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I have two or more sets of dates and times for which I am interested in the location of the Zenith at my location and whether they are coinciding. So far I have been comparing pairs of them on two Stellarium windows with the azimuthal grid but its laborious. Can any one tell me of an easier way. I expect I could do it in excel with a bit more knowledge.

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What @JohnHoltz says is correct. If you need a cook book formula for 2020: The vernal equinox starts the year at 100.1167 degrees greenwich hour angle (GHA). Every second after that it's 0.0041780744757 degrees greater. The sidereal hour angle (SHA) of your zenith is your west longitude minus the GHA of the vernal equinox. If you're an excel user you probably know how to convert D-M-S to decimal degrees, and handle modulo arithmetic. If you're only interested in comparing times, the SHA of your zenith repeats every 86164.0935534009 seconds, so, if the difference in time is an integer multiple of that (plus/minus some tolerance) the zeniths are coincident.

Making a spreadsheet: Start with a column of all the date/times you want to compare. (You might find it easier to have separate date/time columns, if that's the way your data is given.) Make a column of time differences. One way to do that would be to calculate the elapsed time in seconds for each date/time since the start of 2017. Make another column calculating the time difference mod 86164. (The Excel formula would be "=MOD([cell number],86164)", where the cell number would be for the adjacent cell in the time difference column) Then sort the entire spreadsheet on that column. ("Sort" is one of the choices in the "Data" pull-down menu.) At this point you're done. The times with the same zeniths will be bunched in adjacent rows.

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  • $\begingroup$ Thanks both these answers are useful but I think I am going to make a zenith orbit map by print screen from Stellarium and put each data set on separate layers in paint.net then I can see which are co-located and examine them in more detail. $\endgroup$ – user36093 Nov 18 '20 at 23:11
  • $\begingroup$ How do you know the GHA for each year? I need it for 2017 to 2020. $\endgroup$ – user36093 Nov 25 '20 at 13:36
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    $\begingroup$ I somehow lost a comment-answer to your comment-question about how to know GHA. I use the nautical almanac. www.thenauticalalmanac.com has the GHA of the vernal equinox for every hour in for the current year in the daily column "Aries" and many earlier years, as free downloadable .pdf files. I think you should also consider a simple Excel file. I'm running out of space for a comment so I've appended a paragraph to my answer outlining what to do. $\endgroup$ – stretch Nov 27 '20 at 2:15
  • $\begingroup$ I did as you suggested and it is very useful. I keep getting pairs, though, that very similar remainder and should be be very close but when I plot them they are 1hr apart. All the pairs that plot together are in BST or UT and all the pairs that plot 1hr apart one time point is in each. I have corrected for BST in the data but they are selected and plotted based on final their values. Do you have any insight how these data points could get through the MOD division, have v similar remainder values and still be one hour apart when plotted? $\endgroup$ – user36093 Dec 26 '20 at 9:48
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    $\begingroup$ Stellarium apparently assumes your times are all local civil time, and subtracts an hour from any datum taken during the BST seasonal clock change. You corrected for BST and Stellarium over-corrects. When both times in a pair are BST they'll plot together, but the plot will be off by one hour. When UT and BST data points are paired, they'll always plot an hour apart. $\endgroup$ – stretch Dec 27 '20 at 14:47
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Technically, the location of the zenith is constant: 90 degrees altitude, undefined azimuth. :-) I assume that you want it in terms of the coordinates declination and right ascension instead of altitude and azimuth. Declination and Right Ascension are the coordinate system used to locate objects in the sky. They are similar in concept to latitude and longitude. The difference is that one is "locked" to the sky, and the other is "locked" to the Earth.

  • The declination of the zenith is the same as the observer's latitude. If your location is the same, then the declination is the same. (If the latitude is different, the coordinate of the zenith is different.)
  • The right ascension of the zenith can be calculated by using the formulas for the local sidereal time (LST or LMST for the mean local sidereal time.) See the post Local Sidereal Time.
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