**NIGHT PHOTOGRAPHY AND 500-RULE**

In this tutorial, I would like to explain and mathematically prove that the so-called “night photography 500-rule” (or frequently used “600-rule”) is no longer applicable with current high-density sensor DSLRs or mirrorless camera bodies. The rule, designed to achieve clear images with stars represented as dots when shooting at night, is found to be ineffective and breaks down due to advancements in camera sensor technology.

#### What is 500-rule, and how it’s used in photography

The 500-rule is a guideline used in astrophotography to determine the maximum exposure time for capturing clear images of stars without noticeable trailing. The rule states that you should divide 500 by the focal length of the lens used to take the shot. For example, with a 14 mm lens, the calculation would be 500 / 14, resulting in approximately 35 seconds of “maximum allowed” exposure time before star trails become visible.

The rationale behind this rule is to prevent the Earth’s rotation from causing stars to appear as streaks during long exposures. By limiting the exposure time based on the focal length, photographers aim to capture pinpoint stars rather than elongated trails.

#### Pixel Density

It’s true that many modern cameras come equipped with high-density sensors, with resolutions often exceeding 24 megapixels. Leading camera manufacturers such as Nikon, Canon, Fujifilm, Hasselblad, and Sony have embraced sensors with 45 megapixels or more. Even medium format cameras from Hasselblad and Fujifilm now boast sensors with 100 megapixels or higher.

Despite the variations in sensor sizes and resolutions among different camera brands and formats (medium format, full-frame, or APS-C), the fundamental principles of astrophotography calculations remain consistent. Formulas such as the 500-rule or 600-rule can be applied uniformly across these diverse systems to estimate the maximum exposure time before stars begin to exhibit noticeable trails.

In essence, the relationship between focal length, sensor resolution, and exposure time remains relevant, allowing photographers to adapt these calculations to a wide range of contemporary camera setups. However, as previously mentioned, the effectiveness of these rules may be influenced by other factors such as sensor technology, noise reduction algorithms, and image stabilization mechanisms that have evolved with advancements in camera technology. As such, it’s essential for photographers to test and adjust their techniques based on real-world results, especially when working with high-density sensors.

#### What do we need to perform the calculation

Indeed, for accurate astrophotography calculations, obtaining the GPS coordinates of the location where the picture is taken is crucial. While the pixel count and lens focal length are readily available from camera specifications, GPS data is often obtained through one of the following methods:

- Built-in GPS Unit: Many modern cameras come equipped with a built-in GPS unit. This feature automatically records the GPS coordinates of the location where the photo is taken and embeds this information in the image’s metadata.
- Mobile Applications: Some cameras can pair with mobile applications that utilize the GPS unit in a connected smartphone. The app retrieves GPS coordinates and embeds them into the photo’s metadata.
- Web Browser Approximation: If GPS data is not directly available, web browsers can be used to approximate the location. While this method may not be as precise as GPS coordinates obtained from a dedicated unit, it can provide an acceptable level of accuracy for astrophotography calculations. Browsers often provide both latitude and longitude, even though only latitude may be needed for these calculations.

#### Calculations

Before I lay it all down, I need to state some simple facts first, to understand what is being calculated and why. Our Earth’s circumference at the equator is exactly 24,901.55 miles (or 40,075.15 kilometers). 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. However, this value is good only at the equator. To be more precise we need to use some trigonometry to get it anywhere in the world. Knowing it at the equator will help us so all we have to do now is divide it by the cosine of the approximate latitude your picture will be taken at At the equator the angle is zero and Cosine of zero is one so the 240 calculated above will not change when we shoot at the equator.

Earth’s Rotation = 240 seconds/cosine (latitude in degrees).

I have written an iPhone application for photographers downloadable via iTunes store for free called Photos Master that will show you your longitude and latitude. I live at around 33 degrees north so for me the Earth’s Rotation would be:

Earth’s Rotation = 240 seconds/Cosine (33 degrees) = 240/0.838 = 286.4 sec/deg.

As you can see Earth rotates slower at latitude 33 degrees and therefore requires more time to rotate one degree. This should not come as a surprise because we know that the larger the radius of a circle, the greater the speed.

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 Z9/Z8 body along with my Nikon 14-24mm f/2.8 S lens at 14mm. So, let’s 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 Z9/Z8 produces images with resolution of 8256 (horizontally) x 5504 (vertically) pixels. The diagonal therefore is 9922 pixels. This means that within the 114 degrees of angle of view we will have 9922 pixels. We need to know how many pixels there is per single degree:

Pixels per degree = 9922 pixels / 114 degrees = 87.03 pixels/degree

As calculated above at the altitude of 33 degrees I live at, the Earths rotation speed is 286.4 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 = (286.4 seconds/degree) / (87.03 pixels/degree) = 3.29 seconds/pixel.

So 3.29 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. Since the 500-rule would allow to expose for 35 seconds the star trail would be 9 pixels long!