Previously, I described a way of orientating an image so that the edges of the frame are parallel to the North-South and East-West directions. In a lot of situations this perhaps does not need to be that accurate but in the case of binary stars where you are trying to determine the position angle (PA) of one star relative to another, then it does become a bit more of an issue. For example, I had been taking some measurements of the separation and PA of the binary star Xi Bootis, and I wondered if my method for orientating the image could be improved. After all, I was using Photoshop and a chart to determine the angle that a line between two field stars made with the horizontal side of the frame. This was potentially inaccurate because I had to place a cursor on where I thought the centre of the field stars were on the chart.
Let me now revisit the problem. Below is the image I took of the binary star after I had rotated it to what I thought was the correct orientation:-
As well as the two components of the binary star Xi Boo A and B you can faintly see two field stars in the image. To show this better and get an idea of the problem have a look at the diagram below:-
The field stars have been marked as 1 and 2 and there is a dashed line between them. I have also indicated a dashed line that travels due north from 1 until it meets the line drawn due east from 2. In the situation where the angular distance between field stars 1 and 2 is small (less than a degree or thereabouts) then the right angle triangle approximately lies on a flat plane. Let the angular separation between the two field stars be S. This forms the hypotenuse of the right angled triangle. The vertical dashed line forms the adjacent side to the angle θ marked. This side represents a separation which is the difference in the declinations of the two stars. Let this be δ² - δ¹. It follows from basic trigonometry that the angle θ is to a good approximation θ = arccos (δ² - δ¹ / S).
Now this is all very well but we have to determine the equatorial coordinates of the field stars. If you view the page on the BAA website where observations of Xi Boo have been made you will see that, not only is there a link to In-The-Sky.org, but also a link to the SIMBAD astronomical database for this object. This is a very cool page! If you look near the top of the page there is a 'submit query' button and on the same line just before it there is a field where you can adjust the radius of the search. Changing this radius to 12 arc minutes gives a wider view of the locality round Xi Boo and on the left an image with circled objects. Each of these circled objects (in different colours) relates to a line in the table to the left. If you hover over an object in the image it highlights it in the table (very neat).
This enables us to determine what our field stars are. It turns out that star 1 has a designation of V* EO Boo and star 2 has a designation of BD+19 2872 (they have V magnitudes of 8.43 and 9.06, respectively). Furthermore the table lists the epoch 2000 coordinates of these two stars as RA 14 51 42.2211380760, Dec. +19 05 21.217940052 and RA 14 51 38.0435867520, Dec. +19 10 23.589683616, respectively. I am amazed that they are quoting the accuracy of the positions to 9 decimal places in seconds of arc! Using my routine to calculate the separation of these two stars I find that S = 0.085586862 degrees. Further, δ² - δ¹ can be determined from the stored variables of the program and this corresponds to Z-X. I find that δ² - δ¹ = 0.083992154. Hence θ = arccos (δ² - δ¹ / S) = 11.07775584 degrees.
If I now compare this value of θ with some measurements of this angle from my image above using Photoshop I get a value of 11.3 degrees, as best as I can determine, so I think I did a pretty good job in orientating my image!
All text and images © Duncan Hale-Sutton 2025
