Outer Convection Zone Research Articles

Gravitational Distortions of the Shape of the Sun: Constraints on the Models

Due to its own rotation, it is expected that the visual figure of the Sun is a spheroid; this is not truly the case because the solar rotation is not constant both over all heliographic latitudes and in depth. The photospheric shape is thus sensitive to the interior structure: accurate measurements of both limb shape distortions and solar rotation rates determination provide useful constraints on the internal layers (density, shear zones, …). We show why and how the implication of the successive gravitational moments are important to probe the solar interior, and we compare measurements obtained either from space (SOHO/MDI) or from ground-based experiments (scanning heliometer at the Pic du Midi). The found faint departures to the sphericity, not exceeding 22 mas, could explain fluctuations that are not yet taken into account in the classical modelling of the solar irradiance. A crude model could explain the asphericities which is based on a core rotating at a nearly uniform rate combined with a prolate tachocline and an oblate outer convective zone.

ROSATView ofHipparcosF Stars

For a set of 11,900 F stars that have uvby photometry and are in the Hipparcos catalog, we have found 1980 X-ray counterparts (~17%) in the ROSAT All-Sky Survey (RASS). In this paper we analyze X-ray properties of these ROSAT F stars and compare that sample with the entire set of our F stars. The comparison has shown that the ROSAT F stars are on average younger than stars in the whole set, consistent with previous studies that indicate lower X-ray emission in older stars. However, for the ROSAT stars only, the age-X-ray activity relation has proved to be more complicated, with a group of strong X-ray emitters being the oldest rather than the youngest stars. That group determines the shape of the overall relationship between the kinematics and X-ray luminosity. The relationship is in the sense that the mean tangential velocity and velocity dispersion are higher for stars that are more luminous at the X-ray wavelengths, implying that these stars are on average the oldest rather than the youngest. The group of strongest X-ray emitters, log LX > 30.4, which constitutes only a small fraction of the sample stars, ~2.8%, has been found to include a variety of stellar types. It is dominated by very young objects, followed by RS CVn-type stars and other active binaries. As to the sample as a whole, it seems to consist of at least three distinct groups of F stars that differ in their X-ray luminosity and evolution of their X-ray emission. The first one is represented by normal zero-age main-sequence (ZAMS) and post-ZAMS stars whose X-ray emission declines with age, just as expected in the conventional picture of stellar X-ray evolution (roughly half of the sample). The second group consists of strong X-ray emitters that appear to be very young, possibly pre-main-sequence stars. These stars are characterized by high metallicity, high temperature, and high overluminosity (the latter indicating that the star's optical luminosity is too high for the star's temperature and surface gravity). Many of them are located in the distance range ~130-200 pc, near the known regions of ongoing star formation or OB associations. The third group also consists of strong X-ray emitters. However, those are very old rather than very young stars, and they happen to be extremely overluminous as well. Their strong X-ray emission implies that many F stars are capable of increasing rather than decreasing their X-ray luminosity as they get older, thus challenging the idea that X-ray emission continuously declines with age. Most of these old X-ray emitters have significantly evolved away from the ZAMS. If they increase their outer convective zone as they evolve, they would probably be able to maintain a high level of coronal activity, and hence enhanced X-ray emission, despite decaying rotation. Other relationships found for the ROSAT F stars include the correlation of X-ray luminosity with hardness ratio, optical absolute magnitude, effective temperature, overluminosity, metallicity, and reddening. There is also a difference between binary and single stars, with X-ray luminosity being on average slightly higher for unresolved binaries. There is a distinct difference between the groups of soft and hard X-ray emitters in our sample: while the soft sources show a significant correlation between the X-ray luminosity and hardness ratio, no such correlation is seen for hard sources. The tendency for X-ray luminosity to be higher for harder sources implies that stars more luminous in the X-rays have a hotter corona. This tendency breaks up at large hardness values, which may be due to coronal saturation.

