This past weekend was the summer solstice. Since the sun achieves its highest declination on this day, which is related to the obliquity of the ecliptic, this is a fundamental parameter that I try to observe the altitude on the solstice when possible. I did this back in 2018 using the solar angle dial from Book I of the Almagest. This time, I brought out the quadrant as I did for the autumnal equinox in 2018 as well.
The method here is to use a long, straight pipe along the sighting arm. When pointed directly at the sun, the shadow of the pipe should be a ring as seen here.
Once aligned in both altitude and azimuth1, I could then clamp the quadrant in place while I took the reading:
Here, we can see that the altitude was 74.55º which is 74º34′.
In my previous post regarding the solstice altitude, I didn’t use this value to determine the actual obliquity of the ecliptic and instead decided to wait until I had a measure at the solstice as the difference between the two can give the desired value. However, the obliquity can also be determined by using this angle and the latitude of the observer.
To explain, let’s start with a diagram:
Here, I’ve drawn out the angles involved.
To start, let’s look at the angle between the horizon and the altitude of the celestial equator. From the diagram, we can easily see that this is $90º – \phi$2. However, that same angle can also be described as $a – \delta$3. So we can set these two equal:
$$90º – \phi = a – \delta$$
This can easily be rearranged to determine delta as:
$$\delta = a + \phi – 90º$$
Plugging in:
$$\delta = 74º34′ + 38º36′ – 90º = 23º10’$$
This is a bit low as the true value should be 23º24′. So I’m off by about 14 arcminutes or just shy of a quarter of a degree. A tiny portion of this could be due to the fact that this observation was taken at local noon (which was just before 1pm here) and the actual moment of the solstice wasn’t for a few hours yet, but it was more so due to just simple error aiming. If you look at the picture of the shadow, you can clearly see there’s a hint of shadow along the inner right side of the ring indicating it was a bit low, but no matter how much I fussed with it, I couldn’t get it any better.
Comparing this to the value I got for the altitude on the solstice 2 years ago using the solar dial, it appears to be fairly close. The solar dial was only read to whole degree increments, and so I got quite lucky that it was only 10 arcminutes off, but anywhere within about a half a degree would have been acceptable on that instrument.
Applying the same method as above, using only the altitude on the solstice and the latitude, the solar disk would have given an obliquity of 23º36’22” which is 12 arcminutes high. So averaging these results, we’re exceptionally close!
Hopefully, that will continue to increase as the years pass and I’m able to take further measurements.
- The azimuth ring is obviously not present here. It wasn’t necessary since there’s no visible north star to align on. Instead, I determined when the sun was on the meridian by knowing what time it should be via stellarium.
- Where $\phi$ is the latitude of the observer.
- Where a is the altitude and $\delta$ is the declination which will match the obliquity of the ecliptic when the sun is at its maximum declination (i.e., on the solstice).