A binary system is two stars gravitationally bound orbiting a common center of mass. The stars can be of the same or differing sizes and orbits can be as short as a few days or as long as thousands of years. The short ones are easy to detect, the long ones difficult, some probably impossible to detect because of the very long observation period required.
While there is no obvious visible companion star to our Sun, there could be a dark binary, such as a brown dwarf or possibly a relatively small black hole, either of which might be very difficult to detect, without accurate and lengthy analysis.
There is also the possibility that our sun might be in a binary or complex gravitational relationship with one of several nearby “visible” stars. This scenario may require thinking beyond standard Newtonian dynamics to embrace MOND or MOG or some similar theory (that suggests that the constant of G might be stronger between stellar objects than between planetary objects within the solar system). There could be many types of unknown and unidentified masses that might cause our solar system to curve through space, including the local stellar cluster and even the galactic center to some small degree, each producing some small effect within the total precession observable. Consequently, at this point our work is primarily focused on understanding the precession observable and its nuances as the likely signature of our solar system’s angular velocity around some common center of mass. We believe that this approach of analyzing the precession observable (the sun’s motion relative to the fixed stars as seen from earth) will provide valuable and helpful data regarding the sun’s most likely stellar companion (if one exists).
In summary, beyond direct detection – one way to determine if we are in a binary or multiple star system is to see if the Sun is curving through space. To us on Earth that means we should experience a gradual “changing orientation to inertial space.” Such a phenomenon is observed as the precession of the equinox.