j-sepinsky, b-willems, [email protected], and [email protected] A Preprint typeset using LTEX style emulateapj v. 08/22/09 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 j-sepinsky, b-willems, [email protected], and [email protected] 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 momen- tum. 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 magni- tude 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 −9 −1 mass-transfer rate of 10 M⊙ yr 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. Com- parison 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 instanta- neous 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. Subject headings: Celestial mechanics, Stars: Binaries: Close, Stars: Mass Loss 1. INTRODUCTION 2003). Mass transfer between components of close binaries is Furthermore, assumptions of instantaneous circular- a common evolutionary phase for many astrophysically ization immediately before or at the onset of mass trans- interesting binary systems. Indeed, mass ejection and/or fer are in clear contrast with observations of eccentric accretion is responsible for many of the most recogniz- mass transferring systems. In the most recent catalog of able phenomena associated with close binaries, such as eccentric binaries with known apsidal-motion rates com- persistent or transient X-ray emission, neutron star spin- piled by Petrova & Orlov (1999), 26 out of the 128 listed up, and orbital contraction or expansion. Theoretical systems are semi-detached or contact binaries. Among considerations of these and other associated phenomena these mass-transferring systems, 9 have measured eccen- in the literature probe these systems quite effectively, tricities greater than 0.1. In addition, many high-mass X- yet they often do not consider the effects of any eccen- ray binaries are known to have considerable orbital eccen- tricity associated with the binary orbit. This can be of tricities (Raguzova & Popov 2005). While mass transfer particular importance for binaries containing a neutron in these systems is generally thought to be driven by star or a black hole, where mass loss and natal kicks the stellar wind of a massive O- or B-star, it has been occurring during compact object formation may induce suggested that some of them may also be subjected to a significant eccentricity to the binary (e.g. Hills 1983; atmospheric Roche-lobe overflow at each periastron pas- Brandt & Podsiadlowski 1995; Kalogera 1996). After the sage of the massive donor (e.g. Petterson 1978). formation of the compact object, tides tend to circularize Huang (1956), Kruszewski (1964), and Piotrowski the orbit on a timescale which strongly depends on the (1964) were the first to study the effects of mass transfer arXiv:0706.4312v1 [astro-ph] 28 Jun 2007 ratio of the radius of the compact object’s companion on the orbital elements of eccentric binaries. However, to the orbital semi-major axis. Because of this, orbits their treatment was restricted to perturbations of the or- are usually assumed to circularize instantaneously when bital motion caused by the variable component masses. a binary approaches or begins a mass transfer phase. Matese & Whitmire (1983, 1984) extended these early Despite our generally well developed understanding of pioneering studies to include the effects of linear momen- tidal interactions in close binaries, quantitative uncer- tum transport from one star to the other, as well as any tainties in tidal dissipation mechanisms propagate into other possible perturbations caused by the mass trans- the determination of circularization timescales. For ex- fer stream in the system. However, these authors derived ample, Meibom & Mathieu (2005) have shown that cur- the equations governing the motion of the binary compo- rent theories of tidal circularization cannot explain ob- nents with respect to a reference frame with origin at the served degrees of circularization of solar-type binaries in mass center of the binary, which is not an inertial frame. open clusters. Circularization of high-mass binaries, on Their equations therefore do not account for the accel- the other hand, is currently thought to be driven predom- erations of the binary mass center caused by the mass inantly by resonances between dynamic tides and free transfer (see § 3.3). oscillation modes, but initial conditions play an impor- More recent work on mass transfer in eccentric binaries tant role and an extensive computational survey of rel- has mainly focused on smoothed particle hydrodynamics evant parts of the initial parameter space has yet to be calculations of the mass transfer stream over the course of undertaken (Witte & Savonije 1999, 2001; Willems et al. a few orbits, without any consideration of the long-term 2 Sepinsky, Willems, Kalogera, & Rasio evolution of the binary (Layton et al. 1998; Reg¨os et al. this point to lie on the line connecting the mass centers 2005). of the stars, even though non-synchronous rotation may Hence, there is ample observational and theoreti- cause it to oscillate in the direction perpendicular to the cal motivation to revisit the study of eccentric mass- orbital plane with an amplitude proportional to the de- transferring binaries. In this paper, our aim is to derive gree of asynchronism (Matese & Whitmire 1983). Since the equations governing the evolution of the orbital semi- the donor’s rotation axis is assumed to be parallel to the major axis and eccentricity in eccentric mass-transferring orbital angular velocity, we can safely assume that the binaries, assuming conservation of total system mass and transferred mass remains confined to the orbital plane. orbital angular momentum. In a subsequent paper, we We furthermore assume that all mass lost from the will incorporate the effects of mass and orbital angular donor is accreted by its companion, and that any orbital momentum losses from the system. angular momentum transported by the transferred mass Our analysis is based on the seminal work of is immediately returned to the orbit. The mass transfer Hadjidemetriou (1969b) who was the first to derive the thus conserves both the total system mass and the orbital equations of motion of the components of eccentric mass- angular momentum. transferring binaries while properly accounting for the We also neglect any perturbations to the orbital motion effects of the variable component masses on the stars’ other than those due to mass transfer. At the lowest mutual gravitational attraction, the transport of linear order of approximation, these additional perturbations momentum from one star to the other, the accelerations (e.g., due to tides, magnetic breaking, or gravitational of the binary mass center due to the redistribution of radiation) are decoupled from those due to mass transfer, mass in the system, and the perturbations of the orbital and can thus simply be added to obtain the total rates motion caused by the mass-transfer stream. While the of secular change of the orbital elements. equations of motion derived by Hadjidemetriou (1969b) are valid for orbits of arbitrary eccentricity, the author 3. EQUATIONS OF MOTION restricted the derivation of the equations governing the 3.1. Absolute Motion of the Binary Components evolution of the semi-major axis and eccentricity to or- bits with small initial eccentricities. Following Hadjidemetriou (1969b), we derive the equa- The paper is organized as follows. In § 2 and § 3 we tions of motion of the components of an eccentric mass- present the basic assumptions relevant to the investiga- transferring binary with respect to a right handed inertial tion and derive the equations governing the motion of frame of reference OXYZ which has an arbitrary posi- the components of an eccentric mass-transferring binary tion and orientation in space (see Fig. 1). We let Mi be under the assumption of conservative mass transfer. The the mass of star i at some time t at which mass is trans- associated equations governing the rates of change of the ferred from the donor to the accretor, and Mi + δMi the semi-major axis and the orbital eccentricity are derived mass of the same star at some time t + δt, where δt > 0 in § 4, while numerical results for the timescales of or- is a small time interval. With these notations, δMi < 0 bital evolution due to mass transfer as a function of the corresponds to mass loss, and δMi > 0 to mass accretion. initial binary mass ratio and orbital eccentricity are pre- We furthermore denote the point on the stellar surface sented in § 5. For comparison, timescales of orbital evo- at which mass is lost or accreted by Ai. For the donor lution due to dissipative tidal interactions between the star, Ai corresponds to the inner Lagrangian point L1, binary components are presented in § 6. § 7 is devoted to while for the accretor, Ai can be any point on the star’s a summary of our main results and a discussion of future equator.