tides in asynchronous binary systems

Astronomy & Astrophysics manuscript no. paper˙v3 c! ESO 2006September 26, 2006

Tides in asynchronous binary systems

O. Toledano1, E. Moreno2, G. Koenigsberger1 , R. Detmers3, and N. Langer3

1 Centro de Ciencias Fısicas UNAM, Apdo. Postal 48-3, Cuernavaca, Mor. 62251e-mail: [email protected]; [email protected]

2 Instituto de Astronomıa, UNAM, Apdo. Postal 70-264, Mexico D.F. 04510e-mail: [email protected]

3 Sterrenkundig Instituut, Universiteit Utrecht, Postbus 80.000, Utrecht, The Netherlandse-mail: [email protected]; [email protected]

Received :

ABSTRACT

Context. Stellar oscillations are excited in non-synchronously rotating stars in binary systems due to the tidal forces. Tangential components ofthe tides can drive a shear flow which behaves as a di!erentially forced rotating structure in a stratified outer medium.Aims. The aims of this paper are to show that our single-layer approximation for the calculation of the forced oscillations yields results that areconsistent with the predictions for the synchronization timescales in circular orbits, !sync "a6, thus providing a simplified means of computingthe energy dissipation rates, E. Furthermore, by calibrating our model results to fit the relationship between synchronization timescales andorbital separation, we are able to constrain the value of the kinematical viscosity parameter, ".Methods. We compute the values of E for a set of 5 M#+4 M# model binary systems with di!erent orbital separations, a, and use these toestimate the synchronization timescales.Results. The resulting !synch vs. a relation is comparable to that of Zahn (1977) for convective envelopes, providing a calibration method forthe values of ". For the 4+5 M# binary modeled in this paper, " is in the range 0.0015 – 0.0043 for orbital periods in the range 2.5 – 25 d.In addition, E is found to decrease by "2 orders of magnitude as synchronization is approached, implying that binary systems may approachsynchronization relatively quickly but that it takes a much longer timescale to actually attain this condition.Conclusions. The relevance of these results is threefold: 1) our model allows an estimate for the numerical value of " under arbitrary conditionsin the binary system; 2) it can be used to calculate the energy dissipation rates throughout the orbital cycle for any value of eccentricity andstellar rotational velocity; and 3) it provides values of the tangential component of the velocity perturbation at any time throughout the orbit andpredicts the location on the stellar surface where the largest shear instabilities may be occurring. We suggest that one of the possible implicationof the asymmetric distribution of E over the stellar surface is the generation of localized regions of enhanced surface activity.

Key words. interacting binaries – oscillations — stellar rotation —

1. IntroductionA binary system is said to be in equilibrium when the orbit iscircular, the stellar rotation period and the orbital period aresynchronized and the spin axes of both stars are aligned andperpendicular to the orbital plane. In any other case, oscilla-tions are excited and the tidal deformations which are time-dependent become inevitably dissipative. We will focus on thesituation in which the stellar rotation period di!ers from theorbital period, that is, a non-synchronously rotating system. Todescribe the degree of non-synchronicity, we define # = $ 0/",where $0 and " are the rotational and orbital angular veloci-ties, respectively.

As in the case of the terrestrial tides, the resulting asym-metrical shape of the star is due to a second order gravitationale!ect, the di!erential force. Thus, the rising tidal bulges, oppo-

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site to each other and comparable in size, are associated withthe so called, equilibrium tide. Additionally, a set of surfaceoscillations commonly known as the dynamical tide, character-ized by a wide variety of harmonic frequencies, is established.This oscillatory response of the star depends on # and on thestellar and orbital parameters. In addition, it depends on the ca-pability of the stellar material to transfer angular momentumand energy between its di!erent layers, which is representedby the viscosity, ".

The tidally induced oscillations have both radial and az-imuthal components. If we focus on the azimuthal componentand keep in mind that the tidal forces decrease as r3, with r thedistance between a mass element and the center of the perturb-ing star, it is evident that since an external layer of a perturbedstar is subjected to a larger force than a layer below it, a di!er-entially rotating structure will be produced in any # !1 binarysystem. If " !0, the interaction between the di!erent layers

2 Toledano et al.: Asynchronous binary systems

takes the form of a shearing flow, thereby leading to energydissipation. The viscosity and the azimuthal velocity are thecrucial parameters defining the amount of energy that is dissi-pated, E.

