Nonlinear Pulsation Research Articles

The Hertzsprung progression of classical Cepheids in the Gaia era

ABSTRACT A new fine grid of non-linear convective pulsation models for the so-called ‘bump Cepheids’ is presented to investigate the Hertzprung progression (HP) phenomenon shown by their light and radial pulsation velocity curves. The period corresponding to the centre of the HP is investigated as a function of various model assumptions, such as the efficiency of superadiabatic convection, the mass–luminosity relation, and the metal and helium abundances. The assumed mass–luminosity relation is found to significantly affect the phenomenon but variations in the chemical composition as well as in the stellar mass (at fixed mass–luminosity relation) also play a key role in determining the value of the HP centre period. Finally, the predictive capability of the presented theoretical scenario is tested against observed light curves of bump Cepheids in the ESA Gaia data base, also considering the variation of the pulsation amplitudes and of the Fourier parameters R21 and Φ21 with the pulsation period. A qualitative agreement between theory and observations is found for what concerns the evolution of the light curve morphology as the period moves across the HP centre, as well for the pattern in period–amplitude, period–R21, and period–Φ21 planes. A larger sample of observed Cepheids with accurate light curves and metallicities is required in order to derive more quantitative conclusions.

A theoretical framework for BL Her stars. II. New period-luminosity relations in Gaia passbands

In the era of the Hubble tension, it is crucial to obtain a precise calibration of the period-luminosity ($PL$) relations of classical pulsators. Type II Cepheids (T2Cs; often exhibiting negligible or weak metallicity dependence on $PL$ relations) used in combination with RR Lyraes and the tip of the red giant branch may prove useful as an alternative to classical Cepheids for the determination of extragalactic distances. We present new theoretical $PL$ and period-Wesenheit ($PW$) relations for a fine grid of convective BL Her (the shortest period T2Cs) models computed using mesa-rsp in the $Gaia$ passbands and we compare our results with the empirical relations from $Gaia$ DR3. Our goal is to study the effect of metallicity and convection parameters on the theoretical $PL$ and $PW$ relations. We used the state-of-the-art 1D non-linear radial stellar pulsation tool mesa-rsp to compute models of BL Her stars over a wide range of input parameters:\ metallicity ($-2.0\; dex Fe/H dex $), stellar mass ($0.5M_ odot -0.8M_ odot $), stellar luminosity ($50L_ odot -300L_ odot $), and effective temperature (across the full extent of the instability strip; in steps of 50K). We used the Fourier decomposition technique to analyse the light curves obtained from mesa-rsp and $Gaia$ DR3 and then compared the theoretical and empirical $PL$ and $PW$ relations in the $Gaia$ passbands. The BL Her stars in the All Sky region exhibit statistically different $PL$ slopes compared to the theoretical $PL$ slopes computed using the four sets of convection parameters. We find the empirical $PL$ and $PW$ slopes from BL Her stars in the Magellanic Clouds to be statistically consistent with theoretical relations computed using the different convection parameter sets in the $Gaia$ passbands. There is a negligible effect coming from the metallicity on the $PL$ relations in the individual $Gaia$ passbands. However, there is a small but significant negative coefficient of metallicity in the $PWZ$ relations for the BL Her models using the four sets of convection parameters. This could be attributed to the increased sensitivity of bolometric corrections to metallicities at wavelengths shorter than the $V$ band. Our BL Her models also suggest a dependence of the mass-luminosity relation on metallicity. We found the observed Fourier parameter space to be covered well by our models. Higher mass models ($> 0.6\ odot $) may be needed to reliably model the observed light curves of BL Her stars in the All-Sky region. We also found the theoretical light curve structures (especially the Fourier amplitude parameters) to be affected by the choice of convection parameters.

