Interacting Binaries with Eccentric Orbits

# Interacting Binaries with Eccentric Orbits. Secular Orbital Evolution Due To Conservative Mass Transfer

J. F. Sepinsky, B. Willems, V. Kalogera, F. A. Rasio Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208
###### Abstract

We investigate the secular evolution of the orbital semi-major axis and eccentricity due to mass transfer in eccentric binaries, assuming conservation of total system mass and orbital angular momentum. Assuming a delta function mass transfer rate centered at periastron, we find rates of secular change of the orbital semi-major axis and eccentricity which are linearly proportional to the magnitude of the mass transfer rate at periastron. The rates can be positive as well as negative, so that the semi-major axis and eccentricity can increase as well as decrease in time. Adopting a delta-function mass-transfer rate of at periastron yields orbital evolution timescales ranging from a few Myr to a Hubble time or more, depending on the binary mass ratio and orbital eccentricity. Comparison with orbital evolution timescales due to dissipative tides furthermore shows that tides cannot, in all cases, circularize the orbit rapidly enough to justify the often adopted assumption of instantaneous circularization at the onset of mass transfer. The formalism presented can be incorporated in binary evolution and population synthesis codes to create a self-consistent treatment of mass transfer in eccentric binaries.

Celestial mechanics, Stars: Binaries: Close, Stars: Mass Loss
slugcomment: To be submitted to the Astrophysical Journalslugcomment: j-sepinsky, b-willems, vicky@northwestern.edu, and rasio@northwestern.edu

## 1. Introduction

Mass transfer between components of close binaries is a common evolutionary phase for many astrophysically interesting binary systems. Indeed, mass ejection and/or accretion is responsible for many of the most recognizable phenomena associated with close binaries, such as persistent or transient X-ray emission, neutron star spin-up, and orbital contraction or expansion. Theoretical considerations of these and other associated phenomena in the literature probe these systems quite effectively, yet they often do not consider the effects of any eccentricity associated with the binary orbit. This can be of particular importance for binaries containing a neutron star or a black hole, where mass loss and natal kicks occurring during compact object formation may induce a significant eccentricity to the binary (e.g. Hills, 1983; Brandt & Podsiadlowski, 1995; Kalogera, 1996). After the formation of the compact object, tides tend to circularize the orbit on a timescale which strongly depends on the ratio of the radius of the compact object’s companion to the orbital semi-major axis. Because of this, orbits are usually assumed to circularize instantaneously when a binary approaches or begins a mass transfer phase.

Despite our generally well developed understanding of tidal interactions in close binaries, quantitative uncertainties in tidal dissipation mechanisms propagate into the determination of circularization timescales. For example, Meibom & Mathieu (2005) have shown that current theories of tidal circularization cannot explain observed degrees of circularization of solar-type binaries in open clusters. Circularization of high-mass binaries, on the other hand, is currently thought to be driven predominantly by resonances between dynamic tides and free oscillation modes, but initial conditions play an important role and an extensive computational survey of relevant parts of the initial parameter space has yet to be undertaken (Witte & Savonije, 1999, 2001; Willems et al., 2003).

Furthermore, assumptions of instantaneous circularization immediately before or at the onset of mass transfer are in clear contrast with observations of eccentric mass transferring systems. In the most recent catalog of eccentric binaries with known apsidal-motion rates compiled by Petrova & Orlov (1999), 26 out of the 128 listed systems are semi-detached or contact binaries. Among these mass-transferring systems, have measured eccentricities greater than . In addition, many high-mass X-ray binaries are known to have considerable orbital eccentricities (Raguzova & Popov, 2005). While mass transfer in these systems is generally thought to be driven by the stellar wind of a massive O- or B-star, it has been suggested that some of them may also be subjected to atmospheric Roche-lobe overflow at each periastron passage of the massive donor (e.g. Petterson, 1978).

Huang (1956), Kruszewski (1964), and Piotrowski (1964) were the first to study the effects of mass transfer on the orbital elements of eccentric binaries. However, their treatment was restricted to perturbations of the orbital motion caused by the variable component masses. Matese & Whitmire (1983, 1984) extended these early pioneering studies to include the effects of linear momentum transport from one star to the other, as well as any other possible perturbations caused by the mass transfer stream in the system. However, these authors derived the equations governing the motion of the binary components with respect to a reference frame with origin at the mass center of the binary, which is not an inertial frame. Their equations therefore do not account for the accelerations of the binary mass center caused by the mass transfer (see § 3.3).

More recent work on mass transfer in eccentric binaries has mainly focused on smoothed particle hydrodynamics calculations of the mass transfer stream over the course of a few orbits, without any consideration of the long-term evolution of the binary (Layton et al., 1998; Regös et al., 2005).