Observations of the upper solar chromosphere with SUMER

The structure and dynamics of the solar chromosphere are still matters of debate. The chromospheric network reflecting the supergranulation of the outer convection zone of the Sun is a prominent feature of the lower solar atmosphere that extends into the transition zone between chromosphere and corona. In particular, the physics of the so-called “nonmagnetic” chromosphere in internetwork regions as well as the physics of the magnetic network are not yet fully understood. Here we present observations of the H i Lyman continuum obtained in areas of the undisturbed Sun by the Solar Ultraviolet Measurements of Emitted Radiation (SUMER) instrument on the Solar and Heliospheric Observatory (SOHO). The observing sequences are unique in the sense that they cover the spectral range from 67 nm to 93 nm with the highest cadence the SUMER spectrometer can achieve operating near the limit of its mechanism performance, telemetry allocation, and memory capabilities. In this wavelength range not only the Lyman continuum but also many extreme-ultraviolet emission lines (N ii, N iii, S iv, O ii, O iii, O iv, O v, Ne viii, and Mg ix) are prominent, allowing the investigation of radiation formed at temperatures representative of regions from the chromosphere to the corona. Brightenings have been identified that are presumed to be related to the well-known 3 min oscillations as seen, for instance, in Ca ii H2v and K2v observations. The relative temporal variations of the continuum radiance near 77 nm were typically 20% to 40%, whereas simultaneously recorded transition-region lines varied by about 40% of their lowest values in phase with the continuum. In the corona, the Ne viii and Mg ix lines with formation temperatures of 620 000 K and 950 000 K, respectively, experienced relative changes of 10% and displayed no phase relationship with the transition-region lines or the continuum. Radiance variations in the spatial regime across the solar disk show a higher correlation between the chromosphere and the corona than between the transition region and the corona. The observations will be discussed with a view towards providing constraints for modelling chromospheric structure and dynamics. chromosphere – Sun: transition region – Sun: corona – Sun: UV radiation

Orbital period modulation in close binaries due to cyclic $\alpha^2$-dynamo activity

The question is answered whether dynamo-generated magnetic fields are able to produce such quadrupole terms in the gravity potential which can explain the observed cyclic orbital variation of RS CVn stars. We start with spherical dynamo models with outer convection zones but without any dierential rotation, i.e. with 2 -dynamos which are known as nonoscil- lating. With the known anisotropic -tensor of rapidly rotating stars the magnetic modes with the lowest dynamo numbers are nonaxisymmetric with a slow azimuthal drift. We also find, however, stable (i.e. with the lowest dynamo number) axisymmet- ric oscillating modes but only for a very special, highly inhomogeneous -tensor. The dynamo model is a linear one with an arbitrary field amplitude which can be scaled in order to reproduce the observations. The star proves to be as prolate during the maximum of the toroidal field energy and it proves to be as oblate during the maximum of the poloidal field energy. In the time average the influence of the toroidal field dominates and the star is slightly prolate. From the computed temporal variations of the gravitation quadrupole moment a magnetic field of more than 10 5 G is found in order to produce the period modulation of order 10 5 which has been observed.

Relations between star-spot distribution and stellar parameters

Star spots have been shown to exist on a variety of stars. The growing number of observed star spot distributions on different stars - though snapshot-like - may allow the derivation of a fundamental dependency of spot patterns on stellar parameters. This paper tries to establish a connection of generic stellar parameters with the latitude of star spots based on a numerical thin-flux-tube-evolution model. The model is applied to stars with masses between 0.4 and 4.0 M⊙ at evolutionary stages from PMS to early giant stages. All model stars considered have an outer convection zone with a radiative core underneath. It is shown that the rotational period is the dominant parameter in determining the star-spot latitude. Another important factor is the fractional size of the radiative core, while the evolutionary stage and the stellar mass play only a minor role.

Meridional flow and differential rotation by gravity darkening in fast rotating solar-type stars

An explanation is presented for the rather strong total surface differential rotation of the observed very young solar-type stars like AB Dor and PZ Tel. Due to its rapid rotation a nonuniform energy flux leaves the stellar core so that the outer convection zone is nonuniformly heated from below. Due to this `gravity darkening' of the equator a meridional flow is created flowing equatorwards at the surface and thus accelerating the equatorial rotation. The effect linearly grows with the normalized pole-equator difference, \epsilon, of the heat-flux at the bottom of the convection zone. A rotation rate of about 9 h leads to \epsilon=0.1 for a solar-type star. In this case the resulting equator-pole differences of the angular velocity at the stellar surface, \delta\Omega, varies from unobservable 0.005/day to the (desired) value of 0.03 day$^{-1}$ when the dimensionless diffusivity factors $c_\nu$ and c_\chi vary between 1 and 0.1 (standard value c_\nu \simeq c_\chi \simeq 0.3, see Table 1.) In all cases the related temperature differences between pole and equator at the surface are unobservably small. The (clockwise) meridional circulation which we obtain flows opposite to the (counterclockwise) circulation appearing as a byproduct in the \Lambda-theory of the nonuniform rotation in outer convection zones. The consequences of this situation for those dynamo theories of stellar activity are discussed which work with the meridional circulation as the dominant magnetic-advection effect in latitude to produce the solar-like form of the butterfly diagram. Key words: Hydrodynamics, Star: rotation, Stars: pre-main sequence, Stellar activity

