magellanic Cloud ($Z$=$0.004$ and $Y$=$0.25$), Large Magellanic Cloud ($Z$=$0.008$ and $Y$=$0.25$), and M31 ($Z$=$0.03$ and $Y$=$0.28$), and self-consistent updated evolutionary predictions, we derived Period-Age and multi-band Period-Age-Color relations that also take into account variations in the Mass-Luminosity relation. These results, combined with those previously derived for Galactic Cepheids, were used to investigate the metallicity effect when using these variables as age indicators. In particular, we found that a variation in the metal abundance affects both the slope and the zero point of the above-mentioned relations. The new relations were applied to a sample of Gaia Early Data Release 3 Classical Cepheids. The retrieved distribution of the individual ages confirms that a brighter Mass-Luminosity relation produces older ages and that First Overtone pulsators are found to be concentrated towards older ages with respect to the Fundamental ones at a fixed Mass-Luminosity relation. Moreover, the inclusion of a metallicity term in the Period-Age and Period-Age-Color relations slightly modifies the predicted ages. In particular, the age distribution of the selected sample of Galactic Cepheids is found to be shifted towards slightly older values, when the F-mode canonical relations are considered, with respect to the case at a fixed solar chemical composition. A marginally opposite dependence can be found in the noncanonical F-mode and canonical FO-mode cases.

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arXiv2021

Electron conduction opacities at the transition between moderate and strong degeneracy: Uncertainties and impact on stellar models

Electron conduction opacities are one of the main physics inputs for the calculation of low- and intermediate-mass stellar models, and a critical question is how to bridge calculations for moderate and strong degeneracy, which are necessarily performed adopting different methods. The density-temperature regime at the boundary between moderate and strong degeneracy is in fact crucial for modelling the helium cores of red giant branch stars and the hydrogen/helium envelopes of white dwarfs. Prompted by recently published new, improved calculations of electron thermal conductivities and opacities for moderate degeneracy, we study different, physically motivated prescriptions to bridge these new computations with well established results in the regime of strong degeneracy. We find that these different prescriptions have a sizable impact on the predicted He-core masses at the He-flash (up to 0.01$M_{\odot}$ for initial total masses far from the transition to non-degenerate He-cores, and up to $\sim 0.04M_{\odot}$ for masses around the transition), the tip of the red giant branch (up to $\sim$0.1~mag) and the zero age horizontal branch luminosities (up to 0.03~dex for masses far from the transition, and up to $\sim$0.2~dex around the transition), and white dwarf cooling times (up to 40-45\% at high luminosities, and up to $\sim$25\% at low luminosities). Current empirical constraints on the tip of the red giant branch and the zero age horizontal branch absolute magnitudes do not allow yet to definitely exclude any of these alternative options for the conductive opacities. Tests against observations of slowly-cooling faint WDs in old stellar populations will need to be performed to see whether they can set some more stringent constraints on how to bridge calculations of conductive opacities for moderate and strong degeneracy.

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arXiv2019

Horizontal branch morphology: A new photometric parametrization

Theory and observations indicate that the distribution of stars along the horizontal branch of Galactic globular clusters mainly depends on the metal content. However, the existence of globular clusters with similar metal content and absolute age but different horizontal branch morphologies, suggests the presence of another parameter affecting the star distribution along the branch. To investigate the variation of the horizontal branch morphology in Galactic globular clusters, we define a new photometric horizontal branch morphology index, overcoming some of the limitations and degeneracies affecting similar indices available in the literature. We took advantage of a sample of 64 Galactic globular clusters, with both space-based imaging data and homogeneous ground-based photometric catalogues in five different bands ($U$,$B$,$V$, $R$, $I$). The new index, $τ_{HB}$, is defined as the ratio between the areas subtended by the cumulative number distribution in magnitude ($I$) and in colour ($V-I$) of all stars along the horizontal branch. This new index shows a linear trend over the entire range in metallicity (-2.35 $\leq$ [Fe/H] $\leq$ -0.12) covered by our Galactic globular cluster sample. We found a linear relation between $τ_{HB}$ and absolute cluster ages. We also found a quadratic anti-correlation with [Fe/H], becoming linear when we eliminate the age effect on $τ_{HB}$ values. Moreover, we identified a subsample of eight clusters that are peculiar according to their $τ_{HB}$ values. These clusters have bluer horizontal branch morphology when compared to typical ones of similar metallicity. These findings allow us to define them as the 'second parameter' clusters in the sample. A comparison with synthetic horizontal branch models suggests that they cannot be entirely explained with a spread in helium content.

