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Astron. Astrophys. 326, 257-262 (1997)

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1. Introduction

The analysis of eclipsing binary (EB) light curves (LC) involves a large number of parameters. To avoid multiple solutions, as many parameters as possible are kept fixed either at the observed values (e.g. mass ratio) or at the theoretical ones (limb-darkening coefficients, bolometric albedos, gravity-brightening exponents). LC generation models are able to directly adjust such parameters, but this may introduce correlations that make the convergence to physically acceptable solutions difficult or impossible.

The limb-darkening effect on EBLC is moderate (Popper 1984). Its coefficients for different laws (e.g. Al-Naimiy 1978, Wade & Ruciski 1985, Van Hamme 1993, Claret et al. 1995, amongst other) are satisfactorily used in the EB models, but the Strömgren u theoretical values seem not to correctly reproduce the observed u LC of early systems (Vaz et al. 1995 and references therein).

The bolometric albedos affect the LC of all types of close EB (Vaz 1984, 1985). The albedos for convective grey (Vaz & Nordlund 1985) and non-grey (Nordlund & Vaz 1990) atmospheres have been modeled. In non-grey models no convenient expression for the study of EBLC was found, and each case should be treated individually.

1.1. Basic works on the gravity-brightening effect

Von Zeipel (1924) showed that the local emergent flux of a star with an atmosphere in hydrostatic and radiative equilibrium and distorted by rotation or tidal effects is proportional to the local gravity acceleration ([FORMULA]):


Lucy (1967) found that Eq.(1) would hold for convective atmospheres if [FORMULA] 0.32, a value obtained by coupling convective envelopes of models with different surface gravities at depths where the temperature gradient was adiabatic. He recognized that the entropy at the bottom of different models should be the same, in order to represent the same star: "inward integrations of the equations governing the atmospheric structure must always end up on the same adiabat" (Lucy 1967). Lucy made his calculations for the effective temperatures corresponding to stars with [FORMULA] ([FORMULA]) and [FORMULA] [FORMULA] and [FORMULA], using atmosphere models by Baker & Temesváry (1966).

Anderson & Shu (1977) presented another formulation for convective atmospheres arguing that, for a contact star in hydrostatic equilibrium having a common convective envelope (the contact discontinuity model, Shu et al. 1976, Lubow & Shu 1977), the convective flux depends on the effective gravitational potential alone, and not on its gradient (i.e. the local acceleration of gravity). As for late-type stars [FORMULA] and the photosphere must ultimately radiate [FORMULA], this flux should be constant on equipotentials, with consequently [FORMULA] =0.

1.2. Empirical determinations and modern works

Rafert & Twig (R&T,1980) determined [FORMULA] by analysing a uniform sample of detached, semi-detached and contact EB with the WD (Wilson 1979) program. They found [FORMULA] for convective envelopes ([FORMULA]) and [FORMULA] for the radiative ones ([FORMULA]). For [FORMULA] they found empirically [FORMULA].

Hilditch (1981) found very small values of [FORMULA] (four contact EB), and that imposing [FORMULA] =0.32 yields a photometric mass ratio which is significantly different from the spectroscopic one, what does not happen with [FORMULA] =0. However, the obvious presence of star spots in all the 4 systems may render very difficult, and even spoil, attempts to determine temperature variations over the surfaces (Ruciski 1989).

Nakamura & Kitamura (1992 and references therein) did empirical studies on the [FORMULA] exponent (their distortion theory extended to a second order treatment, Kitamura & Nakamura, K&N 1983). For detached EB their values agree with those of von Zeipel (1924) and Lucy (1967), but for early-type both semi-detached (K&N 1987b, 1992) and contact EB (K&N 1988a, b), they obtained [FORMULA].

Sarna (1989), with a model of convective envelope for Roche lobe filling stars (W UMa-type), computed the distribution of effective temperatures and surface gravities for the stars and directly determined [FORMULA] 0.32.

Observation and theory thus generally agree with each other, with [FORMULA] 1.0 for radiative atmospheres and [FORMULA] 0.32 for the convective ones, albeit exponents [FORMULA] 1.0 were found by K&N. There is observational evidence for a transition between these two regimes (K&N and R&T), suggesting that [FORMULA] =0.32 is valid only for temperatures close to what Lucy (1967) used. "Whereas [FORMULA] =1 is a solid, physically well understood upper limit (...), [FORMULA] =0.32 is an approximate relation based on a small set of envelope models which used very crude physics" (Ruciski 1989). Further, there is only one empirical determination (Ruciski 1976) and no theoretical study for [FORMULA] K up to now, all this being the motivation for the present study.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998