{"id":1327,"date":"2017-06-12T20:27:53","date_gmt":"2017-06-13T03:27:53","guid":{"rendered":"https:\/\/partofthething.com\/thoughts\/?p=1327"},"modified":"2024-04-11T07:59:07","modified_gmt":"2024-04-11T14:59:07","slug":"making-a-cheap-and-simple-barn-door-star-tracker-with-software-tangent-correction-for-astrophotography","status":"publish","type":"post","link":"https:\/\/partofthething.com\/thoughts\/making-a-cheap-and-simple-barn-door-star-tracker-with-software-tangent-correction-for-astrophotography\/","title":{"rendered":"Making a cheap and simple barn-door star tracker with software tangent correction for astrophotography"},"content":{"rendered":"\n

I like to mix hobbies, so naturally I’ve been eying astrophotography for a while. I’ve taken a time-lapse here and a moon picture there but, inspired by the folks over at \/r\/astrophotography<\/a>,  I wanted to take it to the next level. Since the Earth is spinning, any long exposure of the night sky has star trails, so you have to make your camera counter-spin if you want clear shots. In this post, you can read about how I made a simple barn door sky tracker to do this.<\/p>\n\n\n\n

Barn door sky trackers<\/a> have been made at home by lots of people for a long time. There are a variety of designs with different levels of complexity and precision required. I thought I’d make the simplest-to-construct one, a Haig mount. To correct he tangent error, I decided to use a cheap microcontroller (MCU) and have it speed up appropriately via software. Fun!<\/p>\n\n\n

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With\/without tracker<\/figcaption><\/figure>\n<\/div>\n\n\n

The Math<\/h2>\n\n\n\n

The math behind this is fun mostly because it’s straight out of high school and you finally at long last get to use it. Here’s the basic design:<\/p>\n\n\n\n

\"Cartoon<\/a><\/figure>\n\n\n\n\n\n\n\n

The angle must increase at a constant rate based on the rotation of the Earth. We all know the rate the world turns so we can get explicit expressions for the angle as a function of time.<\/p>\n\n\n

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\"\"<\/a><\/figure>\n<\/div>\n\n\n

Since the Earth is also rotating around the Sun, it turns out that it actually takes more like 23 hours, 56 minutes, 4.0916 seconds for the stars in the sky to rotate all the way around. This is called the Mean Sidereal Day<\/a> and is the value we’ll actually use in this system. As you can now calculate, the constant angular velocity we need this thing to open at is 7.292115e-05 radians per second.<\/p>\n\n\n\n

Now we are faced with a classic word problem: “What rate should we spin the motor to maintain a constant angular velocity in the above triangle?” First let’s just see how fast the screw has to rise. Bust out some trig to get an expression for y(t) and then take its derivative (don’t forget the chain rule!) and you’ll find:<\/p>\n\n\n

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Ok, now we just have to convert that to a rotation of the screw. The threaded rod I got at Hero Ace downtown is a 1\/4-20 bolt, where the 20 means “there are 20 threads in 1 inch,” or in other words, “there are 20 rotations per 2.54 cm of insertion = 7.874 rotations\/cm.”  Adjust accordingly if you have a different screw thread.<\/p>\n\n\n\n

Let’s plot the rotations per minute required to match the Earth’s rotation for varying values of L my multiplying dy\/dt [cm\/minute] by 7.874 [rotations\/cm] for 90 minutes.<\/p>\n\n\n

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\"Rotation<\/a><\/figure>\n<\/div>\n\n\n

Note: You can check out the Python code used to do these calculations here<\/a>.<\/p>\n\n\n\n

As you can see, the rate changes with time! This is called “the tangent error” and it’s caused by the fact that the acorn nut is sliding along the hypotenuse as the “barn door” opens (see schematic above). There are a bunch of ways people have dealt with it. One is to use a curved screw. Another is to put a specially-shaped cam where the acorn nut contacts the top board. But for this project, I’m simply going to speed up the motor as time goes on if I do a very long exposure. Not having a proper shop, I prefer to use simple physical construction and deal with it in code. If you’re doing 5-10 minute exposures, this issue won’t get to you. Note also that if you have a constant 1 RPM motor, a 29cm L is about right for you.<\/p>\n\n\n\n

Putting it together<\/h2>\n\n\n
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\"ACE<\/a><\/figure>\n<\/div>\n\n\n

I bought some stuff at the local ACE hardware store to put this together. They had pretty much everything I needed in the hardware department. <\/p>\n\n\n

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\"Picture<\/a><\/figure>\n<\/div>\n\n\n

My steps:<\/p>\n\n\n\n