There might be occasions when triple stars are observed. Unfortunately the components are not always spaced sufficiently to measure from a single position. It is not always possible to measure the position of B with respect to A and then rotate the micrometer around and measure the position of C with respect to A. This is because multiple star systems tend to preserve their binary nature. If there are three stars then two of them form a binary while the third component usually orbits the other two as though they were a single star. Likewise, if there were a forth component it would normally be paired with the third component making a binary system where each component was itself a binary.
Measuring a triple star, then, usually entails measure B with respect to A and then measuring C with respect to the pair AB, or more specifically, with respect to the centre of AB. The observation is made this way because when sufficient magnification is used to separate A and B the field of view is usually too small to include C and conversely when the field of view contains C, A and B are usually too close to be separated.
The problem, then, is to find the position angle of C with respect to A. Fortunately the calculations involve nothing more than some simple plane geometry.
Let A 6 = 6C - 6b where 6C is the position angle of C with respect to the mid-point of AB and 6B is the position angle of B with respect to A. Then 6B
where the subscripts B and C are as for the position angles.
The separation of C from A is
Pac =V i PB +PC +n and the position angle of C with respect to A is found from pBpr sin AO tan °0 = ^-
Measurements of zeta Cancri for 2001 are
AB 6 = 78°3 and p = 0'.'86 (2001.205, 8 nights) 2AB-C 6=72°9 and p = 5'.'79 (2001.250, 7 nights)
(1AB means the mid-point of A and B). Performing the calculations gives
The position angles, in this case, are all close to the same value, which suggests that the three components lie close to a straight line.
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