Hence, there is ample observational and theoretical motivation to revisit the study of eccentric mass-transferring binaries. In this paper, our aim is to derive the equations governing the evolution of the orbital semi-major axis and eccentricity in eccentric mass-transferring binaries, assuming conservation of total system mass and orbital angular momentum. In a subsequent paper, we will incorporate the effects of mass and orbital angular momentum losses from the system.

Our analysis is based on the seminal work of Hadjidemetriou (1969b) who was the first to derive the equations of motion of the components of eccentric mass-transferring binaries while properly accounting for the effects of the variable component masses on the stars’ mutual gravitational attraction, the transport of linear momentum from one star to the other, the accelerations of the binary mass center due to the redistribution of mass in the system, and the perturbations of the orbital motion caused by the mass-transfer stream. While the equations of motion derived by Hadjidemetriou (1969b) are valid for orbits of arbitrary eccentricity, the author restricted the derivation of the equations governing the evolution of the semi-major axis and eccentricity to orbits with small initial eccentricities.

The paper is organized as follows. In § 2 and § 3 we present the basic assumptions relevant to the investigation and derive the equations governing the motion of the components of an eccentric mass-transferring binary under the assumption of conservative mass transfer. The associated equations governing the rates of change of the semi-major axis and the orbital eccentricity are derived in § 4, while numerical results for the timescales of orbital evolution due to mass transfer as a function of the initial binary mass ratio and orbital eccentricity are presented in § 5. For comparison, timescales of orbital evolution due to dissipative tidal interactions between the binary components are presented in § 6. § 7 is devoted to a summary of our main results and a discussion of future work. In the appendices, lastly, we derive an equation for the position of the inner Lagrangian point in eccentric binaries with non-synchronously rotating component stars (Appendix A), and present an alternative derivation for the equations governing the secular evolution of the orbital semi-major axis and eccentricity assuming instantaneous mass transfer between two point masses (Appendix B).

## 2. Basic Assumptions

We consider a binary system consisting of two stars in an eccentric orbit with period , semi-major axis , and eccentricity . We let the component stars rotate with angular velocities and parallel to the orbital angular velocity , and assume the rotation rates to be uniform throughout the stars. We also note that the magnitude of varies periodically in time for eccentric binaries, but its direction remains fixed in space. Because of this, the stars cannot be synchronized with the orbital motion at all times.

At some time , one of the stars is assumed to fill its Roche lobe and begins transferring mass to its companion through the inner Lagrangian point . We assume this point to lie on the line connecting the mass centers of the stars, even though non-synchronous rotation may cause it to oscillate in the direction perpendicular to the orbital plane with an amplitude proportional to the degree of asynchronism (Matese & Whitmire, 1983). Since the donor’s rotation axis is assumed to be parallel to the orbital angular velocity, we can safely assume that the transferred mass remains confined to the orbital plane.

We furthermore assume that all mass lost from the donor is accreted by its companion, and that any orbital angular momentum transported by the transferred mass is immediately returned to the orbit. The mass transfer thus conserves both the total system mass and the orbital angular momentum.

We also neglect any perturbations to the orbital motion other than those due to mass transfer. At the lowest order of approximation, these additional perturbations (e.g., due to tides, magnetic breaking, or gravitational radiation) are decoupled from those due to mass transfer, and can thus simply be added to obtain the total rates of secular change of the orbital elements.

## 3. Equations of Motion

### 3.1. Absolute Motion of the Binary Components

Following Hadjidemetriou (1969b), we derive the equations of motion of the components of an eccentric mass-transferring binary with respect to a right handed inertial frame of reference which has an arbitrary position and orientation in space (see Fig. 1). We let be the mass of star at some time at which mass is transferred from the donor to the accretor, and the mass of the same star at some time , where is a small time interval. With these notations, corresponds to mass loss, and to mass accretion. We furthermore denote the point on the stellar surface at which mass is lost or accreted by . For the donor star, corresponds to the inner Lagrangian point , while for the accretor, can be any point on the star’s equator. For the remainder of the paper, we let correspond to the donor and to the accretor.

Because of the mass loss/gain, the center of mass of star at time is shifted from where it would have been had no mass transfer taken place. To describe this perturbation, we introduce an additional right-handed coordinate frame with a spatial velocity such that its origin follows the unperturbed orbit of star , i.e., the origin of follows the path the center of mass of star would have taken had no mass transfer occurred. Thus, at time , the center of mass of star lies at the origin of , while at time it has a non-zero position vector with respect to . We furthermore let the -axis of the frame point in the direction of the orbital angular momentum vector, and let the frame rotate synchronously with the unperturbed orbital angular velocity of the binary in the absence of mass transfer. The direction and orientation of the -axes are then chosen such that at time the -axis points along the direction from the mass center of star to the mass center of its companion.

