Demystifying the 500 "rule"

When we take photos of the stars, the earth's rotation about its axis causes the star's image to move on the photographic sensor or film, causing an undesirable elongation of star images. In order to minimize this elongation (trailing), photographers utilize the 500 "rule" to calculate the longest exposure time limit for a given lens focal length that minimizes star trailing.

Using the 500 "rule", I used a Sony A7S coupled to an 85mm lens with an exposure of 5 seconds to prevent star elongation.

The 500 "rule" is simple to use, just take 500 and divide by the full frame effective focal length of a lens, and what we'll get is an estimate of the longest exposure time limit (in seconds) for the given focal length. For example, if we are using a full frame camera with a 50mm lens, the 500 "rule" will give an upper exposure time limit of 500 ÷ 50 =10 seconds.
If we are using a cropped sensor camera, we must remember to convert to the full frame equivalent focal length utilizing the crop factor (1.5 for APS-C, 2 for micro-4/3) 

Here's where it gets interesting, if we "broke" the 500 "rule", we may discover that sometimes the stars in our photos would turn out alright (not elongated). For example, the photo below titled “The Starry Night Boat” was taken with an Olympus EM5 Mark 2 (micro-4/3) coupled to a 12mm f2 lens. The 500 rule combined with the crop factor would give me an exposure limit of 500÷(12×2) = 20 seconds, however, my exposure for the photo below is 40s, why are the stars not elongated? 

Starry night boat – Olympus EM5 Mark 2 + Zuiko 12mm f2，40s f2 ISO800

Actually this is no mystery because the 500 "rule" is intimately related to a star's angular velocity across the sky and across a lens field of view. The 500 "rule" is merely an estimate and not a real mathematical rule with any form of generality. In fact, if one demands the highest precision in one's image, or if for instance if one owns a high megapixel count camera, the 500 "rule" may have to be revised to a "300" rule or even a "200" rule!

A star's position (distance from the celestial poles) in the sky determines the applicability of the 500 rule. If we look at photos of star trails below, stars closer to the celestial poles (the North Star Polaris or the south star Sigma Octantis) move less over the course of time compared to stars further away from the celestial pole. In fact the stars at the celestial equator moves the most for a given amount of time. As a result, we can actually “break” the 500 rule for stars nearer the celestial poles.

And that is why in “Starry Night Boat”, I was able to pull off a 40s exposure without trailed stars. The region of the Milky Way in the photo is quite far from the celestial equator, almost the southernmost portion of the Milky Way, as a result, I could break the 500 “rule”.

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Geek Zone (enter at your own risk)

As indicated by the diagram below, the 500 “rule” is built upon the concept of how large is a single star’s image on a camera sensor. The time at which the earth’s rotation causes a star’s to increase by a specific dimension (usually about one pixel) gives the 500 “rule” limit.

Let’s assume that a star has to move an angle of α degrees across the sky at the minimum limit of stellar “elongation” on the image sensor, then we can write the equivalence:

α/360= p/2πf = t/24h

Where p is the pixel dimension of the star on the sensor (in millimeters); f is the full frame (35mm) equivalent focal length in millimeters; t is the exposure time limit in seconds and h = 3600 seconds (1 hour);

If we rearrange the equivalencies, and consider a star that has an angular distance θ from the celestial equator, we get:

t = (p×24×h/2π) ÷ (f⋅cosθ) = (p×86,400/2π) ÷ (f⋅cosθ)

Which gives the time limit for which we can expose for a star with minimal elongation on the sensor image. The time limit is found to be equal to the terms in the bracket p×86,400/2π divided by f⋅cosθ, where θ is the angular distance of the star from the celestial equator. 

Assuming we are imaging a star that is right at the celestial equator, then cosθ = cos(0) = 1, and let’s assume p to have a value of 0.035mm, then the term p×86400/2π gives a value close to 500 resulting in what we familiarly call the 500 “rule”.

It is helpful to note that the value of p depends on the pixel size of the sensor, and higher megapixel count cameras (smaller pixels) will have a more stringent 300 “rule” or even 200 “rule”. In addition, since atmospheric conditions also affect the size of star images on the sensor, the value of p also depend on atmospheric/seeing conditions (as well as lens resolution/quality etc) 

(By Mr Wooihou Chan & Dr Derrick Lim)