Points of the stellar surface are considered to belong to an equipotential surface. The mathematics of such level surfaces is similar to that of the zero velocity curves in the restricted three-body problem [cf. Szebehely (1967)], in which a particle of negligible mass is subject to gravitational forces of two massive orbiting bodies. Within that framework two cases are distinguished: circular orbits and elliptic or eccentric8 orbits. We treat them separately because the circular and the eccentric cases require different techniques. More importantly, there are eccentric effects on the components and on the light curves beyond those of the circular case.
We distinguish between absolute and relative orbits. Orbits with absolute orbital semi-major axes a1 and a2 have the origin of coordinates at the system barycenter, whereas orbits with the relative orbital semi-major axis a describe the motion with respect to the center of mass of its companion star. Absolute and relative orbits are coupled by a = ai + a2 (3.1.17)
8 We use the terms eccentric and elliptic orbits synonymously throughout this book.
and the moment equation a1 M1 = a2 M2.
In the next two subsections we use the following symbols appropriate for the general case of an elliptic orbit with the relative orbital semi-major axis a and the eccentricity e, 0 < e < 1; we have the orbital quantities v, u, and 0, and the orbital elements m and i:
• v, (true) longitude in orbit, measured from some reference point to the star's position in the orbit; 0° < v < 360°.
• u, true anomaly, measured from periastron to the star's position in the orbit; 0° < u < 360°.
• 0, (true) phase angle or "geometrical phase," i.e., the angle in the orbital plane measured from conjunction in the direction of motion; 0° < 0 < 360°.
• m, the argument of periastron, i.e., the angle from the ascending node to periastron in the orbital plane (see Fig. 3.6); 0° < m < 360°.
• i, orbital inclination, i.e., the angle by which the plane of the true orbit plane tilts out of the plane-of-sky9 (Fig. 3.6). Note that an edge-on orbit has i = 90°. The
Fig. 3.6 Orbital elements of a binary system. Q is the position angle (measured in the plane-of-sky) of the ascending node or the position angle of the line of nodes, respectively. m is the angular distance in the orbital plane between the line of nodes and the periastron in the direction of the motion of the component. N is used to orientate the plane-of-sky and points to North
9 The correct definition of the inclination is an intricate matter related to the orientation of the coordinate system discussed in Sect. 3.1.1. Although i = 85° and i = 95° lead to the same situations concerning light and radial velocity curves, differences appear for modeling polarimetry and interferometry.
Fig. 3.6 Orbital elements of a binary system. Q is the position angle (measured in the plane-of-sky) of the ascending node or the position angle of the line of nodes, respectively. m is the angular distance in the orbital plane between the line of nodes and the periastron in the direction of the motion of the component. N is used to orientate the plane-of-sky and points to North observer observer inclinations i = 85° and i = 95° can be distinguished by whether the motion as projected onto the plane-of-sky is counter-clockwise or clockwise.
The symbols v and 0 should not be confused with the same symbols used to establish spherical coordinates on the component surfaces introduced in Sect. 3.1.1.
Finally, we use the symbol 0 for the orbital (sometimes also called photometric) phase measured from primary conjunction; 0 < 0 < 1. For circular orbits primary minimum ordinarily, or by convention, coincides with superior conjunction of the primary component, so that 0 = 0 at 0 = 0. Whereas in the circular case, photometric phase and true phase angle are simply connected by
the geometrical phase in the eccentric case is related to the photometric phase by
as shown in Fig. 3.7. Note that u + o is the angle from the node to the star.
If the binary's orbit changes10 in time it might be possible to derive the change, P, of the orbital period, and the apsidal motion parameter, o, if observation times (rather than phase) are available. If the argument of the periastron, o0, is known
10 The physical cause can be apsidal motion, orbit around a third body, mass loss and mass transfer, and solar-type magnetic cycles (Hall 1990). Algol itself is a good example. It undergoes a 1.783 year cycle as it revolves around Algol C and it also has a 32-year magnetic cycle (S0derhjelm 1980). For more details on apsidal motion see page 132.
for a reference time, T0, the instantaneous argument of periastron, o = o(t) , in first-order approximation is given by
Many EB systems have circular orbits due to the accumulated effects of tidal forces. The tidal evolution [cf. Hut (1981)] will continually change the orbital and rotational parameters. Ultimately, either an equilibrium state will be reached asymptotically or the two stars will spiral in toward each other at an increasing rate leading to a collision. An equilibrium state is characterized by coplanarity (the equatorial planes of the two stars coincide with the orbital planes), circularity of the orbit, and corotation (the rotation axes and periods of the components equal those of the orbital motion).
In the circular orbit case, the position of each star is a simple function of the phase ^. Photometric and geometrical phase angles are connected by (3.1.19). The linear distance d between components is then independent of ^ and commonly normalized to d = 1. The light curve minima occur at phases ^ = 0 and ^ = 0.5, or 0 = 0° and 0 = 180°, respectively.
Although the orbits of many EBs are circular (Lucy & Sweeney 1971), some have elliptic orbits and sometimes even high eccentricities [e.g., HR 6469 with e = 0.672 in Scarfe et al. (1994), see Fig. 3.8, showing the orbit]. This is not a great surprise because circularization is a relatively slow process as shown by Hut (1981).
Eccentric orbits have several light curve effects. The eclipse occurring nearer to apastron has the longer duration. In addition, the minima are in general not arranged symmetrically. If tI and tII denote the times of successive primary and secondary minima, respectively, tII > tI, we have
Only when the line of apsides coincides with the line-of-sight is eclipse symmetry reestablished. Primary conjunction occurs at a phase <P1 which can be much different from zero. If the orbit is circular the plane-of-sky distance 5 between centers takes its minimum at This statement is approximately true in the eccentric orbit case.
The phase shift or displacement of the minima depends on e and o and is approximately [cf. Binnendijk (1960, Eq. 384) or Tsesevich (1973)] given by
and describes how much the time interval between primary and secondary conjunction differs from a half-period. For e = 0 the relation tII - tI = P/2 is reproduced.
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