In my previous post I described an observation of the binary star Xi Bootis that I made on the 1st May. From a photograph I was able to make a measurement of the separation and position angle (PA) of the pair and found this to be about 5.5 +/- 0.4 arcseconds and 291 +/- 3 degrees, respectively. This has lead me to wonder if I could get a prediction for these quantities using measured orbital properties of the binary. In my copy of Norton's 2000.0 star atlas and reference handbook (Longman Scientific and Technical 1989 edited by Ian Ridpath, 18th edition) there is a table of elements of some visual binaries (p146, table 46) and in this list is an entry for Xi Bootis (ADS number 9413 in the New General Catalogue of Double Stars). The orbital parameters are as follows:-
Orbital Period P (years) : 151.505
Date of periastron T : 1909.361
Semi-major axis of orbit a (arc seconds): 4.904
Eccentricity of orbit e : 0.512
Inclination of orbit to plane of sky i (degrees) : 140.04
Argument of periastron ω (degrees) : 203.92
PA of ascending node Ω (degrees) : 348.1
The first thing to note is the date of periastron which is when the two stars are closest. The last periastron was in 1909, the next one will be in 1909.361 + 151.505 = 2060.866 which I reckon is the 11th November 2060.
So how do you predict the relative position of these stars using this data? Well, there is a computer program written in Basic that you could use written on page 145 of Norton's 2000.0 but the alternative is to search the web for someone who has provided an online calculator. Such a person is Roger Wesson who is an astronomer at Cardiff University and his calculator can be found here. Unfortunately, Roger does not list the orbital parameters for Xi Boo in his star name look up but you can type in the orbital parameters given here.
The 1st May 2024 as a fraction of a year is, I think, (31 + 29 + 31 + 30 + 1)/366 = 122/366 = 1/3 (or 0.3 recurring as a decimal), so we can write this date approximately as 2024.333. Using Roger's calculator the predicted separation and PA of Xi Boo on this date was 4.81 arcseconds and 289.8 degrees, respectively. The PA is in good agreement with what I observed but the predicted separation is marginally smaller than what I saw but I wouldn't read a lot into this.
If you look at the bottom of Roger's page there is a nice little animation that you can play which shows how Xi Boo B moves around A. I have used the calculator to predict the separation and PA of this binary for the next 150 years in 10 year intervals starting at 2024.0. The plot is shown below:-
As you can see from the graph, Xi Boo B is moving round from a westerly position to a more southerly one over the next 30 years. Also, the PA and separation are decreasing more rapidly over this period (in 2064.0 the separation is just 2.16 arcseconds). As a result, it is a good time to start observing this pair as the changes will become more noticeable!
The question remains as to how reliable are the parameters for the orbit for Xi Boo? I have found another website by Gianluca Sordiglioni which lists these quantities for this binary and they are a little different to the ones presented here. However, I don't expect the predictions for separation and PA to be radically different.
Here are the tabulated values for the points in the graph above using the oribtal parameters listed here.
2024.0 : 4.85 : 290.4
2034.0 : 3.90 : 270.5
2044.0 : 3.08 : 238.7
2054.0 : 2.63 : 191.3
2064.0 : 2.16 : 125.5
2074.0 : 2.76 : 53.5
2084.0 : 4.23 : 21.3
2094.0 : 5.47 : 5.2
2104.0 : 6.36 : 354.4
2114.0 : 6.92 : 346.0
2124.0 : 7.19 : 338.5
2134.0 : 7.20 : 331.2
2144.0 : 6.96 : 323.8
2154.0 : 6.50 : 315.5
2164.0 : 5.83 : 305.6
2174.0 : 4.98 : 292.7
All text and images © Duncan Hale-Sutton 2024
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