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# How to Calculate the Focal Length of an SCT

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### #1 EdZ

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Posted 08 January 2010 - 02:00 PM

This question comes up fairly often, in several forums, particularly when people are interested in knowing the actual magnification they are using while viewing with an SCT. It has come up in the Double Star forum in threads where people compare lowest powers to see a particulr pair split. It comes up in binoviewers when people can't figure out why sometimes the view is muddy, but they don't realize they are underestimating power by 25%. And it comes up in eyepieces when eyepieces are being compared between different scopes but the user may not be aware of the true power in the SCT.

Often people miscalculate the power when using an SCT. WHY? Because an SCT has a movable mirror and a secondary mirror that dramatically changes the shape (and therefore the effective length) of the light cone. Every time you move the mirror in an SCT (or a Mak) to achieve focus, you change the operating focal length. Because of that changing shape of the light cone, you cannot simply add the length of the accessories you've added on the back end of your SCT. The operating focal length must be recalculated.

There is a great article by Chris Lord on his website that provides a great deal more in formulas, charts and explanation. I have outlined here a simpler do it yourself at home for your configuration check of the focal length.

Why would you need to do this?
Well, here are some examples:

my C5 with a standard 1.25" visual back and a 1.25" TV dielectic diagonal operates at a focal length of F=1367mm, not the nominal f/10=1250mm.

my C5, as it is currently configured with a 1.5" long extension tube (to get clear of the focus knob) and a 2" Astrotech dielectric diagonal, operates at a focal length of F=1540mm, f/12.3.

My C8 with a screw-on Crayford focuser and a 2" diagonal operates at about 2550mm approx f/12.5.

Determining the Operating Focal Length of an SCT

Use two different scopes,
your SCT needing testing (in the configuration you would normally use it)
and a refractor of known focal length (wouldn't hurt to check it). A refractor focused focal length can easily change by 50-60mm from close focus to infinity.
We'll assume the refractor has a shorter focal length than the SCT.

Pick 3-4 eyepieces with a variety of focal lenghts and fs diameters (for example, 10,15,20, 25 or 30). To be sure that vignette is not an issue, try not to select eyepieces with very wide field stops that might possbly be vignetted by some portion of the SCT tube. I'd expect youd be fine with plossls up to 32mm.

Hang a tape measure on a telephone pole, perhaps 100 yds away.

Record the dimension across the fov for each eyepiece, first in the refractor.
Then, record for each eyepiece in the SCT.

I did this with a TV85 and my C5. I picked 4 eyepieces a 7.5, 12.5, 18 and 24mm. Set up a measureing tape. Using the eyepieces in a TV85 refractor of known focal length, recorded the extent of visible field (to as fine a division as could be seen) for each eyepiece. Next I brought out my C5 with the standard 1.25" diagonal. ran all 4 eyepieces through the C5/1.25"diag and recorded the extent of fov again.

I then compared the reading for each eyepiece in the TV85 to the reading in the C5. Got near exactly the same ratio of drop in fov for all 4 eyepieces. This is good consistent data for the 4 eyepieces. So what that shows so far is only one thing changed, the POWER. And because only power changed, all 4 eyepieces have exactly the same ratio reduction in the fov. Variances here were on the order of 0% to 1%.

So, Determine factor for each eyepiece
dimension across fov in ref is to dimension across fov in SCT
equals
power in ref is to power in SCT.
Smaller dimension reading in the SCT(assuming SCT is longer) is due entirely to higher power, nothing else.

Multiply refractor power by factor to get SCT power.
Having calculated operating powers in SCT calc backwards to get operating focal length.

The factors should be near exactly the same for all eyepieces. If not, recheck. If so, you can establish the operating power of your SCT. Very minor differences may be seen due to few mm difference in operating focal length of SCT to accomodate placement of field stops. Parfocalizing would eliminate any differences, but is not necessary as error is very small, on the order of 5-10mm.

May seem like a lot of work, but takes perhaps only 15 minutes, and you are done forever.

The accuracy of this method is well within 1%. Focal length will be slightly (not significantly) different at infinity. The farther away you put your test scale, the closer you will be to infinity, but you are probably within 1-2% even when testing at just a few hundred feet.

Post in the Cats Forum related to
Calculating the Focal Length of a C5 with a spacer and a2" diagonal installed on back end

edz

### #2 Lane

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Posted 08 January 2010 - 07:01 PM

Thanks for posting this edz.

As soon as the weather gets a bit warmer I am going to try this at our club's site in the daytime. Out there I should be able to get as far away from the tape measure as I want.

I will use my new AT106 and calculate the true focal length of my SCT's in various configurations. I don't own any long plossls but I assume using some UO Orthos will work ok, I have the 25, 18, 12, 9, 7, and 6 to work with.

