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01-12-2014, 09:00 PM   #1
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K5 Scale Factor - Arcseconds per pixel for several lenses

Some scale factors (arcseconds/per pixel) for lenses on a K5.

I have taken some pictures of the sky at night with my K5 and various lenses. With a lens fairly wide open (f/2.8 or 4) and with moderate ISOs (400-800), short exposures of a few seconds will result in images that have lots of stars in a frame.

If you know the positions of the stars - available from various sky display software (I like Stellarium) - you can measure the star (x,y) positions in pixels (for instance: turn on the “info” window in Photoshop Elements, after making sure the units are pixels in “preferences”) and compare the pixel distance to the actual angular distance between the stars. To get the actual angular separations, you must make the proper calculations for the star positions (you can’t just take the difference in Right Ascension and Declination). If you are trying to measure the scale for wide angle lenses, you also have to make a correction to the pixel separation values because the sky is not “flat” over a wide angle.

I have done this for three of my lenses: an old (70's) SMC Pentax-M 1:1.7 50mm, and my much more modern Pentax-D FA Macro 100mm WR and Pentax DA* 300mm F4. Typically I measured the separation between 4 to 6 stars for each lens. The results, with formal standard deviations, for these three lenses are:

50 mm - - - - - - - - 18.80 +/- 0.04 arcseconds / pixel

100mm Macro - - - 10.15 +/- 0.02 arcseconds / pixel

300mm DA* - - - - - 3.380 +/- 0.005 arcseconds / pixel

(I was rather astounded at the internal consistency of each set of measurements!)

The pixels per arcsecond should scale inversely as the focal length. For the 100mm and 300mm lenses, this is very nicely the case: the scale ratio is a factor of 10.15/3.38 = 3.003, exceedingly close to the 3.00 factor expected for focal length ratio. For the 50mm lens compared to the 100mm macro, the factor is not quite so close: 10.15/18.80 = 0.5399, compared to the expected 0.500 . Since the shorter the focal length, the more uncertain (at least in my mind) the actual location of the “focal point” of the lens will be compared to the sensor plane, I am not surprised by this difference. (If you doubt that statement, then where would you put the “focal point” of, say, a 10mm lens. My Sigma 10-20mm lens is some 80mm long, and the rearmost lens element is around 20mm from the sensor when set to 10mm focal length.)

01-12-2014, 10:57 PM   #2
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Thanks for this, Dave. I have always wondered if the developers of Stellarium (which is also my main planetarium app) would ever code up a "micrometer" type of tool as found in GoogleEarth for measuring distances. Put in the variables (focal length, pixel pitch, screen size, etc) and it can do the math to get the distance in arcseconds. I expect that the formula for my K-3 would scale by the difference in pixel pitch.

Jack
01-13-2014, 04:17 AM   #3
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QuoteOriginally posted by AstroDave Quote
For the 50mm lens compared to the 100mm macro, the factor is not quite so close: 10.15/18.80 = 0.5399, compared to the expected 0.500 . Since the shorter the focal length, the more uncertain (at least in my mind) the actual location of the “focal point” of the lens will be compared to the sensor plane,
It appears to me that you overlook the fact that equal angular distances will not cover the same amount of pixels (= the same linear extent) near the centre and near the corners of your image.

Take any pair of stars with a given angular separation: If you place the image of the pair of stars at the corner of your sensor, then the distance between the images of the two stars - as measured on a linear scale (: say, in milimeters) - will be longer than if you place the image of the pair near the centre of your sensor. For wide field lenses this difference will be quite significant. For long FL lenses you will hardly note the effect.

Let RA0 and Dec0 be the co-ordinates of the centre point of your image (i.e.: the co-ordinates at which your telescope / lens was pointing during exposure). Then, a star with co-ordinates RA1 and Dec1 will be projected on your focal plane in an x-y co-ordinate system with origo (x=0, y=0) at the centre of your image as follows:
x1 = FAC * (sin(RA1-RA0) * cos(Dec1)
y1 = FAC * (cos(RA1-RA0) * cos(Dec1 * sin (Dec0) - sin(Dec1) * cos(Dec0))
where FAC is a constant depending upon the focal length of your optical system and the magnification of your image on the sensor, (if applicable)
01-13-2014, 05:12 AM   #4
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QuoteOriginally posted by jbinpg Quote
Put in the variables (focal length, pixel pitch, screen size, etc) and it can do the math to get the distance in arcseconds.
In order for the math to work, you will need the exact Right Ascension and Declination values for the centre of your image. There isn't really any direct way to establish that, but there are tools that can help you if you can measure x,y-positions for stars with known celestial co-ordinates:

Astrometry Calculator

From there, the calculation of angular distances is fairly straightforward. The general formula reads as follows:

A = cos(Angular Distance) = sin(Decl.1)sin(Decl.2) + cos(Decl.1)cos(Decl.2)cos(RA.1 - RA.2)
and thus, Angular Distance = arccos(A)

See for example:

Anglular Distance Between Two Stars Calculator

01-13-2014, 07:35 AM   #5
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What you are worrying about is what I meant by "If you are trying to measure the scale for wide angle lenses, you also have to make a correction to the pixel separation values because the sky is not “flat” over a wide angle" in my original post.

The result is that my scale factors above are indeed correct, but only for near the center of the field for a wide angle lens. To make my correction you do not need to know the astronomical center of the field, but rather the center in pixel value and then correct for the pixel distance of each star from the center.

I am an astronomer (hence my forum name), and I do know how to calculate the angular distance between stars, given their RA and Dec. It is a bit unfortunate that Stellarium does not do this for a pair of stars. I actually used another program when I was looking up star positions, and it does do this calculation and gives you the true separation between the two last stars you clicked on - saved me a few columns in my analysis spreadsheet!!
01-13-2014, 07:50 AM   #6
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Very cool results Dave. As for the 50mm results, many 50mm lenses are in reality closer to 52mm in focal length. I don't know if your Pentax-M is one of them, but the results do seem to suggest a longer real focal length.
01-13-2014, 03:47 PM   #7
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Thanks, Steen. Link bookmarked..

Jack
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