To describe the shift in the mass center of star due to the mass loss/gain, we denote the position vector of at times and with respect to the inertial frame by and , respectively. The position vector of the center of mass of star at times and is then given by and , where is the position vector of the center of mass of star at time with respect to . Moreover, we denote by and the vectors from to the position where the point on the stellar surface would be at times and , respectively, had no mass been lost/accreted. The various position vectors at time are related by

 (Mi+δMi)(→R′i+δ→r′i)=Mi→R′i+δMi(→R′i+→r′Ai), (1)

which, at the lowest order of approximation in and , yields

 δ→r′i=δMiMi→r′Ai. (2)

As expected, the displacement of the center of mass of star due to the mass loss/gain is directed along the line connecting the center of mass of the star and the mass ejection/accretion point.

We furthermore denote with the vector from the center of mass of star at time to the position where the point would be at time , had no mass been transferred between the binary components, and with the perturbation of this vector caused by the mass transfer. It then follows that and thus, by definition, . At the lowest order of approximation in and , equation (2) therefore also yields

 δ→ρ′Ai=δMiMi→r′Ai. (3)

The definitions of and the relations between these various position vectors are illustrated schematically in Fig. 1.

Next, we denote the absolute velocity of the center of mass of star with respect to the inertial frame of reference at times and by and , respectively, and the absolute velocity of the ejected/accreted mass element by . The linear momentum of star  at time is then given by

 →Q1=M1→V1, (4)

and the total linear momentum of star 1 and the ejected mass element at time by

 →Q′1=(M1+δM1)→V′1−δM1→WδM1. (5)

Similarly, the total linear momentum of star 2 and the mass element to be accreted at time is given by

 →Q2=M2→V2+δMi→WδMi, (6)

and the linear momentum of star 2 at time by

 →Q′2=(M2+δM2)→V′2. (7)

At time the velocity of the center of mass of star can be written as

 →V′i=→V′Oi+(→Ω′orb+δ→Ω′orb)×δ→r′i, (8)

where is the absolute velocity of the origin of at time , is the orbital angular velocity of the binary at time in the absence of mass transfer, and is the perturbation of the orbital angular velocity at time due to the mass loss/gain of the binary components.

In the limit of small , taking the difference between the linear momenta and , dividing the resulting equation by , and noting that the absolute velocity of the origin of at time is equal to , yields

 Mid→VOidt=→Fi+˙Mi→UδMi. (9)

Here, is the sum of all external forces acting on star , is the mass loss/accretion rate of star , and

 →UδMi=→WδMi−→VOi−→Ωorb×→rAi (10)

is the relative velocity of the ejected/accreted mass element with respect to the ejection/accretion point . In the derivation of equation (10), we have made use of equations (2) and (8) and restricted ourselves to first-order terms in the small quantities and .

The absolute acceleration of the center of mass of star with respect to the inertial frame is given by

 d2→Ridt2=→γOi+→γrel,i+→γcor,i, (11)

where is the acceleration of the origin of with respect to , is the relative acceleration of the center of mass of star with respect to , and is the Coriolis acceleration of the center of mass of star with respect to 111The centrifugal acceleration does not play a role since it is proportional to which vanishes for small .. The expressions for and follow from the observation that , which one obtains by dividing equation (3) by in the limiting case of small . The equation of motion for the mass center of star with respect to the inertial frame then becomes

 Mid2→Ridt2 = →Fi+˙Mi(→UδMi+2→Ωorb×→rAi) (12) + ¨Mi→rAi.

### 3.2. Relative Motion of the Binary Components

We can now obtain the equation describing the relative motion of the accretor (star 2) with respect to the donor (star 1) by taking the difference of the equations of motion of the stars with respect to the inertial frame of reference. For convenience, we first decompose the sum of the external forces acting on each star as

 →Fi=−GM1M2|→r|2→Ri|→Ri|+→fi, (13)

where is the Newtonian constant of gravitation, and the total gravitational force exerted on star by the particles in the mass-transfer stream. It follows that

 d2→rdt2 = −G(M1+M2)|→r|3→r+→f2M2−→f1M1 + ˙M2M2(→vδM2+→Ωorb×→rA2) − ˙M1M1(→vδM1+→Ωorb×→rA1) + ¨M2M2→rA2−¨M1M1→rA1,

where is the position vector of the accretor with respect to the donor, and is the velocity of the ejected/accreted mass element with respect to the mass center of the mass losing/gaining star.

