A&A 384, 441-451 (2002)
DOI: 10.1051/0004-6361:20020021
B. Willems1 - C. Aerts2
1 - Department of Physics and Astronomy, The Open University,
Walton Hall, Milton Keynes, MK7 6AA, UK
2 -
Instituut voor Sterrenkunde, Katholieke Universiteit Leuven,
Celestijnenlaan 200B, 3001 Leuven, Belgium
Received 25 July 2001 / Accepted 4 January 2002
Abstract
A theoretical framework for the determination of tidally
induced radial-velocity variations in a component of a close binary
is presented. Both the free and the forced oscillations of the
component are treated as linear, isentropic perturbations of a
spherically symmetric star. Resonances between dynamic
tides and free oscillation modes are taken into account by means of
the formalism developed by Smeyers et al. (1998). The amplitude of the
tidally induced radial-velocity variations seen by the observer
depends on the orbital eccentricity and on the orbital inclination.
The amplitude increases with increasing orbital eccentricity and is
most sensitive to the value of the orbital inclination when
.
In the case of a
ZAMS
star with a
compact companion, it is shown that
resonant dynamic tides can lead to radial-velocity variations with
amplitudes large enough to be detected in observations. The shape of
the tidally induced radial-velocity curves varies from very irregular for
orbital periods away from any resonances with free oscillation modes
to sinusoidal for orbital periods close to a resonance with
a free oscillation mode. Our investigation is concluded with an
application to the slowly pulsating B star HD177863 showing the
possibility of resonant excitation of a high-order second-degree
g+-mode in this star.
Key words: binaries: close - stars: oscillations - methods: analytical - stars: individual: HD177863
In close binary systems of stars, each component is subject to the time-dependent tidal force exerted by its companion. A commonly used approach for the study of tidal effects in close binaries is based on the expansion of the tide-generating potential in Fourier series in terms of multiples of the companion's mean motion. Through these expansions, the tidal action of the companion induces an infinite number of partial dynamic tides in the star, each with its own forcing angular frequency.
In binaries with shorter orbital periods, the forcing angular frequencies may be close to the eigenfrequencies of the free oscillation modes g+ of the component stars (Cowling 1941). These proximities lead to resonances which enhance the tidal motions of the mass elements and can have significant consequences for the observational properties of the binary.
Zahn (1970) studied resonances of dynamic tides with low-frequency g+-modes in non-rotating stars consisting of a convective core and a radiative envelope. For his purpose, he used an asymptotic representation for the eigenfunctions of the g+-modes which he established neglecting the perturbation of the gravitational field due to the star's tidal distortion. Zahn's treatment was later generalised by Rocca (1987) to include the effects of the Coriolis force in slowly rotating binary components.
The luminosity variations associated with tidally excited oscillation modes in close binaries were investigated by Kumar et al. (1995). The authors proposed a new method for the determination of the orbital inclination based on fitting the shape of theoretically derived light curves to the shape of observationally determined light curves. They applied their results to various polytropic stellar models and to the binary pulsars PSRJ0045-7319 and PSRB1259-63.
More recently, Smeyers et al. (1998) (hereafter referred to as Paper I) derived semi-analytical solutions for the components of a resonant dynamic tide by means of a two-time variable expansion procedure in which both the free and the forced oscillations of the star were treated as linear, isentropic oscillations of a spherically symmetric star. The authors concluded that, at the lowest-order of approximation, the oscillation mode involved in the resonance is excited with the forcing frequency of the resonant dynamic tide. Their investigation was subsequently extended by Willems et al. (2001) to take into account the effects of radiative damping in the nonadiabatic surface layers of a star.
The duration of a resonance is determined by the combined effect of stellar and orbital evolution. Witte & Savonije (1999, 2001) have shown that when both effects are taken into account, a dynamic tide can easily become locked in a resonance for a prolonged period of time. Such a long-term resonance can have a significant influence on the secular evolution of the orbital elements and increases the probability of detecting the tidally induced oscillations in the star.