### #3 dvb

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Posted 09 January 2010 - 01:03 AM

Something I'd like to do - but it would involve climbing over my fence, going downhill a ways, and then shinnying up someone elses's large fir tree!
Doesn't seem likely to happen!

### #4 Lane

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Posted 09 January 2010 - 02:04 AM

Our site is very large and has some big trees around the edges. I am going to throw a rope around a tall tree limb and tie the end to a end of a tape measure then pull the tape up so it is about 20 feet high and use that as my target. I think if I use a Fat Max tape measure it will stay straight as long as there is no wind.

### #5 EdZ

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Posted 09 January 2010 - 08:01 AM

An important note here.

My TV85 is closest to the full 600mm specified focal length when focused very close at about 100 feet. When focused at 200yds, the focal length is closer to 580mm. At infinity the TV85 operates at F=575mm.

Measuring the focal length of a refractor is fairly easy. Just make sure you know what yours is.

edz

### #6 EdKiefer

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Posted 09 January 2010 - 08:36 AM

Hi Edz

You posted this
"my C5 with a standard 1.25" visual back and a 1.25" TV dielectic diagonal operates at a focal length of F=1367mm, not the nominal f/10=1250mm.

my C5, as it is currently configured with a 1.5" long extension tube (to get clear of the focus knob) and a 2" Astrotech dielectric diagonal, operates at a focal length of F=1540mm, f/12.3.

My C8 with a screw-on Crayford focuser and a 2" diagonal operates at about 2550mm approx f/12.5. "

Are you basically saying SCT stated FL is without diagonal , eyepiece straight into visual back ?

Also since the secondary on most SCT are same mag (5x ? ) couldn't we get some rough formula like distance extended from visual-back *5 plus FL is real FL ?

I understand doing your FOV measurements with EP would be accurate way, I am just trying to visualize in my head whats going on to raise FL so much .

### #7 hudson_yak

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Posted 09 January 2010 - 09:21 AM

For an 8" SCT, the nominal focal length occurs when the focal plane is 90-100 mm behind the rear cell.

The ray tracing report on Yahoo sct-user group indicates 2550 FL occurs at around 10 inches backfocus. Ed's numbers are in the ballpark though seem a little high unless the crayford is a really long one.

The method described above is ok if none of your eyepieces have a known field stop diameter, however, if you have any that do I'd consider drift-timing a star close to declination 0 to be a much simplier method, with no requirements for another scope nor concerns about not focusing at infinity.

FL=field-stop-diameter/seconds-across-field*240*57.3

Mike

### #8 vahe

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Posted 09 January 2010 - 09:29 AM

Often people miscalculate the power when using an SCT. WHY? Because an SCT has a movable mirror and a secondary mirror that dramatically changes the shape (and therefore the effective length) of the light cone. Every time you move the mirror in an SCT (or a Mak) to achieve focus, you change the operating focal length. Because of that changing shape of the light cone, you cannot simply add the length of the accessories you've added on the back end of your SCT. The operating focal length must be recalculated.

Just to expand on your comment, my TEC6, a 6” F/12 Mak Cass, has a moving mirror, according to Yuri the F ratio varies from F/11 to F/14 as the primary is moved in and out, that is a huge change to the optical parameters and in addition to increase and decrease to the focal length it also affects the overall optical figure, making it better at one extreme and not so good at the other.

The TEC has a F/3 primary where the commercial SCT’s come with F/2 primaries which will make the focal length variations even larger with moving primary and in some cases result in loss of actual aperture.

Vahe

### #9 EdZ

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Posted 09 January 2010 - 09:33 AM

I've actually done drift timing to confirm some of my measurements. Sure you can do that, but that requires that you have a caliper to accurately measure the field stop of the eyepiece to at least 0.1mm. Otherwise you don't know what the expected fov is. So that's not really a simpler method, it just requires different instruments, a precise caliper instead of another scope.

edz

For a complete explanation of how various movable mirror scopes act see Chris Lord's article. Position of nominal F and change in focal length with mirror movement varies with the scope and I would not rely on a point of 100mm behind the backplate as a reference point. This simple method allows you to determine focal length of ANY movable mirror scope.

edz

http://www.brayebrookobservatory.org/

### #10 hudson_yak

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Posted 09 January 2010 - 09:49 AM

would not rely on a point of 100mm behind the backplate as a reference point.

That's correct, which is why I mentioned the 8" specifically. For that one, anyway, it's not at the rear cell. The ray-tracing report shows none of the bigger ones have it there either, though they are at least 90mm or more. I'd be curious to know what drift-timing reveals for the C5 as your number there seems a bit questionable. One would expect the nominal focal length to be possible in a typical visual configuration. A prism diagonal shortens the visual path a bit so that could explain some of the difference.

