A&A 398, 1111-1115 (2003)
DOI: 10.1051/0004-6361:20021726
Research Note
B. Willems1 - A. Claret2
1 - Department of Physics and Astronomy, The Open University,
Walton Hall, Milton Keynes, MK7 6AA, UK
2 -
Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004,
18080 Granada, Spain
Received 26 September 2002 / Accepted 18 November 2002
Abstract
The paper is devoted to the analysis of the systematic
deviations between the classical formula for the rate of secular
apsidal motion in close binaries and the formula established within
the framework of the theory of dynamic tides. The basic behaviour is
found to be governed by the dependency of the forcing frequencies on
the orbital period and the rotational angular velocity, irrespective
of the stellar model. For binaries with short orbital periods
a good agreement between the classical apsidal-motion formula and
the formula established within the framework of the theory of
dynamic tides is only possible for certain values of the
rotational angular velocity.
Key words: binaries: close - stars: oscillations - celestial mechanics
In close binary systems of stars, the perturbation of the external gravitational field due to the tidal and rotational distortions of the component stars yields a periodic shift of the position of the periastron. The effect was first studied by Russell (1928), and subsequently by Cowling (1938) and Sterne (1939). The latter author derived a formula for the rate of secular apsidal motion based on the assumption that the orbital period is long in comparison to the periods of the free oscillation modes of the component stars. The validity of this assumption has been investigated extensively by, e.g., Papaloizou & Pringle (1980), Smeyers et al. (1991), Quataert et al. (1996), Smeyers et al. (1998), and Smeyers & Willems (2001). Smeyers & Willems (2001) found the classical apsidal-motion formula to be valid up to high orbital eccentricities as long as the orbital and rotational angular velocities remain small. For binaries with higher orbital and rotational angular velocities, the apsidal-motion rates predicted by the classical formula deviate from those predicted by the theory of dynamic tides due the effects of the compressibility of the stellar fluid and the possibility of resonances between dynamic tides and free oscillation modes of the component stars.
One of the most uncertain parameters in the analysis of the dynamics of a close binary is the rotational angular velocity of the component stars. Claret & Willems (2002) therefore considered the effects of dynamic tides on the apsidal-motion rate as a function of the rotational angular velocity for a carefully selected sample of eclipsing binaries with accurate determinations of the stellar masses and radii and the apsidal-motion rate. The authors found that even for binaries with short orbital periods, a good agreement between the classical apsidal-motion formula and the formula established within the framework of the theory of dynamic tides is possible for certain values of the rotational angular velocity.
In this research note, we aim to shed more light on the results found by
Claret & Willems (2002) by examining more closely the role of the forcing
frequencies and the effects of the compressibility of the stellar
fluid on the rate of secular apsidal motion due to the tidal distortions
of close binary components. In Sects. 2 and 3, we recall the
basic assumptions and the equations governing the rate of secular
change of the longitude of the periastron in the classical framework
and in the framework of the theory of dynamic tides. In Sect. 4, we
illustrate the behaviour of the relative deviations between the
classical apsidal-motion formula and the formula established within
the framework of the theory of dynamic tides as a function of the
rotational angular velocity in the case of an evolved
main-sequence star. In Sects. 5 and 6, we unravel this
behaviour by looking at the effects of the forcing frequencies and the
stellar model separately. In the final section, we briefly summarise
our conclusions.
Consider a close binary system of stars with masses M1 and M2that are orbiting around each other under the influence of their
mutual gravitational force. Let
be the orbital period,
a the semi-major axis, and e the orbital eccentricity. We assume
the first star to rotate uniformly around an axis perpendicular to the
orbital plane with an angular velocity
whose magnitude
is small in comparison to the stellar break-up speed
.
The companion star is treated as a point mass.
Furthermore, we denote by
a system of spherical coordinates with respect to an orthogonal frame
of reference that is corotating with the star. The potential governing
the tidal force exerted by the companion can then be expanded in terms
of unnormalised spherical harmonics
and in
Fourier series in terms of multiples of the companion's mean motion n as
For the remainder of the paper, we restrict ourselves to the
contributions of the dominant terms associated with
in the
expansion of the tide-generating potential.
The most commonly used formula for the rate of secular change of the
longitude of the periastron due to the tidal deformations of the
components of a close binary was derived by Cowling (1938) and
Sterne (1939) on the assumption that the orbital period is long in
comparison with the free harmonic periods of the component
stars. The rate of secular apsidal motion is then given by
For binaries with short orbital periods or rapidly rotating component
stars, the rates of secular apsidal motion predicted by the classical
formula deviate from the corresponding rates predicted by the formula
established in the framework of the theory of dynamic tides due to the
increased role of the stellar compressibility at higher forcing
frequencies and due to resonances of dynamic tides with free
oscillation modes of the component stars (Smeyers & Willems 2001). The
extent of the deviations depends on the evolutionary stage of the star
due to the increase of the stellar radius and the redistribution of
the mass as the star evolves on the main sequence (Willems & Claret 2002). In
the present investigation, we focus on the role of the forcing
frequencies and the compressibility of the stellar fluid. In analogy
to previous investigations (Smeyers & Willems 2001; Willems & Claret 2002; Claret & Willems 2002), we study the
deviations between the apsidal-motion rates predicted by the classical
formula and the rates predicted by the formula taking into account the
effects of dynamic tides by means of the relative difference
We illustrate the behaviour of the relative differences
as a
function of the rotational angular velocity
in the case
of a
main-sequence star with a central hydrogen
abundance
and a radius
.
Since the
rotational angular velocity
is assumed to be small, we
neglect the effects of the Coriolis force and the centrifugal
force for the determination of the constants F2,m,k. The tides
then correspond to those of a non-rotating spherically symmetric
equilibrium star (for more details see, e.g., Willems & Claret 2002).
