A unified normal modes approach to dynamic tides and its application to rotating stars with realistic structure

P. B. Ivanov and S. V. Chernov, PN Lebedev Physical Institute

and J. C. B. Papaloizou,

University of Cambridge details in

2013MNRAS.tmp.1263I

Dynamic versus quasi-static tides in the normal mode approach

• There are two contributions, which can be separated by the requirement that wave-like phenomena play either major or minor role.

• 1) Quasi-static tides. These may be thought as due to excitation of normal modes with eigen frequencies much

• larger than forcing frequencies. The energy and angular momentum are transferred to stars on a time scale of order of inverse forcing frequencies by viscous forces.

• The main uncertainty is the value of “tidal Q” defined as • the ratio of the energy stored in tidal bulge to an• amount of energy dissipated per one orbital period.

Dynamic tides• There are two types of dynamic tides: • 1) the ones appropriate for highly eccentric orbits• (DT in the sense of Press and Teukolsky). During periastron

passage normal modes of planet and stellar pulsations are exited. These are: fundamental modes with frequencies ~ *, g-modes in case of the presence stably stratified sizable regions as expected in stars, low frequency inertial waves with frequencies ~ , etc.

• 2) Dynamic tides in the sense of Zahn, operating • in rotating stars with any eccentricity. In this case it is

assumed that eigen frequencies are of the same order as forcing frequencies. Energy and angular momentum are transferred to the star by the action of viscosity.

• In all cases it can be shown that dynamic tides are associated with resonances between normal modes of

• stellar pulsations and external forcing.

Dynamic tides operating in systems with periodic orbits

Let me assume that the orbit is strictly periodic. In this case we can decompose all quantities of interest in Fourier series in time:

Equations of motion for the case of uniformly rotating star

Canonical energy

It is assumed below that the modes have ‘dense’ spectrum (it is appropriate for, say, g-modes) and the resonance condition is nearly satisfied for a particular mode:

We will need an expression for the mode norm

where ξ is the mode eigen vector, and η ≈ ξ when viscosityis small, and an expression for the mode coupling with the perturbing potential.

For eigen frequencies of neighboring modes we use Taylor decomposition of the form

Change of canonical energy in the case of dense spectrum

Overlap integralsThe forcing term can be represented as a product of two parts: the one determined by the amplitude of perturbing potential and the one describing properties of an individual resonant eigen mode. The latter term is called the overlap integral.

Regime of moderately large viscosity

Regime of very small viscosity

Analogous expressions for parabolic encounters

Evaluation of the overlap integrals, the case of sun-like stars

The overlap integrals can be evaluated analytically using the WKBJ technique. This can be done in the so-called ‘traditional’approximation for rotating stars. Calculations are especially simplefor sun-like stars with a radiative core and convective envelope.

Overlap integrals in the traditional and Cowling approximations

There are two important contributions, one is determined by the convective envelope, another – by the region close to the base ofconvective zone. These are important at small and intermediate values of eigen frequencies, respectively.

Rotationally modified gravity modes

Properties are determined by Brunt-Vaisala frequency

Eigen frequencies in WKBJ approximation may be shown tobe given by the expression

and it is assumed that close to be base of convective zone BV frequency scales approximately as a power of distance x from thebase

υ=2/ω

Almost synchronous rotation

Non-rotating star

New results for parabolic encounters

• The strength of interaction is characterised by the transferred energy expressed in units

• Gmp2/R*, where mp – is the perturber mass

assumed to be one Jupiter from now on.

• Additionally we plot evolution time scale

• of semi-major axis as a function of orbital

• period after circularisation.

Other stars in comparison with the Sun-like ones

BV frequency and density distributions for stars with M=1.5M☼

BV frequency and density distributions for stars with M=2M☼

BV frequency and density distributions for a star with M=5M☼

Conclusions• In all tidal problems the overlap integrals play a

fundamental role. Provided that the action of viscosity is specified they fully determine tidal interactions. The formalism based on the overlap integrals allows one to study tidal interactions in all possible regimes.

• When either parabolic tidal encounters are considered or dynamic tides in the regime of moderately large viscosity it is enough to know the overlap integrals and eigen frequencies to calculate all quantities determining tidal interactions.

• The transfer of energy as a result of tidal interaction is larger for stars having convective envelopes. Parabolic tidal encounters are also stronger for stars with smaller average density. Say, when a sun-like model is compared with a n=3 polytrope of the same mass and radius,

• The transfer of energy is much larger in the case of stars with realistic structure when η is sufficiently large.