The long term trend of the angular momentum transfer anddissipative e!ects in binary systems is one in which the rotationfrequency becomes synchronous with the orbital frequency.The theoretical framework for the calculation of tidal synchro-nization was developed by Zahn (1966, 1977, 1989), who in-voked the tidal friction as a physical mechanism asociated withtidally excited oscillations which contribute to the angular mo-mentum transfer. In stars with convective envelopes, redistri-bution of angular momentum occurs through dissipation pro-cesses like turbulent viscosity, retarding the equilibrium tide.In stars with radiative envelopes, the mechanism found by Zahn(1975, 1977) is the action of radiative damping taking place onthe dynamical tide. Zahn (1977, 1989) found the characteris-tic timescale for synchronization as a function of a, the majorsemi-axis of the orbit; !sync " a6 for stars with convective en-velopes and !sync " a8.5 for stars with radiative envelopes.

On orbital timescales, the tidal oscillations can produceobservable photometric and absorption line-profile variability(see, for example, Smith 1977; Gies & Kullavanijaya 1988;Willems & Aerts 2002). Moreno & Koenigsberger (1999) pre-sented a simplified model for the calculation of the surface os-cillations of a star driven by the tidal interaction in a binarysystem. The calculation is performed for a thin surface layer ofsmall surface elements distributed along the equatorial belt ofthe star. The solution of the equations of motion for these sur-face elements provides the velocity perturbations in the radialand azimuthal directions as a function of time within the orbitalcycle of the binary. In the non-inertial reference frame used tosolve the equations of motion the model includes the gravita-tional forces of both stars, gas pressure, centrifugal, coriolis,and viscous forces. In the model we compute only the motionof the centero fmass of each surface element, thus ignoringthe details of the inner motions within these elements. In thisapproximation, the buoyancy force is not taken into account.Finally, we note that the results in this paper are obtained usinga polytropic index n=1.5, corresponding to a rigorous convec-tive envelope, instead of n=1 as was used previously (Morenoet al. 2005; Koenigsberger et al. 2006), although the di!erencebetween the results obtained for these two values of n is notsignificant.

Moreno et al. (2005) extended the model to include thecomputation of the photospheric absorption lines that wouldbe produced by the perturbed stellar surface. This ab initio cal-culation of the line profile variability due to the forced tidaloscillations was applied to the case of % Per, showing thatfrom a qualitative standpoint, the predicted line profile vari-ability is comparable to the observations. More recently, themodel was used to analyze the peculiar line-profile variabil-ity in the optical counterpart of the X-ray binary 2S0114+650(Koenigsberger et al. 2006), leading to the conclusion that thetidal perturbations could be the “trigger” for the periodic ejec-tion of wind material at densities that are higher than the av-erage wind density. They hypothesize that these e!ects resultfrom the combined action of the shear energy dissipation, the

radial oscillations and the radiation pressure. The significanceof this result is that, in addition to providing a tool for studyingline profile variability in binary systems, the model providesa means of constraining the value of the viscosity, which isthe parameter involved in the energy dissipation mechanisms.Thus, although the model is designed to study the variability inbinary systems on orbital timescales, it is of interest to explorewhether it yields results that are consistent with the long-termbehavior, specifically, the tidal evolutionary theories. That is,using the value of viscosity and the angular velocity perturba-tions obtained from the calculation, an estimate of the rate ofdissipated mechanical energy, E, can be derived. The timescaleover which E becomes negligible is a measure of the synchro-nization timescale of the binary system. Thus, the results ofthis one-layer calculation of the tidal oscillations may be com-pared with the predictions of tidal synchronization theory suchas those of Zahn (1977, 1989) yielding, on the one hand, a testof theMoreno et al. (2005)model and, on the other, a constrainton the numerical values of the viscosity. These are two of theobjectives of this paper.

The expression for the calculation of the viscous energydissipation rate is derived in Sections 2 and 3; in Section 4 wepresent the results of our model calculations for a 5 M#+ 4 M#binary system at di!erent orbital separations and compare ourpredictions with the synchronization timescales of Zahn; and inSection 5 we discuss the implications of this result and presentthe conclusions.

