Hydrostatic equilibrium is equivalent to constant density and pressure on equipo-tential surfaces. If we assume that density p, temperature 26 T, and pressure p are related to each other by an equation of state, e.g., the ideal gas law,
26 Note that this temperature is the local thermodynamic temperature which differs conceptually from the effective temperature defined as a function of bolometric flux.
with the universal gas constant R, then on equipotentials we see constant temperature as well. So, the star in hydrostatic equilibrium is homogeneous on each equipo-tential.
A rotating star differs from a nonrotating star in shape, local surface gravity acceleration, and surface brightness. It develops oblateness and a pole-to-equator variation in surface brightness. This variation is called gravity brightening (sometimes, especially in the older literature, called gravity darkening). This phenomenon, for radiative envelopes, has its origin in the temperature gradient in and near the surface. The atmosphere is assumed to be locally plane-parallel (that implies we need to consider only one geometrical dimension), but irradiated from below by a radiative flux varying across the stellar surface. According to von Zeipel's theorem 27 this flux and, due to the Stefan-Boltzmann law (3.2.15), the temperature are also determined by the gradient (w.r.t. the optical depth t) of the source function in the subphotospheric layers. The result is that this flux is proportional to the effective surface gravity acceleration g at the given point of the surface. A modern derivation of this result is found in Kippenhahn & Weigert (1989, Eq. 42.6, p. 436) and reads
4ac 3 dT
where F is the vector of radiative energy flux and g is the effective gravitational acceleration consisting of gravitational and centrifugal acceleration. The proportionality factor k(U) describes the conduction of this radiative transport and depends only on the potential U because the temperature, T = T(U), and opacity k(p, T) = k(U) depend only on potential (Kippenhahn & Weigert, 1989, p. 436). If we want to compute the radiative flux on a given equipotential it varies only with g and is antiparallel to g.
Gravity brightening is thus described by a relation between the local effective temperature Tl (or the local bolometric flux) and the local surface gravity acceleration. For our purpose is sufficient to consider only the modulus of g, i.e., according to (3.1.58) we get g =!gl = ivU|= go ivQI, (3.2.11)
where g0 is a proportionality factor. Thus, for each surface point rs e S' we express the local surface gravity acceleration in terms of potential gradient components of the gradient of the Roche potential,
27 There are three papers by von Zeipel (1924a, b, c) related to the radiative equilibrium of distorted stars. Relevant to our problem is Eq. (36) in the first paper, and Eqs. (90) and (91) in the third paper.
//9Q\2 /9Q\2 /9Q\2 gi = g(rs):= g0lVQ(rs)l=g^(+ (-dy) + (. (3.2.12)
Further in the book the subscript l will indicate "local." Once gl is known, the local bolometric flux Fl = F(rs) is computed according to i7 J? (glY \ 1-00, von Zeipel theorem, m
where the index p refers to the pole of a star, and the exponent g should not be confused with the modulus of g. The upper part of (3.2.13) summarizes the von Zeipel theorem by von Zeipel (1924a) for stars in radiative equilibrium. Lucy's law established by Lucy (1967) gives the relation for stellar envelopes in convective equilibrium. The exponent 0.32 is an estimate derived numerically from tables of convective stellar envelopes.
Let us make a few more remarks on the von Zeipel theorem. The radiative equilibrium in regions with different g (and consequently also the optical depths of particular equipotentials) gives rise to temperature gradients along the equipotential surface. We should expect that this temperature gradient leads to a mass flow (meridional28 circulation) parallel to the surface which tends to homogenize the physical conditions in the layer. Thus, strictly speaking, it is not possible for a rotating and tidally distorted star to be in hydrostatic and radiative equilibrium simultaneously. The complicated dependence of the temperature on optical depth in the photosphere in radiative equilibrium immediately violates the underlying assumptions of the homogeneity on equipotentials. If we give up the radiative equilibrium assumption and assume that the horizontal homogenization would be effective in the photosphere and lead to weaker temperature variations on equipotentials, then we can use the result due to Hadrava (1987, 1988), who showed that the flux would vary, under these circumstances, such that
and that limb darkening would also depend on g. To close the discussion on von Zeipel's theorem we conclude that a rigid analysis of the problem should include hydrodynamic calculations of meridional circulations in the atmosphere and also the computation of three-dimensional radiative transfer (giving up the assumption of a local plane parallel atmosphere), keeping in mind that the theorem is an approximation which in many binaries seems to represent the situation appropriately.
As is obvious from (3.2.13), gravity brightening is strong in distorted stars hot enough to have radiative envelopes. Early-spectral-type close binaries such as TU Muscae [cf. Andersen & Gr0nbech (1975)] are the best examples. In these systems,
28 Meridional circulation is for instance described by Tassoul (1978, Chap. 8) or Kippenhahn and Weigert (1989, pp. 437-443).
the bolometric flux is directly proportional to local gravity. Because bolometric flux and local effective temperature are coupled by the Stefan-Boltzmann law
from (3.2.13) with p := g/4 we can derive a similar relation for the local effective temperature r rr I o» / a g I 0.25, von Zeipel-theorem, Tl = Tp( — ) - P = 7 = 1 (3.2.16)
4 I ^ 0.08, Lucy's law, where Tp is the polar effective temperature. The polar temperature is the highest temperature on the star and therefore higher than the "spectroscopically observed" effective temperature Teff, which is some kind of average (weighted with aspect effects). If the stars move in elliptic orbits, in contrast to Teff the polar effective temperature, Tp, varies with phase. Therefore, Wilson (1979) recommends use of the mean surface effective temperature Teff as input parameter. For a star with surface S, Teff can be defined through the bolometric luminosity L and the Stefan-Boltzmann law,
L = aSTe4ff, a = 5.6705 ■ 10—8 J ■ m—2s—1K—4. (3.2.17)
On the other hand, Teff may be computed from the average flux over the surface
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