Graphing my eyepiece search

[Zermelo]

Just a caution before you waste any more of your time — this was just a bit of fun to while away the cloudy evenings — I expect the more experienced readers to squint slightly, suck in air through their teeth and say “I’d just go for an XYZ”, and they’ll probably be right. My own guess was that I’d end up reverting to type and buying another Starguider.
But if anyone else is getting bored of waiting for a clear night, I’d welcome opinions (even if you’re pointing out some howling mistake).

 

I’ve been looking to identify my optimum “finder” eyepiece. It would give me the widest possible true field of view and be the first one I’d use after each GoTo operation. I’d also use it for viewing the larger objects, so while it doesn’t have to be sharp right to the edge, it mustn’t be too bad (I have an F/5 scope).

Some abbreviations used in the following:

FoV_T                            true field of view
FoV_A                    apparent field of view
FL_OTA                            focal length of the telescope
FR_OTA                    focal ratio of the telescope
EP                          eyepiece
S                            eyepiece field stop diameter
FL_EP                       eyepiece focal length

I believe the most accurate way of calculating the FoV_T of an EP uses its field stop diameter S and the focal length of the telescope, FL_OTA :

FoV_T= S*57.296/ FL_OTA                   - (1)

with the constant converting from radians to degrees.

The maximum field stop diameter is obviously limited by the size of the EP barrel. In a 1.25” format eyepiece (which is all my scope will take) I measured the internal barrel diameter as 28.1mm, but realistically the largest stop will be smaller than that, for reasons of mechanical practicality and image quality. In the latest edition of the “Buyer’s Guide to Eyepieces” spreadsheet, the largest I can find is 27.3mm, so I have used this. Combining it with my OTA focal length of 750mm in (1) gives a maximum FoV_T of 2.15°.

For actual eyepieces, the potential problem with the FoV_T formula is that stop diameters are often not published, and may not be easily accessible to measure, either. An alternative, slightly less accurate, formula uses the EP focal length and apparent field of view, both of which are intrinsic to the EP:

FoV_T = FoV_A*FL_EP/FL_OTA                  - (2)

Apparent fields of view are published more often than stop diameters. (2) also shows the (well known) fact that, for a given OTA, the FoV_T can be increased either by selecting a longer eyepiece or, for a given focal length, substituting it with one of greater FoV_A (perhaps of a different design).   

To compare the properties of different eyepieces against each other, and against the ideal maximum FoV_T, I drew up a chart of focal length against apparent field of view, which are the two values by which most are defined, and which are independent of each other:

(all graphs are plotted in Desmos)

First, how is the limiting value of FoV_T calculated above represented here?

Re-arranging (2) gives:

FoV_A = FoV_T*FL_OTA/ FL_EP 

when again setting FL_OTA =750mm, and also FoV_T =2.15° for the maximum, it gives:

FoV_A = 1612.5/ FL_EP                        - (3)

So FoV_A is inversely proportional to EP focal length, and we get a hyperbola as the boundary between the “possible” and “impossible” (shaded) zones of EP configuration:

Similar hyperbolae will result for any fixed value of FoV_T and FL_OTA (like a family of contour lines). Others could be plotted to represent the constraints of 2” or 0.965” barrels.

Next, I want to add another constraint: exit pupil.
I haven’t gotten around to measuring my actual dark-adapted dilation, but at my age, and in Bortle 4 skies, I would want to restrict the EP exit pupil to no larger than about 5.5mm. Any more and I’ll not use all the light. Using the relationship between exit pupil, EP focal length and OTA focal ratio, this means  

FL_EP/FR_OTA < 5.5 , or for my F/5 scope:

FL_EP < 27.5mm                   - (4)

On the graph, this is represented by an area bounded on the right by a straight, vertical line:

So all the allowable EP configurations must appear in the intersection that remains unshaded.

As above, because the FoV_T is proportional to the product of FoV_A and EP focal length, increasing either (or both) will increase FoV_T. On the graph, that means that better FoV_T is to be had in the “up-right” direction, while keeping out of the two “no go” areas. So we’re looking for an EP on or close to the curve, but to the left of the vertical line. The further we follow the hyperbola to the left, the higher the FoV_A and the shorter the EP focal length. The latter means higher magnification and smaller exit pupil, which is better for seeing fainter objects in light pollution; but the former usually entails a more exotic design and — assuming a decent attempt to maintain image quality across the field — more expense. Conversely, aiming for the intersection of the hyperbola and vertical line will likely find an EP that is well-corrected at a lesser cost, but at lower magnification.

Here my other constraint comes in – I’m on a budget and don’t want to go into three figures sterling. So the area of the graph that interests me is around the intersection point, and a little to the left of it.