2. Viscous angular momentum transport in ashearing medium

The approximate treatment of angular momentum transfer be-tween the equatorial surface layer and the rigidly-rotating innerregion of the star is made assuming that the equatorial exter-nal layer behaves as a shearing thin disk, in analogy with thedissipation mechanisms described for accretion disks (see forexample, Frank et al. 1985, Lynden-Bell and Pringle, 1974). Inthe assumed di!erentially rotating medium, tangential stressesbetween adjacent layers are assumed to be the mechanism oftransport of angular momentum. Included in these is the angu-lar momentum transport due to magnetic torques, which hasbeen shown to be the dominant mechanism in recent stellarmodel calculations (Petrovic et al. 2005) where the formula-tion of Spruit (2002) is used to treat the magnetic field. In ourcalculations, the mechanisms by which transfer of angular mo-mentum in the shear flow occurs is englobed in the viscosity.

We adopt the approach for the transport of angular momen-tum that consists of imagining an idealized situation in whichparts of a ring-shaped outer layer slide over a rigidly-rotatinginner region. If friction between adjacent layers is assumed toexist and if the angular velocity increases or decreases out-wards, a net torquewill be exerted by the outer ring on the innerregion. The resulting torques will work towards synchronizingthe outer and inner region, dissipating energy in the process.

For this mechanism to be active, the particles in the externalring have to be able to interact with those in the inner rotatingregion. However, contrary to the case of many accretion diskmodels where the particle velocity is Keplerian, the velocity of

Toledano et al.: Asynchronous binary systems 3

di!erent layers on a stellar surface is determined by the stellarrotation, the tidal forcing and the viscosity of the material.

Let us assume that the viscosity in the gaseous shearingmedium is a turbulent one. Gas elements with small randommotions at a typical turbulent velocity vturb, travel a distance&turb (the characteristic length and velocity scales of the turbu-lence), in all directions before mixing with the surroundings.There exists a typical turbulent viscosity,

" " &turbvturb (1)

However, both &turb and vturb are unknown parameters.A clever solution to this problem was proposed in the early1970’ s by Shakura and Sunyaev (1973), consisting of a sim-ple parametrization of the two turbulence scales. For the lengthscale, they assumed isotropy and characterized it by the typi-cal size of the largest turbulent eddies which cannot exceed thethickness H of the assumed ring &turb < H. For the velocityscale, a more straightforward scaling could be made. That is, ifthe turbulent velocity were supersonic, emerging shock waveswould dissipate the energy and reduce the velocity to that of, orbelow, the sound speed. This leads to the choice of a v turb thatis subsonic, thus vturb < Cs, and the viscosity can be written;

" " 'HCs (2)

with ' < 1. This is the well-known '-prescription of Shakura& Sunyaev (1973).

The physical idea of the viscous torque g is to assume that,if the gas flow is turbulent to some extent, gas particles of ad-jacent layers will be exchanged in the radial direction (Frank1985). Since the two di!erent radial flows of material originat-ing in both layers will have di!erent specific angular momenta,this will cause a net transfer of angular momentum betweenthem.

We can then calculate the magnitude of the viscous torqueon the equatorial stellar region in terms of a kinematic turbu-lent viscosity ", which can be expressed as " = µ/(. Here (is the local density of the gas and µ is the coe#cient of shearviscosity which is related to the component !r) of the viscousstress tensor in cylindrical coordinates (r, ), z) by (Landau andLifshitz 1984):

!r) = µ(1r*ur*)+*u)*r$ u)

r) (3)

where ur, u) are the radial and azimuthal components of thevelocity field. Now, inserting u) = $)r with$) as the rotationalangular velocity of the tidal shear flow,

*u)*r=*(r$))*r

= $) + r*$)*r, (4)

and neglecting the shear e!ects in the radial direction, the vis-cous stress tensor becomes,

!r) = µr*$)*r

(5)

This is a tangential force per unit area, and multiplying!r) by the lever arm r, gives the viscous torque per unit area

exerted by the sliding ring-shaped outer layer on the rigidly-rotating inner region. Integrating over the area of interaction(dA = rd)dz),

g =!

r!r)dA =!µr3*$)*r

d)dz (6)

= "

2+!

0

r3*$)*r

d)H/2!

$H/2

(dz (7)

= "$

2+!

0

r3*$)*r

d) (8)

where we have used that the density per unit area on the equa-torial plane of the equatorial stellar region is

$ =

H/2!

$H/2

(dz (9)

with H the width of the equatorial stellar region. With $ = (H,the torque can also be expressed as

g = "(H2+!

