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Register a new userZahns Theory of Dynamical Tides and Its Application to Stars
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S.V. Chernov
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Zahns theory of dynamical tides is analyzed critically. We compare the results of this theory with our numerical calculations for stars with a convective core and a radiative envelope and with masses of one and a half and two solar masses. We show that for a binary system consisting of stars of one and a half or two solar masses and a point object with a mass equal to the solar mass and with an orbital period of one day under the assumption of a dense spectrum and moderately rapid dissipation, the evolution time scales of the semimajor axis will be shorter than those in Zahns theory by several orders of magnitude
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We determine the response of a uniformly rotating star to tidal perturbations due to a companion. General periodic orbits and parabolic flybys are considered. We evaluate energy and angular momentum exchange rates as a sum of contributions from normal modes allowing for dissipative processes. We consider the case when the response is dominated by the contribution of an identifiable regular spectrum of low frequency modes, such as gravity modes and evaluate it in the limit of very weak dissipation. Our formalism may be applied both to Sun-like stars with radiative cores and convective envelopes and to more massive stars with convective cores and radiative envelopes. We provide general expressions for transfer of energy and angular momentum valid for an orbit with any eccentricity. Detailed calculations are made for Sun-like stars in the slow rotation regime where centrifugal distortion is neglected in the equilibrium and the traditional approximation is made for the normal modes. We use both a WKBJ procedure and direct numerical evaluation which are found to be in good agreement for regimes of interest. Finally we use our formalism to determine the evolution time scales for an object, in an orbit of small eccentricity, around a Sun-like star in which the tidal response is assumed to occur. Systems with either no rotation or synchronous rotation are considered. Only rotationally modified gravity modes are taken into account under the assumption that wave dissipation occurs close to the stellar centre.
Zahn (1975) first put forward and calculated in detail the torque experienced by stars in a close binary systems due to dynamical tides. His widely used formula for stars with radiative envelopes and convective cores is expressed in terms of the stellar radius, even though the torque is actually being applied to the convective core at the core radius. This results in a large prefactor, which is very sensitive to the global properties of the star, that multiplies the torque. This large factor is compensated by a very small multiplicative factor, $E_{2}$. Although this is mathematically accurate, depending on the application this can lead to significant errors. The problem is even more severe, since the calculation of $E_{2}$ itself is non-trivial, and different authors have obtained inconsistent values of $E_{2}$. Moreover, many codes (e.g. BSE, StarTrack, MESA) interpolate (and sometimes extrapolate) a fit of $E_{2}$ values to the stellar mass, often in regimes where this is not sound practice. We express the torque in an alternate form, cast in terms of parameters at the envelope-core boundary and a dimensionless coefficient, $beta_{2}$. Previous attempts to express the torque in such a form are either missing an important factor, which depends on the density profile of the star, or are not easy to implement. We show that $beta_{2}$ is almost independent of the properties of the star and its value is approximately unity. Our formula for the torque is simple to implement and avoids the difficulties associated with the classic expression.
We study the effect of dynamical tides associated with the excitation of gravity waves in an interior radiative region of the central star on orbital evolution in observed systems containing Hot Jupiters. We consider WASP-43, Ogle-tr-113, WASP-12, and WASP-18 which contain stars on the main sequence (MS). For these systems there are observational estimates regarding the rate of change of the orbital period. We also investigate Kepler-91 which contains an evolved giant star. We adopt the formalism of Ivanov et al. for calculating the orbital evolution. For the MS stars we determine expected rates of orbital evolution under different assumptions about the amount of dissipation acting on the tides, estimate the effect of stellar rotation for the two most rapidly rotating stars and compare results with observations. All cases apart from possibly WASP-43 are consistent with a regime in which gravity waves are damped during their propagation over the star. However, at present this is not definitive as observational errors are large. We find that although it is expected to apply to Kepler-91, linear radiative damping cannot explain this dis- sipation regime applying to MS stars. Thus, a nonlinear mechanism may be needed. Kepler-91 is found to be such that the time scale for evolution of the star is comparable to that for the orbit. This implies that significant orbital circularisation may have occurred through tides acting on the star. Quasi-static tides, stellar winds, hydrodynamic drag and tides acting on the planet have likely played a minor role.
We investigate the change in the orbital period of a binary system due to dynamical tides by taking into account the evolution of a main-sequence star. Three stars with masses of one, one and a half, and two solar masses are considered. A star of one solar mass at lifetimes $t=4.57times10^9$ yr closely corresponds to our Sun. We show that a planet of one Jupiter mass revolving around a star of one solar mass will fall onto the star in the main-sequence lifetime of the star due to dynamical tides if the initial orbital period of the planet is less than $P_{rm orb}approx2.8$ days. Planets of one Jupiter mass with an orbital period$P_{rm orb}approx2$ days or shorter will fall onto a star of one and a half and two solar masses in the mainsequence lifetime of the star.
We review our recent results on a unified normal mode approach to dynamic tides proposed in Ivanov, Papaloizou $&$ Chernov (2013) and Chernov, Papaloizou $&$ Ivanov (2013). Our formalism can be used whenever the tidal interactions are mainly determined by normal modes of a star with identifiable regular spectrum of low frequency modes. We provide in the text basic expressions for tidal energy and angular momentum transfer valid both for periodic and parabolic orbits, and different assumptions about efficiency of normal mode damping due to viscosity and/or non-linear effects and discuss applications to binary stars and close orbiting extrasolar planets.
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