Equation (3.2) can be written in the form of a perturbed two-body problem as

 d2→rdt2=−G(M1+M2)|→r|3→r+S^x+T^y+W^z, (15)

where is a unit vector in the direction of , is a unit vector in the orbital plane perpendicular to in the direction of the orbital motion, and is a unit vector perpendicular to the orbital plane parallel to and in the same direction as . The functions , , and are found by taking the dot product of the perturbing force arising from the mass transfer between the binary components and the unit vector in the , , and directions, respectively. These vector components are

 S = f2,xM2−f1,xM1+˙M2M2(vδM2,x−|→Ωorb||→rA2|sinϕ) (16) − ˙M1M1vδM1,x+¨M2M2|→rA2|cosϕ−¨M1M1|→rA1|, T = f2,yM2−f1,yM1+˙M2M2(vδM2,y+|→Ωorb||→rA2|cosϕ) (17) − ˙M1M1(vδM1,y+|→Ωorb||→rA1|)+¨M2M2|→rA2|sinϕ, W = f2,zM2−f1,zM1, (18)

where is the angle between and the vector from the center of mass of the accretor to the mass accretion point , and the subscripts , , and denote vector components in the , , and directions, respectively. In working out the vector products , we assumed that is located on the equator of the accreting star222For brevity, we refer to the point as lying on the stellar surface. Though, in practice, it can lie at any point near the star where the transferred mass can be considered to be part of the accretor. For instance, if an accretion disk has formed around the accretor, it would be equally valid to write as the point where the transferred mass impacts the outer edge of the accretion disk.. The terms contributing to the perturbed orbital motion can be categorized as follows: (i) term proportional to represent gravitational perturbation on the binary components caused by mass elements in the mass-transfer stream; (ii) terms proportional to represent linear momentum exchange between the mass donor and accretor; and (iii) terms proportional to represent shifts in the position of the mass centers of the mass donor and accretor due to the non-spherical symmetry of the mass loss or gain. In the limiting case where both stars are treated as point masses ( and ), the only non-zero terms in the perturbed equations of motion are those due to gravitational perturbations of the mass transfer stream and the transport of linear momentum.

### 3.3. Comparison with Previous Work

The most recent study on the orbital evolution of eccentric mass-transferring binaries has been presented by Matese & Whitmire (1983, 1984, hereafter MW83 and MW84, respectively). These authors extended the work of Huang (1956), Kruszewski (1964), and Piotrowski (1964) by accounting for the effects of linear momentum transport between the binary components, as well as possible perturbations to the orbital motion caused by the mass transfer stream. However, they also derived the equations describing the motion of the binary components with respect to a frame of reference with origin at the mass center of the binary, which, for mass transferring systems, is not an inertial frame of reference. The equations therefore do not account for the accelerations of the binary mass center caused by the mass transfer. Here, we demonstrate that if the procedure adopted by Matese & Whitmire is developed with respect to an inertial frame of reference that is not connected to the binary, the resulting equations are in agreement with those derived in § 3.2.

The core of Matese & Whitmire’s derivation is presented in Section II of MW83. While the authors choose to adopt a reference frame with origin at the binary mass center early on in the investigation, the choice of the frame does not affect the derivation of the equations of motion up to and including their equation (24). In particular, equation (13) in MW83, which, in our notation, reads

 →pi=Mi˙→Ri−˙Mi→rAi, (19)

is valid with respect to any inertial frame of reference with arbitrary position and orientation in space. The same applies to equation (1) in MW84:

 ˙→pi=−GM1M2→Ri−→R3−i|→Ri−→R3−i|3+→fi+→Ψi. (20)

In these equations, is the linear momentum of star , and is the amount of linear momentum transported by the transferred mass per unit time (see equation (3) of MW84). Substitution of equation (19) into equation (20) then yields

 Mi¨→Ri= − GM3−i→Ri−→R3−i|→Ri−→R3−i|3 (21) + →fiMi+˙MiMi(→vδMi+˙→rAi)+¨MiMi→RAi,

and thus

 d2→rdt2= − G(M1+M2)|→r|3→r+→f2M2−→f1M1 (22) + ˙M2M2(→vδM2+˙→rA2)−˙M1M1(→vδM1+˙→rA1) + ¨M2M2→rA2−¨M1M1→rA1,

where . Setting , this equation is in perfect agreement with equation (3.2) derived in § 3.2.

