Tuesday, March 1, 2016
132 AD Summer Solstice Calculation
The determination of the time of the summer solstice in 132 AD has an added complication since the Sun's altitude at noon passes through a maximum at this time of year.
The changes in altitude are nearly linear and the time at which change is zero can be used to find the time of the solstice. Using central differences gives a better estimate of the peak value.
The calculation is similar to that used to find the time of the equinox.
A plot shows that this procedure gives a good estimate for the time of the peak.
The times of the solstices are not affected by the latitude of the place of the observation but those of the equinoxes are so the length of the seasons, the interval for the quadrant passages, are subject to error. The computed seasonal intervals may vary from year to year so an averaged value may work better. This may be why Ptolemy used Hipparchus' seasonal lengths for his anomaly.
Ptolemy would have had no problem doing the interpolations since differences in times are proportional to the differences in altitude for the equinoxes and second differences for the solstices. He would also have been familiar with finding maxima and minima since Hero of Alexandria had shown in the 1st Century AD that light travels the shortest path between two points.
Ptolemy appears to have done his observations in Alexandria I am not aware of the exact location. The value for the latitude of Alexandria suggests a location somewhat south of the location for the Museum which were the coordinates assumed for Alexandria. Ptolemy's altitude for the Celestial Equator would be subject to observation errors for his latitude and that of the direction of his Zenith. There is also a slight correction for geodetic latitude that he would have ignored as well as that for atmospheric refraction. If we emulate Ptolemy correctly it may be possible to correct his time for the Autumnal Equinox. The time of day would probably be within two or three hours of the time that Ptolemy gave. As we saw calendar creep complicates determining what he meant by the date given, Athyr 7.
Supplemental (Mar 1): On p. 134 of Toomer, Ptolemy's Almagest, Ptolemy states that the error in the observed times of the solstices and equinoxes may have been up to 1/4 of a day. From our calculations an error of 1/24th degree produced a 2 hour error. Three random errors of about 1/24th degree with 3 standard deviation bounds would result in a error of 3√3 times that or about 10 hours and 4 sources of error of error would result in 6 times that or 12 hours of error. With the Vernal Equinox moving along the ecliptic the length of the seasons change over long periods of time. I don't recall Ptolemy stating how he determined the time of day of an equinox or a solstice. In both cases it may have been by direct observation either of the crossing of the assumed equator or of the altitude of the Sun respectively.