From an observational point of view, firm evidence of tidally excited
oscillations is still scarce, possibly due to the absence of any
systematic observational study on the nature of the pulsations
discovered in close binaries. Studies of individual stars are reported
in the literature, e.g. Sco (Fitch 1967), 14AurA
(Fitch & Wisniewski 1979), Spica (Smith 1985a,b), 16Lac
(Chapellier et al. 1995), V539Arae (Clausen 1996), among others, but
the resonant nature of the modes was hardly ever proven. An
introduction to a systematic observational study in the case of
early-type binaries is given by Aerts et al. (1998). Their initiative
resulted from a more general long-term systematic study of
line-profile variations in
Cep stars and in slowly pulsating
B stars, which led to the assessment that many of these types of
pulsators belong to a close binary. Further evidence for the presence
of B-type g-mode pulsators in close binaries was found by
De Cat et al. (2000). An additional systematic observational project in
this respect is the SEarch for FOrced Nonradial Oscillations (SEFONO)
by Harmanec et al. (1997).
In this investigation, our aim is to determine the radial-velocity variations associated with resonantly excited oscillation modes in binaries with short orbital periods. To this end, we use semi-analytical solutions for the components of a resonant dynamic tide derived by Smeyers et al. (1998). We conclude our investigation by applying our results to the slowly pulsating B star HD177863.
The plan of the paper is as follows. In Sects. 2 and 3, we present the
basic assumptions adopted in our investigation and we decompose the
tide-generating potential in terms of spherical harmonics and in
Fourier series in terms of the companion's mean motion. In Sects. 4
and 5, we derive an expression for the radial-velocity variations
associated with a resonant dynamic tide.
The contributions of non-resonant dynamic tides to the tidally induced
radial-velocity variations are determined in Sect. 6. In Sect. 7, we
determine the total variation of a star's radial
velocity due to the various resonant and non-resonant dynamic tides
and we illustrate the role of the orbital eccentricity and the orbital
inclination. In Sect. 8, we apply our results to a binary consisting
of a
zero-age main sequence star and a
compact companion. In Sect. 9, the possibility of a
resonantly excited oscillation mode in the slowly pulsating B
star HD177863 is investigated. The final section is devoted to
concluding remarks.
Consider a close binary system of stars that are orbiting around each
other in an unvarying Keplerian orbit with semi-major axis a and
orbital eccentricity e. The first star, with mass M1 and
radius R1, is rotating uniformly around an axis perpendicular to
the orbital plane in the sense of the orbital motion. The angular
velocity
is assumed to be low so that the effects of
the Coriolis force and the centrifugal force can be neglected. The
second star, with mass M2, is treated as a point mass.
We start from the right-handed orthogonal frame of reference
introduced in Paper I. The
origin of the frame of reference coincides with the mass centre C1of the uniformly rotating star and the
-plane corresponds to the orbital plane of the binary. The
directions of the
- and the
-axis
coincide with the direction from the star's mass centre to the
periastron in the companion's relative orbit and to the direction
of the star's angular velocity
,
respectively.
As in Paper I, we pass on to an orthogonal frame of reference whose origin
and
-axis correspond to the origin and the
-axis of the frame of
reference
,
but whose
-axis and
-axis are
corotating with the star. We use this frame of reference to describe
the nonradial oscillations induced in the star by the orbiting
companion. With respect to the corotating frame of reference, we
introduce the spherical coordinates
.
The
transformation formulae from the Cartesian coordinates
,
,
to the spherical coordinates r,
,
are given by
In order to study the radial-velocity variations due to the tidal motions of the mass elements located at the star's surface, it is convenient to pass on to a frame of reference whose origin coincides with the mass centre of the tidally distorted star and whose polar axis is directed from the mass centre to the observer. We perform this transformation in two steps.
First, we introduce
an orthogonal frame of reference
whose origin and
-axis coincide with the origin
and the
-axis of the frame of reference
.
The direction of the
-axis corresponds to the direction from the star's mass
centre to the ascending node in the relative orbit of the companion.
