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Mathematics behind the barn door tracker

Let’s look at a bit of math behind how the barn door does it’s magic. This article is specifically about the “isosceles” barn door tracker. This is the only type of tracker considered in this site. Consider the diagram below. The barn door tracker has two arms:

  • A moving arm which carries the camera (AC in drawing below)
  • A fixed, stationary arm (AB in drawing below).

The moving or camera arm with the camera is connected to the fixed arm through a hinge. The aim is to make the camera arm rotate about the fixed arm at exactly the same speed at which the earth rotates.

The speed of earth’s rotation

  • The earth takes 23 hours 56 minutes 4 seconds to rotate around it’s axis.
  • Expressed in minutes, it is approximately 23×60 + 56 = 1,436 minutes
  • One rotation is equal to 360°
  • Implies speed of earth’s rotation : 360°/1,436 minutes ~ 0.25° per minute ~ 15° per hour.

Neutralizing the earth’s rotation

The primary goal of the barn door tracker is to rotate the camera arm at the rate of 0.25° every minute to neutralize the effect of earth’s rotation. That is, increase θ in the diagram below at the rate of 0.25° every minute.

Mechanically this is achieved by increasing the length ‘S’ with the help of a screw connected to a motor. When S increases, the angle θ increases. By increasing the length S at a precise calculated rate, the angle θ can be increased at 0.25° per minute, our goal. Let us see how this speed is arrived at.

We know that θ has to increase at 0.25° per minute. And to increase θ, we need to increase S. Let us derive the relation between S and θ.

△ ABC is an isosceles triangle. This is because both arms AB and AC are of the same length. To make our math simpler, let’s divide the  △ ABC into two equal parts separated by the dotted line. The two new triangles are right angle triangles. Let’s apply some basic trigonometry:

sin(θ/2) = (S/2)/L
sin(θ/2) = S/2·L
S = 2·L·sin(θ/2)

Let us calculate the required increase in S every hour for tracking (approximately).

At the beginning S = 0 mm, θ = 0°
After 1 hour, θ should be 15° (earth’s rotation speed)
S = 2·L·sin(θ/2) per hour
S = 2·L·sin(15/2) per hour
S = 2·L x 0.1305 per hour
S = 0.261·L per hour

Note: Earth’s rotation is not exactly 15° per hour. Remember earth takes 23h 56m 4s to make one rotation. This translates to 15.04° per hour. We will use this more accurate speed in the software we would write later.

As an example, if we had chosen the arm length L = 300mm in the design, S = 78.3mm at the end of 1 hour of tracking. See below:


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