This question mentions the "Rule of 600" for avoiding star-trails in astrophotography.
What is this rule?
How was it derived?
How should it be applied?
Stars move. Like with any other movement, what we care about is how much they move on the sensor during exposure: A movement that occurs only within a single pixel is not a movement the sensor can capture, i.e. the movement appears frozen.
But when movement takes a point across several pixels during the exposure, it will be visible as movement blur, in this case star trails. A rule like the "rule of 600" is similar in spirit to the "rule of 1/focal length" for handheld exposure, in that it attempts to give exposure times that yield approximately the same movement blur for most focal lengths.
The derivation is fairly simple:
By the 600 rule, those 8.5 pixels represent the maximum acceptable movement blur before star points turn into star trails. (That's what the rule says. Whether an 8-pixel smear is acceptable for a particular purpose is a different discussion.)
If we plug a 400mm lens into the same formulas, we get max 1.5 seconds exposure time and a 7.3 pixels movement during the exposure. So it's not an exact rule - the blur is slightly different for different focal lengths - but as a rule of thumb it's pretty close.
If we were using a 1.5x crop sensor with the same 24Mpx resolution (e.g. Nikon D3200) and used focal lengths to give equivalent angles of view, we would have e.g. 16mm focal length, 37.5 seconds exposure time and 12.7 pixels blur. That's 50% more blur.
In this case a "rule of 400" for the crop sensor camera would give the same blur as the "rule of 600" for the full frame example.
I suggest using "rule of 600" (or a stricter version with a smaller numerator) with the equivalent rather than the actual focal length, that way the rule gives the same results for smaller sensors. (E.g. 16mm on a 1.5x crop sensor is equivalent to 24mm on a full frame; use the "24mm equivalent" rather than the "16mm actual" focal length to calculate the max exposure time.)
Different stars move at different speeds relative to the Earth. The fastest movement is along the celestial equator, while the Pole star (Polaris for the Northern hemisphere) at the celestial pole hardly moves at all.
The effect can be seen in this picture from wikimedia commons: Polaris appears as a fixed point in the middle while other stars revolve around it, and the length of the star trails increase with their distance from Polaris.
The calculation above is for the worst-case scenario, when the picture includes stars that move along the celestial equator.
I guess the takeaway message is that the 600 in the "rule of 600" depends on camera resolution, sensor size, where in the sky you point the camera, and what you consider acceptable blur.
Use a smaller number if you want less blur.
Conversely, a higher number might be acceptable if you shoot a close crop of Polaris, use a low-resolution camera and/or target a low-resolution output format.
The rule of 600 states that to 'eliminate' star trails the exposure time in seconds should be 600 divided by the focal length of the taking lens. 20mm lens could go to 30 seconds, 300mm lens could go to 2 seconds.
Of course (like any motion blur) you will never eliminate star trails- you merely reduce the trail to an acceptable level for a given enlargement. The only perfect solution is a "perfectly aligned tracking equatorial mount" and there ain't no such thing.
The etiology is difficult if not impossible- it's sort of like 'Handhold no slower than 1/focal-length shutter speed'- a rule of thumb or common wisdom that works in many but not all cases.
A discussion of the pros and cons (and math) can be found here: http://blog.starcircleacademy.com/2012/06/600-rule/
An interesting and more general discussion of star trails can be found here: http://blog.starcircleacademy.com/startrails/
This rule applies to the shutter speed you should use when taking photographs of the night sky. The rule is as follows:
For example, if using a 300mm lens, if you use a shutter speed of (600/300) = 2s or shorter, you should avoid seeing the stars as lines, rather than points, of light.
As far as I can tell there is no record of who came up with the rule or how it was derived, however it would most likely have been based on trial and error using 35mm film, with the inherently lower resolution (grain) and lower tolerance (frame size) than today's cameras, and rounded up (or down) to a nice round 600.
As for application, care should be taken. Modern digital sensors are much sharper than 35mm film, meaning there is less tolerance when it comes to motion blur. Additionally, most digital cameras these days have smaller sensors than the 36mm x 24mm of 35mm film, meaning there is EVEN LESS tolerance, so it should probably be adjusted to be more like a 400 rule when using these cropped-sensor cameras (that is, if you think 600 is still a valid value for full frame cameras, which is arguable). Conversely, if using medium format cameras, a larger number could be used.