How to measure the length of a sidereal day from star trails

Here is a picture of star trails taken with a camera on a very long exposure setting. (about 30 mins)

pic D. Drumm

We can use this image to calculate how long it takes the Earth to rotate, i.e. the length of a sidereal day. Remember that this is 4 minutes shorter than an average solar day.

Basically, if it takes time T for the Earth to rotate 360⁰ and it takes time t for it to rotate θ degrees then, because the angular velocity (the number of degrees swept out per minute) is constant then;

We need to know the time over which the exposure was taken and we need to measure the angle that a trail (pick a big one) makes with the celestial pole. Then we can bung these numbers into the equation above and get a value for T.

I suggest you do all the time measurements in minutes for simplicity.

Example

The angle on the diagram above = 7.5 ⁰. The exposure time was 30 minutes.

So T = (360/7.5) x 30 = 1440 minutes.  (The actual time is 1436 minutes, or 23 hours and 56 minutes. Why is our answer slightly different?)