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This is a backyard astronomy question.

My middle-school-aged son and I would like to measure the apparent angle between two easily-visible stars more precisely than measuring with our outstretched hands and fingers. Our current technique gives measurements in fingers instead of degrees, and is really inconsistent depending on how we need to twist our arms, and his arms seem to be growing longer!

What are some inexpensive ways we can get a better measurement of the angle between two stars?

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    $\begingroup$ "his arms seem to be growing longer!" But his fingers should be getting thicker to compensate. $\endgroup$
    – Graham Nye
    Commented Mar 15, 2022 at 20:24
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    $\begingroup$ @GrahamNye kids' growth isn't uniform though. They may, for example, grow out then up. As an example, mine gets hungrier in advance of a (height) growth spurt, enough that we can tell it's coming. $\endgroup$
    – Chris H
    Commented Mar 16, 2022 at 14:27

3 Answers 3

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A tool such as Jacob's staff or a cross-staff can be used. This is essentially two pieces of wood in a cross shape, one of which can slide on the other.

enter image description here
By Original: Fantagu Vector: Majo statt Senf - Own work based on: Jakobsstab3.jpg, CC BY-SA 4.0

By aligning the arms of the cross to the two stars you can find the angle between them.

A practical variation of this is the Sky Ruler (from BBC Sky at Night magazine). This is marked in degrees, so you don't need to do any trigonometry, and is made of card so you don't need carpentry skills.

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    $\begingroup$ +1, I made one of these for an astronomy workshop, and if I remember correctly we reached a ~1 arcmin precision. $\endgroup$
    – pela
    Commented Mar 15, 2022 at 8:19
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My smartphone takes 4096x3072 photographs that are about 60 degrees wide. That comes to about 68 pixels per degree, or a little less than one arcminute per pixel.

You can calibrate this by taking a picture of the moon, which is about 30 arcminutes wide. You can calibrate more accurately by finding out exactly how far away the moon is today and how big it is in the sky.

Take a picture and count the pixels. You'll need a mount to hold it steady for a dark shot.

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The classic tool for such a measurement is the sextant, which is more precise than the cross-staff mentioned in another answer.

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    $\begingroup$ A sextant is designed for handheld use on a rocking, moving boat. The more accurate, cheaper, and less complicated instrument for land use is a theodolite or transit. Today, of course, we could just use a telescope with digital setting circles. But a cross-staff is a good, cheap, and easy way to do it. $\endgroup$
    – Greg Miller
    Commented Mar 16, 2022 at 1:42
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    $\begingroup$ @GregMiller Hmm, my understanding is that a theodolite measures angles in a horizontal or vertical plane. This would require more work and calculation to measure a general oblique angle between two stars. A sextant can directly measure oblique angles, such as lunar distance. $\endgroup$
    – nanoman
    Commented Mar 16, 2022 at 3:11
  • $\begingroup$ It usually is, but you can tilt a theodolite just like you can a sextant. It isn't very common, mostly because navigation on land is much easier done with a map and landmarks. $\endgroup$
    – Greg Miller
    Commented Mar 16, 2022 at 13:46
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    $\begingroup$ Sextant is much cheaper than a theodolite. Both are out of range of a middle school project budget. Previous, deleted comment had theodolite cheaper. $\endgroup$
    – stretch
    Commented Mar 17, 2022 at 19:24

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