Final answer:
To determine if two stars are part of a binary star system, observe changes in relative positions over time, check for periodic dimming indicating eclipses, and analyze orbit sizes, periods, and orbital velocities in relation to their masses.
Step-by-step explanation:
To determine whether two stars observed close together are part of a binary star system, one could make several types of measurements using a good observatory. One crucial observation would be to look for changes in their relative positions over time. If one of the components appears to be moving around the other in a consistent pattern, this could indicate gravitational interaction, and thus that the stars are in orbit around their common center of mass. Historically, observations like these led to the discovery that the Castor system in the constellation of Gemini was indeed a visual binary.
Additionally, if the binary stars are aligned such that they eclipse each other from our vantage point on Earth, it would result in periodical dimming of the system as one star passes in front of the other. This dimming would produce a characteristic light curve, which is a graph of brightness over time. However, such eclipses only occur in a minority of binary systems where the orbital plane is nearly edge-on relative to our line of sight.
In cases where we are looking at the system face-on, we could also analyze the orbit sizes, periods, and orbital velocities of the stars. A more massive star would orbit closer to the center of mass and have a smaller orbit, while the less massive one would orbit farther away, according to the principles of gravitational interaction in a binary system.