The resulting relative deviations
are presented in
Fig. 1 for the orbital eccentricity e=0.4 and for
orbital periods ranging from 5.0 to 12.5 days. For convenience, the
rotational angular velocity
is expressed in units of the
orbital angular velocity
at the periastron of the relative
orbit. The peaks in the curves correspond to resonances between
dynamic tides and free oscillation modes. Since we are here
interested in the systematic trend caused by the effects of the
forcing frequencies and the compressibility of the stellar fluid
rather than in a detailed treatment of resonant dynamic tides, we did
not attempt to find all possible resonances for each orbital
configuration. For a detailed treatment of the resonances,
we refer the reader to Smeyers & Willems (2001) and Willems & Claret (2002).
![]() |
Figure 1:
The relative differences ![]() ![]() ![]() |
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In the limiting case where
,
the forcing
frequencies
are determined solely by the mean motion
n. As shown by Smeyers & Willems (2001), the relative differences
are
then generally negative and larger in absolute value for shorter
orbital periods. For larger values of
,
the relative
differences
increase, reach a positive maximum, and
subsequently decrease to become negative. Hence, we find two values
of the rotational angular velocity for which the relative differences
become zero. These values depend on both the period and the
eccentricity of the orbit, and do not necessarily correspond to
synchronised rotation with the orbital angular velocity at the
periastron of the relative orbit. We will further scrutinise this
behaviour in the following section.
Due to the expansion of the tide-generating potential in Fourier
series in terms of multiples of the companion's mean motion, the
tidally distorted star is subjected to an infinite number of forcing
angular frequencies
(see Eq. (1)). For a given
orbital eccentricity, the largest differences between the classical
apsidal-motion formula and the formula established within the
framework of the theory of dynamic tides can be expected to occur when
high forcing frequencies are associated with large Fourier
coefficients c2,m,k. Vice versa, the smallest differences can be
expected to occur when the dominant terms in the expansion of the
tide-generating potential all have small forcing frequencies.
A similar reasoning applies to Expansion (4) which renders
the apsidal-motion rate within the framework of the theory of dynamic
tides. Since the system of differential equations governing
non-resonant dynamic tides in components of close binaries depends on
the square of the forcing frequency, the deviations of the constants
F2,m,k from the classical apsidal-motion constant k2 are of
the order of
.
Because of this, Smeyers & Willems (2001) and
Claret & Willems (2002) were able to approximate the systematic deviations
between the classical apsidal-motion formula and the formula
established within the framework of the theory of dynamic tides by
second-degree polynomials in the companion's mean motion n or the
star's rotational angular velocity
.
In the present context,
this property implies that the differences between the two formulae
can be expected to be large when high forcing frequencies are
associated with large coefficients G2,m,k.
In order to shed some more light on this, we examine the behaviour of
the function
The variations of S1 as a function of the rotational angular
velocity
are displayed in Fig. 2 in the case of the
orbital eccentricity e=0.4 and orbital periods ranging from 5.0 to 12.5 days. For given values of e and
,
the function S1 initially decreases with increasing values of
.
The decrease is caused by the decrease of the forcing
frequencies associated with m=-2, which generally provide the
dominant contributions to S1. As
becomes larger, S1reaches a minimum when the forcing frequencies associated with the
largest coefficients
G2,-2,k are close to zero. The frequencies
associated with m=-2 subsequently become more and more negative,
resulting in an indefinite increase of the function S1.
![]() |
Figure 2:
The variation of
![]() ![]() |
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As a preliminary conclusion we can thus say that the opposite of the
function S1 shows the same qualitative behaviour as the
relative differences
shown in Fig. 1. In the next
section, we will make this even clearer by adding some information on
the stellar model to the definition of S1.
The response of a particular stellar model to the tidal forcing of a
close companion is determined by the constants F2,m,k.
For low-frequency, non-resonant dynamic tides in stars with a radiative
envelope, these constants can be approximated by their
asymptotic representation
The asymptotic approximation for the constants F2,m,k is valid
for forcing frequencies
.
Higher
forcing frequencies, such as those associated with large values of k, require additional terms in the asymptotic
approximation. However, the second-order approximation given by
Eq. (7) is sufficient to further clarify the qualitative
behaviour of the relative differences
observed in
Fig. 1.
We proceed in a similar way as for the introduction of the function S1 in the previous section, but now take into account the structure
of the stellar model by defining a function S2 as
The variations of S2 as a function of the rotational angular
velocity
are displayed in Fig. 3 for the same
main-sequence model used in Sect. 4. The qualitative
behaviour of S2 is essentially the same as that of -S1. In
addition, the inclusion of the model dependent factors introduces two
zeros in the curves, similar to the behaviour of the relative
differences
displayed in Fig. 1.
![]() |
Figure 3:
The variation of
![]() ![]() |
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In this research note, we scrutinised the behaviour of the systematic relative deviations between the classical formula for the rate of secular apsidal motion and the formula established within the framework of the theory of dynamic tides as a function of the often uncertain rotational angular velocity of the component stars. We found that when the effects of resonant dynamic tides are neglected, the basic behaviour of the relative differences is governed by the dependency of the forcing frequencies on the orbital period and the rotational angular velocity, irrespective of the stellar model. For a given value of the orbital eccentricity, a good agreement between the classical apsidal-motion formula and the formula established within the framework of the theory of dynamic tides is possible even for binaries with short orbital periods, but only for certain values of the rotational angular velocity.
Acknowledgements
BW and AC acknowledge the support received by the British Particle Physics and Astronomy Research Council (PPA/G/S/1999/00127) and the Spanish DGI (AYA2000-2559), respectively.