Next, we pass on to an orthogonal frame of reference
with the same
origin and the same
-axis as the frame of reference
,
but with its
-axis directed from the star's mass centre to the observer. The
Cartesian coordinates
,
,
are related to the
Cartesian coordinates
,
,
by the transformation formulae
The geometry involved in the introduction of the various frames of reference is illustrated in Fig. 1.
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Figure 1: The geometry involved in the introduction of the various frames of reference. The ascending node and the periastron of the relative orbit of the companion are denoted by the points labeled A and P, respectively. |
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The tidal force exerted by the companion is derived from the
tide-generating potential
,
where
is a small dimensionless parameter defined as
Following Polfliet & Smeyers (1990), we expand the tide-generating potential in
terms of unnormalised spherical harmonics
and
in Fourier series in terms of multiples of the companion's mean motion
,
where
is the orbital period. The expansion can be written as
The Fourier coefficients
decrease with increasing
values of the multiple k of the companion's mean motion. The
decrease is slower for higher orbital eccentricities, so that the
number of terms that has to be taken into account in
Eq. (4) of the tide-generating potential increases with
increasing values of the orbital eccentricity. For
,
the only
non-zero coefficients c2,m,k are those associated with
m=-2,0,2. A more elaborate discussion of these coefficients can be
found in Willems (2000).
For realistic orbital periods, the forcing angular frequencies
may be close to the eigenfrequencies of the star's free
oscillation modes g+. The tidal action exerted by the
companion is then enhanced and the oscillation mode involved is
resonantly excited with the forcing frequency of the
dynamic tide (Paper I). The effects of resonant excitations of modes in close binaries are
particularly important in the cases of resonances of dynamic tides
with g+-modes of a lower radial order.
We consider a single partial dynamic tide generated by the term
Furthermore, let the relative frequency difference
The components of the resonant dynamic tide with respect to the local
coordinate basis
,
,
are obtained
from Solutions (9) by
application of the transformation formulae for contravariant vector
components. The transformation formula takes the form
The tidal motions of the mass elements located at the star's surface
contribute to the radial-velocity variations seen by the observer.
Since the mass elements of the undistorted equilibrium star are assumed
to be at rest with respect to the corotating frame of reference,
the radial component of the velocity field associated with the tidal
motions is given by
In the case of a resonance between a dynamic tide and a free oscillation mode, the tidally induced radial-velocity variations due to the mass elements located at the star's surface take the form
In addition to the partial dynamic tide with forcing angular frequency
,
we also take into account the partial dynamic tide
with forcing angular frequency
,
which is also resonant.
By the use of the symmetry properties
and
,
the global
solution for the variation of the star's radial velocity due to a
resonance of a dynamic tide with a free oscillation mode takes the form
We again consider a single partial dynamic tide generated by a term of
the form
where
and
are solutions of the homogeneous
fourth-order system of linear differential equations
Here
is the radial part of the total perturbation of the
gravitational potential (for a definition see,
e.g., Polfliet & Smeyers 1990), g the local
gravity, c2 the square of the isentropic sound velocity, and
the
square of the Brunt-Väisälä frequency.
The functions
,
,
and
depend on
the values of the azimuthal number m and on the Fourier index k through
the forcing angular frequency
.
The solutions of Eqs. (29)-(31) must
satisfy boundary conditions at the star's centre and at the star's surface.
At r=0, the radial component of the tidal displacement must remain finite.
At r=R1, the Lagrangian perturbation of the pressure must vanish, and the
continuity of the gravitational potential and its gradient requires that
By comparison of Eqs. (28) with Eqs. (9), it
follows that the contribution of a non-resonant dynamic tide to the
tidally induced radial-velocity variations takes the form
In accordance with Eq. (4) of the tide-generating potential, the total variation of the star's radial velocity due to the tidal motions of the mass elements located at its surface, is obtained by adding the contributions stemming from the various resonant and non-resonant partial dynamic tides.
When one restricts the contributions to those resulting from the
tides associated with the second-degree spherical harmonics, the total
variation of the star's radial velocity due to its tidal response can
be written as
The contributions of the various partial dynamic tides to the observed
tidally induced radial-velocity variations depend on the orbital
inclination and on
the orbital eccentricity through the coefficients
a2,m,0(i) and
c2,m,k(e), respectively.