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arXiv2017

The brightness of the Red Giant Branch tip: Theoretical framework, a set of reference models, and predicted observables

The brightness of the tip of the Red Giant Branch is a useful reference quantity for several fields of astrophysics. An accurate theoretical prediction is needed for such purposes. Aims. We intend to provide a solid theoretical prediction for it, valid for a reference set of standard physical assumptions, and mostly independent of numerical details. We examine the dependence on physical assumptions and numerical details, for a wide range of metallicities and masses, and based on two different stellar evolution codes. We adjust differences between the codes to treat the physics as identical as possible. After we have succeeded in reproducing the tip brightness between the codes, we present a reference set of models based on the most up to date physical inputs, but neglecting microscopic diffusion, and convert theoretical luminosities to observed infrared colours suitable for observations of resolved populations of stars and include analytic fits to facilitate their use. We find that consistent use of updated nuclear reactions, including an appropriate treatment of the electron screening effects, and careful time-stepping on the upper red giant branch are the most important aspects to bring initially discrepant theoretical values into agreement. Small, but visible differences remain unexplained for very low metallicities and mass values at and above 1.2 Msun, corresponding to ages younger than 4 Gyr. The colour transformations introduce larger uncertainties than the differences between the two stellar evolution codes. We demonstrate that careful stellar modeling allows an accurate prediction for the luminosity of the Red Giant Branch tip. Differences to empirically determined brightnesses may result either from insufficient colour transformations or from deficits in the constitutional physics. We present the best-tested theoretical reference values to date.

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arXiv2016

On the determination of the He abundance distribution in globular clusters from the width of the main sequence

One crucial piece of information to study the origin of multiple stellar populations in globular clusters, is the range of initial helium abundances $Δ{Y}$ amongst the sub-populations hosted by each cluster. These estimates are commonly obtained by measuring the width in colour of the unevolved main sequence in an optical colour-magnitude-diagram. The measured colour spread is then compared with predictions from theoretical stellar isochrones with varying initial He abundances, to determine $Δ{Y}$. The availability of UV/optical magnitudes thanks to the {\sl HST UV Legacy Survey of Galactic GCs} project, will allow the homogeneous determination of $Δ{Y}$ for a large Galactic globular cluster sample. From a theoretical point of view, accurate UV CMDs can efficiently disentangle the various sub-populations, and main sequence colour differences in the ACS $F606W-(F606W-F814W)$ diagram allow an estimate of $Δ{Y}$. We demonstrate that from a theoretical perspective the ($F606W-F814W$) colour is an extremely reliable He-abundance indicator. The derivative d$Y$/d($F606W-F814W$), computed at a fixed luminosity along the unevolved main sequence, is largely insensitive to the physical assumptions made in stellar model computations, being more sensitive to the choice of the bolometric correction scale, and is only slightly dependent on the adopted set of stellar models. From a theoretical point of view the ($F606W-F814W$) colour width of the cluster main sequence is therefore a robust diagnostic of the $Δ{Y}$ range.

