# Gear math

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In this article series, I'm trying to build a working equatorial mount for a 10" Dobsonian telescope.

This is just a quick update post to talk about the mount's gear ratios.

The mount has to rotate in sync with the Earth's rotation (albeit in the opposite direction to counteract it), so naturally I thought it would have to rotate once every 24 hours. But that sounds a bit too easy, doesn't it? I soon discovered, that instead of solar time, which we use in our daily lives, the mount has to rotate by sidereal time. So instead of 24 hours, each rotation should last approximately 23h56min.

That doesn't sound that different, until you start to do the math. 24 hours equates to a nice, round number, 86400 seconds (60s * 60min * 24h). 23h56min on the other hand, gives us 86164s. This becomes a problem as you start working with the mount's gear ratios.

Assuming the motor is rotating at 1RPS (rotations per second), according to WolframAlpha we would need the gear ratios 1:2 (7 times), 1:3 (3 times) and 1:5 (2 times) to get 86400 (2^7*3^3*5^2=86400). Nice and even. But when we look at the factors of 86164, we get to the problem. One of the factors of 86164 is 1657, which is an insanely large gear ratio. Assuming a 1:2 gear in our mount would be 5cm in diameter, a 1:1657 gear would have a diameter of 41.2m. Totally unfeasible.

I almost decided to go with the 86400s rotation period and suffer the minor accuracy problems, but I wanted to give raw computing power a chance. So I wrote a simple C script, which takes a desired rotation period and the largest allowed gear ratio, and returns a number closest to the desired period with acceptable gear ratios. It is basically just a glorified least-factors calculator.

I ran the script with the desired period of 86164 seconds, and a maximum gear ratio of 7, since that would give us a gear of 17.5cm in diameter, which is large but borderline acceptable.

```\$ ./optimalfactors 86164 7
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 7
86016 (diff: -148)
```

86016 seconds! While it still has an error of over 2 minutes (148 seconds), it is 37.2% more accurate than the 236 seconds of error with 86400s. So I've decided to go with 86016s as the mount's rotation period.

Here are some alternative option with different gear ratios:

```\$ ./optimalfactors 86164 4
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3
82944 (diff: -3220)
```

```\$ ./optimalfactors 86164 6
2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 5, 5
86400 (diff: 236)
```

```\$ ./optimalfactors 86164 10
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 7
86016 (diff: -148)
```

```\$ ./optimalfactors 86164 11
2, 2, 2, 2, 2, 5, 7, 7, 11
86240 (diff: 76)
```

```\$ ./optimalfactors 86164 17
2, 3, 5, 13, 13, 17
86190 (diff: 26)
```

```\$ ./optimalfactors 86164 19
2, 2, 2, 3, 3, 3, 3, 7, 19
86184 (diff: 20)
```

```\$ ./optimalfactors 86164 50
3, 13, 47, 47
86151 (diff: -13)
```

Building an equatorial mount for a Dobsonian telescope - Math and lasers (Part 4)

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