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Using 2 stars to determine the actual focal length of a lens (at a distance)
Posted By: dosdan, 12-21-2011, 04:59 PM

This technique is based on the "easy method" described by Bob Atkins here. I've simplified it, using on-line star maps & an angular separation calculator, and used a K-5 as the example.

We use any two stars, as star positions are known very accurately. Pick 2 stars that have a diagonal angle-of-view within the diagonal AOV of your lens. Using a star separation that fills most of your camera's sensor diagonal will give greater accuracy. Brighter stars are better, but not critical. Bob recommends a 1/10s or faster shutter speed to prevent earth rotation movement blurring, but even that's not that critical unless you want very high accuracy. You don't need a very bright exposure either. As long as you can see the centre of the 2 stars of interest, that's good enough. So pick any 2 widely-spaced named stars that are reasonably bright in a constellation.

I used high sharpening to tighten up the captured star points. The old lens I used for this article, a Pentax-A 70-200/F4 (non-SMC version), has some coma aberration. In normal usage you'd rarely notice this, but it becomes obvious when you shoot stars. Even with the fan of light caused by this aberration which blurred the star point (covering a 20x by 20px box in my case), the accuracy of FL determination was still about 0.5%.

Step 1. Use an AOV calculator to determine a good max. diagonal separation to aim for.

Angle of View Calculator

For a 200mm FL on a Pentax APS-C sensor, the diagonal AOV is 8.2. For a 300 mm FL, it is 5.5.


Step 2. Go to Your Sky, to the Sky Map section, pick a nearby city, create a map and zoom in/pan to find a suitable constellation. The current FOV is shown under the map.

Your Sky

I used the Crux constellation (Southern Cross). The 3 stars in Orion's belt are also a good choice for both hemispheres.





I used Acrux & Gacrux as the 2 points in the shot to be used for the angular separation measurement:




A full-sized version of this shot is available here if you want to use it to try out the measurement process mentioned in Step 5.


Step 3. On the screen, below the FOV figure, there is a link to the object catalogues

I went to the named stars list and searched for crux to find Acrux & Gacrux. Acrux is located at:

Right Ascension (RA): 12h 26.598m
Declination: -63 5.950' (objects in the Southern sky have a negative declination).

Unfortunately, the Angular Separation calculator I used wanted the values in a different form. The easiest way to change the values to the required format is to click on the link for that star's name in the catalogue. So, when I clicked on the link for Acrux, the map redrew with this star at the centre. The correctly formatted values are shown in the "Aim Point":

RA: 12h 26m 35s
Dec: -63 5' 57"

The values for Gacrux are:

RA: 12h 31m 9s
Dec: -57 6' 48"

(Alternatively, you can convert 12h 26.598m to full hh mm ss format. Just multiply the decimal part of the minutes by 60 i.e. 0.598 x 60 = 35.88s, close enough to the 35s used above.)

Step 4. Next I used an Angular Separation Calculator.

I entered-and-tabbed the top line only of this calculator (Acrux on the left). When you click on CALCULATE all the other fields are filled out:




We use the distance in radians: 0.1049391


Step 5. Next, determine the distance in pixels between the 2 stars on the shot image. I used the centre for Gacrux (100% crop):




And the hinge centre point of the coma "fan" for Acrux:




The pixel x,y points (co-ordinates from the top left) in my K-5 image are:

Acrux: 4172, 178
Gacrux: 596, 2951

From Pythagoras' Theorem, a2 + b2 = c2, with the distance between the two stars being the hypotenuse. In this example, we use: (xacrux - xgacrux)2 + (yacrux - ygacrux)2 = pixel distance2

= (4172 - 596)2+ (178 - 2951)2 = 20477305. The pixel distance is the square root of this: 4525px


Step 6. Now we work out the scaling px/mm figure for the K-5 sensor. The specifications in the back of the manual show that the sensor width is 23.7mm & that there are 4928px in this width. So the value is 4928px / 23.7mm = 207.93px/mm (The reciprocal of this value is the pixel pitch: 4.8μm/px)


Step 7.
Then determine the separation distance in mm on the sensor: 4525px / 207.93px/mm = 21.76mm


Step 8. Finally, use the equation: FL [in mm] = Separation Distance [in mm] / Separation Angle [in radians]
= 21.76mm / 0.1049391 = 207.4mm


So the max FL of this 70-200mm lens is 207mm.

Dan.

Last edited by dosdan; 04-13-2016 at 03:38 PM.
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12-21-2011, 06:02 PM   #2
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Nice presentation imho!

Note that the calculation gives the actual focal length of the lens; the result does not include any 'crop factor'.
12-25-2011, 08:21 PM   #3
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Last night was a clear enough sky in the last few days to see some stars again. So I decided to measure the max. FL of the DA*60-250/F4. I used 2 shots of the separation between Acrux (Alpha Crucis) & Mimosa (Beta Crucis). The angular separation between them is 4.243376 .

The 2 FL results I got were:
240.9mm
241.6mm

I then used the 2 outside stars in the line-of-three in Orion's belt, Alnitak (Zeta Orionis) & Mintaka (Delta Orionis). The angular separation is 2.735777. The FL result I got was:
242.3mm

However the larger angular separation in the first two shots compared with the third should be a bit more accurate due to the greater pixel separation in the image: 3709.4px, 3721.2px, 2405px.

So I think this zoom lens has a 241mm max. FL.

Why the interest in determining the actual FL of a lens? I think the marketing department sometimes plays loose with the truth when specifying the FL of a lens. Knowing the actual FL makes it a bit more accurate when you're trying to work out the distance to a subject in a shot based on subject height and the percentage of the frame height that the subject occupies, as described here:

https://www.pentaxforums.com/forums/photography-articles/143778-subject-heigh...ml#post1747344

With zoom lenses this distance determination is most useful at the max. FL, due to the granularity and questionable accuracy of intermediate FLs reported by the lens. But with sports, a lot of shots end up at the max. FL.

Dan.

Last edited by dosdan; 12-25-2011 at 09:11 PM.
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