Theoretical Period–Radius and Period–Luminosity Relations for Mira Variables with Solar Metallicity

Evolutionary sequences of AGB stars with initial masses on the main sequence MZAMS =1.5 M , 2 M and 3 M were computed for the initial metallicity Z = 0.014. Selected models ofevolutionary sequences with envelopes under thermal equilibrium were used as initial conditions forcalculation of nonlinear stellar pulsations. The hydrodynamic models of each evolutionary sequence areshown to concentrate along the continuous line in the period–radius and period–luminosity diagrams.The theoretical period–radius and period–luminosity relations differ from one another for different main–sequence star masses because the stellar luminosity of AGB stars depends on the degenerate carboncore mass which increases with increasing MZAMS. In hydrodynamic models of evolutionary sequencesMZAMS = 2 M and MZAMS = 3 M the periods of the first overtone pulsators are 86 d ≤ Π ≤ 123 dand 174 d ≤ Π ≤ 204 d, whereas all models of the evolutionary sequence MZAMS = 1.5 M oscillate inthe fundamental mode. Fairly regular radial oscillations exist in stars with pulsation periods Π 500 d.In models with longer periods the amplitude rapidly increases with increasing Π and oscillations becomeirregular.

Steady-State Amplitude of Nonlinear Pulsations of a Gas Bubble in Liquid under the Action of Periodic External Pressure in Resonance

Steady-State Amplitude of Nonlinear Pulsations of a Gas Bubble in Liquid under the Action of Periodic External Pressure in Resonance

A multiphase study of theoretical and observed light curves of classical Cepheids in the Magellanic Clouds

ABSTRACTWe present an analysis of the theoretical and observed light-curve parameters of the fundamental mode (FU) classical Cepheids in the Magellanic Clouds in V- and I- photometric bands. The state-of-the-art 1D non-linear radial stellar pulsation (RSP) code in mesa (mesa-rsp) has been utilized to generate the theoretical light curves using four sets of convection parameters. Theoretical light curves with two chemical compositions: Z = 0.008 and Z = 0.004 appropriate for the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC), respectively, covered a wide range of periods ($3\lt P (\rm {d})\lt 32$). The observed light curves are taken from the OGLE-IV data base. We compare theoretical and observed Fourier parameters (FPs), and investigate the period–luminosity (PL), period–colour (PC), and amplitude–colour (AC) relations as a function of pulsation phase for short (log P < 1), long (log P > 1), and all periods. The multiphase relations obtained from theoretical and observed light curves in the PL/PC/AC plane are found to be dynamic in nature, with the effect more pronounced at Φ ∼ 0.75–0.85. Furthermore, a contrasting behaviour of the theoretical/observed multiphase PL and PC relations between the short and long periods has been found for both LMC and SMC. The analysis shows that multiphase PL relations are more stringent to test the models with observations over the FPs. Distances to the LMC/SMC determined using long period Cepheids are found to be in good agreement with the literature values when the term R21 is added to the PL relation.

Mechanism of multiphase coupling transport evolution of free sink vortex

In the evolution of confluence sink vortex with a free surface, there exists some physical processes , such as multiphase coupling, mass transfer, and intensive energy exchange. Here, the transport mechanism of multiphase coupling is a complex dynamic problem with highly nonlinear characteristics. The mechanical modeling and numerical solution of multiphase viscous coupled transport are facing a significant challenge. To address the above problem, a method of modeling and solving multiphase coupling transport of the free sink vortex is proposed. Based on the coupled level set and volume-of-fluid (CLSVOF) method, a multiphase coupling transport model of the free sink vortex is set up with a continuous surface tension model and a realizable (<i>k</i>-<i>ε</i>) turbulence model. By using an effective volumetric correction scheme, the high-speed rotating flow is calculated, and the mass conservation of flow field and the velocity field without divergence are ensured. Then, an interphase coupling solution approach accurately traces the multiphase fluid distribution and multiphase interface. The multiphase coupling interface and cross-scale vortex cluster transport laws are obtained according to the multi-characteristic physical variables. The interaction mechanism between the multiphase coupling transport process and the pressure pulsation characteristics is revealed. The results show that the multiphase coupling transport is the critical state of the fluid medium transition. The vortex microclusters are subjected to different spatiotemporal disturbance modes and form the layered threaded waveforms at the interface. With the increase of the nozzle sizes, the multiphase coupling process is strengthened, and the coupling energy shock causes nonlinear pressure pulsation. This study can offer valuable references to the researches of the vortex transport mechanism, cross-scale solution of vortex cluster, and flow pattern tracking.