The TV Pan 24 is my go-to EP for this. Known 27mm FS and no optics ahead of it that might confuse things. It's also wide enough to make the timing long enough to minimise stop-watch reaction errors. Accuracy can be in low single-digits percentage-wise.

Mike

### #11 EdZ

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Posted 09 January 2010 - 10:07 AM

I reported on this probably 5-6 years ago, and many times since. In fact, I wrote a review using my C5 back in 2003, in which already I had reported the operating f# was f/11.

I drift timed my C5 with the original standard 1.25" visual back and a 1.25" Televue dielectric 90° mirror diagonal. I must have recorded the drift times from about 15 different eyepieces. I did each eyepiece 3 times. The readings ranged from 1356 to 1383.

It is normal for the SCT focal length to vary with eyepieces, since eyepieces have the field stop at different positions forward or reward from the eyepiece shoulder. This requires moving the mirror in/out to acheive focus and therefore changes focal length. So a range of 1356 to 1383 does not signify an error in measurements, it signifies a change in focal length for the eyepieces used.

edz

### #12 EdZ

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Posted 09 January 2010 - 11:41 AM

Our site is very large and has some big trees around the edges. I am going to throw a rope around a tall tree limb and tie the end to a end of a tape measure then pull the tape up so it is about 20 feet high and use that as my target. I think if I use a Fat Max tape measure it will stay straight as long as there is no wind.

You probably don't need to do that. An 18mm (60°) eyepiece even at a very low power of 33x in a widefield scope, can see only an fov of 1 meter wide at a distance of 125'. Even if you were to extend the distance to 100 yards, it would still only see a fov of 2.5 meters. That's my baseline refractor. Any f/10 SCT is going to see less than half that fov

edz

### #13 JimK

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Posted 09 January 2010 - 11:56 AM

Here's a CN thread where I posted another method of determining SCT focal length (using a Celestron Microguide eyepiece). In thinking about it, I'm just not sure what the 82506 conversion number is based on.

And yes, I measured different focal lengths for different viewing configurations and found the telescope manufacturer's information to be "nominal." It was an interesting exercise for me.

Perhaps at some point I will do a comparision with my 80mm refractor, following your method. Given it's nominal focal length of 555mm, I'll have to figure out exactly how to establish a more accurate value, for the refractor objective is significantly recessed. I also would need to figure out its thickness to establish a focal length "starting point." Maybe at some other time...

Thanks for the post/information, Ed!

### #14 Doug76

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Posted 09 January 2010 - 01:03 PM

I'll use my 60/1000 refractor as the comparison scope. Makes the translation easy.

### #15 dvb

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Posted 09 January 2010 - 01:08 PM

Thanks, Jim. I appreciated that you had gone to the trouble to post your results.

It would be terrific if those who are able to undertake the testing recommended by Ed could post their results (perhaps on a "stickie" thread?) for the benefit of those of us who aren't able to do such testing.

Anybody with a C9.25, a refractor diagonal and a Televue shorty diagonal, and a BinoVue care to do some testing?

### #16 wh48gs

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Posted 09 January 2010 - 01:30 PM

Also since the secondary on most SCT are same mag (5x ? ) couldn't we get some rough formula like distance extended from visual-back *5 plus FL is real FL ?

You can simply add the extra distance to the secondary-to-primary separation, and multiply the system f.l. with the ratio of the sum vs. secondary-to-primary alone. This is probably as close as most people need, since the change of marginal ray height on the secondary is small.

For accurate result, a couple of formulas will do. For given extra distance E beyond the original focus, secondary magnification "m" and the original secondary-to-primary separation "t", the needed mirror shift "s" is given by:

s=Et/[(m-1)E+tm^2)]

The new secondary magnification is then:

m'=r/[r-(t+s)/f1],

where "r" is the ratio of mirror radii of curvature, r=R2/R1, and "f1" the primary f.l. New system f.l. is f'=m'f1.

No one knows exactly these parameters, unless have them measured (they probably vary somewhat), but it can be assumed that for a typical f/2/10 SCT they are in proximity of r~0.312 and t~f1/4, giving secondary magnification ~5. Substituting those gives:

s~f1E/(16E+25f1) and m'~1/[0.2-3.3(s/f1)]

Unless you measure the radii and mirror separation, the simplest way from the beginning is probably just as good.

Vla

### #17 Rick

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Posted 09 January 2010 - 05:17 PM

### #18 Lane

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Posted 09 January 2010 - 08:07 PM

Our site is very large and has some big trees around the edges. I am going to throw a rope around a tall tree limb and tie the end to a end of a tape measure then pull the tape up so it is about 20 feet high and use that as my target. I think if I use a Fat Max tape measure it will stay straight as long as there is no wind.