0

r3*$)*r

d) (10)

According to this model, no net angular momentum istransported unless there is shearing di!erential rotation; that is,the condition for net angular momentum transport is, *$)/*r !0.

3. The energy dissipation caused by thedifferential viscous torque

A natural consequence of the shearing di!erential rotation isthe dissipation of energy, E. Let us consider the energy dissi-pation caused by the viscous torque acting on the stellar equa-torial external layer between r and r + %r. The net di!erentialviscous torque on this layer is

g(r + %r) $ g(r) " *g*r%r (11)

Since the torque is acting in the direction of the angularvelocity $), the rate of work exerted onto the external layer is;

$)*g*r%r =

*

*r(g$))%r $ g

*$)*r%r (12)

where the first term of the right-hand side represents the rate oftransported mechanical energy through the gas by the torque,and the term $g(*$)/*r)%r, the corresponding local rate ofloss of mechanical energy converted into heat in the gas. Weare interested in the dissipation caused by the viscous torqueswithin the gas at a rate g(*$)/*r)%r on the external layer ofwidth %r.


The local dissipation rate on an equatorial surface elementwith an azimuthal width %) is

Eelem = g*$)*r%r = "(Hr3

"*$)*r

#2%r%) (13)

= "(

"r*$)*r

#2r%)%rH (14)

Since we require an estimate of the dissipation rate on theentire surface of the star, the previous description must be ex-tended to include the dissipation produced at polar angles (,).An approximate generalization can be performed consideringthat the same dissipation mecanisms apply to a series of ringsof thickness r%,, situated across the star from the north poleto the south pole, and the elements in these rings with an az-imuthal angle ) have the same angular velocity as that of theequatorial element with the same azimuth ). This approxima-tion was used by Moreno et al. (2005) for the line-profile cal-culation.

From Eq. (14), the local dissipation rate on any surface el-ement is

Eelem = "("r*$)*r

sin,#2r(sin,)%)%r(r%,) (15)

with $) the angular velocity of the equatorial surface elementin the meridian of the element. Integrating in polar angle, weobtain the dissipation rate on a meridional shell,

Emerid.shell = "("r*$)*r

#2r2%)%r

+!

0

sin3,d, (16)

=43"(

"r*$)*r

#2r2%)%r (17)

To obtain the total dissipation rate E, this equation must beintegrated over the azimuthal coordinate ). We use the fol-lowing approximation: (*$)/*r) " (%$)/%r)i, where %$) "%V()i)/ri, with ri the radial position of the i-th equatorial sur-face element, and %V()i) the di!erence of azimuthal velocityof the element and the inner stellar region. Then, with (%r) i theradial thickness of the element, the total energy dissipation canbe obtained as a sum over the n equatorial surface elements,

E " 43"(

n$

i

"r2%$)%r

#2

i%)i(%r)i (18)

" 43"(

n$

i

"rili(%r)i

(%V()i))2#

(19)

where additionally, ri%)i is simplified by the factor li, whichrepresents the azimuthal length of the element.

4. Synchronization timescalesFor a super-synchronously rotating star, the transfer of angu-lar momentum tends to brake the outer stellar layers, causingthe total energy of the system to gradually decline. This, inturn, drives the system towards synchronization and, therefore,|%V()i)|% 0.

Zahn (1977, 1989) found two di!erent relations betweenthe synchronization timescale and the orbital separation in bi-nary systems, depending on whether the star’s outer envelope isradiative, !Zahn " (a/R1)8.5, or convective, !Zahn " (a/R1)6. Heproposed that for stars with convective envelopes, the most e#-cient synchronizing mechanism is due to turbulent viscosity. Inwhat follows, we will concentrate on this formulation, a choicethat is motivated a posteriori by the results we derive and bythe possibility that, even in stars with radiative envelopes, tur-bulence is induced by the di!erential rotation.

Zahn (1977) gives in his Eq. 6.1 an approximate character-istic synchronization timescale

!Zahn " q$2(a/R1)6 " 104((1 + q)/2q)2P4 years (20)

where q = M2/M1 is the mass ratio of the secondary to theprimary star and a is the orbital separation in solar units, R1is the equilibrium radius of the star in solar units and P is theorbital period in days.