Anomalous Cepheids: Updated Theoretical Period–Luminosity–Color and Period–Wesenheit Relations

AbstractIn the context of a project aimed to provide an updated theoretical scenario for various classes of radially pulsating stars, we present the first results obtained for anomalous Cepheids. By adopting reliable and updated evolutionary prescriptions concerning the luminosity levels for various core He-burning stellar models with masses suitable for entering the instability strip, we have computed nonlinear convective pulsation models for both fundamental and first-overtone mode anomalous Cepheids by exploring the impact of varying the metal abundance as well as the efficiency of super-adiabatic convection. These numerical simulations have allowed us to retrieve the boundaries of the instability strip and all relevant pulsation properties, namely period, amplitude, bolometric light and radial velocity curves. This theoretical scenario has been transformed into the Gaia photometric system to derive the first theoretical Period–Luminosity–Colour and Period–Wesenheit relations in the Gaia bands.

Asteroseismic inferences from the study of non-linearities in δ Sct stars

Many pulsating star light curves must be interpreted strictly within the framework of a non-linear theory. The detection of non-linear interactions between pulsation modes has increased due to the ultra-precise photometric data provided by space missions. For this reason, it is now possible and essential to continue the effort of building data-driven non-linear pulsation models. The analysis of non-linear stellar light curves has become a fundamental challenge in asteroseismology in the era of space missions. In this article, a diagnostic method is presented. It is based on an observational characterization of the non-linear behavior of high-amplitude delta Scuti (HADS) stars. It diagnoses the nature of non-linear pulsations in δ Sct stars of lower amplitudes. It potentially provides an extra hint for identifying a radial mode and determine its radial order n. Additionally, it could show signatures of moderate to rapid rotation in these stars. Overall, this article shows the capabilities for making asteroseismic inferences from studying combination frequencies in δ Sct stars.

Period–colour and amplitude–colour relations for OGLE-δ Scuti stars in the Galactic bulge and LMC

ABSTRACT We present an analysis on the behaviour of the Galactic bulge and the Large Magellanic Cloud (LMC) δ Scuti stars in terms of period–colour and amplitude–colour (PCAC) relations at maximum, mean, and minimum light. The publicly available Optical Gravitational Lensing Experiment-IV (OGLE-IV) light curves for Galactic bulge and OGLE-III light curves for LMC δ Scuti stars are exploited for the analysis. It has been found that the Galactic bulge δ Scuti stars obey flat PC relations at maximum/mean/minimum light, while the LMC δ Scutis have sloped, sloped, and flat PC relations at maximum, mean, and minimum light, respectively. Both the Galactic bulge and the LMC δ Scutis have sloped, flat, and sloped AC relations at maximum, mean, and minimum light, respectively. These relations also show that Galactic δ Scutis are hotter as compared to their LMC counterparts. The period–amplitude (PA) relations for δ Scutis exhibit different behaviour in the Galactic bulge and the LMC. The LMC variables are found to have higher amplitudes at a given period. The amplitude of the Galactic bulge δ Scuti shows a bimodal distribution which can be modelled using a two-component Gaussian Mixture Model: one component with a lower amplitude and another with a higher amplitude. The observed behaviour of the δ Scuti PCAC relations can be explained using the theory of the interaction of hydrogen ionization front (HIF) and stellar photosphere as well as the PA diagram. We use mesa-rsp to calculate theoretical non-linear hydrodynamical pulsation models for δ Scuti stars with input metallicities of Z = 0.02 and Z = 0.008 appropriate for the Galactic bulge and LMC, respectively. The observed PCAC relations and theoretical calculations support the HIF–photosphere interaction theory.

New Theoretical Period–Luminosity–Metallicity Relations for RR Lyrae in the Rubin-LSST Filters

The revolutionary power of future Rubin-LSST observations will allow us to significantly improve the physics of pulsating stars, including RR Lyrae. In this context, an updated theoretical scenario predicting all the relevant pulsation observables in the corresponding photometric filters is mandatory. The bolometric light curves are based on a recently computed extensive set of nonlinear convective pulsation models for RR Lyrae stars, covering a broad range of metal content, and have been transformed into the Vera C. Rubin Observatory Legacy Survey of Space and Time (Rubin-LSST) photometric system. Predicted Rubin-LSST mean magnitudes and pulsation amplitudes have been adopted to build the Bailey diagrams (luminosity amplitude versus period) and the color–color diagrams in these bands. The current findings indicate that the g LSST–r LSST and r LSST–i LSST colors obey to a well-defined linear relation with the metal content. Moreover, the period–luminosity relations display in the reddest filters (r LSST, i LSST, z LSST, y LSST) a significant dependence on the assumed metal abundance. In particular, more metal-rich RR Lyrae are predicted to be fainter at a fixed period. Metal-dependent period–Wesenheit relations for different combinations of optical and near-infrared filters are also provided. These represent powerful tools to infer individual distances independently of reddening uncertainties, once the metal abundance is known and no relevant deviations from the adopted extinction law occur. Finally, we also derived new linear and quadratic absolute magnitude metallicity relations (g LSST versus [Fe/H]) and the metallicity coefficient is consistent with previous findings concerning the B and the V band.

