Pre-visualizing star trails

Last weekend I had the chance to make my first attempt at photographing star trails. The idea of this type of shot is to point your camera at a fixed location in the night and make an exposure long enough so that the stars will appear to move across the sky as the Earth rotates.  Making star trails can be fun and gratifying but it requires a certain amount of commitment on the part of the photographer.  For best results, you need a dark, clear sky, devoid of light pollution. This means you'll likely have to travel far from your home and you may have to get up late at night after the moon has set. Making the exposures also requires a fair amount of time. A single image can require a cumulative exposure time of an hour or more. For these reasons, it's helpful to know what your shots are likely to look before you invest time and energy making start trails. This is not a situation where "chimping" and correcting is practical. In this blog post, I provide some help for pre-visualizing star trails. I hope this information will help you make the star trail shots you expect, with a minimum costly trial and error.

The shape of star trails

If you put your camera on a tripod, point it at a dark sky and hold the shutter open for a few seconds, you'll make a lovely image of some stars.  If you hold the shutter open longer, the stars will appear to streak across the sky in the opposite direction of the Earth's rotation. The shape of the streaks depends on the direction your camera is pointed. To explain how, I'll first need to say a few words about coordinates.

There are two standard systems for expressing locations in the night sky: horizon coordinates, and equatorial coordinates.  Horizon coordinates are undoubtedly more intuitive for people who spend most of their time navigating on the Earth's surface, but astronomers have good reasons for preferring equatorial coordinates. Under the horizon coordinate system, directions are expressed as compass headings (either N.E.W.S. or degrees clockwise from due north) and elevation is expressed in degrees above the horizon.  Under the equatorial system, locations are expressed in terms of declination (degrees above or below the celestial equator) and right ascension (hours, minutes, and seconds off the vernal equinox). The equatorial coordinate system is conceptually similar -- and indeed very closely related to -- the system of latitudes and longitudes used for fixing terrestrial locations. If you're comfortable working with equatorial coordinates then you probably already know everything else I'm about to say, so I'm going to express things in terms of horizon coordinates wherever possible.  In a few cases, we will have to think of the camera position in terms of declination, however.

The celestial equator is an imaginary circle that lies directly above the Earth's equator, way, way out it space.  A star that lies on the celestial equator will appear to make a straight-line track across the sky as the Earth rotates.  If you live in the northern hemisphere and look due south, the celestial equator runs east to west at a particular angle above the horizon.  The farther north you live, the closer will be the celestial equator to the horizon.  On the other hand, if you happen to live on the Earth's equator, then the celestial equator will lie directly above you. If you live in the northern hemisphere and know your latitude, finding the location of the celestial equator is easy.  Simply subtract your latitude from 90 degrees to get the angle of the celestial equator above the horizon when facing south. This also works in the southern hemisphere, but you'll have to face north.

OK, let's suppose you live in on mid northern latitude and you're facing south. Stars that lie above or below the celestial equator will trace arcs that bend away from it. Any star lying between the celestial equator and the horizon will appear to trace a concave path, while a star lying above it will trace a convex path.

Now turn around and face north.  Can you see the pole star?  It's a fairly bright star, between 30 and 60 degrees above the horizon, at the end of the the Little Dipper. It just so happens that that star lies directly above the north pole. It won't appear to move as the Earth rotates, but all other stars in the northern sky will trace circles around it. Once again, knowing your latitude helps.  Polaris, the aptly named pole star, is located due north at an angle above the horizon equal to your latitude.

Can you visualize the pattern yet?  If you look east, stars will rise at the horizon and launch themselves overhead, tracing paths that arc away from the celestial equator.  If you face west, the stars will appear to crash toward the horizon in a similar pattern.  The figure below sketches the shapes of star trails for each of the four primary compass headings, assuming you're located in a mid norther latitude.  If you live below the equator, simply reverse the compass headings.

Shape of star trails from a mid-northern latitude

The length of the trails

The length of the star trails that appear in your photograph depend on the duration of your exposure, the position of your camera, and the angle of view of your camera.  The camera's angle of view, in turn, depends on the focal length of your lens and the size of your image sensor.  With the help of a little trigonometry, one can express the length of a star trail as a percentage of the image frame.  This will allow you to calculate, for example, how long of an exposure you would need to make star trails that stretch across one-quarter of your image frame when using a 50mm lens on a crop sensor (APS-C) camera body.

First, we need to express the distance that a star appears to move through the sky as a function of time. If you looked northward from a point above the arctic circle in wintertime, you could follow the path of a star as it appeared to trace a circle around the pole star. It would take one day (roughly 86,400 seconds) for a star to complete a 360 degree circuit.  The relative radius of the circle traced by the star depends on it's angle above the celestial equator (it's declination).  As I've already mentioned, the poll star (declination: 90 degrees north) itself does not appear to move.  At the other extreme, the stars that appear to cover the most distance through the sky are those that lie on the celestial equator (declination 0 degrees).  If we normalize the radius of a circle made by a star on the celestial equator to be one, then the length of its circle is 2*pi and it covers a distance of 2*pi/86,400 per second.  Stars that lie above the celestial equator trace smaller circles.  The length of the radius of the circle traced by a star at declination THETA degrees above the celestial equator is equal to cos(THETA*pi/180). (Note that the bit inside the parenthesis simply converts from degrees to radians.) So the distance covered per second by a star at declination THETA is

(1)   D/S = cos(THETA*pi/180)*2*pi/86,400

where D is the distance traveled in radians (recall that 2*pi radians = 360 degrees) and S is the duration of the exposure in seconds.  Unfortunately, if you aren't into astronomy, it probably won't be easy to determine your camera's declination (THETA) directly, although there are apps that can help.  If you happen to have a sextant handy, then you're golden.  Otherwise, you can get pretty close by triangulating between the celestial horizon (THETA = 0) and the pole star (THETA = 90), both of which I've already told you how to find.  Here's a diagram that might help.