A Critical Examination of Li Pollution and Giant‐Planet Consumption by a Host Star

We examine the overall likelihood that the consumption of a giant planet could pollute its host star. First, we show that the detection of 6Li in SWPs could be used to distinguish between giant planet formation theories, and can be used to detect the consumption of giant planets independent of planet mass. Second, we examine the probability that giant planets between 1 and 3 M_J could be destroyed in the outer convection zone of stars slightly more massive than the Sun, finding that heated giant planets would be efficiently destroyed near the surface of the star, while the cores of cold giant planets may be able to survive to the base of the star's convection zone. Third, we examine the tidal interaction between a planet and its host star, showing that this interaction can lead to tidal disruption of the planet and accretion of its material, or orbital decay followed by hydrodynamical interaction. Throughout, we consider the case of HD 82943, a star known to have two planets and having a preliminary detection of 6Li. Using stellar models including diffusion, we estimate the mass of the HD 82943 to be about 1.2 M_sun and its age to be about 0.5-1.5 Gyr. The observed 7Li abundance for HD 82943 is consistent with stars of similar T_eff and age in the open cluster NGC 752. We describe a possible dynamical history for a hypothetical planet in the presence of the two resonant planets currently known. (abridged)

Future Doppler Ground-Based Search for Solar-Like Oscillations with HARPS

Observations of solar-like oscillation frequencies allow us to constrain the theory of structure and evolution of stars with an outer convective zone. Recent improvements in radial velocity measurements have led to p-mode detection on Procyon (Martic et al., 1999) and β Hydri (Bedding et al., 2001; Carrier et al., 2001a). Very recently, characterization of the p-modes of α Cen A has been made with the spectrograph CORALIE mounted on the 1.2-m Swiss telescope at the ESO La Silla Observatory (Bouchy & Carrier, 2001, Carrier et al., these proceedings). The power of the radial velocity technique promises the development of ground-based search for solar-like oscillations on a large sample of stars especially with the future spectrograph HARPS.

The composition of the solar wind

Accurate knowledge of the composition of the solar wind and of solar system abundances allows us to improve our understanding about various processes affecting the elemental, isotopic, and charge-state abundances of the solar wind. During the evolution of the Sun, isotopic and elemental abundances have been affected by migration processes at the interface between the well mixed outer convective zone and the radiative zone. In the solar atmosphere, the isotopic and elemental composition can be affected by Coulomb drag and wave-particle interactions. In addition, elemental abundances are clearly fractionated by some process that is controlled by the first ionization potential (FIP) of the elements. The abundances of the charge states are determined by processes in the corona, and, in interplanetary space, serve as valuable tracers for the coronal origin of the solar wind. The abundances of certain key elements and their isotopes can safely and accurately be determined from their meteoritic values. Their abundances in the solar wind can be used to obtain estimates for the degree of fractionation undergone by other elements and their isotopes, for which the meteoritic abundances do not necessarily reflect photospheric values, or for which it is not clear which meteoritic component corresponds to solar values. In short, the composition of the solar wind can yield valuable information about processes in the early solar system and its evolution, the evolution of the Sun, processes in the solar interior, atmosphere, and corona, as well as in interplanetary space.

The composition of the solar photosphere

The Sun is unique because chemical composition data can be derived from very different layers, from the center to the outermost coronal layers, using very different techniques like helioseismology, spectroscopy, particle collection techniques, … Differences in chemical composition are observed allowing to discover how the different solar layers evolve. The composition of the solar photosphere, which represents the composition of the outer convective zone, is the basic reference source of elemental abundances. We review the current status of our knowledge of the chemical composition of this layer and compare it with data from other solar sources as well as with meteoritic data.