In MW83 and MW84, the authors incorrectly set and in equation (21), which is valid only when the origin of the frame of reference coincide with the mass center of the binary. For a mass-transferring binary, such a frame is, however, not an inertial frame and can therefore not be used for the derivation of the equations of motion of the binary components. Instead of equation (22), Matese & Whitmire therefore find equations (7)–(8) in MW84, which lack the terms associated with the acceleration of the binary mass center due to the mass transfer.

## 4. Orbital Evolution Equations

### 4.1. Secular Variation of the Orbital Elements

In the classical framework of the theory of osculating elements, the equations governing the rate of change of the orbital semi-major axis and eccentricity due to mass transfer are obtained from the perturbing functions and as (see, e.g., Sterne, 1960; Brouwer & Clemence, 1961; Danby, 1962; Fitzpatrick, 1970)

 dedt=(1−e2)1/2na (24) ×{Ssinν+T[2cosν+e(1+cos2ν)1+ecosν]},

where is the mean motion and the true anomaly. These equations are independent of the perturbing function which solely appears in the equations governing the rates of change of the orbital inclination, the longitude of the ascending node, and the longitude of the periastron.

After substitution of equations (16) and (17) for and into equations (23) and (24), the equations governing the rates of change of the semi-major axis and eccentricity contain periodic as well as secular terms. Here we are mainly interested in the long-term secular evolution of the orbit, and so we remove the periodic terms by averaging the equations over one orbital period:

 ⟨dedt⟩sec≡1Porb∫Porb/2−Porb/2dedtdt. (26)

The integrals in these definitions are most conveniently computed in terms of the true anomaly, . We therefore make a change of variables using

 dt=(1−e2)3/2n(1+ecosν)2dν. (27)

For binaries with eccentric orbits, the resulting integrals can be calculated analytically only for very specific functional prescriptions of the mass-transfer rate (e.g., when is approximated by a Dirac delta function centered on the periastron, see § 5). In general, the integrals must be computed numerically.

### 4.2. Conservation of Orbital Angular Momentum

Since the perturbing functions and depend on the properties of the mass transfer stream, calculation of the rates of secular change of the orbital semi-major axis and eccentricity, in principle, requires the calculation of the trajectories of the particles in the stream (cf. Hadjidemetriou, 1969a). As long as no mass is lost from the system, such a calculation automatically incorporates the conservation of total angular momentum in the system. Special cases of angular momentum conservation can, however, be used to bypass the calculation of detailed particle trajectories. Here, we adopt such a special case and assume that any orbital angular momentum carried by the particles in the mass-transfer stream is always immediately returned to the orbit, so that the orbital angular momentum of the binary is conserved.

The orbital angular momentum of a binary with a semi-major axis and eccentricity is given by

 Jorb=M1M2[Ga(1−e2)M1+M2]1/2, (28)

so that

 ˙JorbJorb=˙M1M1+˙M2M2−12˙M1+˙M2M1+M2+12˙aa−e˙e1−e2, (29)

where a dot indicates the time derivative.

In the case of eccentric orbits, substitution of equations (23) and (24) into equation (29) leads to

 ˙JorbJorb = ˙M1M1+˙M2M2−12˙M1+˙M2M1+M2 (30) + (1−e2)1/2na(1+ecosν)T.

As we shall see in the next section, by setting and and substituting equation (17) for , equation (30) allows us to calculate the -component of the final velocities of the accreting particles as a function of their initial velocities without resorting to the computation of the ballistic trajectories of the mass transfer stream.

In the limiting case of a circular orbit, equation (29) is usually used to derive the rate of change of the semi-major axis of circular binaries under the assumption of conservation of both total mass () and orbital angular momentum ():

The assumption of orbital angular momentum conservation over secular timescales () is a standard assumption in nearly all investigations of conservative mass transfer in binary systems (e.g., Soberman et al., 1997; Pribulla, 1998), which is valid over long timescales provided there is no significant storage of angular momentum in the spins of the components stars, the accretion flow, and/or the accretion disk. In future work, we will investigate the consequences of both mass and orbital angular momentum losses from the binary on the evolution of the orbital elements.

## 5. Orbital Evolution Timescales

In order to assess the timescales of orbital evolution due to mass transfer in eccentric binaries, we observe that, for eccentric binaries, mass transfer is expected to occur first at the periastron of the relative orbit, where the component stars are closest to each other. We therefore explore the order of magnitude of the timescales assuming a delta function mass transfer profile centered at the periastron of the binary orbit

 ˙M1=˙M0δ(ν), (32)

where is the instantaneous mass transfer rate, and is the Dirac delta function.