Adopting our one-layer model for calculating the tidal os-cillations, and assuming that all of the energy dissipation asso-ciated with the shearing forces occurs on this surface layer, wecan estimate the timescale required for the rotation period ofthe star to be synchronized with the orbital period of its com-panion as follows. If Einitrot and E

syncrot are the present and synchro-

nization stellar rotational energies, computed with the momentof inertia of a rigid sphere and the azimuthal velocity at thestellar equator, Erot "(1/2)I($)2 " (1/5)M1 V2rot, then a roughestimate of the timescale needed to synchronize the system is

!sync "15M1(Vinit

rot )2 $ 15M1(Vsync

rot )2

43"(%ni

&rili(%r)i (%V()i))

2' (21)

4.1. Comparison with Zahn’s synchronizationtimescales

In order to compare the above synchronization rates with thosethat are inferred from Zahn’s (a/R1)6 prediction, our code wasused to compute values of %V() i) and ri for a set of binarymodels with di!erent orbital periods.

The test binary system that was selected consists of a MainSequence intermediate-mass system with M1 =5 M#, M2 =4M# and R1 =3.2 R# in a circular orbit. Observational evidence(Tassoul 2000; Pan 1997; Mathews & Mathieu 1992; Witte& Savonije 1999) provides constraints for the circularizationtimescales of such systems, and !sync is found to be shorter than!circ (Claret & Cunha 1997; Zahn 1977; Hut 1981).

We chose to fix the value of # = $0/" =2, which meansthat for decreasing orbital separations, the rotational angularvelocity, $0, increases proportionately with the increasing or-bital angular velocity". Although this might seem like a some-what arbitrary choice, it is important to note that it is # rather


Fig. 1. Top: The di!erence of azimuthal velocity of each surface ele-ment and the rigidly rotating inner stellar body, %V()) along the equa-torial region for a Porb =5 d binary system. The angle ) is measuredwith respect to the line connecting the two stars. Bottom: The com-ponent of the velocity in the radial direction of each surface elementalong the equatorial belt for the same model. The star is rotating attwice the orbital frequency.

than Vrot that plays a fundamental role in defining the frequen-cies and amplitudes of the oscillations. An intuitive way thiscan be understood is to consider two binary systems with dif-ferent $0, but both with # =1. Because # = 1 corresponds tothe equilibrium condition, no oscillations are present, regard-less of how much larger one of the values of $0 may be withrespect to the other one.

Figure 1 illustrates the example of %V()) and V r along theequatorial belt for the Porb =5 day binary system. The abscissacorresponds to the location of the surface element along theequator with respect to the line connecting the two stars. Thelargest di!erences between the surface azimuthal velocity andthe inner rigid-body rotating region are in general associatedwith the tidal bulges () "0 and 180&) and the locations per-pendicular to the axis of the orbit, and it is at these locationswhere the largest energy dissipation occurs. Note that the tan-gential component is "2 orders of magnitude larger than theradial component, making it the dominant source of energy dis-sipation.1 When the orbital separation decreases, the azimuthalvariation is no longer smooth, with shorter scale oscillationsappearing, as illustrated in Figure 2 where we plot %V()) forthe Porb =2.5 day binary system. Because we are consideringcircular orbits, %V()) shown in Figures 1 and 2 do not changesignificantly as a function of orbital phase and it is possible toassign a single %V()) curve to each of the models. This would

1 It is also the dominant source of photospheric absorption line pro-file variability as shown by Moreno et al. 2005.

Fig. 2. The azimuthal velocity %V()) along the equatorial region ofsurface elements with angle ) at a given time, for the Porb =2.5 daybinary system and with -R=0.01 R1 and " =0.0043 R2# d$1.

not be the case if the orbit were eccentric, where strong changesoccur between periastron and apastron phases.

The value of the external oscillating layer’s density, (, wasobtained by computing stellar structure models using the codedescribed in Yoon & Langer (2005). The value ( =10 $7 g cm$3that we adopted corresponds to the average density in the ex-ternal 73 layers of the theoretical 5 M# star. These 73 layerscomprise approximately -R/R1 =0.01, which is the thicknessof the outer layer that we adopted for the oscillations calcula-tion.

