## Visual Binaries

We consider a visual binary, assuming initially that the brighter primary component is stationary and the fainter secondary component is orbiting around it. The angular separation of the stars and the angular direction to the secondary can be directly observed. Making use of observations extending over many years or decades, the relative orbit of the secondary can be determined. The first binary orbit to be determined was that of f UMa in 1830 (Fig. 9.2).

The observations of visual binaries only give the projection of the relative orbital ellipse on the plane of the sky. The shape and position of the true orbit are not known. However, they can be calculated if one makes use of the fact that the primary should be located at a focal point of the relative orbit. The deviation of the projected position of the primary from the focus of the projected relative orbit allows one to determine the orientation of the true orbit.

The absolute size of the orbit can only be found if the distance of the binary is known. Knowing this, the total mass of the system can be calculated from Kepler's third law.

The masses of the individual components can be determined by observing the motions of both components relative to the centre of mass (Fig. 9.3). Let the semimajor axes of the orbital ellipses of the primary and the secondary be a 1 and a2. Then, according to the definition of the centre of mass, ai = mi, (9.1)

a2 m1

Fig. 9.2. In 1830 the orbit of f Ursae Majoris was the first binary orbit determined observationally
Fig. 9.3. The components of a binary system move around their common centre of mass. Ai, A 2 denote the positions of the stars at a given time A, and similarly for B and C

where m1 and m2 are the component masses. The semimajor axis of the relative orbit is a = ai + a2. (9.2)

For example, the masses of the components of f UMa have been found to be 1.3 and 1.0 solar masses.

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