Spectrum synthesis for radially pulsating stars with shocked atmospheres

ABSTRACT spec_puls describes a suite of computer programs to simulate the emergent spectrum from a radially pulsating star. It combines a Christy-type non-linear pulsation code with classical stellar atmosphere codes. The principal aim is to interpret the dynamical spectrum of the radially pulsating extreme helium star V652 Herculis (V652 Her), which shows a strong shock at minimum radius. The components are general enough to treat other classes of radial pulsation. The theoretical spectrum from a shocked pulsation model shows line doubling, with the blue component emerging at standstill velocity and accelerating blueward. The doubling phase depends on line depth and parent ion. The behaviour of line cores post-shock points to a drop in the ionization temperature, although the gas temperature in the model remains high. Shock compression leads to phase-dependent strengthening of Stark-broadened line wings, with the far wings responding first. With velocity-, temperature-, and pressure-sensitive diagnostics, detailed tomography of the pulsation-driven shock in V652 Her seems possible. Even when no shock is present, the dynamical spectrum is significantly different from a model in hydrostatic equilibrium. Using the quasi-static approximation (e.g. at maximum radius) may lead to a considerable underestimate of the star’s mean effective temperature and surface gravity.

Influence of rigid wall on the nonlinear pulsation of nearby bubble

This paper mainly focuses on the nonlinear pulsation of a bubble near the rigid wall. Dynamics of near-wall bubble and free bubble are discussed and compared in details. Investigation reveals as the driving acoustic pressure amplitude increases, nonlinear pulsation of bubble becomes intense gradually. Besides, decreasing the viscosity of host liquid is advantageous for the nonlinear pulsation of bubble. Bifurcation diagrams of bubble radius show acoustic reflection of the rigid wall makes the initial bifurcation appear at low driving acoustic amplitude and on bubble with small ambient radius, and makes the bifurcation still exist for bubble in high-viscosity liquids. That indicates the rigid wall will produce enhancement on the nonlinearity of nearby bubble. As the bubble approaches the wall, the enhancement becomes strong. Moreover, research on the influence of driving frequency shows the rigid wall makes the frequency band corresponding to chaos around the resonant frequency of free bubble shift downward.

The period-age relation of long-period variables

Context. Pieces of empirical evidence suggest the existence of a period-age relation for long-period variables (LPVs). Yet, this property has hardly been studied on theoretical grounds thus far. Aims. We aim to examine the period-age relation using the results from recent nonlinear pulsation calculations. Methods. We combined isochrone models with theoretical periods to simulate the distribution of fundamental mode LPV pulsators, which include Miras, in the period-age plane, and we compared it with observations of LPVs in Galactic and Magellanic Clouds’ clusters. Results. In agreement with observations, models predict that the fundamental mode period decreases with increasing age because of the dominant role of mass in shaping stellar structure and evolution. At a given age, the period distribution shows a non-negligible width and is skewed toward short periods, except for young C-rich stars. As a result, the period-age relations of O-rich and C-rich models are predicted to have different slopes. We derived best-fit relations describing age and initial mass as a function of the fundamental mode period for both O- and C-rich models. Conclusions. The study confirms the power of the period-age relations to study populations of LPVs of specific types, either O-rich or C-rich, on statistical grounds. In doing so, it is recommended not to limit a study to Miras, which would make it prone to selection biases, but rather to include semi-regular variables that pulsate predominantly in the fundamental mode. The use of the relations to study individual LPVs, on the other hand, requires more care given the scatter in the period distribution predicted at any given age.