We calculate the rates of secular change of the orbital semi-major axis and eccentricity from equations (23)–(27), and neglect any gravitational attractions exerted by the particles in the mass-transfer stream on the component stars. Hence, we set

 f1,x = f2,x=0, (33) f1,y = f2,y=0. (34)

Substituting equations (17), (27), and (32) – (34) into equation (30) for then yields a relationship between the initial and final -component of the velocities of the transferred mass and the initial and final positions of the transferred mass given by

 qvδM2,y + vδM1,y=na(1−q)(1+e1−e)1/2−|→Ωorb,P||→rA1,P| (35) − q|→Ωorb,P||→rA2|cosϕP(1−dϕdν∣∣∣ν=0),

where the subscript indicates quantities evaluated at the periastron of the binary orbit, is the binary mass ration, and we have used the relation . Assuming the transferred mass elements are ejected by star 1 at the point with a velocity equal to the star’s rotational velocity at , we write

 vδM1,x=0, (36) vδM1,y=−|→Ωorb,P||→rA1,P|. (37)

Moreover, under the assumption that each periastron passage of the binary components give rise to an extremum of , the derivative is equal to zero in equation (35), so that

 vδM2,y=|→Ωorb,P|[|→rA1,P|(1−q)q+|→rA2|cosϕP]. (38)

For a binary with orbital period , eccentricity , and component masses and , with the distance from star 1 to (See Appendix A), and (the circularization radius around a compact object for these binary parameters; see Frank, King, & Raine (2002)), the accreting matter has a -velocity component of the order of .

After substitution of equation (27) and equations (33)–(35), the integrals in equations (25) and (26) for the rates of secular change of the orbital semi-major axis and eccentricity can be solved analytically to obtain

 ⟨dadt⟩sec = aπ˙M0M11(1−e2)1/2[qe|→rA2|acosϕP (39) + e|→rA1,P|a+(q−1)(1−e2)],
 ⟨dedt⟩sec = (1−e2)1/22π˙M0M1[q|→rA2|acosϕP (40) + |→rA1,P|a+2(1−e)(q−1)].

We note that for a delta-function mass transfer rate given by equation (32) the -component of the velocities and does not enter into the derivation of theses equations due to the term in equations (23) and (24). Furthermore, in the limiting case of a circular orbit, equation (39) reduces to equation (31), provided that in that equation is interpreted as the secular mean mass transfer rate . In Appendix B, we present an alternative derivations to equations (39) and (40) in the limiting case where the stars are treated as point masses.

The rates of secular change of the semi-major axis and orbital eccentricity are thus linearly proportional to the magnitude of the mass transfer rate at periastron. Besides the obvious dependencies on , , , and , the rates also depend on the ratio of the donor’s rotational angular velocity to the orbital angular velocity at periastron through the position of the point, . A fitting formula for the position of the point accurate to better than over a wide range of , , and is given by equation (A15) in Appendix A. While the fitting formula can be used to obtain fully analytical rates of secular change of the semi-major axis and eccentricity, we here use the exact solutions for the position of obtained by numerically solving equation (A13) in Appendix A. For a detailed discussion of the properties of the point in eccentric binaries, we refer the interested reader to Sepinsky et al. (2007).

To explore the effects of mass transfer on the orbital elements of eccentric binaries, we calculate the rates of secular change of the semi-major axis and eccentricity and determine the characteristic timescales and . While the actual timescales are given by the absolute values of and , we here allow the timescales to be negative as well as positive in order to distinguish negative from positive rates of secular change of the orbital elements. We also note that since (see Appendix A), the timescales do not explicitly depend on the orbital semi-major axis except through the ratio of the radius of the accretor to the semi-major axis. For convenience, we therefore assume the accretor to be a compact object with radius . The timescales are found to be insensitive to terms containing in equation (39) and (40). Varying from to changes the timescales by less that 10%. In what follows, we therefore set . An implicit dependence on may then still occur through the amplitude of the mass transfer rate at periastron. Since incorporating such a dependence in the analysis requires detailed modeling of the evolution of the donor star, which is beyond the scope of this investigation, we here restrict ourselves to exploring the timescales of orbital evolution for a constant . The linear dependence of and on in any case allows for any easy rescaling of our results to different mass transfer rates.

In Fig. 2, we show the variations of and as functions of for and . In all cases, the donor is assumed to rotate synchronously with the orbital angular velocity at the periastron, and the accretor is assumed to be a neutron star of mass . The timescales of the secular evolution of the semi-major axis show a strong dependence on , and a milder dependence on , unless . The timescales for the secular evolution of the orbital eccentricity always depend strongly on both and . These timescales can furthermore be positive as well as negative, so that the semi-major axis and eccentricity can increase as well as decrease under the influence of mass transfer at the periastron of the binary orbit.

