The Orbital Elements of a Visual Binary

Abstract

The orbit of a visual pair describes, in the form of seven so-called orbital elements, the motion of the companion relative to the primary star, such as is obtained from the relative observations. In the two-body problem with the masses M₁ and M₂, the radius vector r, and the constant of gravitation k², the relative motion is represented by the second-order differential equation

$${d^2}r/d{t^2} = - {k^2}\left( {{M_1} = {M_2}} \right)r/|r{|^3}$$

((4))

. Its six scalar constants of integration correspond to six elements which suffice to specify a particular solution if the total mass M₁ + M₂ is assumed to be known. This holds for the solar system but not for double stars. The Kepler laws describe the solutions of Equation (4), and the 1st and 2nd laws are readily utilized: the motion proceeds in a conic section (an ellipse, if periodic) with the primary star in the focus, and the area swept by the radius vector in the conic section is constant per unit of time. The 3rd law (formula 1), although equally valid, cannot be applied until the orbit is known, and a determination of the unknown total mass is attempted. The semiaxis major, therefore, is an independent element and, moreover, from visual and all positional data known only in arc sec, not in linear measure such as astronomical units or kilometers.