You probably don't need to do that. An 18mm (60°) eyepiece even at a very low power of 33x in a widefield scope, can see only an fov of 1 meter wide at a distance of 125'. Even if you were to extend the distance to 100 yards, it would still only see a fov of 2.5 meters. That's my baseline refractor. Any f/10 SCT is going to see less than half that fov

edz

You are right of course, for some reason I was thinking it would really be wide, but that is based on views from much further than 100 yards. Anyway too cold to try this out right now, maybe in couple of weeks.

### #19 Brian Risley

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Posted 09 January 2010 - 08:28 PM

My understanding that the FL of the C-8 was based on focus on a standard T mounted camera. Any other configuration would change the fl.
Brian

### #20 hudson_yak

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Posted 09 January 2010 - 09:22 PM

I drift timed my C5 with the original standard 1.25" visual back and a 1.25" Televue dielectric 90° mirror diagonal. I must have recorded the drift times from about 15 different eyepieces. I did each eyepiece 3 times. The readings ranged from 1356 to 1383.

Ok, don't know what to say other than it seems unusual compared to the 8" and above Celestron models. I certainly agree with the comment about different eyepiece focal locations affecting the scope FL somewhat.

FWIW I drift-timed my new TV85 tonight with the Pan 24 and got a FL of 598. I don't remember exactly what I got for the ED100 a couple years ago other than I remember it being similarly remarkably close to the nominal spec.

Mike

### #21 David Knisely

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Posted 10 January 2010 - 02:05 AM

Here is my data:

NexStar 9.25 inch SCT eyepiece Appox. True Field of Views
with Astrophysics 2" visual back and Williams Star Diagonal.

40mm Konig:................1.0875 deg (65.25' arc). EFL: 2424mm (f/10.32)
30mm WideScan III:......1.0719 deg (64.23' arc). EFL: 2352mm (f/10.01)
14mm Ultrawide (2"):....0.5031 deg (30.18' arc). EFL: 2312mm (f/9.84)
14mm Ultrawide (1.25"): 0.4825 deg (28.95' arc). EFL: 2411mm (f/10.26)
40mm Elux Plossl:........0.6608 deg (39.65' arc). EFL: 2341mm (f/9.96)
30mm Ultrascopic:........0.6194 deg (37.17' arc). EFL: 2412mm (f/10.27)
24mm Panoptic:...........0.6564 deg (39.38' arc). EFL: 2357mm (f/10.03)
10mm Ultrascopic:........0.1950 deg (11.70' arc). EFL: 2447mm (f/10.40)

Mean f/ratio for NexStar 9.25GPS: f/10.14 (maximum deviation from f/10: 4%).

### #22 dvb

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Posted 10 January 2010 - 02:19 AM

Thanks, David! All quite close to f/10.

### #23 Dan McConaughy

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Posted 10 January 2010 - 08:45 AM

How do the abberations vary when the back focus is moved outward from the original design?

### #24 vahe

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Posted 10 January 2010 - 10:46 AM

How do the abberations vary when the back focus is moved outward from the original design?

OK, regarding the aberrations when the primary is moved away from the optimum design point.
This is specifically applicable to my TEC6, here is what Yuri said;
Moving the mirror 5mm towards the meniscus/secondary the theoretical error of 1/8th wave is reduced to 1/10th wave, F/12 becomes F/14.
Moving the mirror in opposite direction, away from meniscus/secondary the theoretical error of 1/8th wave becomes 1/7.5th wave, F/12 becomes F/11.
In both cases coma & astigmatism will not change.

Vahe

### #25 wh48gs

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Posted 10 January 2010 - 02:50 PM

How do the abberations vary when the back focus is moved outward from the original design?

Since only aberration contribution of the secondary changes appreciably, the general two-mirror relation still applies:

W=(1-m^2)s/512(m^3)(F1^4)

where W is the PV wavefront error at best focus (overcorrection for extending the focus), s is the mirror shift, m the original secondary magnification and F1 the primary's focal ratio.

Back focus extention in terms of mirror shift is:

E=sm^2/[1-s(m-1)/t],

with t being the secondary-to-primary's-focus separation
(approx. 1/4 of primary's f.l. in a typical f/2/10 SCT). For smaller extentions, up to 2-3 inches, simple aproximation E~sm^2 is good enough.

With m~5 and F1~2, that gives about 1/23 wave PV of overcorrection induced with every inch of focus extention (i.e. with every mm of primary's shift). Magnitude of final error depends on the correction level - and sign - at the original focus.

Vla

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