The remaining input parameter that needs to be defined forthe oscillations code as well as for computing the synchroniza-tion times using Eq. (21) is the viscosity, ". Our practice (seeMoreno et al. 2005) has been to use the smallest value of " al-lowed by the code. That is, when the oscillation amplitudes arevery large, the surface elements may either overlap with or be-come detached from their neighboring elements, at which timethe computation can no longer proceed. This is prevented byassigning a su#ciently large value to the viscosity parameter.Our initial set of calculations were performedwith " =0.005R2#day$1, which is slightly larger than the minimum " allowed forthe Porb =2.5 day binary system. The results yielded a a least-means-square fit with !sync '(a/R1)5.7, which is similar to thatof !Zahn ' (a/R1)6, but displaced vertically on the log! sync vsa/R# plane. Thus, the next step consisted in adjusting the valuesof " to make !sync coincide with !Zahn. The resulting values of" and energy dissipation rates are listed in the last two columnsof Table 1, and the synchronization timescales are plotted inFigure 3. Note that the 0.0015–0.0043R2#/d range in " suggeststhat part of the turbulent viscosity is itself produced through the


Fig. 3. Synchronization timescales (squares) calculated from our tidaloscillation model for a fixed # =2, and -R=0.01 R1, and for values of" as listed in Table 1 compared with Zahn’s (1977) relation for starswith convective envelopes (continuous line).

tidal perturbations processes, since the closest binaries requirethe largest values of ".

We thus find that the shear energy dissipation due to thetangential velocity components of the surface layer used in ourapproximate treatment of the tidal interaction leads to synchro-nization timescales that are consistent with those predicted byZahn’s treatment of the star’s entire structure for the case inwhich the outer envelope is convective.

Table 1. Binary system models for the !sync vs. a calculation.

Porb a Vorb Vrot " E(d) (R#) (km s$1) (km s$1) (R2/d) erg s$12.5 16.11 326.26 130.00 0.0043 1.38'10345.0 25.58 258.96 64.80 0.0015 2.59'10327.5 33.52 226.22 43.20 0.0024 2.20'103110.0 40.60 205.53 32.40 0.0028 3.99'103012.5 47.12 190.80 25.90 0.0030 1.06'103015.0 53.21 179.55 21.60 0.0031 3.45'102917.5 58.97 170.55 18.50 0.0031 1.39'102920.0 64.46 163.13 16.20 0.0031 6.16'102822.5 69.72 156.85 14.40 0.0031 3.02'102825.0 74.80 151.44 13.00 0.0031 1.60'1028

4.2. Dependence on " and -R/R1The viscosity and the thickness of the oscillating layer are freeparameters that are not a priori constrained. We find that for agiven orbital separation, the energy dissipation rates may span a

Fig. 4. Energy dissipation rates calculated for di!erent viscosities "with a fixed layer thickness -R=0.01 R1 (left column) and for a chang-ing layer thickness with a fixed viscosity " =0.025 R2# d$1 (right col-umn). Three binary systems with di!erent orbital periods are illus-trated, from top to bottom: Porb =2.5, 10 and 25 d.

range of up to 2 orders of magnitude, depending on the combi-nation of " and -R/R1. In Figure 4 we illustrate the dependenceof the energy dissipation rate on viscosity (left panels) and onthe thickness of the layer (right panels). For Porb = 2.5 days,the energy dissipation rates decrease with increasing viscosity.Since the explicit dependence of E on the viscosity is linear,(see equation 17) the energy dissipation would be expected toincrease for increasing viscosity. On the other hand, becausethere is also a strong dependence of the dissipation rates on theamplitudes of the oscillations, and larger viscosities tend to re-duce the amplitude of oscillation, the net result is a decrease indissipation rates for larger viscosities. We thus see that for theclosest binary systems, it is this decrease in oscillation ampli-tudes with increasing viscosities that dominates the behavior ofE.

For the longer orbital periods (Porb = 10 days), the inter-play between the e!ect of increasing viscosity and decreasingoscillation amplitudes becomes more clear. In the middle panelof Figure 4 we see how E first increases with increasing " andthen decreases as the oscillation amplitudes rapidly decline,due to the continued increase in ".


The thickness of the layer also has a strong influence on thevalue of E. In the binary with the shortest orbital period, E in-creases systematically by up to 2 orders of magnitude when in-creasing -R from R1/200 to R1/50. For longer period binaries,however (middle and bottom panels, right), there is a criticalpoint at which E reaches a maximum value and then starts todecrease with increasing -R/R1. For the Porb = 10 days systemthis critical value is -Rc "1/75 R1 while for the Porb = 25 dayscase the turn around point appears at -Rc "1/100 R1. Hence,the dependence of dissipation on the thickness of the layer isapproximately linear up to the thickness -Rc/R1, with E in-creasing monotonically.However, once the layer is thicker than-Rc/R1, the surface layer has more inertia and its azimuthal ve-locity amplitudes become smaller and, once again, it is this ef-fect that dominates the trend in E.