Hydrodynamic Model of Decaying Radial Oscillations in RU Cam

Calculations of stellar evolution up to the early white dwarf stage were carried out for stars with mass on the main sequence $M_0=0.82M_\odot$, $0.85M_\odot$, $0.9M_\odot$ and with initial abundances of helium and heavier elements $Y=0.25$ and $Z=10^{-3}$, respectively. For each value of $M_0$ the AGB and post--AGB evolutionary phases were computed with three values of the mass loss parameter in the Bl\"ocker formula: $\eta_B=0.02$, 0.05 and 0.1. The variable star RU~Cam with pulsation period $\Pi\approx 22$ day is shown to be in the post--AGB stage and the pulsation amplitude decrease in years 1962--1963 is due to movement of the star across the HR diagram beyond the pulsation instability region. Theoretical estimates of the mass and the luminosity of RU~Cam are $0.524M_\odot\le M\le 0.532M_\odot$ and $2.20\times 10^3L_\odot\le L\le 2.33\times 10^3L_\odot$, respectively. Hydrodynamic calculations of nonlinear stellar pulsations show that while the star approaches the instability boundary a significant reduction ($\approx 90\%$) in the pulsation amplitude occurs for nearly two years with subsequent slow decay of low--amplitude oscillations. Solution of the equations of hydrodynamics with time--dependent inner boundary conditions describing evolutionary changes in the radius and the luminosity at the bottom of the pulsating envelope allows us to conclude that decay of radial oscillations in RU~Cam is accompanied by the effect of oscillation hysteresis. In particular, the stage of large--amplitude limit cycle oscillations extends by $\approx 12$ years and the subsequent stage of decaying small--amplitude oscilations spreads beyond the formal boundary of pulsation instability.

Effect of nonlinear translations on the pulsation of cavitation bubbles

The pulsations and translations of cavitation bubbles obey combined ordinary differential equations, and their nonlinearities are studied by the bifurcation diagram and the phase diagram in a strong ultrasonic field. Bubble pulsation can change regularly or irregularly with changing driving pressure in the time domain. The bifurcation diagrams of the pulsation versus driving pressure show that the pulsations and translations of bubbles have nonlinear characteristics, and the nonlinear translations of bubbles can disorder the pulsations for certain parameters. Disorder of the pulsation can also be caused by nonlinear pulsation itself. In addition, the phase diagrams also show that the nonlinear translations make a large contribution to the pulsations. The same result can also be obtained when the ambient radii of two bubbles are different.

Nonlinear pulsations of horizontal jets

A shallow-water model with horizontally nonuniform density is used to study the dynamics of jet flows that arise under the influence of buoyancy and the Coriolis force. Within this approach, the jet is described by a self-similar compactly-localized solution and interpreted as a band of shear flow having a temperature contrast with the ambient fluid. In addition to stationary states, the dynamics of such jets admit cyclonic rotation with a constant angular velocity and transverse nonlinear pulsations. The phase portrait corresponding to this model shows that regimes with pulsating jets develop along closed trajectories bounded by the separatrix loop. The theory predicts that the period for warm jet pulsations is longer than the inertial oscillation period caused by the Earth’s rotation, while for cold jet pulsations, it is shorter. Thus, only warm jets can have a noticeable effect on the atmospheric dynamics in the synoptic range. In particular, they may well be responsible for additional spectral peaks that appear in this range of wind speed fluctuations.

A theoretical scenario for Galactic RR Lyrae in the Gaia data base: constraints on the parallax offset

ABSTRACT On the basis of an extended set of non-linear convective RR Lyrae pulsation models we derive the first theoretical light curves in the Gaia bands G, GBP, and GRP and the corresponding intensity-weighted mean magnitudes and pulsation amplitudes. The effects of chemical composition on the derived Bailey diagrams in the Gaia filters are discussed for both Fundamental and first overtone mode pulsators. The inferred mean magnitudes and colours are used to derive the first theoretical Period–Wesenheit relations for RR Lyrae in the Gaia filters. The application of the theoretical Period–Wesenheit relations for both the Fundamental and first overtone mode to Galactic RR Lyrae in the Gaia Data Release 2 data base and complementary information on individual metal abundances allows us to derive theoretical estimates of their individual parallaxes. These results are compared with the astrometric solutions to conclude that a very small offset, consistent with zero, is required in order to reconcile the predicted distances with Gaia results.