From Fig. 2, as well as equations (39) and (40), it can be seen that, for a given ratio of the donor’s rotational angular velocity to the orbital angular velocity at periastron, the line dividing positive from negative rates of secular change of the orbital elements is a function of and . This is illustrated further in Fig. 3 where the timescales of orbital evolution are displayed as contour plots in the -plane. The thick black line near the center of the plots marks the transition values of and where the rates of secular change of and transition from being positive (to the left of the thick black line) to negative (to the right of the thick black line). Varying between and changes the position of the transition line by less than in comparison to the case displayed in Figures 2 and 3.

In the limiting case of a circular orbit, the orbit expands when and shrinks when , in agreement with the classical result obtained from equation (31). For non-zero eccentricities, the critical mass ratio separating positive from negative values of decreases with increasing orbital eccentricities. This behavior can be understood by substituting the fitting formula for the position of the point given by equation (A15) in Appendix A into equation (39) and setting . However, we can fit the critical mass ratio separating expanding from shrinking orbits with a simpler formula given by

 qcrit≃1−0.4e+0.18e2. (41)

The critical mass ratio separating positive from negative values of is largely independent of . Proceeding in a similar fashion as for the derivation of equation (41), we derive the critical mass ratio separating increasing from decreasing eccentricities to be approximately given by

 qcrit≃0.76+0.012e. (42)

Last, we note that a more quantitative numerical comparison between the above approximation formulae for and the exact numerical solutions shows that equations (41) and (42) are accurate to better than 1%.

## 6. Tidal Evolution Timescales

A crucial question for assessing the relevance of the work presented here is how the derived orbital evolution timescales compare to the corresponding timescales associated with other orbital evolution mechanisms such as tides. In Fig. 4, we show the secular evolution timescales of the semi-major axis and orbital eccentricity of a mass-transferring binary due to tidal dissipation in the donor star as a function of , for different values of the eccentricity, . The timescales are strong functions of and are determined as in Hurley et al. (2002)333Note that there is a typo in equation (42) of Hurley et al. (2002). The correct equation for for stars with radiative envelopes is (J. Hurley, Private Communication) (see also Zahn, 1977, 1978; Hut, 1981). The radius is determined by assuming the donor is on the zero-age main sequence and that the orbital separation is then obtained by equating the radius of the donor (given by Tout et al., 1996) to the volume-equivalent radius of its Roche lobe at the periastron of the binary orbit (see Sepinsky et al., 2007). As before, we assume the donor rotates synchronously with the orbital motion at periastron and that the accretor is a neutron star.

The timescales of orbital evolution due to tides range from a few Myr to more than a Hubble time, depending on the binary mass ratio and the orbital eccentricity. The discontinuity in the timescales at corresponds to the transition from donor stars with convective envelopes () to donor stars with radiative envelopes () which are subject to different tidal dissipation mechanisms. It follows that tides do not necessarily lead to rapid circularization during the early stages of mass transfer, especially for orbital eccentricities . Furthermore, for the adopted system parameters, the orbital eccentricity always decreases, while the orbital semi-major axis can either increase or decrease. Hence, in some regions of the parameter space, the effects of tides and mass transfer are additive, while in other regions they are competitive. This is illustrated in more detail in figure 5 where we show the orbital evolution timescales due to the combined effect of tides and mass transfer. In the calculations of the timescales, we have assumed that, at the lowest order of approximation, the effects of tides and mass transfer are decoupled. The total rate of change of the orbital elements is then given by the sum of the rate of change of the orbital elements due to tides and mass transfer.

When and , the effects of tides and mass transfer on the orbital semi-major axis are always opposed, with the orbital expansion due to mass transfer dominating the orbital shrinkage due to tides. In the case of the orbital eccentricity, the increase of the eccentricity due to mass transfer dominates the decrease due to tides for mass ratios smaller than some critical mass ratio which depends strongly on the orbital eccentricity. Since the timescales of orbital evolution due to mass transfer are inversely proportional to the magnitude of the mass-transfer rate at periastron, the parameter space where and the extent to which mass transfer dominates increases with the rate of mass transfer at periastron.