Fixing declination from a northern latitude

We need to compare equation (1) to the angle of view of your camera.  The angle of view of a camera frame is equal to

(2)   A = 2*arctan(C/(2*F))

where C is the size of the image sensor (in millimeters) and F is the lens focal length (also in mm).  If you know the crop factor of your camera, you can compute the size of your image sensor by dividing 35 (the dimensions of a full frame sensor) by the crop factor.  For example, APS-C cameras have image sensors that are about 22 mm wide on their long axes (35/1.6 ~= 22).

The length of a star trail expressed as a percentage of your image frame is L = D/A, or, by substitution

(3) L = S*cos(THETA*pi/180)*pi / (86,400*arctan(C/(2*F))

where

S = exposure time in seconds,
THETA = camera declination in degrees above the celestial equator,
C = sensor size in mm, and
F = lens focal length in mm.

You can plug this formula into a spreadsheet and do the calculation for any combination of declinations, and camera/lens configurations.  But if you'd rather not, here's a helpful table for typical situations. It should get you in the ballpark of what you're looking for.  It reports the length of a star trail as a percentage of your image frame produced by a one minute exposure. To use it, look up the percentage associated with your image sensor type (full frame or crop APS-C sensor), your lens focal length, and the declination where your camera is pointed.  Then figure out how long an exposure you need to get the look you want.  For instance, suppose I'm using a crop sensor camera wearing a 50mm lens.  I've decided on a composition with the camera pointed 20 degrees above the celestial equator and I'd like each star trail to extend about a quarter of the length of the frame.  The table above tells me that a one minute exposure will produce a star trail that spans about 0.95 percent of the frame, so to span 25 percent of the frame I'd need about a 30 minute exposure time.

Crop Frame (APS-C)
	Focal Length
Dec.	10	20	50	90
-20	0.25%	0.41%	0.95%	1.69%
0	0.26%	0.43%	1.01%	1.79%
20	0.25%	0.41%	0.95%	1.69%
40	0.20%	0.33%	0.77%	1.37%
60	0.13%	0.22%	0.50%	0.90%
80	0.05%	0.08%	0.17%	0.31%
90	0.00%	0.00%	0.00%	0.00%
Full Frame (35mm)
	Focal Length
Dec.	10	20	50	90
-20	0.19%	0.29%	0.61%	1.07%
0	0.21%	0.30%	0.65%	1.14%
20	0.19%	0.29%	0.61%	1.07%
40	0.16%	0.23%	0.50%	0.87%
60	0.10%	0.15%	0.32%	0.57%
80	0.04%	0.05%	0.11%	0.20%
90	0.00%	0.00%	0.00%	0.00%

Putting it all together

The next time you try to make star trial I suggest you use this information to do a little pre-planning.  You'll want to check the weather, of course, and make sure the moon won't be bothering you.  But I also suggest you spend some time thinking about the image you want to create.  For arcing trails, roughly parallel to the horizon, plan to point your camera south (or north if you live in the southern hemisphere).  For diagonal trails, plan to face east or west.  And for circles, point your camera toward the pole star.  For short trails that look like comets, use a relatively short exposure and a relatively wide lens.  if you're facing south you'll have to use a shorter duration and/or a wider lens than if you're facing north. You can use the formula or table above to determine the exposure time that matches the look you're trying to achieve.

Finally, let me add a brief comment on technique.  The simplest approach to making star trails is to set your camera for a single very long exposure. While this is the simplest approach and one which makes a lot of sense if you're shooting film, it probably isn't the best option for people using digital cameras.  Instead, I strongly recommend you take a sequence of shorter exposures and then combine them in post using image staking software such as StarstaX. It will help to reduce sensor noise which can build up in longer exposures, and it will give you the option of making shorter star trails by stacking fewer frames if you decide you don't like your long ones after the fact.  Also, if you bump your tripod nine-tenths of the way through your planned exposure, the data you've already captured won't be ruined.

The photo up top was a stacked composite of 217 exposures of 25 seconds each taken at 30 second intervals for a total exposure time of 106 minutes.  I used a 17mm lens set to f/3.8 on a crop frame body at ISO 400.  The camera was pointed south-southeast, angled upward about 40 degrees.

The photo below was taken at the same location.  It was a composite of 114 frames of 15 seconds each, taken at 20 second intervals for a total exposure time of 38 minutes.  I used the same camera/lens combination, this time set to f/2.8 and ISO 400.  The camera was facing east-northeast angled up about 10 degrees.