Tropical year
The 'tropical' year is the time between successive passages of the Sun through the same ecliptic longitude. The ecliptic longitude and the right ascension of the Sun are both zero degrees at the spring equinox, and 180 degrees at the autumnal equinox, by definition. The 'tropical year' is the fancy name for the common 'length of the year', the time to orbit the Sun once.
The altitude of the celestial equator for me, as worked out earlier, was 50.98 degrees. At the autumnal equinox, the Sun's meridian altitude was 50.98 degrees. (The equinox occurred when the Sun passed through the celestial equator.) Here are plots of the solar meridian altitude in the fall of both 2020 and 2021, with linear fits to the data:
Using the fits, for the 2020 data (on the left) set 50.98 = 153.68 - 0.3868x and solve for x to get 265.512 for the time of passage. For the 2021 data (on the right) set 50.98 = 295.01 - 0.38687x and solve for x to get 630.078. The difference is 365.268 days, my rough estimate of the length of time it takes the planet that I am standing on, to orbit its star, using only one year's worth of meridian altitudes measured from my rooftop observatory.
365.268 days converts to 365 days, 6 hours, and 26 minutes; about 38 minutes too long compared to the currently accepted value. To get a better value, I needed to connect my crossing times with older ones. In honor of Tycho, I used his estimate of the autumnal equinox of 1588. Here I provide portions of the emails between me and Dr. Rosa that let me refine my estimate of the tropical year. Here are actual translated excerpts from Tycho's Latin text, with comments by Dr. Rosa:
Note: The boxed data above, plus the upper and lower notes to those boxes, are Tycho's original comments. Rosa's comments detail the additional factors needed to convert to a modern time estimate.
Adding the reference value of my modified Julian Date (2458849.5) to my autumn 2020 data value of 265.512, I find the equinox at JD 2459115.012.
Here is the calculation of the tropical year based on the 432 years between Tycho's datum and mine in a note to Dr. Rosa.
This value of 365.24219 days is almost exactly the currently accepted value of 365.242018 days. By using Tycho's estimate of the equinox in 1588, and my data, I was able to obtain the length of the year to within 15 seconds, even though Tycho's estimate and mine were both off by an hour or more.