Solar measurements
To build a star catalog in celestial coordinates (right ascensions and declinations), the Sun is critical. The meridian altitude of the Sun defines right ascension, and its north-south movement during the year reveals the latitude of the observer and the tilt of the Earth relative to its orbit plane.
Latitude is essential because it allows conversion of a meridian altitude into a celestial declination. The tilt of the Earth (its 'obliquity') is essential because it allows the meridian altitude of the Sun to be converted to right ascension.
Details of the geometry, and presentation of my data are on the solar page.
Because of my restricted camera FOV, I was limited to measurements of the upper and lower limb of the Sun. But the desired quantity is the altitude of the center of the Sun. The Sun is a circular target about 0.5 degrees in diameter, my FOV is about 0.33 x 0.25 degrees. I first thought that measuring the altitudes of the solar limbs would be problematical because I could not measure their altitude simultaneously. After some experimentation, I found that as long as I took repeated measurements of the solar limb altitudes as they passed through the meridian, I could get not only a good altitude measurement of the solar center, but I could also get an estimate of the solar diameter. Tycho could get the solar center altitude with his instruments, but not the solar diameter.
The solar diameter intrigued me because it is an indirect proxy for the Sun-Earth distance. The day-to-day motion of the Sun along the ecliptic also gives an idea of the Earth's velocity around the Sun. Maybe, with a distance estimator and a velocity estimator, I could check on Kepler's law of areas, and maybe estimate the ellipticity of the Earth's orbit. I also wanted to see if my data could give me a value for the Earth's orbital period. All this from measurements of meridian altitudes!
As a reminder, the measurement consisted of bring the solar limb (the upper limb in this photo) into tangency with the software reticle horizontal axis, then recording the altitude.
Here is a sample data collection for one solar meridian passage, with 34 data points total. Data gaps are caused by clouds:
The upper limb and lower limb curves can be fit with 2nd order curves. The peaks of these mathematical fits give an excellent estimate of the maximum limb altitudes, even if there are few data points near the actual maximum. In a curve fit, each data point contributes to the final peak value estimate, so the error in the estimate is the nominal error in each point divided by the square root of the number of points in the fit. The encoder counts are 0.009 degrees (34 arcsec), so the nominal error is about 1/2 an encoder count, or about 17 arcsec. Divided by the square root of 18, the error in the estimate of the peak is about 4 arcsec.
The lower limb is similar, and the entire data run has typically 30 data points over 30 minutes.
This accuracy doesn't hold for the measurement of the absolute altitude of the limb because there are zero-point errors in the altitude axis, typically one or two encoder counts or about one arc minute.
The limb peak altitude accuracy does hold for the solar diameter measurement because the difference in limb peak altitudes is not affected by zero-point errors. So the diameter estimates (the difference between the two limb measurements), should be good to about 5 or 6 arcsec, one sigma.
Here are shown the upper and lower limb data fits, and the residuals for the upper limb fit:
For this particular day, my data summary looked like this: (all units degrees or UT days)
Results_solar_transit_2021-06-06 MJD 522 MJD(0)= 2458849.5
Upper limb: Tpeak= 0.71317 Alt_peak= 73.968 (Upper limb peak at JD = 2458849.5 + 522 + 0.71317)
Lower limb: Tpeak= 0.71333 Alt_peak= 73.443
Solar center peak alt= 73.706 [73.726 from JPL Horizons, delta= -0.021] (this is my sanity check)
diameter = 0.525 deg, 31.5 arc minutes
I collected solar meridian altitude data on 180 separate days. With about 30 data points per day, this means that I made about 5400 individual, hand-guided, altitude measurements of the solar limbs. No image processing was used.