[This is just one of many articles in the author’s Astronomy Digest.]
The image below is that of the Orion Constellation taken using a Sony A7S full frame, 12 megapixel, camera mounted on a fixed tripod. A 50 mm focal length lens was used stopped down to f/4 and an ISO of 800 was used. The exposure length was 10 seconds as determined by the ‘500’ rule for determining the maximum exposure length that can be used without significant star trailing being caused. It was taken as a test of this simple rule which is discussed below along with more accurate, but more complex, varients.
If a camera is used on a fixed tripod then, due to the Earth’s rotation, the stars being imaged will move across the camera’s sensor. The result is to cause ‘star trailing’. The 500 rule is used to determine the longest exposure that one can use with a given focal length lens so that this is prevented. One simply divides the focal length of the lens into 500. The rule applies to a lens mounted on a full frame camera. If, for example, the lens is mounted on an APS-C sensor camera the actual focal length has to be multiplied by 1.6 for a Canon camera and 1.5 for all other cameras. For a Micro 4/3 camera the factor is 2. This is called the ‘crop factor’. So, for example, a 35 mm focal length lens mounted on a Nikon or Sony APS-C sensor camera has an effective focal length of 52.5 mm and so the maximum exposure time would be ~10 seconds.
I think an easier way to consider this is to use instead; 350 for cameras with APS-C sensors and 250 for Micro 4/3 cameras.
The motion of the stars across the sky (and hence sensor) is greatest at the celestial equator and least at the celestial poles. So the basic time given by the 500 rule can be increased by a factor of 1/Cos(Dec) – the declination of the constellation. For example, Cassiopeia is at Dec 60 so Cos(Dec) is 0.5 and 1/Cos(Dec) is 2, so the exposure time could be doubled.
So the ‘improved’ 500 rule is:
T = 500 / ((focal length ) x Cos(Dec)) for full frame cameras
T = 350 / ((focal length ) x Cos(Dec)) for APS-C cameras
T = 250 / ((focal length ) x Cos(Dec)) for Micro 4/3 cameras
Though the 500 rule could just be an empirical rule that ‘just’ works it can, however, be justified theoretically.
1) It is determined by the Earth’s rotation which, at Dec Zero is 15 arc seconds per second.
2) The aperture of the lens which determines the size of the Airy Disk. Using a 50 mm lens at f/4, the aperture is 12.5 mm and the resolution ( ~size of the Airy Disk) is 9 arc seconds.
3) The atmospheric ‘seeing’ – typically 2 – 4 arc seconds.
From 2 and 3, the size of a stellar image on the sensor will be around 10 – 12 arc seconds.
4) The pixel size of your camera which, typically, could be from ~4 to 8 microns in size. For the Sony A7S, 12 Megapixel full frame sensor camera this is ~8 microns = 0.008 mm. If we now assume a 50 mm lens, each pixel subtends ((0.008/50) x 57.3 x 3,600) arc seconds. This is 33 arc seconds.
Let us suppose we are using this camera and 50 mm lens to image the Orion Constellation. Orion lies at Dec zero so one simply divides 50 into 500 to get the suggested exposure time of 10 seconds. During this exposure, the stars will have moved 10×15 = 150 arc seconds so the star’s image will have moved across 4.5 pixels but, as the stellar image is just over 1 pixel in size we might expect it to be elongated over 5 to 6 pixels. This is what is observed in the top left image of Figure 1 which is a single raw frame processed in Affinity Photo. The colours could be scintillation effects as Salph, the lower left of the Orion bright stars, was at quite a low elevation. The upper right image is the aligned stack of 100 frames. The colour effects have nicely integrated out but the stellar image is larger.
Trailing is apparent in both images. It is quite easy to correct for slight trailing as in this case. In Photoshop or Affinity Photo (a superb low cost program) simply duplicate the image to give an upper layer. Choose ‘Darken’ as the blending mode. Click on the ‘Move’ tool and move over to the image and click again. (Otherwise the blending mode may change.) Then, using the arrow keys move the top layer over the bottom layer and observe how the star trailing is reduced. Flatten or Merge the two layers. In the case of the single frame a move of 2 pixels gave the roundest image but, in the case of the stacked image, just one.
So, in this case, using a camera with a low resolution sensor employing a large pixel sensor, the 500 rule has worked pretty well.
With a high resolution sensor, one could well need to use what might be called the 400 or 300 rule instead as the trailing will be more evident. However, as a little star trailing is easy to correct, it may be better to use the 500 rule and correct the star trailing in post processing.
More complex formula
Frédéric Michaud of the Astronomical Society of Le Havre has studied this problem extensively:
Using Google Chrome, right clicking on the page allows it to be translated into English.
He has produced formula which take into account the aperture of the lens – which will determine the size of the Airy Disk for a stellar image on the sensor – and the pixel size of the sensor – smaller pixels will make star trailing more apparent.
There are several versions of increasing accuracy. The simplest is:
100 x (4 – crop factor ) / ( focal length x cos(dec))
For a full frame camera this is simply the ‘300’ rule and so gives an exposure of 6 seconds with a 50 mm lens at dec(0).
This seems quite conservative.
A more complex version is:
(35 x Aperture) + 30 x pixel size) / (focal length x cos(dec)) where the pixel size is in microns.
For a full frame sensor with 8 micron pixels this gives 8.2 seconds with a 50 mm focal length at an aperture of f/4.
This is almost the same as the most complex formulation which gave an exposure of 7.8 seconds.
The website includes a built in calculator to derive the maximum exposure time for any camera/lens/aperture combination.