Star Angles

published 2025-11-20

By Christopher Howard

We've had quite a few clear nights this last week or so, but I haven't been able to get out much, due to tiredness and scheduling restrictions related to my work schedule, family commitments, and church commitments. I saw a bit of green aurora for a few minutes, during that big solar storm recently, but was only able to stay outside for a few minutes before needing to drag my exhausted self off to bed.

One idea that I have been interested in recently is producing accurate star sketches using angular distance measurements. The basic idea is, if I can measure accurately the angular distance between any two arbitrary stars, then I can reproduce the exact shape of a group of stars on paper, either by calculating the angles with trigonometry, or just using a drafting compass.

The first question was how to measure the angles. I have a sextant, which is a very accurate tool for measuring angles, but unfortunately it doesn't work so well for pairs of arbitrary stars. The problem is that you have to get the two stars you want lined up next to each other inside the two tiny viewing windows. With that small FoV, it is hard to get them pointed in the right direction, and of course most of the stars look very similar, so you don't even know if you are star in the window is the one you meant to point at. This is a different use case of a sextant from the usual one where you measuring the angle between the horizon and some bright navigational star.

There is the old hand/finger span trick, but that method is not precise enough for my purposes.

Something I tried, one night, was just holding a ruler in my hand, perpendicular to my outstretched arm. I would record the distance in centimeters between various stars, and then later divide those numbers by the length of my arm, to get numbers I could convert into angles, using trigonometry. I also attach a red LED light to my arm, to illuminate the tic marks on the ruler.

Getting the angles is fairly easy to do using the T and S scales on my slide rule, but a bit tedious over a large number of measurements. So I wrote a little C program to read in the SSS length measurements and spit out the angles.

I realized later that it really isn't necessary to calculate the angles, if working with paper. One can just convert the angular distances to some scale of distance on the paper, and then use a drafting compass to draw circles and make the triangles from intersections. On the other hand, knowing the actual angle values does allow you to compare the angles you recorded against the angles in Stellarium — using the angle-measuring plugin — which is presumed to be highly accurate.

The arm and ruler approach was, however, not satisfactory for a few reasons. One is that my angle measurements were still coming out quite imprecise, vs. the Stellarium values. Generally my angles were smaller by 0.5-1.5 degrees that what Stellarium showed. Also, the ruler I was using was a highly reflective color, so that it made it difficult to see the stars next to it.

I have an idea to build a more accurate version of this, with some kind of angle ruler, in conjunction with a yard stick or something similar. I would want a black ruler so that not so much light is reflected back at me from the LED. It would be ideal to have a curved ruler at the end, so you could measure radians directly, but I'm not sure how I would go about getting or making one of those.

Another approach would be to measure the actual AZ/EL of the stars, and then just plot those values. This idea has appeal, but I'm not quite sure how to go about this in an inexpensive way. It is fairly easy to make an inexpensive inclinometer, to measure the elevation, but not so easy to measure differences in Azimuth. I think I would need to get a small telescope or spotting scope, and a good az/el mount with dials, which I don't currently have. I would think some kind of surveying transit might work, but those aren't cheap either, even on Ebay.

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CC BY-SA 4.0 Deed