## Third Body Effects on Light and Radial Velocity Curves

EBs are sometimes members of multiple star system [cf. Mayer (2005) and references therein]. Among 728 multiple systems with 3-7 components contained in an extended 6/1999 edition of the Tokovinin (1997) catalogue of physical multiple stars, there are 83 eclipsing binaries. In the simplest case a third body orbits with the binary around the system barycenter. Observational evidence of third bodies comes from spectroscopy (disturbances of the radial velocity curve), or from analysis of times of minima. A linear relation between Observed-Computed times of minima and time indicates constant period3; a parabola implies a constant rate of period change; and a sinusoid implies apsidal motion4 or variation in arrival time (lighttime effect) due to orbital motion of theclose binary around the system barycen-ter (binary plus third body). EB programs ideally should consider three short-term effects imposed by the third body: Third light as described in Sect. 3.4.1, the lighttime effect to all observable phenomena, and radial velocities.

Although they are not short-term effects, here we want to summarize a few results on the dynamics and the effects of the third body on the binary, referring the reader to Eggleton (2006) for a more detailed discussion.

Apsidal motion can be due to gravitational perturbations by other bodies, finite nonspherical mass distributions of the stars, and generalrelativistic effects [cf. Quataert et al. (1996)]. If due to a third body, apsidal motion may be accompanied by precession. Both, apsidal motion and precession, result from the rotation of the binary's orbital frame around the barycenter. If the third body orbit is the outer orbit, both effects are on a timescale (Eggleton 2006, p.203)

a2C3 M3 2n P

where P and P3 are the period of the inner and outer orbit, e3 is the eccentricity of the outer orbit, h is the angular momentum of the third body, and

Neither precession nor apsidal motion is expected to have a significant effect on the long-term orbital evolution of a binary. Given that P and a are purely functions of the orbital energy but not of angular momentum, they remain constant in the lowest approximation of the Keplerian orbit perturbation. The dynamic cause is that the third body's force is derivable from a potential and thus does no work around

3 The period is constant but incorrect if the slope of the line is different from zero.

4 The term apsidal motion refers to the rotation of lines of apsides of an eccentric binary orbit, or the rotation of the periastron. Apsidal motion is caused by perturbations to the 1/r gravitational potential and indicates deviations from Keplerian elliptic motion.

a closed curve. It thus has no net effect on the binary's orbital energy during a complete cycle. As the binary eccentricity and angular momentum can fluctuate, a third body can, nevertheless, strongly influence a binary's orbital evolution. One would not intuitively expect this behavior, particularly if the third body moves in a wide orbit with a period of the order of 104 years or has only very low mass. If the inclination between the outer and inner orbit is larger than about 39°, large cyclic variations in e become possible at high inclinations (Kozai 1962). These cycles are known as Kozai cycles and are related to Kozai's (1962, Sect. 4.8) analysis of the interaction between Jupiter and solar system asteroids with high inclination. The cyclic eccentricity variations, coupled with the approximate constancy of a, means that tidal friction can become important at periastron during part of the cycle, even if it is unimportant during the small eccentricity part of the cycle. Due to this friction, over many Kozai cycles the inner orbit will shrink as well as become circularized, the final period being roughly the period when the stars are close enough for apsidal motion due to their distortion to dominate over apsidal motion due to the third body. As other perturbations of the third body can reduce the effect of Kozai cycles, there is a maximum outer orbit size that can generate Kozai cycles, but this may still be several thousand times larger than the inner orbit with periods up to 103-104 years. Eggleton & Kiseleva-Eggleton (2001) applied this theory to SS Lac, a binary, now lacking eclipses due to a rotating orbital plane.

Having discussed the dynamic consequences of a third body qualitatively, we present a few quantitative relations from Van Hamme & Wilson (2005,2007) involving six third-body parameters that can be also derived from light and radial velocity curve fitting. These parameters are semi-major axis, a', of the outer relative orbit; eccentricity, e3; the argument of periastron, d, of the close orbit's center of mass; period, P'; inclination, i' (angle between the plane of sky and the third-body orbit), and superior conjunction, T', of the EB center-of-mass with the barycenter. Note that T ' is well defined, whereas the EB periastron passage, T'peri, is undefined for circular orbits and weakly defined for small eccentricities.

As the binary star components orbit the barycenter, the periodically varying light-travel time leads to the light-time effect, i.e., the difference, At = tobs - tsys, between the barycentric time, tobs, and the EB center-of-mass time, tsys, which sets the orbital phase in the third-body orbit. The light-time effect time difference effects all observables and is given by

1 + e3 cos u with the true anomaly, u, and the light-time semi-amplitude for circular third-body orbits a' sini'

vfl where a' = a12 + a3 is the semi-major axis of the outer relative orbit, and c is the vacuum speed of light. Thetrue anomaly u and the eccentric anomaly E are related by (3.1.27), which here gives u E /1 + e3 tan- = B tan-, B :=/--3. (5.2.6)

The mean anomaly

is related to the eccentric anomaly E through Kepler's equation (3.1.28)

E - e3 sin E = M. (5.2.8) Alternatively, the mean anomaly can be expressed as

with the mean anomaly Mc at the time of superior conjunction, T', obtained from Kepler's equation with E = Ec. The latter can be obtained from the true anomaly at conjunction n uc = - - (5.2.10)

with

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