Transcript Document

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:
For eigen frequencies of neighboring modes we use Taylor
decomposition of the form
We will need an expression for the mode norm
where ξ is the mode eigen vector, and η ≈ ξ when viscosity
is small, and an expression for the mode coupling with
the perturbing potential.
Change of canonical energy in the
case of dense spectrum
Overlap integrals
The 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 simple
for 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 of
convective 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 to
be 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 the
base
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.