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Calculating the Resolution of Solar System Images
Submitted: Wednesday, 15th November 2006 by Alan Sheehan B.E.

(Note: the following method for calculating image resolution assumes that the Dawes Limit or Raleigh Criteria for the scope is not exceeded. In such cases the resolution is not limited by the pixel size of the camera, but by the optics of the scope and different resolution criteria apply. While the distance represented by a pixel will still be valid, the information inside each pixel is not fully resolved. Poor seeing and/or focus can also result in poorer resolution than calculated here.)

Calculating the resolution of a planetary or solar image is not really all that difficult. You just need to know 3 things and be able to multiply and divide. Consider the following diagram. We can calculate the resolution of the image, R, by applying the method of similar triangles to this diagram.

resolution.jpg

Click to Enlarge
Calculating Resolution

where:

  • R = resolution per pixel in metres
  • d = distance to the subject (planet or sun) in metres
  • L = focal length of the telescope in metres
  • p = pixel spacing (or size) in metres.

For simplicity I’ve assumed all measurements are in SI standard units of metres, but this doesn’t matter as long as L and p are in the same units, and R and d are also in the same units. For example, we could have L and p measured in mm and R and d measured in kilometres, and everything will still be OK.

Applying similar triangles, we get:

P   R
- = -
L   d

And by rearranging the equation we get:

R = (p x d) / L

Example

I have a Celestron C8 scope, a ToUcam camera, a 2x Barlow and a 2.5x Powermate. What will the resolution of the images I take be?

Prime Focus

Let’s assume I take a lunar image at prime focus of the C8.

  • The focal length of a C8 is:
    • L = 2032mm = 2.032m
  • The pixel size or spacing of a ToUcam CCD chip is:
    • p = 5.6microns = 5.6 x10-6m ( = 0.0000056m)
  • The average distance to the moon is:
    • d = 3.84401x108m ( = 384401000m = 384401km)

Therefore R = (5.6x10-6m x 3.84401x108m) / 2.032m = 1059m per pixel.

2x Barlow
  • L = 4064mm = 4.064m
  • p = 5.6microns = 5.6 x10-6m
  • d = 3.84401x108m

Therefore R = (5.6x10-6m x 3.84401x108m) / 4.064m = 530m per pixel.

Alternatively, once I know the resolution for prime focus, I could simply divide it by the magnification of the Barlow:

R = (1059m per pixel) / 2 = 530m per pixel.

2.5x Powermate
  • L = 5800mm = 5.8m
  • p = 5.6microns = 5.6 x10-6m
  • d = 3.84401x108m

Therefore R = (5.6x10-6m x 3.84401x108m) / 5.8m = 424m per pixel.

Alternatively, once I know the resolution for prime focus, I could simply divide it by the magnification of the Powermate:

R = (1059m per pixel) / 2.5 = 424m per pixel.

Now because telescope focal lengths can vary a little bit either way from the nominal design value, and because the distance to the moon varies due to the elliptical nature of the orbit, it is reasonable to round off the calculated resolution. I tend to err to the conservative side and round up so that I can’t be accused of over exaggerating the resolution. So I would say the resolution of my images are as follows:

  • Prime focus of C8:  R = 1km / pixel.
  • 2x Barlow with C8:  R = 500m / pixel.
  • 2.5x Powermate with C8: R = 400m per pixel.

These are nice easy round numbers within 10% of the calculated values, and that’s reasonable as a rough guide to the scale of your images.

The method is accurate, however, so if you accurately know the focal length of your scope and the distance to the planet or sun at the time of capture, you can calculate the resolution of your image accurately, which you may want to do to calculate the size of storms on Jupiter, for instance. Once you know the resolution, measuring the size of features is just a matter of counting pixels (and maybe a little bit more maths if there’s some perspective involved).

Article by Alan Sheeny B.E (sheeny). Discuss this article on the IceInSpace Forum.

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