With the Vernal Equinox is less than a month away it is a good time to use the HORIZONS data to show how one can find the time of an equinox. This example starts with hourly azimuth values for the latter half of September 132. The time is represented by the day of the month plus a fraction which includes the hour of the day in UT and the difference in longitude of Alexandria from the prime meridian. A cubic spline fit is used to obtain functions which will calculate the azimuth and altitude of the Sun for an arbitrary time. The time of midday, t_md, is defined as when the Sun reaches a azimuth of 180° and is found by linear interpolation. This is equivalent to finding the time when the Sun crosses the meridian. The functions previously found are used to check the azimuth and calculate the altitude, alt_md, of the sun at noon.
The positions of the altitude at noon appear as a set of points along a line decreasing by about the same amount each day as shown by the red line in the figure below. The blue line represents the altitude of the equator which is assumed to be 90 degrees minus the latitude of Alexandria. The coefficients of the line passing through the altitudes for midday are found with the line function which does a least squares fit. The equinox occurs when the Sun crosses the equator which is easily computed using the coefficients of the line.
If the value for the altitude is 3 minutes of arc too high the calculated time for the equinox will occur about 2 hours earlier.
One can see how sensitive the results are to errors in measurement.
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