A theoretical framework for BL Her stars – I. Effect of metallicity and convection parameters on period–luminosity and period–radius relations

ABSTRACT We present a new grid of convective BL Herculis models using the state-of-the-art 1D non-linear radial stellar pulsation tool mesa-rsp. We investigate the impact of metallicity and four sets of different convection parameters on multiwavelength properties. Non-linear models were computed for periods typical for BL Her stars, i.e. 1 ≤ P(d) ≤ 4 covering a wide range of input parameters – metallicity (−2.0 dex ≤ [Fe/H] ≤ 0.0 dex), stellar mass (0.5–0.8 M⊙), luminosity (50–300 L⊙), and effective temperature (full extent of the instability strip; in steps of 50 K). The total number of BL Her models with full-amplitude stable pulsations used in this study is 10 280 across the four sets of convection parameters. We obtain their multiband (UBVRIJHKLL′M) light curves and derive new theoretical period–luminosity (PL), period–Wesenheit (PW), and period–radius (PR) relations at mean light. We find that the models computed with radiative cooling show statistically similar slopes for PL, PW, and PR relations. Most empirical relations match well with the theoretical PL, PW, and PR relations from the BL Her models computed using the four sets of convection parameters. However, PL slopes of the models with radiative cooling provide a better match to empirical relations for BL Her stars in the Large Magellanic Cloud in the HKS bands. For each set of convection parameters, the effect of metallicity is significant in U and B bands and negligible in infrared bands, which is consistent with empirical results. No significant metallicity effects are seen in the PR relations.

Modelling long-period variables – II. Fundamental mode pulsation in the non-linear regime

ABSTRACT Long-period variability in luminous red giants has several promising applications, all of which require models able to accurately predict pulsation periods. Linear pulsation models have proven successful in reproducing the observed periods of overtone modes in evolved red giants, but they fail to accurately predict their fundamental mode (FM) periods. Here, we use a 1D hydrodynamic code to investigate the long-period variability of M-type asymptotic giant branch stars in the non-linear regime. We examine the period and stability of low-order radial pulsation modes as a function of mass and radius, and find overtone mode periods in complete agreement with predictions from linear pulsation models. In contrast, non-linear models predict an earlier onset of dominant FM pulsation, and shorter periods at large radii. Both features lead to a substantially better agreement with observations that we verify against OGLE and Gaia data for the Magellanic Clouds. We provide simple analytical relations describing the non-linear FM period–mass–radius relation. Differences with respect to linear predictions originate from the readjustment of the envelope structure induced by large-amplitude pulsation. We investigate the impact of turbulent viscosity on linear and non-linear pulsation, and probe possible effects of varying metallicity and carbon abundance.

Standing on the Shoulders of Giants: New Mass and Distance Estimates for Betelgeuse through Combined Evolutionary, Asteroseismic, and Hydrodynamic Simulations with MESA

Abstract We conduct a rigorous examination of the nearby red supergiant Betelgeuse by drawing on the synthesis of new observational data and three different modeling techniques. Our observational results include the release of new, processed photometric measurements collected with the space-based Solar Mass Ejection Imager instrument prior to Betelgeuse’s recent, unprecedented dimming event. We detect the first radial overtone in the photometric data and report a period of 185 ± 13.5 days. Our theoretical predictions include self-consistent results from multi-timescale evolutionary, oscillatory, and hydrodynamic simulations conducted with the Modules for Experiments in Stellar Astrophysics software suite. Significant outcomes of our modeling efforts include a precise prediction for the star’s radius: . In concert with additional constraints, this allows us to derive a new, independent distance estimate of pc and a parallax of mas, in good agreement with Hipparcos but less so with recent radio measurements. Seismic results from both perturbed hydrostatic and evolving hydrodynamic simulations constrain the period and driving mechanisms of Betelgeuse’s dominant periodicities in new ways. Our analyses converge to the conclusion that Betelgeuse’s ≈400 day period is the result of pulsation in the fundamental mode, driven by the κ-mechanism. Grid-based hydrodynamic modeling reveals that the behavior of the oscillating envelope is mass-dependent, and likewise suggests that the nonlinear pulsation excitation time could serve as a mass constraint. Our results place α Orionis definitively in the early core helium-burning phase of the red supergiant branch. We report a present-day mass of 16.5–19 M ⊙—slightly lower than typical literature values.