Digital Camera Astrophotography Observations

	Digital Camera Astrophotography Formulae by Gregory Pruden and Markus Ridder	Deutsch

In the pursuit of the perfect image, we have created a web page to assist digital camera astrophotographers with calculating various optical and other characteristics of their setup. We are looking for more helpful formula so if you have any suggestions drop us a line or post a message to the [digital_astro] yahoo group.

After combining formula from various sources and even working some problems out ourselves we wanted to then test these formula observationally or more precisely photographically. We are currently working on the Field of view, or piece of sky that falls on our cameras' ccd chip, calculations. We are proposing two methods for the FOV calculation; the simulation method and the drift method. For each method we have two tests; one with a Schmidt-Cassegrain telescope (SCT) and one with a Fraunhofer refractor.

The Simulation Method
The simple idea is to take some star field images with different scope/eyepiece/zoom combinations and to compare the obtained FOVs with the theoretical calculations. The comparison e.g.. can be done with a sky charting software which allows to include a certain FOV into the picture.

Test 1 (SCT)

The first measurable star field we tested is Eta Cassiopeia and its neighbors. Using the spreadsheet we anticipated that the field of view using a Nikon Coolpix 995 at 8.2mm (the widest zoom setting) and a Meade LX90 2032mm/203mm would be 40.8' x 30.6'.

At first glance, this dark subtracted image looks like our task might be harder than we thought.

Inverting the image and using GammaLog Histogram scaling the stars really stand out.

This chart shows the computed positions of Eta Cas and its surrounding stars using Starry Night Pro.

Combining the two images from above, the Starry Night Pro stars are gray and have been offset to show the two starfields. We can see that we are definitely looking at the same star field.

Now for the measurement. Using the distance tool in AIP4WIN we measured the distance in pixels from the center bright star Eta Cas to the surrounding stars. [This could also be done in any package using a rectangle diagonally corner to corner and then computing the diagonal length using the base and height.]

Photograph Pixel Distance
Star 1	Star 2	Number of Pixels
Eta Cas	HIP3809	635.27
ETA Cas	TYCO3663-2650-1	914.86
ETA Cas	TYCO3663-1792-1	80.33
ETA Cas	TYCO3663-672-1	827.84
ETA Cas	TYCO3663-1502-1	807.62
ETA Cas	TYCO3663-2002-1	589.51
ETA Cas	TYCO3663-1096-1	365.78

The details for these measurements are available here.

Next using the angular distance tool in Starry Night Pro we determined the distance between the same stars and computed the pixel per arcminute by dividing the pixel distance from above by the angular distance below.

Starry Night Pro Sky Chart values
Star 1	Star 2	Angular Distance	Pixel per arcminute
ETA Cas	HIP3809	11.58333333	54.84345324
ETA Cas	TYCO3663-2650-1	16.48333333	55.50212336
ETA Cas	TYCO3663-1792-1	1.516666667	52.96483516
ETA Cas	TYCO3663-672-1	14.91666667	55.49765363
ETA Cas	TYCO3663-1502-1	14.6	55.31643836
ETA Cas	TYCO3663-2002-1	10.6	55.61415094
ETA Cas	TYCO3663-1096-1	6.85	53.39854015

Taking the average of Pixel per arcminute values calculated above we obtain an average pixel per arcminute value of 54.73388498

The above photograph was taken using the maximum non-interpolated resolution of the Nikon Coolpix 995 digital camera or 2048x1596.

Using our average pixel per arcminute value the field of view of the photograph is:

FOV = pixels / pixels per arcminute = arcminutes
FOVx = 2048/54.73388498 = 37.41740607'
FOVy = 1596/54.73388498 = 28.06305455'

Comparing these to our initial values from the spreadsheet we see that using the Simulation method we are off by 4' arcminutes or 10%. It is not clear that this error is in the spreadsheet or Starry Night Pro or a combination, but tests by another software package on the same images would verify this. If you own a package or know of a source that will tell the angular distance between stars in a field please let us know.

Test 2 (SCT)

For this test we used an object for which the size from earth is well know; our home star, the Sun. At this time of year the Sun is 1902 arc seconds(") in apparent diameter. The photograph below was taken using an 80/500 refractor and a 19mm panoptic EP by a Nikon Coolpix 950 digital camera with the zoom set at 16.8mm.

