## The True Orbit

Whilst astronomers regard the brighter component as fixed and map the motion of the fainter one around it, in reality, both stars in a binary system move in ellipses around the common centre of gravity. The size of the ellipse is directly proportional to the mass of the star, so in the Sirius system, for instance, the primary has a mass of 1.5 M0, the white dwarf companion 1.0 M0 and so the size of ellipses traced out on the sky are in the ratio 1.0 to 1.5 for the primary and secondary (Figure 7.1). The ratio of the masses is inversely proportional to the size of the apparent orbits (see eqn 1.1 in Chapter 1), so this gives one relation between the two masses. To get the sum of the masses requires the determination of the true orbit from the apparent orbit and this is what this chapter will describe.

We regard the primary star as fixed and measure the motion of the secondary star with respect to it, and in Chapter 1 we saw that in binary stars the motion of the secondary star with respect to the primary is an ellipse. This is called the apparent ellipse or orbit and is the projection of the true orbit on the plane of the sky. Since the eccentricities of true orbits can vary from circular to extremely elliptical (in practice the highest eccentricity so far observed is 0.975), then the range of apparent ellipses is even more varied because the true orbit can be tilted in two dimensions at any angle to the line of sight. We need the true orbit in order to determine the sum of the masses of the two stars in the Figure 7.1. The real orbits of the stars in the Sinus system.

binary. This is still the only direct means of finding stellar masses.

On the face of it then the measurements that we make of separation and position angle at a range of epochs are all the information that we have to try and disentangle the true orbit from the apparent orbit. We do, however also know the time at which each observation was made much more precisely than either of the measured quantities. There are other clues, for instance, in the way that the companion moves in the apparent orbit.

In Figure 7.2 I plot the apparent motion of the binary OX 363. In this case (x, y) rectangular coordinates are used rather than the 9, p polar coordinates which are more familiar to the observer. Each dot on the apparent ellipse represents the position of the companion at two-year intervals. It is immediately clear that the motion is not uniform but it is considerably faster in the third quadrant i.e. between south and west. The point at which the motion is fastest represents the periastron (or closest approach) in both the true and the apparent orbits.

Kepler's second law tells us that areas swept out in given times must be equal and this also applies to both the true and the apparent orbit. In Figure 7.2 although the three shaded areas are shown at different points in the apparent orbit because they are all traced out over a ten-year interval, the areas are the same. We also know that the centre of the apparent orbit is the projected centre of the true orbit. In most cases the motion is described by the fainter star relative to the brighter star that is fixed in the focus of the ellipse as if the total mass were concentrated in the fixed centre of attraction.

In the true orbit the centre of the ellipse is called C, the focus, and where the brighter star is located is called

 / 1 ) • V —a--^ J \ Orbital ellipse with excentricity e = 0.8

Figure 7.3. The true elements of a visual binary star.

A. The periastron P is the closest point of the ellipse to A. The geometry of the motion suggests use of polar coordinates. The elements of the real orbit are as follows (Figure 7.3):

P the revolution period in years; alternatively the mean motion per year (n = 360/P or ^ = 2n/P is given); T the time passage through periastron; e the numerical eccentricity e of the orbit; the auxiliary angle 0 is given by e = sin 0; a the semiaxis major in arcseconds.

Figure 7.3. The true elements of a visual binary star. 