In figure 6, we show the total orbital evolution timescale due to the sum of tidal and mass transfer effects as a contour plot in the -plane. The thick black lines indicate the transitions from positive (left of the thick black line) to negative (right of the thick black line) rates of change of the semi-major axis and eccentricity. The white dividing line near corresponds to the transition between tidal dissipation mechanisms in stars with convective envelopes () and stars with radiative envelopes (). It follows that there are large regions of parameter space where the combined effects of mass transfer and tidal evolution do not rapidly circularize the orbit. In particular, for and orbital circularization always takes longer than 10 Gyr, while for to the left of the thick black line the orbital eccentricity grows rather than shrinks. For a given left of the thick black line, the timescales for eccentricity growth increase with increasing though, so that there is no runaway eccentricity growth. Hence, for small , mass transfer at the periastron of eccentric orbits may provide a means for inducing non-negligible eccentricities in low-mass binary or planetary systems. The orbital semi-major axis, on the other hand, always increases when , but can increase as well as decrease when , depending on the binary mass ratio . We recall that both the tidal and mass transfer orbital evolution time scales depend on the ratio of the donor’s rotational angular velocity to the orbital angular velocity and that we have set in all figures shown.

## 7. Concluding Remarks

We developed a formalism to calculate the evolution of the semi-major axis and orbital eccentricity due to mass transfer in eccentric binaries, assuming conservation of total system mass and orbital angular momentum. Adopting a delta-function mass-transfer profile centered at the periastron of the binary orbit yields rates of secular change of the orbital elements that are linearly proportional to the magnitude of the mass-transfer rate at the periastron. For , this yields timescales of orbital evolution ranging from a few Myr to a Hubble time or longer. Depending on the initial binary mass ratio and orbital eccentricity, the rates of secular change of the orbital semi-major axis and eccentricity can be positive as well as negative, so these orbital elements can increase as well as decrease with time.

Comparison of the timescales of orbital evolution due to mass transfer with the timescales of orbital evolution due to tidal dissipation shows that the effects can either be additive or competitive, depending on the binary mass ratio, the orbital eccentricity, and the magnitude of the mass-transfer rate at the periastron. Contrary to what is often assumed in even the most state-of-the-art binary evolution and population synthesis codes, tides do not always lead to rapid circularization during the early stages of mass transfer. Thus, phases of episodic mass transfer may occur at successive periastron passages and may persist for long periods of time. As a first approximation, the evolution of the orbital semi-major axis and eccentricity due to mass transfer in eccentric binaries can be incorporated into binary evolution and population synthesis codes by means of equations (39) and (40) in which the mass-transfer rate is approximated by a delta-function of amplitude centered at the periastron of the binary orbit.

In future papers, we will relax the assumption of conservation of total system mass and orbital angular momentum, and examine the effects of non-conservative mass transfer on the orbital elements of eccentric binaries. We also intend to study the onset of mass transfer in eccentric binaries in more detail, adopting realistic mass-transfer rates appropriate for atmospheric Roche-lobe overflow in interacting binaries as discussed by Ritter (1988). We will consider individual binary systems that are known to be eccentric and transferring mass during periastron passage, as well as populations of eccentric mass-transferring binaries and their descendants.

## Appendix A Equipotential Surfaces and the Inner Lagrangian point in Eccentric Binaries

A crucial element in the description of mass transfer in any binary system is the location of the inner Lagrangian point () through which matter flows from the donor to the accretor. While a solution for the location of is not analytic, the case of circular orbits with synchronized components can be approximated by the formula

 XL1=0.5+0.22logq, (A1)

where is the distance of the point from the mass center of the donor star in units of the distance between the stars, and is the mass ratio of the binary defined as the ratio of the donor mass to the accretor mass . For mass ratios in the range , this formula is valid to an accuracy of better than 2% (e.g. Drobyshevski & Reznikov, 1974). The above formula has been generalized by Pratt & Strittmatter (1976) to include the effect of a non-synchronously rotating donor star:

 XL1=[0.53−0.03(Ω1Ωorb)2](1+49logq). (A2)

Here, is the rotational angular velocity of the donor, and the orbital angular velocity of the binary. For and , the formula is valid to an accuracy of better than 3%. It is to be noted though that Pratt & Strittmatter (1976) derived the position of the point under the assumption that the orbital and rotational periods of the binary and its component stars are much longer than the dynamical timescale of the donor. Despite the better than 3% accuracy of the fit given by equation (A2), the formula may therefore still break down because the underlying formalism is no longer valid.

In this paper, it is necessary to generalize the formula for the position of the point even further to account for a non-zero eccentricity as well as non-synchronous rotation. For this purpose, we determine the equipotential surfaces describing the shape of the components of a non-synchronous, eccentric binary. Our procedure is a generalization of the steps outlined by Limber (1963) for the derivation of the equipotentials of a non-synchronous, circular binary.

Here, we consider an eccentric binary system where the stars are considered to be centrally condensed and spherically symmetric, and thus can be well described by Roche models of masses and . Their orbit is assumed to be Keplerian with semi-major axis and eccentricity