The variation of the product of the coefficients
a2,m,0(i) and
c2,m,k(e) is displayed in Fig. 2 as a function of k,
for the orbital inclinations
and
,
and the
orbital eccentricities e=0.3 and e=0.5. The product
decreases with increasing values of k due to the rapid decrease of the
Fourier coefficients
c2,m,k(e) for higher-order harmonics in
Eq. (4) of the tide-generating potential.
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Figure 2:
Logarithmic representation of the absolute value of the
products
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In the case of the lower orbital inclination ,
the products
of the coefficients
a2,m,0(i) and
c2,m,k(e) are largest in
absolute value for the coefficients associated with the azimuthal
number m=0. The prevalence is more pronounced for the eccentricity
e=0.5 than for the eccentricity e=0.3 due to the increase of
the coefficients
c2,0,k(e) with increasing values of the orbital
eccentricity. For larger values of the orbital inclination, the
products of the coefficients
a2,m,0(i) and
c2,m,k(i)associated with m=-2 rapidly become the
dominant products. In the case of the orbital inclination
,
they are approximately two orders of magnitude larger in
absolute value than the products associated with the azimuthal number
m=0. The products of the coefficients
associated with m=2
are generally two or more orders of magnitude smaller
than the products associated with the azimuthal numbers m=-2 and m=0.
We have applied the expressions derived in the previous sections to a
binary consisting of a
zero-age main sequence star and a
companion which we approximate by a point mass.
We considered orbital periods ranging
from 2 to 6 days and the orbital eccentricities e=0.3 and
e=0.5. The rotation of the star is assumed to be synchronised with
the orbital motion of the companion in the periastron of its relative
orbit. In addition, we have set both the longitude of the periastron
and the time of periastron passage
equal to zero. For
the limb-darkening coefficient we adopted the value u=0.36, which is
appropriate for main-sequence B-type stars.
The observed amplitude of the tidally induced radial-velocity
variations is displayed in Figs. 3 and 4 as a
function of the orbital period. The orbital inclination varies from
in the upper panels of the figures to
in the
lower panels. The numerous peaks appearing at shorter orbital periods
correspond to resonances of dynamic tides with free oscillation
modes. Bearing in mind that, in the perturbation theory used in
Paper I, the relative frequency
difference
is assumed to be of the order of
,
the calculations near the resonances are restricted
to values of
larger than or equal to
in absolute value.
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Figure 3:
The observed amplitude of the tidally induced radial-velocity
variations as a function of the orbital period for the
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Figure 4:
The observed amplitude of the tidally induced radial-velocity
variations as a function of the orbital period for the
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When the orbital plane is observed face-on, i.e. when ,
the
tidally induced radial-velocity variations are determined solely by
the dynamic tides associated with the azimuthal number m=0. For
higher values of the orbital inclination, additional contributions
arise from the dynamic tides associated with the azimuthal numbers
,
although the contributions associated with m=2 are
usually negligible. The amplitude of the tidally induced
radial-velocity variations is largest when the orbital plane is seen
edge-on, i.e. when
.
The resonances are then almost
exclusively due to the dynamic tides associated with the azimuthal
number m=-2.
For close resonances, the amplitudes of the tidally induced radial-velocity variations can be quite large so that they are certainly detectable in observations. They increase with increasing values of the orbital eccentricity due to the smaller periastron distances associated with higher orbital eccentricities. The number of resonances is also larger for the orbital eccentricity e=0.5 than for the orbital eccentricity e=0.3 due to the larger number of terms that has to be taken into account in Eq. (4) of the tide-generating potential. For some cases our results point towards very high amplitudes which are above the typical sound speed in the atmosphere of the star. This would imply shock waves to occur, but our linear theory breaks down at these high velocities. The study of the nonlinear effects is currently beyond the scope of our investigation and the applications are limited to radial-velocity variations that can be well described by the linear theory.
A detailed representation of the effects of resonances in the range of
orbital periods from 3.3 to 3.7 days is given in Fig. 5 for
the orbital eccentricity e=0.5 and the orbital inclinations
and
.
