Wide Field Astrophotography
Wide field Astrophotography usually involves the use of a camera with a normal camera lens and not coupled to a telescope optically. There are at least three ways to use a digital camera for wide field astrophotography. One is simply using a tripod and pointing at the sky. The second is mounting the camera "piggyback" on a telescope and third is by using an equatorial mount or barn-door tracker. The latter are methods to avoid star trails or capturing the movement of the earth. For this first part we will focus on Tripod Astrophotography and avoiding the star trails by selecting subjects and shutter speeds.
The above image includes, from right to left, Jupiter, Saturn, Hyades, and Pleiades and was taken with a Nikon Coolpix 995 with a Wide Angle Lens attachment.
Part 1 Tripod Images
With maximum digital camera exposures getting longer and longer with each new model of camera the effects of the Earth's movement can be seen in exposures of as little as two seconds at the celestial equator in a typical telephoto zoom position and nine seconds at a typical wide zoom position. If your camera has a maximum of eight seconds or less simply leaving it in the widest zoom position should avoid star trails. If your camera permits exposures of longer than 8 seconds or if you wish to use telephoto zoom in your images then some method of star trail avoidance is required.
The stars appear to move and the earth rotates at 15º per hour and the effects of this motion on images are more pronounced near the celestial equator and negligible at the celestial poles. Knowing how many seconds of exposure can be used at a particular area of the sky then is important to using only a tripod for wide field astrophotography.
In order to calculate the maximum shutter speed to avoid star trails we must decide how large of a star trail will be detectable in your images. The apparent angular size of a star trail is:
Equation 1: A = (t/240) cos d
Where d is the declination or celestial altitude of a star in the center of your image.
We know that the field of view s for a given focal length and angular size is:
s = 2F tan (A/2)
Next replacing the drift length in the field of view formula we get:
Equation 2: d = 2F tan ((t/480) cos d)
Where d is in millimeters and solving for t we get:
Equation 3: t = (480 arctan( d/2f) ) / cos d
Unfortunately, we must now convert this size d to the size in millimeters on the digital camera's ccd detector. This requires a series of additional calculations that we have already performed. See the website Digital Camera Astrophotography for specifics but unlike 35mm and its focal plane each detector type has a different size. First we must decide how much drift is tolerable in pixel units and then convert these pixels to mm.
< 1 pixel of drift
From previous investigations we know that for a Nikon Coolpix 995 the size of a single pixel is 3.45 x 3.45 microns and as a short cut and assuming that we want our maximum drift to be that of a single pixel then the size d is .00345mm or 3.45 microns. Using a single pixel of drift simplifies things dramatically and so we can just replace d in Equation 3 with p the size of a single pixel width.
Equation 4: t = (480 arctan( p/2f) ) / cos d
This equation now is dependant only on the width of a pixel, focal length and declination of the target star. A drift of 10 pixels is objectionable but a drift of 4 pixels is almost undetectable and so by replacing p in Equation 4 with 4p will provide for longer exposures at the expense of 4 pixels of drift. As an example an 8.2mm focal length photograph could image a target at the celestial equator for 23 seconds with 4 pixels drift as opposed to the 6 seconds for a 1 pixel trail.
4 pixels of drift
We can test our formula for maximum shutter speed (Equation 4) by inspecting a photo with drift and counting the number of pixels of drift. For a photo taken during the 2001 Leonids meter shower a drift was seen of ~10.5 pixels. So we can replace p with 10.5p and calculate the t or shutter speed of the photo. The Nikon 995 reported in the EXIF data that the focal length of the zoom lens was 8.2mm and from a Star catalogue we determined that the declination of Regulus is 12º.
10.5 pixels of drift
= 480 arctan( 10.5p/2f ) / cos 12º
Comparing the value t above to the shutter speed from the EXIF data or 60s we see that we are within about two seconds which should be sufficient for avoiding star drift due to the rotation of the earth.
Going back to equation 2 we could calculate the size of single pixel in our digital cameras by taking a photo with a star drift. Let's use the Leonids photo from above and replace d with 10.5p or 10.5 pixels.
10.5p = 2F tan( ( t/480 ) cos d)
(2F tan ( (t/480 ) cos d ))/10.5
The size agrees with our pixel size to within .00003 mm or .03 microns.
The measurement of the pixels of drift above was done counting pixels and since the drift trails are relatively short the 10.5 pixels value is only a guess based on the average measurement of 5 sets of trails. The longest exposure possible with your digital camera of a star near the celestial equator such as Heze in Virgo in the summertime or Mintaka in Orion in the winter would provide longer trails and thus a better estimation of the number of pixels. If a star is used with a declination of 0º then the equations are simplified even further as the cosine of 0º is 1.
t = 480 arctan( p/2f)
p = (2F tan ( t/480 ))/L
where L is the number of pixels measured in the drift of the target star.
Visit the campanion page which will do these calculations for you and produce a table of exposures for your camera based on declinations 0, 15, 30, 45, 60, and 75 degrees.
So by performing this simple test you can determine the size of a pixel in your digital camera and at the same time determine the maximum shutter speed that can be used in a specific area of the sky when taking tripod wide field digital camera astrophotography.
Once we know the size of a single pixel we can then calculate the field of view of any image using the formula:
Equation 5: s = 2F tan( A/2 )
Where s is the size in mm, F is focal length, and A is the angular size of an object
Solving Equation 5 for A we get:
Equation 6: A = 2 arctan( .5s/F)
sx = p
Where sx and sy are the size in mm of the image in the x and y directions, p is the size of a pixel, and w and h are the number of pixels in the width and heighth of the photograph.
Ax = 2 arctan( .5p*w/F)
So any photograph taken with the Nikon 995, not attached to any other lens, at the widest zoom position or Focal Length 8.2 set on the FINE resolution will produce an image that is:
Ax = 2 arctan(.5*.00345*2048/(2*8.2)) = 47º
or 47 x 36 degrees
Adding the Wide Angle Adapter and the image will include sky 70 x 56 degrees
See the companion site with field of view calculations for each focal length based on your camera.
If you know the pixel size of your camera then enter it below along with your maximum uninterpolated resolution and hit submit to see a table of exposures.
Part 2 (coming soon)