5. Discussion

5.1. The kinematical viscosityThe energy dissipation rates computed from the results of theone-layer stellar oscillation model for binary systems lead tothe possibility of constraining the value of the viscosity ". Thiscan be performed by comparing the value of the synchroniza-tion timescale for a given binary system as obtained from ourmodel !sync, with the synchronization timescale curve predictedby Zahn’s relation for the same system, !Zahn. The di!erencebetween !sync and !Zahn, if attributed to viscosity alone, allowsits value to be determined. For the system parameters listed inTable 1 we derive " in the range 0.0015–0.0043R2# d$1 =0.84–2.41'1014 cm2 s$1. If we apply the analogy with accretion diskstudies where the viscosity is incorporated in the '$parameter,an approximate relationship between kinematical viscosity andthe Shakura & Sunyaev ' value is

" " ' kTµmH

1$

(22)

from where

' " 0.0981 (%V)/km/s)µ("/R2#/day)

(T/(10000&K)(R1/R#)(23)

Adopting the results illustrated in Figure 2 for the maxi-mum azimutal velocity, %V()), assuming a mean molecularweight for a fully ionized gas, µ =0.62, T"10000 &K, and using" =0.005 R2#d$1, the above equation yields values of ' (0.002.

It is interesting also to compare this result with the valuesof the corresponding sum of di!usion coe#cients for rotationalmixing that are computed by the Binary Evolutionary Code(BEC; Petrovic et al. 2005, and references therein). In order tomake this comparison, BEC was run for a Porb = 5 d binary sys-tem with masses 5+4 M#. The computation was stopped afteran evolutionary time of "106 years in order to examine the val-ues of these coe#cients. The -R=0.01 R1 layer that we adoptedfor the oscillations calculations is equivalent to 73 layers of theBEC computation, and the sum of the values of the combineddi!usion coe#cients for rotational mixing 2 from these layers2 see Heger (1998) for a summary and description of the di!usion

coe#cients that are computed

Fig. 5. Energy dissipation rates for di!erent values of #, for a binarysystem with M1 =5 M#, M2 =4 M#, R1 =3.2 R#, Porb =5 d, -R=0.01R1 and " =0.005 R2# d$1.

is 1.65'1014 cm2 s$1 (=0.0030 R2#/d). It is not, however, clearhow the values obtained for these 73 layers should be com-bined in order to compare the result with the " parameter thatis used in the one-layer calculation, and the sum is most likelyan overestimate. On the other hand, this value does not includethe e!ects due to a magnetic field, which would tend to increasethe value of the di!usion coe#cient for each layer.

5.2. Evolution of energy dissipation ratesThe energy dissipation rate, which is extracted from the rota-tional energy Erot of the star through the action of de-spinning,is calculated from the di!erence between its initial rotationalenergy and the energy at the time of synchronization dividedby the synchronization timescale. Thus, this represents an av-erage of the energy dissipation rate over the time it takes thestar to reach synchronization. However, the rate of dissipationis expected to be a function of time, gradually vanishing asthe system approaches synchronization. Indeed, Figure 5 illus-trates the dependence on # of the energy dissipation rate dueto azimuthal motions, showing that as # %1, E %0. Figure5 corresponds to the model binary system with Porb =5 days," =0.005 R2# day$1, and other parameters as used for Table 1.

Though we have not addressed the case of excentric bi-nary systems in this paper, it is important to note that due tothe dependence of angular velocity with orbital separation, #is a function of orbital phase. This e!ect, combined with thevariation in orbital separation, leads to strongly varying energydissipation rates over the orbital cycle. An exhaustive studyof the available parameter space is beyond the scope of thepresent paper, but will be the objective of a forthcoming in-vestigation. It is, however, worth noting that within the frame-work of Zahn’s (1977) theory, the synchronization timescalesare several orders of magnitude smaller than the circulariza-tion timescales. Hence, all binary systems should evolve topseudo-synchronization (i.e., average orbital angular velocity=rotational angular velocity) long before the orbit is circular-


ized. Thus, large numbers of asynchronous binary systems incircular orbits are not expected to exist for stars on the MainSequence.

