convective overshooting in white dwarf and zz ceti stars

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convective overshooting in white dwarf and zz ceti stars

RevMexAA (Serie de Conferencias), 14, 20–21 (2002)CONVECTIVE OVERSHOOTING IN WHITE DWARF AND ZZ CETI STARSN. Falcón and J. LabradorDepto de Física, FACYT Universidad de Carabobo, Av. Bolivar Norte, Valencia Ap. 129 CP 2001, Venezuelanelsonfalcon@hispavista. comThe light curves of ZZ Ceti stars exhibit variationsof luminosity with typical intervals betweensuccessive pulses in the range from the hundreds tothe second thousands, and, in general, the “periods”are not constant.The oscillations and pulsations (non radial g modes)in ZZ Ceti stars are usually attributed to temperatureintrinsic variations (Väth et al. 2001). Instandard ZZ Ceti asteroseismology the possibility ofheat propagation by waves is obviated in the energytransport equation. This simplification will be spuriousin the degenerate material because, the relaxationtime is not negligible. The temperature gradientin stars is given, in a good approximation, bythe energy transport equation.The Fourier-Maxwell law leads to a parabolicequation for T, according to which perturbationspropagate with infinite speed. Neglecting the relaxationtime (τ) is, in general, a sensible thing to dobecause for ordinary stellar material it is very small.There are, situations where τ may not be negligible(in the degenerate material) . For example inneutron star interior τ is of the order of 10 2 s for atemperature of 10 6 K (Herrera & Falcon 1995a).dTdr = − 3 κρ ∂F(τ4ac T3∂t +F ) = − 3 κρ ∂L16ac T 3 (τr2 ∂t +L) .(1)The possibility that the time of relaxation is notnegligible, in the nucleus of WD, would bear the existenceof thermal waves. We consider the situation inthe surrounding of the outer boundary of a convectiveenvelope. Consider the temperature excess DTof a moving element adiabatically over the surroundings(gradient ∇). Obviously DT ∼ ∇−∇ ad and DTbecomes negative above the border, i. e. the overshootingelements become cooler than the surrounding,which results in a cooling of the upper layersand an increase of the gradient ∇. But now, thetemperature excess DT is not monotonous, becausethe interior material is degenerate. For times shorterthan the relaxation time the MLT theory (Herrera& Falcon 1995b) demand that DT ∼ DT (d) f(x, ω)f(x,w), where DT (d) denote the “classical” temperaturedifference (i. e. ∇ − ∇ ad ) and f(x, ω) is adamped oscillatory function, time dependent. Underthis consideration the luminosity due to the convectiveflow admit a quasi periodic variation (Herrera& Falcón 1995b).L = L (d) 1[ χω 2 + 1 exp (ω 2 − 1 )] (2)2[(5 + ω 2 ) cos(ωχ) − (3 − ]ω2 )sin(ωχ)ωand ω 2 ≡ 4ττ d−1 and χ ≡ t2τ. Equation (2) connectsthe standard luminosity (L) of MLT and the luminositybefore thermal relaxation (L (d) ) owing to theheat waves, where f is a function of time τ, relaxationtime τ and thermal relaxation time τ d . On the otherhand, the calculation of the relaxation time can bedone starting from the relationship:τ =kV 2 C V≈ 10−3 T −2β 2 , (3)where V, k and C v denote respectively the thermalwave speed, the thermal conductivity for degeneratematerial and the specific heat from the Chandrasekhar’srelationship.The luminosity fluctuation, due to the causalpropagation of the thermal flow (2) suggests an approximatemodel for the study of the ZZ Ceti stars.Inside the WD the matter is degenerate and the thermalconductivity is dominated by electrons, thereforethe use of the Cattaneo Law is justified. Thealeatory movement of certain convective globules,inside the gradient of temperature, would carry afluctuation in the temperature and luminosity. Thedue luminosity variation to the convective flow couldhave a damped oscillatory behavior (2).According to this model the periods of the luminosityfluctuations would be of the order of thethermal relaxation time. Also, because the convectiveflow is only an fraction of the total thermal flow,then the luminosity variations would be damped andsmall relative to the intrinsic luminosity. The behavioras erratic function of the ZZ Ceti light curves,20


OVERSHOOTING IN WHITE DWARF AND ZZ CETI STARS 21would be explained in term of the sum of several convectivefluctuations along time. Several convectivecells flow sequentially, each one due to some specificthermal fluctuation, they could reproduce thelight curves of some ZZ Ceti stars. The results showthat the luminosity had a qualitative behavior as adamped oscillation. The theoretical adjustment allowsto model the curve of light piecewise; it is consistentwith the presented pattern, according to whichthe fluctuations of the luminosity are caused to eachother by independent “convective cells”. However,a detailed analysis requires to consider of several cellsinside the star, each one of which contributes to thetotal convective flow.REFERENCESAlthaus L. G. & Benvenuto 1996, MNRAS, 278, 981Hansen C.J. & Kawaler S. D. 1994, Stellar Interiors,Springer-Verlag, N.Y.Herrera, L. & Falcon, N. 1995a, ApSS, 229, 105Herrera, L. & Falcon, N. 1995b, ApSS, 234, 139McGraw, J. T. 1977, PhD Thesis, University of Texasat AustinRobinson, E. L. . 1979, in IAU Colloquium 53, RochesterN.Y., p 343-358Robinson, E. L. 1983, ApJ, 262, L11-15Väth, H. et al. 1997, in White Dwarfs. Ed. Isern et al.,Kluwer Acad. Press, Dordrecht, 481

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