### Pixel Density & 500/600 Rule

In this tutorial I’d like to explain and mathematically prove that the so-called night photography 500-rule (or frequently used 600-rule) breaks and with current high-density-sensor DSLRs we can’t rely on it when taking pictures at night.

What is 500-rule, and how it’s used in photography?
The 500-rule simply states that to obtain clear with stars-as-dots shots at night we take number 500 and divided it by the focal length of the lens used to take the shot. For example, a 14 mm lens would allow us to expose for about 35 seconds at night and theoretically, still obtain the stars without trails. The math here is trivial. We divide 500 by 14 and we get little over 35, which is a number in seconds of our “maximum allowed” exposure time before the star trails show up. The 600-rule would allow us to expose for even longer, but interestingly when I look at some of my pictures taken at night exposed for just 35 seconds (or even less) no matter what ISO I’ve used I can clearly see that the stars are not as dots, and they do have small trails stretching over several pixels on the screen. How is this possible?

Pixel Density
Many modern DLRS are equipped with high-density sensors nowadays. It is hard to find a camera body with less than 20MP now. Bodies such as Nikon D4/D4s or Canon 1D X may have less than 20MP, but these cameras are normally used for sport and fast action photography, not landscape, so let’s not count them for now. The two most popular and frequently used cameras for landscape photography (as of now) are Canon 5D Mark III or IV with 30+ MP sensor, and Nikon D810 with its 36Megapixels sensor. There are also a medium format camera bodies that can have even more megapixels but again, these are mostly used for studio work cameras, and their steep price tag is a deal killer for many.
Still, no matter what camera brand and sensor format we choose (medium, 35mm FF, 24mm APS-C etc.) the formulas and calculations I’m about to show you will work just the same, and can be applied to calculate the maximum exposure time before star trails become visible.

Calculations
The fun begins now, but before I lay it all down we need to state some simple facts first, to understand what is being calculated and why. Earth rotates once every 24 hours but we need this value in seconds. To get that we simply multiply 24 * 3600, which is 86,400 seconds. We also need to know how many seconds Earth needs to rotate one degree. This equals to: 86,400 seconds/360 degrees = 240 seconds per degree. Since a sphere, such as our Earth, rotates at a constant angular velocity at any point over its surface this value is good anywhere in the world no matter what the latitude on Earth we will be shooting from.

From optics we know that the angle of view of a lens is calculated as:

Angle of view = 2 arctan (d/2f)

where,
d = diagonal of a sensor in mm
(For FF the diagonal is 43.3mm. Why? Applying the Pythagorean theorem on 36 x 24mm sensor we have: Square root of (36 squared + 24 squared) = 43.3mm)
f = focal length of lens in mm

For almost all my night shots I use a full frame Nikon D810 body along with my Nikon 14-24mm f/2.8 lens at 14mm. So lets see what is the angle of view of a 14mm lens on a FF sensor camera.

Angle of view = 2 arctan (43.3mm/2 * 14mm) = 2arctan (1.546) = 2*57deg = 114deg. This is exactly what Nikon reports on their website.

Now we have all the elements of equation to get the actual value of the exposure time in seconds. Since we have used the diagonal for the angle of view, lets also use it to get the number of pixels of my camera’s sensor. Nikon D810 produces images with resolution of 7360 (horizontally) x 4912 (vertically) pixels. The diagonal therefore is 8848 pixels. This means that within the 114 degrees of angle of view we will have 8848 pixels. We need to know how many pixels there is per single degree:

Pixels per degree = 8848 pixels / 114 degrees = 77.61 pixels/degree

As calculated above the rotation speed is 240 seconds per degree. But we already know how many pixels per degree we have for the specified focal length and sensor size. So, to obtain the max exposure time we simply cancel degrees out and substitute as follows:

Maximum exposure time = (240 seconds/degree) / (77.61pixels/degree) = 3.1 seconds/pixel.

So 3.1 seconds is the maximum “allowed” time that we can expose for and still get stars as dots. Anything larger than that will be captured on more than one pixel and will appear as a trail. The 500-rule let us expose for 35 seconds but in 35 seconds we will already have almost a 9 pixels long trail!

The bottom line is, don’t use 500-rule unless you shoot with a full frame 4MP sensor body, which doesn’t exists. If you want to shoot stars-as-dots or Milky Way and obtain great results get Astrotrac Star Tracker (I have no affiliation with the company but I strongly recommend it. It is very accurate and results can be astonishing. Google it if you’d like to know more about it). Here is an example of a 4 pixels long trail made by a 12MP Nikon D3 camera with exposure time of only 30 seconds.