The strongest resonances observed give
rise to radial-velocity variations with amplitudes well above
.
Several orbital periods lead to
resonances with two oscillation modes simultaneously. The peak in the
amplitude near
days, for instance, is caused by
simultaneous resonances with the oscillation modes g12+ and
g16+.
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Figure 5:
Detailed representation of the observed amplitude of the tidally
induced radial-velocity variations of the
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In Fig. 6, the tidally induced radial-velocity variations corresponding to the points labeled (a) - (e) are shown as a function of the orbital phase. The central panel shows that for an exact resonance - point (c), the radial-velocity variations are independent of the orbital phase. The reason is that the tidally induced radial-velocity variations given by Eq. (35) are dominated by the term corresponding to the resonant dynamic tide. Outside the resonance - cases (a) and (e), the radial-velocity variations are markedly less regular functions of the orbital phase since they are now the result of a superposition of purely non-resonant dynamic tides.
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Figure 6:
Observed tidally induced radial-velocity variations of the
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The bright slowly pulsating B star (SPB) HD177863 was discovered to
be a radial-velocity variable by Eggen (1977), while
Waelkens & Rufener (1985) detected intrinsic photometric variability in this
object. Later on, Waelkens (1991) classified the star as an SPB.
De Cat et al. (2000) showed this star to be a single-lined spectroscopic
binary and determined the orbital parameters of the system. They found
an orbital period
days, an
eccentricity e=0.60, a longitude of the periastron
,
and a time of periastron passage
.
The large eccentricity implies that the system is either still very
young or that it has undergone some episodes of mass transfer. We find
no indication of the occurrence of such episodes in the literature,
while the relatively young age is confirmed by the study of
Westin (1985), who finds an age of 64 million years. In what
follows, we therefore assume that the system consists of two
main-sequence stars. From the spectroscopic analysis and the mass
function,
one can furthermore derive that the orbital inclination is between
35 and 90 degrees and that the companion has a mass smaller than
.
We also note that De Cat & Aerts (in preparation) have
found the primary to rotate supersynchronously with a projected
equatorial rotation velocity between 45 and
.
De Cat (2001) also performed a detailed frequency study on both extensive spectroscopic and photometric time series and found two intrinsic frequencies for the star: 0.84059c.d-1 and 0.10108c.d-1. The first one of these differs less than 0.001c.d-1 from 10 times the orbital frequency. This observation, together with the binary configuration, led De Cat (2001) to conclude that one may be dealing with a resonantly excited mode.
In order to investigate if the suggestion by De Cat (2001) is
supported by the theory developed in our paper, we determined the
tidal response of a
stellar model for a fixed orbital
period of 11.9154 days as a function of the rotational angular
velocity
.
The model has an age of 64 million years,
corresponding to a central hydrogen abundance
and a radius
.
The limb-darkening coefficient is assumed to take
the value u=0.36.
The resulting amplitudes of the tidally induced radial-velocity
variations are shown in Fig. 7 for the orbital
inclinations
,
,
and
.
Since the expressions established in our theory are
derived in the linear approximation, we restrict ourselves to the
presentation of radial-velocity variations with amplitudes smaller
than the sound speed in the atmosphere of the star. The range of
rotational angular velocities considered for each inclination is
determined from the range of projected equatorial rotation
velocities. The companion mass for each inclination results from the
mass function of the binary.
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Figure 7:
Observed amplitude of the tidally induced radial-velocity
variations in HD177863, for the orbital inclinations
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A large number of resonances is found for all three orbital
inclinations, among which a whole range are caused by partial dynamic
tides associated with the Fourier index k=10 in
Eq. (4) of the tide-generating potential. The azimuthal
number of the resonant dynamic tide takes the value m=-2 in all
these cases, which is compatible with a preliminary mode
identification performed by De Cat (2001). While the photometric
data did not allow him to identify the mode, he found convincing
evidence from the variations of the moments of the line profiles
(for a definition see, e.g., Aerts et al. 1992) to conclude that the
observed mode is most likely a sectoral
mode. We note
that any resonances caused by partial dynamic tides associated with
the azimuthal number m=0 would have led to large amplitudes of the
tidally induced radial velocity variations for all values of the
rotational angular velocity.