Pluging these values in our spreadsheet we caculate field of view as:

FOVx = 49.6 arcminutes = 2976 arcseconds
FOVy = 37.2 arcminutes = 2232 arcseconds

Tip: We can see from the spreadsheet that the Sun should fit completely in a photograph taken with this setup and these settings.

This photo of the Sun was taken just after a powerful X5-Class solar flare on August 26, 2001.

Using the above image we measured the diameter of the Sun to be 1180 pixels and the Sun is known to be 1902 arcseconds or 31.7 arcminutes. Just as in the first test we determine the pixels per arcminute by dividing the measured pixels by the known diameter in arcminutes.

Pixels per arcminute = pixels / arcminutes = 1180 / 31.7 = 37.224

Now we extrapolate the pixel per arcminutes to the entire photograph which is 1600x1200 pixels

FOVx = 1600/37.224 = 42.98 arcminutes
FOVy = 1200/37.224 = 32.24 arcminutes

Comparing these values to those from our calculator using the known object diameter, which removes the accurracy of the Starry Night software as a possible issue, you can see that the error in x is 6.7' or 13.5% and the error in y is 4.96' or 13.33%.

The Drift Method
The "Drift Method" is a well-known way to measure the real FOV of an eyepiece. A good description of this can be found here.

The simple idea is to make an image of a star with the drives on your mount switched off. Due to the earth's rotation, the star will show up on your image as a short trail. The real length of the trail (in arcseconds) depends on the length of exposure and on the star's declination. The simple rule to use is:

s["] = 15.04" * Texp [sec] * cos (Dec)

Measuring the length of the stars trail (or rather its projection x´/y´ on the x/y axis) in your image gives a relation between the real length and the apparent length that can be used to calculate the FOV of your scope/EP/camera setup.

Test 1 (refractor)

In order to keep the error as small as possible it is best to expose as long as possible. Since digital cameras usually do not allow for very long exposures, we used a simple trick to simulate long exposures. We took three images in a row without touching the scope between the exposures. Between two consecutive exposures, we waited exactly 60sec. This gives three small trails on the combined image. The time between the beginning of the first and the last exposure is then 120sec.

1600x1200 image using 80/500 Fraunhofer Refractor, 25mm Plossl, Nikon CP950

2s, 2s, 2s
@19mm Fc

1600x1200 image using 80/500 Fraunhofer Refractor, 25mm Plossl, Nikon CP950

2s, 4s, 8s
@19mm Fc

The target of our measurements was "Alpha Peg". It is quite close to celestial equator [DE=15,2°] and as such, leaves a long trail on the image. The results of these measurements are:

Alpha Peg
star dec coordinate [°]	15.00
exposure time [sec]	120.00
pixel covered horizontal	853.00
pixel covered vertical	400.00

Calculations
reduced time [sec]	115.91
star motion during exposure [arcmin]	29.06
slope of star motion [°]	25.12
horizontal star motion [arcmin]	26.31
vertical star motion [arcmin]	12.34
FOV horizontal [arcmin]	49.35
FOV vertical [arcmin]	37.01

Using our "afocal spreadsheet" , the Nikon 950's 19mm camera focal length, and the Fraunhofer Refractor's focal length of 500mm yielded a Field of View of 57.69' x 43.27'

The field of view measured using the drift method was 49.35' x 37.01'

Comparing the results we determined that using the drift method we are off by 8' x 6' or 14%.

Compairing these values with the values using the same setup (test 2) in the Simulation method confirms that the FOV measurements are correct as they are nearly identical.

Test 2 (SCT)

In the second test we used the Nikon Coolpix 995 which allows for exposures up to about 1 minute. We took three exposures of the same star, Alp Peg, at 60.1 seconds each. For demonstration purposes we added the three photographs to create a combined picture.

2048x1536 image using 2032/203 SCT, 28mm Plossl, and Nikon Coolpix 995

The average horizontal distance was 672.67 and the average vertical distance was 420.33

Plugging in these values into the equations from test 2 we obtain the following Field of View:

FOVx = 37.29'
FOVx = 27.96'

Using the drift method in test 2 proved to be extraordinarily close (8 arcseconds) to the measured values with the same setup in the Test 1 using the the Simulation Method.

Conclusions
We are still working on the source of our error.