In the rotating frame of reference, equipotential surfaces can be calculated for a given mass ratio of the two stars. The Roche equipotential surfaces are closed near each assumed point-mass component, and are essentially spherical in shape. As the distance from the point mass increases, the shape of the equipotential surface elongates in the direction of the center of gravity, until the two surfaces ``touch'' in a roughly ``hour-glass'' form (refer to Fig. 1.1). The Roche limit or critical Roche lobe for the star represents the outermost closed equipotential surface around each stellar component (Kopal 1955).
In 1772, Joseph Louis Lagrange solved the equations of motions of the restricted three-body problem. The restricted three-body problem deals with the motion of an infinitesimal particle in the gravitational field of two point masses. Lagrange calculated zero-velocity curves around two bodies, which restrict the particle's motion. These calculated zero-velocity curves directly coincide with the equipotential surfaces described in the Roche model. Lagrange calculated five equilibrium points within the system where a particle would remain at rest in the rotating frame of reference. These equilibrium points are the Lagrangian points. The intersection point between the two masses (the neck of the ``hour glass'') is labeled the inner Lagrangian point, or the L1 point. Figure 1.1 displays the positions of the Lagrangian points in the binary system, KU Cygni, as well as the critical Roche lobes around each stellar component and other multiple equipotential surfaces. These surfaces were calculated assuming a mass ratio, q, of 0.125 (Olson, Etzel & Dewey 1995), corresponding to a reduced mass ratio, , of 0.11085. (Note that ).
Zdenek Kopal (1955) identified three general types of binary systems, based on the degree to which each stellar surface fills its critical Roche lobe. Figure 1.2 displays a schematic diagram of these binary systems. If the surface of each star is located beneath its respective Roche lobe, Kopal described the binary to be a detached system. In this case, the components underfill their Roche lobes. If one component fills its Roche lobe, so that the surface of the star coincides with its Roche lobe, then the system is described as a semi-detached binary system. Mass from the contact secondary star can exit through the inner Lagrangian (L1) point and form an accretion disk around the detached compact. If both components fill their Roche lobe or extend beyond their Roche lobes, the binary is a contact system, in which case, a common envelope surrounds both stars.
In semi-detached and contact systems, the surface of one star or surfaces of both stars, respectively, coincide with their respective Roche lobes. Kuiper (1941) realized the importance of Roche lobe geometry in understanding the evolution of binary system components. If the pressure in the stellar photosphere is not in equilibrium near the L1, L2 or L3 points, mass can easily move across these points. In the semi-detached configuration, mass transfer can occur across the L1 point, creating a gas stream that leaves one star and falls towards the other star. In a contact system, matter can exit through the L2 and L3 points, creating a circumbinary shell around both components or creating a binary wind that carries away mass. Evidence exists for these gaseous streams of matter in the spectrum of the binary as spurious spectral features with varying radial velocities that are inconsistent with the Keplerian radial velocities of the components.
The spectral type and evolutionary stage of the binary components describes yet another classification scheme beyond that offered by Kopal. For example, cataclysmic variable (CV) stars contain a degenerate object, namely a white dwarf, and a late-type, main-sequence star (sometimes evolved). Within the CV class, these binaries can be subdivided further into other classes, depending on their light curve properties. X-ray binaries contain a compact object, such as a neutron star or black hole candidate, that can be detected by its gravitational effect on the visible component. The zoo of binary stars reveals pairings of many different stars at various stages of evolution.