In the case of the orbital inclination
,
we indicated the
positions of the resonances associated with m=-2 and k=10 by
dotted vertical lines in Fig. 7. We find a possible
resonant excitation of
sectoral
modes ranging from the mode g27+ up to the
mode g53+. For the orbital inclinations
and
,
the radial orders of the modes excited by the partial
dynamic tide associated with m=-2 and k=10 are even higher.
Upon close inspection, the pattern of resonances in
Fig. 7 can be seen to repeat itself
at equidistantly spaced values of the rotational angular
velocity .
The reason is easily identified when the forcing
angular frequency
is rewritten as
We conclude that due to the large uncertainty in the rotational angular velocity and the orbital inclination of HD177863, a definitive mode identification is not yet possible at this time. In addition, the amplitude of the tidally induced radial-velocity variations can not be determined in the isentropic approximation so that the inclusion of nonadiabatic effects in our treatment becomes desirable. For higher rotation rates, more resonances can also be expected due to the rotational splitting of the stellar eigenfrequencies by the Coriolis force. We plan to explore the influence of the nonadiabatic effects and the Coriolis force in more detail in subsequent investigations. In particular, we will investigate the behaviour of the predicted observable quantities (photometric variations and line-profile variations) resulting from the candidate resonant modes that we have found in this work with the goal to confront them with the data. Such an iterative procedure can perhaps allow us to come to a definite mode identification and detailed modelling with the goal to constrain the internal structure parameters of the primary.
In this paper, we derived semi-analytical expressions for the tidally induced radial-velocity variations in a uniformly rotating component of a close binary. We neglect the effects of the Coriolis force and the centrifugal force and treat both the free and the forced oscillations of the component as linear, isentropic perturbations of a spherically symmetric star. We take into account the possibility of resonances between dynamic tides and free oscillation modes by means of the perturbation theory developed by Smeyers et al. (1998).
The amplitudes of the tidally induced radial-velocity variations
depend on the orbital eccentricity and on the inclination of the
orbital plane with respect to the plane perpendicular to the line of
sight. They increase with increasing values of the orbital
eccentricity and are largest when the orbital plane is seen edge-on.
The amplitude is most sensitive to the values of the
orbital inclination when
.
A similar
conclusion was reached by Kumar et al. (1995) in an investigation on the
stellar luminosity variations associated with tidally excited oscillation
modes.
From the application to a
zero-age main sequence star, it
follows that the amplitude of the tidally induced radial-velocity
variations seen by an observer is small, except when resonances occur
between dynamic tides and free oscillations modes. The resonances
enhance the tidal motions of the mass elements and lead to
radial-velocity variations with amplitudes that are up to an order of
magnitude larger than those observed outside resonances. Some orbital
periods are seen to give rise to simultaneous resonances with two
oscillation modes.
The shape of the tidally induced radial-velocity curves changes markedly with the proximity of a dynamic tide to a resonance with a free oscillation mode. For close resonances, the tidally induced radial-velocity variations are almost exclusively caused by the tide involved in the resonance and therefore exhibit a very sinusoidal-like behaviour. Outside resonance, the radial-velocity variations are less regular functions of the orbital phase.
For conclusion, we applied our results to the slowly pulsating B star HD177863 and showed the possible resonant excitation of a high-order second-degree sectoral g+-mode in this star.
Acknowledgements
The authors express their sincere thanks to Dr. A. Claret for providing them with theoretical stellar models and to an anonymous referee whose valuable comments led to an improvement of the paper. Bart Willems acknowledges the financial support of PPARC grant PPA/G/S/1999/00127.
The functions
and
defined by
Eqs. (23) and (24) render the influence of the
adopted limb-darkening law on the contributions of the radial and the
transverse component of the tidal displacement to the tidally induced
radial-velocity variations. In the case of the limb-darkening law
given by Eq. (21), the functions
and
take the form
In the particular case where ,
the functions
and
take the form