Seismic study of stellar convective cores

It has been shown that a discontinuity in the derivatives of the sound speed at the edge of the convective regions inside a star gives rise to a characteristic oscillatory signal in the frequencies of stellar oscillations. This oscillatory signal has been suggested as a means to study the base of the outer convection zone in low mass stars and possibly the outer edge of the convective core in high mass stars. Using stellar models we show that because of a phenomenon similar to aliasing in Fourier transform, it may not be possible to use this signal to detect the convective core. Nevertheless, it may be possible to determine the size of convective cores using the frequency separation \nu_{n+1,l}-\nu_{n,l}.

The Abundance Distribution in the Extrasolar-Planet Host Star HD 19994

Abundances of 22 elements have been determined from a high-resolution, high signal-to-noise spectrum of HD 19994, a star recently announced as harboring an extrasolar planet. A detailed spectroscopic analysis of this star finds it to have a mass of 1.2 ± 0.1 M⊙. HD 19994 is found to be slightly enriched in "metals" relative to the Sun ([Fe/H] = +0.09 ± 0.05 and an average of all elements of [m/H] = +0.13), as are most stars known with extrasolar planets. In an investigative search for possible signatures of accretion of metal-rich gas onto the parent stars in such systems (using HD 19994 and published abundances for other stars), it is found that a small subset of stars with planets exhibits a trend of increasing [X/H] with increasing condensation temperature for a given element X. This trend may point to the accretion of chemically fractionated solid material into the outer (thin) convection zones of these solar-type stars. It is also found that this small group of stars exhibiting an accretion signature all have large planets orbiting much closer than is found, in general, for stars with planets not showing this peculiar abundance trend, suggesting a physical link between accretion and orbital separation. In addition, the stars showing evidence of fractionated accretion are, on average, of larger mass (1.2 M⊙) than stars not showing measurable evidence of this accretion (1.0 M⊙).

The Mass of the Convective Zone in FGK Main-Sequence Stars and the Effect of Accreted Planetary Material on Apparent Metallicity Determinations

The mass of the outer convective zone in FGK main sequence stars decreases dramatically with stellar mass. Therefore, any contamination of a star's atmosphere by accreted planetary material should affect hotter stars much more than cool stars. If recent suggestions that high metal abundances in stars with planets are caused by planetesimal accretion are correct, then metallicity enhancements in earlier-type stars with planets should be very pronounced. No such trend is seen, however.

Deuterium and Helium-3 in the Protosolar Cloud

New measurements of the isotopic composition of helium in the solar wind obtained from the Solar Wind Ion Composition Spectrometer (SWICS) on Ulysses are presented and compared with earlier SWICS results and previous mass spectrometric determinations of this ratio with the Apollo Solar Wind Composition (SWC) experiment and the Ion Composition Instrument (ICI) on the International Sun Earth Explorer 3 (ISEE 3). The new SWICS data from both the fast and slow solar wind are extrapolated to the photosphere to obtain a representative value of the present-day ratio of3He/4He = (3.75 ± 0.27) × 10−4in the Outer Convective Zone (OCZ) of the Sun. After corrections of this ratio for secular changes caused by diffusion, mixing and3He production by incomplete H-burning (Vauclair 1998), we obtain (D +3He)/H = (3.6 ± 0.38) × 10−5for the Protosolar Cloud (PSC). Adopting the Jovian3He/4He ratio = (1.66 ±0.05) × 10−4measured by the Galileo Probe mass spectrometer (Mahaffy et al. 1998) as representative for the PSC, we obtain (D/H)protosolar= (1.94 ±0.39) × 10−5. Using results of galactic evolution studies (Tosi 1998, 2000) and the D and3He abundances in the Protosolar Cloud and the Local Interstellar Cloud (Linsky 1998; Gloeckler &Geiss 1998), we estimate (D/H)primordial= (2.4 − 4.2) × 10−5. This range corresponds to a universal baryon/photon ratio of (5.9−4.2) × 10−10.