5.3. Non-synchronous rotation and surface activity

An interesting consequence suggested by Figure 5 is that non-synchonous binary systems will tend rapidly towards values of# near unity, but will live for a relatively longer time near theequilibrium value, but without having achieved it. Given thevery large uncertainties in the observable parameters, primarilyVrot sin i and R1, it is not easy to establish whether a particularbinary system has actually achieved equilibrium. Thus, theremay be a large number of non-synchronously rotating binarysystems in which observable e!ects due to the tidal oscilla-tions may be present. In particular, the peculiar X-ray emissionthat is observed in a small number of B-star binary systemsin which there is evidence for non-synchronous rotation (Haroet al. 2003) may be a manifestation of tidally-induced activity.Although the detailed mechanism for converting tidal energyinto X-ray emission is not known, we have previously specu-lated (Moreno et al. 2005; Koenigsberger et al. 2006) that theshear produced by the relative azimuthal motions of the outerstellar layers may lead to magnetic field generation near thestellar surface as well as mass ejection episodes. Thus, the X-ray emission could be associated with these processes. The pos-sible connection between surface activity and tidal interactionshas already been investigated in connection with the distribu-tion of star spots in RS CVn-type binary systems (Holtzwarth& Schussler (2002).

Other observational manifestations of this activity may be“turbulence” above the expected value for the e!ective temper-ature of the star, and photometric and line profile variability.For example, the theoretical absorption line profiles that werecomputed for the optical counterpart of the X-ray binary sys-tem 2S0114+650 required a “macroturbulence” parameter of37.5 km s$1 to match the observations (Koenigsberger et al.2006), significantly larger than the thermal speeds expected ina B1-supergiant star. However, the problem of excessively large“macroturbulent” velocities is widely encountered amongmas-sive stars and, to our knowledge, there has been no systematicstudy made to determine whether the problem is more severein binary stars than in single stars.

6. Conclusions

We have shown that a single-layer approximation for the cal-culation of tidal oscillations yields results that are consistentwith the predictions of Zahn’s (1966, 1977, 1989) theory forthe synchronization timescales in circular orbits, ! sync " a6,thus providing a simplified means of computing the energy dis-sipation rates, E in binary systems. Furthermore, by calibrat-ing our model results to fit Zahn’s relationship, we are able toconstrain the value of the kinematical viscosity parameter, ".For the binary system models considered in this investigation," "0.0015–0.0043 R2/day. This is significantly larger than val-ues typically obtained from detailed computations of the in-

terior structure of stellar convective layers, in the absence ofmagnetic fields.

Our code makes no assumption regarding the energy dissi-pation mechanism except that the dissipation processes can bedescribed through the parameter ". We show in this paper thatif we assume that the energy dissipation mechanism is relatedto turbulent viscosity, we arrive at Equation 19, which allowsus to compute energy dissipation rates using the output of thecode, and we show that the derived energy dissipation rates areconsistent with those predicted by theoretical models of starswith convective envelopes. However, the code includes no apriori assumptions regarding the energy transport mechanism.In order to compare the results of the code with the theory forstars having outer radiative envelopes, the e!ects due to radia-tive damping need to be incorporated in the derivation of thesynchronization timescale, which goes beyond the scope of thepresent investigation.

In summary, however,we have shown that theMoreno et al.(2005) one-layer model yields results that are consistent withother theoretical calculations of the tidal interactions in binarysystems. It has the advantage that it can be used to computethe tidal oscillations for arbitrary stellar rotational velocitiesand orbital excentricities, allowing an estimate for the numer-ical value of " under general conditions in binary systems. Inaddition, the code provides values of the radial and tangentialcomponents of the oscillation velocities at any time throughoutthe orbit, as well as the location on the stellar surface where thelargest shear instabilities may be occurring. This allows a com-parison with observational quantities, such as absorption lineprofiles and diagnostics of surface activity and should allow amethodical exploration of the impact that tidal oscillations mayhave on the surface properties of stars in non-synchronously ro-tating binary systems.

7. AcknowledgementsGK thanks S)ren Meibom and Daniel Proga for help-ful discussions. Support from CONACYT grant 36569 andUNAM/PAPIIT IN119205 are acknowledged.

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