Previously, I obtained a formula for calculating the exact time of the Summer Solstice. Now, let’s test it (the previous derivation was based on data from 2009).
The date of the Summer Solstice in June of year Y is: 21.9938 + 0.2422Y - \lfloor Y/4 \rfloor - \lfloor Y/400 \rfloor + \lfloor Y/100 \rfloor where \lfloor x \rfloor denotes the integer part.
Substituting Y=2011 into the above formula, we get 22.058, which corresponds to June 22nd at 01:23. According to data obtained from the Shouxing Perpetual Calendar, the actual time is June 22nd at 01:16:29.
This indicates that the error is not very large, suggesting the formula has practical utility. Therefore, I attempted to improve the precision of the data. I changed 0.2422 to 0.24219879 and 21.9938 to 21.991346736. That is:
21.991346736 + 0.24219879Y - \lfloor Y/4 \rfloor - \lfloor Y/400 \rfloor + \lfloor Y/100 \rfloor
Having fitted the data for 2011, let’s test a more distant year (2020). Substituting the value, we get 21.232902536, which is June 21st at 05:35:22. Checking the Shouxing Perpetual Calendar again, we find the time is June 21st at 05:43:33. From this, we can see that increasing the precision of the initial data can improve the calculation accuracy (previously, an error of several minutes would occur in just one or two years; now, it takes ten years to accumulate the same error). However, errors still exist because this formula considers an unperturbed model. (Note: The observation location for the corrected data is BoJone’s hometown, Yunfu, Guangdong, which may differ from other locations.)
In reality, the algorithm used here is not a celestial mechanics calculation in the true sense; it merely uses floor functions to express the current Gregorian calendar system. If one only needs accuracy to the day, this formula should be reliable for the next several hundred years.
red A question for everyone: In fact, interpolation methods can be used to further improve accuracy. For example, one could assume the error in year Y, \Delta T_Y, is proportional to (Y-2011)^a, where a is a constant and a \ll 1 (certainly a changes over time, but the change should be slow, otherwise the length of each year would vary significantly). You can use data from other years for comparative correction. Readers are encouraged to try it themselves—let’s see who can fit the most accurate formula.
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