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arXiv2016

On the red giant branch mass loss in 47 Tucanae: Constraints from the horizontal branch morphology

We obtain stringent constraints on the actual efficiency of mass loss for red giant branch stars in the Galactic globular cluster 47 Tuc, by comparing synthetic modeling based on stellar evolution tracks with the observed distribution of stars along the horizontal branch in the colour-magnitude-diagram. We confirm that the observed, wedge-shaped distribution of the horizontal branch can be reproduced only by accounting for a range of initial He abundances --in agreement with inferences from the analysis of the main sequence-- and a red giant branch mass loss with a small dispersion. We have carefully investigated several possible sources of uncertainty that could affect the results of the horizontal branch modeling, stemming from uncertainties in both stellar model computations and the cluster properties such as heavy element abundances, reddening and age. We determine a firm lower limit of ~0.17$Mo for the mass lost by red giant branch stars, corresponding to horizontal branch stellar masses between ~0.65Mo and ~0.73Mo (the range driven by the range of initial helium abundances). We also derive that in this cluster the amount of mass lost along the asymptotic giant branch stars is comparable to the mass lost during the previous red giant branch phase. These results confirm for this cluster the disagreement between colour-magnitude-diagram analyses and inferences from recent studies of the dynamics of the cluster stars, that predict a much less efficient red giant branch mass loss. A comparison between the results from these two techniques applied to other clusters is required, to gain more insights about the origin of this disagreement.

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arXiv2016

The Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters. VII. Implications from the Nearly Universal Nature of Horizontal Branch Discontinuities

The UV-initiative Hubble Space Telescope Treasury survey of Galactic globular clusters provides a new window into the phenomena that shape the morphological features of the horizontal branch (HB). Using this large and homogeneous catalog of UV and blue photometry, we demonstrate that the HB exhibits discontinuities that are remarkably consistent in color (effective temperature). This consistency is apparent even among some of the most massive clusters hosting multiple distinct sub-populations (such as NGC 2808, omega Cen, and NGC 6715), demonstrating that these phenomena are primarily driven by atmospheric physics that is independent of the underlying population properties. However, inconsistencies arise in the metal-rich clusters NGC 6388 and NGC 6441, where the discontinuity within the blue HB (BHB) distribution shifts ~1,000 K to 2,000 K hotter. We demonstrate that this shift is likely due to a large helium enhancement in the BHB stars of these clusters, which in turn affects the surface convection and evolution of such stars. Our survey also increases the number of Galactic globular clusters known to host blue-hook stars (also known as late hot flashers) from 6 to 23 clusters. These clusters are biased toward the bright end of the globular cluster luminosity function, confirming that blue-hook stars tend to form in the most massive clusters with significant self-enrichment.

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arXiv2015

The red giant branch phase transition: Implications for the RGB luminosity function bump and detections of Li-rich red clump stars

We performed a detailed study of the evolution of the luminosity of He-ignition stage and of the red giant branch bump luminosity during the red giant branch phase transition for various metallicities. To this purpose we calculated a grid of stellar models that sample the mass range of the transition with a fine mass step equal to ${\rm 0.01M_\odot}$. We find that for a stellar population with a given initial chemical composition, there is a critical age (of 1.1-1.2~Gyr) around which a decrease in age of just 20-30 million years causes a drastic drop in the red giant branch tip brightness. We also find a narrow age range (a few $10^7$ yr) around the transition, characterized by the luminosity of the red giant branch bump being brighter than the luminosity of He ignition. We discuss a possible link between this occurrence and observations of Li-rich core He-burning stars.

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arXiv2015

Post first dredge-up [C/N] ratio as age indicator. Theoretical calibration

We performed a detailed analysis of the use of [C/N] measured in red giant branch stars between the completion of the first dredge up and the red giant branch bump ([C/N]_{FDU}) as age indicator. [C/N]_{FDU} cannot give accurate ages for individual stars, but may provide a general chronology for the formation of composite populations and add constraints to analyses of red giants from surface gravity-effective temperature diagrams. We provide a theoretical calibration of [C/N]_{FDU} in terms of total metallicity [M/H] and age, for ages greater than 1 Gyr, which we tested against variations in the initial heavy element distribution (scaled-solar vs alpha-enhanced), efficiency of overshooting from MS convective cores and from the convective envelopes, variations in the initial He abundance and in the mixing length parameter. Our calibration is compared with a small sample of available measurements of [C/N]_{FDU} in star clusters and halo field stars, which at least qualitatively confirm the overall trend of the predicted [C/N]_{FDU} with age and [M/H]. The use of [C/N]_{FDU}-[M/H]-age relations obtained from independent sets of stellar evolution calculations cause age differences (for a given [C/N]_{FDU} and [M/H] pair) up to about 2~Gyr. More accurate spectroscopic measurements of [C/N]_{FDU} in star clusters with well-established ages and metallicities are required to better test theoretical calibrations of this age indicator.