Warm Dust in the Cool Brown Dwarf Gliese 229B and Spectroscopic Diagnosis of Dusty Photospheres

Dust should be formed in the photosphere of cool brown dwarfs, but it is not clear in what form the dust exists. We propose a model in which a rather warm dust layer exists deep in the photosphere. At relatively high temperature there, dust grains are in detailed balance with the gas and the homogeneous dust-gas mixture works as an efficient source of opacity. The warm dust layer, which can be optically thick in the optical region, effectively blocks the optical radiation and emits like a blackbody of rather high temperature. The dust layer, however, can be optically thin in the infrared and has little effect on the infrared radiation. In the cool upper region, dust grows too large to be sustained with the gas and segregates from the gaseous atmosphere. Dust no longer works as an opacity source, and volatile molecules such as CH4 and H2O take the role. Also, nonrefractory elements such as alkali metals remain in gaseous form, and the neutral atoms contribute significantly to block the optical radiation (e.g., K I doublet). We show that such a hybrid model can be constructed without any ad hoc assumption except that the transition from the dust-gas detailed balance regime to the dust-gas segregation phase takes place at a certain transition temperature Ttr. By a sudden increase of opacity due to dust at Ttr, an outer convective zone appears, followed by intermediate radiative and inner deep convective zones. The emergent spectrum is also a hybrid of the dust- and gas-dominated cases and offers a natural explanation of the observed spectrum of Gliese 229B through the optical to the infrared.

Planet Consumption and Stellar Metallicity Enhancements

The evolution of a giant planet within the stellar envelope of a main-sequence star is investigated as a possible mechanism for enhancing the stellar metallicities of the parent stars of extrasolar planetary systems. Three-dimensional hydrodynamical simulations of a planet subject to impacting stellar matter indicate that the envelope of a Jupiter-like giant planet can be completely stripped in the outer stellar convection zone of a 1 M☉ star. In contrast, Jupiter-like and less massive Saturn-like giant planets are able to survive through the base of the convection zone of a 1.22 M☉ star. Although strongly dependent on details of planetary interior models, partial or total dissolution of giant planets can result in significant enhancements in the metallicity of host stars with masses in the range 1.0 M≲M≲1.3 M. The implications of these results with regard to planetary orbital migration are briefly discussed.

First determination of the silicon isotopic composition of the solar wind: WIND/MASS results

Silicon is a common material in the solar system. For instance, Si accounts for about 10% of the material in primitive meteorites (CI chondrites). Since silicon is a refractory element, we expect the meteoritic isotopic composition to be very similar to that of the Sun. The isotopic composition of Si in meteorites is well known and varies little. Thus the three stable isotopes of Si may serve as powerful indicators to test fractionation of isotopes in the transition from the solar atmosphere into the solar wind. We present, for the first time, measurements of the isotopic composition of Si in the solar wind. The data were obtained with the MASS instrument aboard the WIND spacecraft and accumulated in exceedingly cold and slow wind. Such wind is often associated with large superradial expansion factors and with current sheet crossings which in turn are associated with the most efficient isotopic fractionation processes in the solar wind acceleration region. We detect little or no isotopic fractionation between the solar surface assumed to be of meteoritic composition and the solar wind. This constrains solar wind acceleration models and puts stringent limits on possible secular changes in the isotopic composition of the outer solar convective zone, the solar atmosphere, and the solar wind.

DB white dwarf evolution in the frame of the full spectrum turbulence theory

We present an analysis of the evolution of carbon-oxygen DB white dwarfs (helium­ rich envelope) for a wide range of effective temperatures and luminosities. To this end, we employ a full stellar evolution code, in which we include a new equation of state for helium plasmas recently developed by Saumon, Chabrier & Van Hom and new OPAL radiative opacities. The most important feature of our models is that the transport of energy by convection is described by the full spectrum turbulence theory. In particular, we have adopted two versions of this theory for stellar convection: the Canuto & Mazzitelli theory and the more recent, self-consistent theory developed by Canuto, Goldman & Mazzitelli. Both theories, which have no free parameters and account for the whole spectrum of turbulent eddies, represent a great improvement compared to the mixing-length theory approach used thus far in almost all white dwarf studies. Neutrino energy losses as well as crystallization were taken into account. In order to explore the sensitivity of our results to various input model parameters, we vary the model mass from 0.5 to 1.0 Mo in intervals of 0.1 Mo, and the helium layer mass in the interval of 10-6 ~ MHe/M* ~ 10- 2 • The emphasis is put mainly on the behaviour of the evolving outer convection zone. In particular, we analyse the dependence of the location of the theoretical blue edge of the instability strip on the various input parameters. We find that the new ingredients we have incorporated in this study - mostly the new formulations for stellar convection -lead to theoretical blue edges in agreement with observations of pulsating DB white dwarfs. In this context, the Canuto, Goldman & Mazzitelli self­ consistent theory yields theoretical blue edges somewhat hotter than those given by the Canuto & Mazzitelli theory, which is more consistent with a recent determination of the effective temperature of the hot DBV GD 358. Contrary to previous results, we find that, according to the new theories for convection, non­ variable DB white dwarfs falling within the instability strip cannot be low-mass configurations. In order to compare with previous computations, we include in our calculations the most common parametrizations of the mixing-length theory usually employed in almost all previous white dwarf studies. In this context, we find that the ML2 parametrization provides a reasonable agreement with the observed blue edge for the DB instability strip. However, the profile of the outer convective zone given by the mixing-length theory is markedly different from that given by both of the new convective formulations.

