Kingdom A&S is in just over a month and coincidentally falls on the one year anniversary of first light of the quadrant. Originally, I’d planned to do a deep dive statistical review of the quadrant, looking at sources of error, but this would be a modern mathematical review of an instrument that isn’t entirely period. Discussing it with friends, we decided it was a little too meta/degrees of separation.
Instead, I’ll do a blog post! So if you really want to get into the accuracy on the quadrant, hold on because I’m about to get mathy.
First off, what I’ll be looking at here is mostly RA and Dec. These are directly equivalent to the sidereal time and altitude respectively. In the case of RA, this is because, when an object is on the meridian, RA = ST. Period. In the case of Declination, the latitude and azimuth also need to be taken into account, but again, observing on the meridian simplifies this such that a degree of error in altitude directly equals a degree of error in declination. Thus, I’ll work in the converted RA and Dec because this is easiest to compare to published values.
Second, a note on data processing. To handle all the observations, I have a program I wrote that handles a good deal of the work for me. It’s written in Java and connects to a MySQL database. It consists of 3 objects/tables: Stars, Locations, and Observations. The Locations table really just stores the Lat/Long so I don’t need to enter that every time. The Observation table is where I enter the altitude, azimuth, and sidereal time, as well as selecting the star and location from drop down lists. The program then goes through the math, and calculates the RA/Dec and appends that to the observation, as well as updating the Star table with both the average RA/Dec and the standard deviation of all observations of that star.
I’ve pulled all that information into Excel to play with, and also added columns for the true RA, Dec, and magnitudes from Stellarium. The end result is that I have a lot of information to play with. So let’s start simple.
To begin, we can look at simple averages of the errors when compared to the true values. For RA, I come up with an average of -0.12º indicating that we tend to be taking observations about 29 seconds early. That’s pretty good because we only record ST/RA to the nearest minute. So an average within the lowest division of measure is excellent.
However, there’s a pretty large standard deviation of 1º which means we’re we’re within ± 4 minutes only ~68% of the time. That’s a pretty big spread. Here’s the histogram.
Overall, it’s a decent bell curve except for that hump at -0.83º to -0.53º which is likely a result of the error in alignment on 6/8 at Lilies. In fact, that single night of misalignment is responsible for almost all of the outliers. The left, 18 of the top 19 points were from that single night, as are the top 2 on the right. Removing that night, the error drops to 0.09º indicating we otherwise take measurements about 15 seconds late, with a standard deviation of 0.66º. In other words, one night of data swung the average by 0.21º and increase the spread by ~50%!
But there’s something else to notice about those most extreme points on the right: The top three are of stars in Ursa Minor which is the constellation Polaris is in. That means, these stars are making a rather small circle around the pole. Thus, the error in RA is magnified immensely if the quadrant is misaligned or the observer simply misjudges how centered the star is near the meridian even if it is aligned properly. The takeaway is that the closer something is to the celestial equator is, the easier it is to get the correct RA, and the closer to the poles, the harder. This shows pretty well in the data as well.
Again, this data includes the night of 6/8 which the quadrant was apparently misaligned meaning we have some pretty extreme outliers in there. But what happens when we take them out?
Now, we don’t see those extremes as well, indicating that on most other nights, alignment was relatively good. That’s reassuring.
Turning our attend to Declination, the error in Dec averages only 0.07º. Recall that the scale on the quadrant is only marked ever 0.10º, but it’s reasonably easy to read between each of the marks to read it down to 0.05º. So this amount of error is reasonably close to what we should expect. The standard deviation is lower here too as it’s only 0.35º.
So what sources of error creep in here? There’s a few that come to mind.
First, let’s consider The effect of incorrectly judging when the object is on the meridian. We can do this by looking at the error in Dec as a function of the error in RA:
Here, we can see something rather surprising: The largest errors tend to happen right around 0 error in the RA. I don’t really have a good explanation for this, but it’s clear that being further off in RA doesn’t make for a worse measurement in Dec.
Next, let’s look at the error in Dec vs altitude.
In this, we can see there’s very little trend. It’s slightly downwards, but that’s largely because of the top 3 outliers.
The first question this answers for me is whether or not we can see any errors that are caused by some viewing angles being more comfortable than others. Personally, I’ve felt that objects near the horizon are also hard because, when viewing objects that are higher up, I can reach with the other hand and steady the quadrant against the box for the plumb so I’m not having to try to hold it perfectly still while the observation is taken. But when observing something low on the horizon, the central column and box are too far away to do this on, so I’m having to try to hold perfectly still while the second person takes the reading. Often, this isn’t an issue, but some people have had trouble taking the reading and it means holding still for a prolonged period of time which is difficult when you’re trying to achieve higher precision.
The second question this could potentially shed light on is whether or not atmospheric refraction were playing any sort of important role. I know that this was something Tycho did try to take into consideration, but did so unsuccessfully as he treated the atmosphere as something that had a hard cutoff with consistent density beneath that point. Obviously, this instrument isn’t as accurate as his, so it’s questionable as to whether or not it should come into play. If it did, we should expect to see more of an error near the horizon. Again, there is that slight trend with higher errors towards the left, but I suspect this is more because of the few outliers than because of atmospheric refraction. However, I popped some numbers into an online calculator (10º Alt, 18º C, 921 mbar), and it says that the error should be in the range of about 4.7 arcmin, which is slightly higher than the average error in declination. So it’s entirely possible that this effect could become apparent with more data and better precision.
Next, let’s look to see if there’s more error on fainter stars. Not I’m going to use the absolute value of the error here.
Here, there’s very clearly a slight upwards trend, but it’s not terrible. If we can see the star, we’re pretty good at getting it sighted. This is somewhat surprising to me as many of the fainter stars have been hard for me to sight because they disappear when not using averted vision.
Lastly, are we getting any better?
Here, I’ve plotted the Dec error vs time. There’s a very slightly downward trend. But the weight on the left side is really driven by the two tall points and the one negative one. They effectively cancel each other out which makes the left side look a lot more steady than it would otherwise. In other words, aside from the overall trend improving, the spread is also getting better which is also nice.
So what are the takeaways?
First off, the error in the quadrant is right about what we should expect given the divisions on both RA and Dec.
However, the spread on RA could be improved significantly. The biggest immediate improvement based on this would be to really work on the alignment. I discussed some ways to potentially find true north better in this post, but haven’t done much with it. Perhaps I should. But even then, just being a little more patient and waiting until the star is really aligned instead of just in front of the width of the front end of the quadrant would help.
In terms of declination, we’re obviously shooting a little high. Being cognizant of that may help, but otherwise, we should keep on doing what we’re doing!