## Geometrical Methods

The well-observed orbit of X1356 = ra Leo (plot from Mason and Hartkopf 1 is used to illustrate a geometrical method (Figure 8.2). The elements are (van Dessel 1976):

P = 118.227 years, T = 1959.40, a = 0.880, e = 0.557, i = 6°05, ra = 302°65, Q = 325°69 ascending.

First the apparent orbit is drawn manually. In Figure 8.3 the primary star is located in the centre O of the coordinate system, P is the periastron, A is the apas-tron, C is the centre of the ellipse, the line connecting the apastron and the periastron is the projected PC*!, y) North I 0 X

North I 0 X

Figure 8.3.

Geometrical method.

semiaxis major, and L and Q are the points where the true anomaly is -90° and +90°. The elements are found as follows:

1. Draw the complete ellipse (the law of areas must be fulfilled). Construct the centre C of the ellipse. After the periastron P is found, the eccentricity is calculated: e = CO / CP.

2. Draw the tangent in P: first find the focal points Fj and F2 of the apparent ellipse. Draw the triangle F1PF2. The straight line perpendicular to the line cutting the angle in P into halves is parallel to the tangent in P.

3. Draw the line LOQ: it is parallel to the tangent in P.

4. Determine the coordinates (x, y) of the points P(xj,yj) and Qfe ,y2).

5. The Thiele-Innes constants are calculated as follows:

6. Calculate the elements i, œ and Q. The relations are:

a2 (1 + cos 2i) = A2 + B2 + F2 G2 = 2u a2 cos i = AG - BF = v a2 = u +-yj (u + v)(u - v).

7. Determine the period and the time of periastron from the observed positions. The areal constant c in the apparent ellipse is twice the area of the sector S swept by the projected radius vector in a time At:

c can be determined graphically from a well observed part of the orbit.

Another definition for the areal constant c is:

57.296

If a and b are the major and minor semiaxes of the apparent ellipse, respectively, the period is:

Now the preliminary elements have to be corrected in a least-square fit. The final result will depend on the weights assigned to the observations. For an easier control of the residuals, it is recommended to separate visual observations, photographic positions and speckle measurements. Given a standard deviation, o for a measurement, the weight, w, is: 