The High Chromospheres of the Late A Stars

We report the detection of N V 1239 Å transition region emission in HST/Goddard High Resolution Spectrograph spectra of the A7 V stars, α Aql and α Cep. Our observations provide the first direct evidence of 1-3 × 105 K material in the atmospheres of normal A-type stars. For both stars, and for the mid A-type star τ3 Eri, we also report the detection of chromospheric emission in the Si III 1206 Å line. At a B-V color of 0.16 and an effective temperature of ~8200 K, τ3 Eri becomes the hottest main-sequence star known to have a chromosphere and, thus, an outer convection zone. We see no firm evidence that the Si III line surface fluxes of the A stars are any lower than those of moderately active, solar-type G and K stars. This contrasts sharply with their coronal X-ray emission, which is more than 100 times weaker than that of the later type stars. Given the strength of the N V emission observed here, it now appears unlikely that the X-ray faintness of A stars is due to their forming very cool, ≤1 MK coronae. An alternative explanation in terms of mass loss in coronal winds remains a possibility, though we conclude from moderate resolution spectra of the Si III lines that such winds, if they exist, do not penetrate into the chromospheric Si III-forming layers of the star, since the profiles of these lines are not blueshifted and may well be redshifted with respect to the star.

Evolution of Helium White Dwarfs of Low and Intermediate Masses

We present detailed calculations of the evolution of low-mass, helium white dwarf models with masses from M = 0.1 to M = 0.5 M? at intervals of 0.05 M? and with a metallicity of Z = 10-3. For this purpose, we have taken fully into account finite-temperature effects by means of a detailed and updated stellar evolutionary code, in which the convective energy transport is described according to the new model for turbulent convection developed by Canuto & Mazzitelli. Furthermore, our code considers the most recent opacity data computed by the Livermore Group (OPAL data), and also the new equation of state for helium plasmas developed by Saumon, Chabrier, & Van Horn. Neutrino emission is fully taken into account as well. For models with M ? 0.3 M? we started our calculations from fully convective models located at the helium-Hayashi line for each configuration, far away from the white dwarf regime. By contrast, the evolutionary sequences corresponding to 0.35, 0.4, 0.45, and 0.5 M? were started from initial models resembling white dwarf structures. This was necessary in order to avoid the onset of helium burning. A consequence of this constraint is the existence of a forbidden region in the HR diagram above log (L/L?) = -0.25 and hotter than log Teff = 4.45, where helium white dwarfs can exist only for brief intervals. All the models were evolved to log (L/L?) = -5. The evolutionary tracks in the HR diagram have been carefully analyzed, and we found that the convective efficiency affects the tracks noticeably only in the high-luminosity (pre-white dwarf) regime. We also examined the evolution of central conditions, neutrino luminosity, radii, surface gravity, and ages. Central densities, radii, and surface gravities asymptotically approach the zero temperature Hamada-Salpeter results, as expected. Neutrino losses are important for the more massive helium white dwarf configurations and should be taken into account in detailed evolutionary studies of these objects. Finally, the structure of the outer convective zone was analyzed in both the framework of the mixing length theory (for different convective efficiencies) and the Canuto & Mazzitelli theory. We found that the profile of the outer convective zone given by the Canuto & Mazzitelli model is very different from that given by any version of the mixing length theory. This behavior is critical for pulsational instability; however, stellar parameters such as radius and surface gravity are not significantly affected in the white dwarf domain. These models should be especially suitable for the interpretation of the data about the recently discovered low-mass white dwarfs in systems containing another white dwarf or a millisecond pulsar.