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arXiv2015

Uncertainties on near-core mixing in red-clump stars: effects on the period spacing and on the luminosity of the AGB bump

Low-mass stars in the He-core-burning phase (HeCB) play a major role in stellar, galactic, and extragalactic astrophysics. The ability to predict accurately the properties of these stars, however, depends on our understanding of convection, which remains one of the key open questions in stellar modelling. We argue that the combination of the luminosity of the AGB bump (AGBb) and the period spacing of gravity modes (DP) during the HeCB phase, provides us with a decisive test to discriminate between competing models of these stars. We use the MESA, BaSTI, and PARSEC stellar evolution codes to model a typical giant star observed by Kepler. We explore how various near-core-mixing scenarios affect the predictions of the above-mentioned constraints, and we find that DP depends strongly on the prescription adopted. Moreover we show that the detailed behaviour of DP shows the signature of sharp variations in the Brunt-Vaisala frequency, which could potentially give additional information about near-core features. We find evidence for the AGBb among Kepler targets, and a first comparison with observations shows that, even if standard models are able to reproduce the luminosity distribution, no standard model can account for satisfactorily the period spacing of HeCB stars. Our analysis allows us to outline a candidate model to describe simultaneously the two observed distributions: a model with a moderate overshooting region characterized by an adiabatic thermal stratification. This prescription will be tested in the future on cluster stars, to limit possible observational biases.

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arXiv2014

On the missing second generation AGB stars in NGC6752

[Abridged] In recent years the view of Galactic globular clusters as simple stellar populations has changed dramatically, as it is now thought that basically all GCs host multiple stellar populations, each with its own chemical abundance pattern and colour-magnitude diagram sequence. Recent spectroscopic observations of asymptotic giant branch stars in the GC NGC6752 have disclosed a low [Na/Fe] abundance for the whole sample, suggesting that they are all first-generation stars, and that all second-generation stars fail to reach the AGB in this cluster. A scenario proposed to explain these observations invokes strong mass loss in second-generation horizontal branch stars possibly induced by the metal enhancement associated to radiative levitation. This enhanced mass loss would prevent second generation stars from reaching the AGB phase, thus explaining at the same time the low value of the ratio between HB and AGB stars (the R_2 parameter) observed in NGC6752. We have critically discussed this scenario, finding that the required mass-loss rates are of the order of 10^{-9} Mo/yr, significantly higher than current theoretical and empirical constraints. By making use of synthetic HB simulations, we demonstrate that our modelling predicts correctly the R_2 parameter for NGC6752, without the need to invoke very efficient mass loss during the core He-burning stage. Our simulations for NGC6752 HB predict however the presence of a significant fraction - at the level of about 50% - second generation stars along the cluster AGB. We conclude that there is no simple explanation for the lack of second generation stars in the spectroscopically surveyed sample, although the interplay between mass loss (with low rates) and radiative levitation may play a role in explaining this puzzle.

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arXiv2013

The BaSTI Stellar Evolution Database: models for extremely metal-poor and super-metal-rich stellar populations

We present an extension of the BaSTI stellar evolution database to extremely metal poor (${\rm Z=10^{-5}}$) and super-metal-rich (Z=0.05) metallicities, with both scaled-solar and $α$-enhanced ([$α$/Fe]=0.4) heavy element distributions. These new tracks (from the pre-main sequence to the early-asymptotic giant branch phase), horizontal branch models and isochrones, will enable the use of the BaSTI database to study, i.e., the most metal poor populations found in Local Group faint dwarf galaxies, and the metal rich component of the Galactic bulge. An overview of several fundamental predictions of stellar evolution over the full metallicity range of BaSTI is presented, together with comparisons with literature calculations at ${\rm Z=10^{-5}}$ and Z=0.05.

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