If the observational material does not allow the entire ellipse to be drawn, the Thiele-van den Bos method is recommended (Figure 8.4). It requires three
North 0 X
North 0 X
Figure 8.4. The
Thiele-van den Bos method.
well-observed places (pi, di) and the corresponding (xi, yi) and an approximate value for the areal constant c or alternatively the mean motion x.
The idea of using the difference ellipse sector - triangle between three positions was found by Gauss. For a long time this method was used for the orbit computation of planets, but then Thiele2 applied the method to binary stars. Van den Bos modified it by inserting the Thiele-Innes constants and was the first to use it in practical work. The following example demonstrates the method. Suppose we are given the following three well-observed positions at three times ti:
t1 = 1880.00, 61 = 79.5, p1 = 0.546, x1 = +0.099, y1 = +0.537
t2 = 192 2.00, 62 = 130.9, p2 = 1.069, x2 = -0.700, y2 = +0.808
t3 = 1949.00, e3 = 158.5, p3 = 0.562, x3 = -0.523, y3 = +0.206.
Note that the three positions have to be determined with the utmost care. If the arc including the three positions does not fulfil the law of areas, no result will be obtained.
The double areal constant c has been determined from the observational material: c = +0.0139. Note that in retrograde orbits the sign of c is negative.
For the time interval ty - tx, Kepler's equation reads as follows:
The double area of the triangles can be calculated from the observed positions:
For our example we get:
The following fundamental relation holds:
Introducing the quantities p and q we get:
E3 - E1 = p + q; t3 - t1 - A13/c = L13 (8.3) i L12 = p - sin p (8.4)
First the quantities Lxy have to be calculated. Then a X has to be found by trial and error which satisfies the equations (8.4), (8.5) and (8.6). If the double areal constant c is not known with the required accuracy, but an approximate value for x is known (for example, because of recurrence of the companion in a position), c has to be found. Our example gives:
If the observed arc shows little curvature, i.e. it is undetermined, the values for the Lxy become very small and the method will give very uncertain results, if at all. Insertion into the equations (8.4), (8.5) and (8.6) gives the following values for ¡, p, q:
Now E2 and e are computed from the following two formulae:
A 23 cos p - A12 cos q - A13 e cos E2 = ——-L——-1-13.
Another critical point: is the result for e reliable? Up to now no definitely parabolic or hyperbolic orbits have been found!
In our example the result is:
Note that e must be positive.
E1 and E3 are obtained from equations (8.1) and (8.2). Equation (8.3) serves as a check. Results:
Now the quantities Mx are calculated:
Inserting the three values for Mx into
1x x we get three different values for T. The mean value is taken: T = 1841.17.
Finally the Thiele-Innes constants are calculated using the relations:
x,- = AX-i + FY; 7,. = BXt + GYt a and i are calculated from
The quadrant of the node is determined by the relations
In case the ascending node is unknown (i.e. no radialvelocity measurements are available), the value 0 < Q <180° is taken.
For the remaining elements we get:
Warning: the Thiele-van den Bos method is instructive, seems elegant and makes full use of the high accuracy of the times ti but it cannot handle all cases. According to Couteau, it "satisfies the spirit, but not always the investigator".
Now the so-obtained initial elements should be corrected by means of a least-square fit.3-6 In case the orbit is too uncertain to correct all elements simultaneously one or several elements have to be fixed, limiting the number of elements to be corrected in one step. As a rule, this procedure is an iterative one. The covariance matrix will show which elements are weakly determined or whether there is a strong coupling between two elements. It also allows us to calculate approximate values for the errors of the individual elements, but this result will depend very much on the weights assigned to the observations.
a2 (1 + cos2 i) = A2 + B2 + F2 + G2 = 2u a2 cos i = AG - BF = v
œ and Q are calculated from:
A + G = 2a cos ( + H) cos2—; A - G = 2a cos (o> - H) sin^ B - F = 2a sin ( + H) cos2 — ; - B - F = 2a sin (o> - H) sin2 —
Another way is to define a set of initial values for P, T and e, in the next step calculating the four Thiele-Innes elements by a least-square fit, and varying P, T and e in a three-dimensional grid search.7
There are many other methods to get initial elements for a binary star orbit.8-13 Whatever method is adopted, there is no single method which can handle all cases equally well or which can deliver the final solution in one step.
Edge-on orbits and other special cases have been described in Double Stars.4
1 Mason, B. and Hartkopf, W.I., 1999, Fifth Orbit Catalogue, US Naval Observatory.
4 Heintz, W.D., 1978, Double Stars, Reidel, Dordrecht, Holland.
5 Bos, W.H. van den, 1926, Union Obs. Circ., 2, 356.
6 Bos, W.H. van den, 1937, Union Obs. Circ., 98, 337.
7 Hartkopf, W.I., McAlister, H.A. and Franz, O.G., 1989, Astron. J., 98, 1014.
8 Kowalsky, M., 1873, Proceedings of the Kasan Imperial University and 1935 in The Binary Stars, by Aitken, R.G., McGraw-Hill.
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