Now I can start adding in specific eyepieces – they appear as points, with co-ordinates defined by their focal length and FoV_A.
First, a few outliers that don’t really qualify, just to see how the land lies:

Here, there’s an OVL Nirvana (= SkyWatcher equivalent) at quite a short 16mm, but with its 82° FoV_A pushing it towards the line. Unfortunately they don’t do any longer EPs in this range. Explore Scientific do, but they’re available in 2” barrels only.
“Plossl32/50” here represents a number of 32mm Plossls from different makers, inexpensive but unfortunately with a largeish exit pupil.
On the right is the (discontinued) 35mm Baader Eudiascopic (= Celestron Ultima, Orion Ultrascopic) that is reputed to provide the largest FoV_T in a 1.25” format (although it’s way too long for me, and too rich). In fact, on my chart, its 2.29° FoV_T pushes it quite a bit above the line – apparently this genuine, but achieved with a quirky design that causes some vignetting.

I’m using the FoV_A from the Buyer’s Guide spreadsheet, which I assume is the manufacturer’s figure, but I see there are also columns for “manufacturer’s field stop” and “calculated field stop”, the latter derived from the EP focal length and FoV_A. In some cases there is a discrepancy between these two field stop values, with the calculated version being higher. I’m guessing that in some cases the advertised FoV_A is “optimistic”, or just plain misquoted, perhaps where there is a number of EPs in a range having varying FoV_A . A case in point:

Here, there’s a 24mm marketed on eBay and elsewhere under the badge “Angeleyes”, claiming a suspiciously impressive 70° FoV_A. When tested by Ernest it turned out to have a more reasonable 53°. In fact this model seems to be a generic sold under various names, often described as “Extra Flat”. There are other focal lengths in the range with larger FoV_A, so it's possibly just a sloppy cut-and-paste.

And now to add in the other contenders I found (enlarging the area of interest):

Here we have:

ES24/68                              Explore Scientific 24mm 68°
ES26/62                              Explore Scientific 26mm 62°
Hyperion21/68                  Baader Hyperion 21mm 68°
Hyperion24/68                  Baader Hyperion 24mm 68°
BST25/60                            BST Starguider 25mm 60°
EF27/53                              Skywatcher ExtraFlat 25mm 53°
                                             OVL ExtraFlat 25mm 53° (£55)
                                             AngelEyes, etc. 25mm 53° (£45)
X-Cel25/60                         Celestron X-Cel LX 25mm 60°
Orion24/65                        Orion Ultra Flat Field 24mm 65° (?)
                                             APM Ultra Flat Field 24mm 65° (£150)
Antares25/70                    Antares W70 25mm 70° (sic)

Filter on price:

the Explore Scientific 68° model is too expensive for me (new), but the 26/62 is just about OK. The Hyperions are also just inside my limit. The BST can be had for £42, the Celestron for £70 and the Antares for £50. The ExtraFlats vary by badge, but around £45-£55. I can’t find a price for the Orion24/65, thought the APM version is too much, by some distance.

And on performance:

Reviews of the ES 62° seem to be mostly positive. Views are described as transparent, contrasty and flat, though some edge astigmatism is present even on medium-speed scopes.

The Hyperion 68° range seems to be less well thought of than the ES 62s, with significant edge aberrations seen on faster scopes. The 24mm seems to be especially unpopular.

The Starguider range as a whole is well corrected down to F/5, though I have seen reports of the 25mm performance being a bit soft at the edges on faster scopes.

Opinions of the various ExtraFlat incarnations (assuming they are essentially the same) are that they are well made for the price, but also suffer edge of field aberration in faster scopes.

The x-cel LX seems to perform very similar to the Starguiders (one reviewer said they were the same, under the covers) with the general LX build quality being slightly better, though its quality control may be a bit variable. Consensus is that they’re not quite as good as the ES24/68.

The Orion 24/65 seems to be the same as the version from APM. A head-to-head test on a fast scope rated the APM better than the BST 25mm, especially on field edge sharpness.

The Antares is recommended for slower scopes only, even on their own web site.

Conclusion (ish)

Still in the race so far: ES26/62 (FoV_T 2.1°, £94),   BST25/60 (FoV_T 2.0°, £42),   EF27/53 (FoV_T 1.9°, £45),   X-Cel25/60 (FoV_T 2.0°, £70)

They are close enough together in FoV_T that I don't think I'd notice the difference. While the ES24/68 would have edged the others for performance, its cheaper cousin seems to be less impressive, probably not enough to justify double the price of the BST? There seems to be too little difference between the performance of the Celestron and BST to justify the extra either.

The one that I can’t find much about is the Extra Flat 27mm (Skywatcher, OVL, Angeleyes, etc.), which can be had for about the same price as the BST.
Anyone got a view between those two, when used on an F/5?

 

 

 

 

 

 

 

 

[Mr Spock]

Fascinating.

I have no clue what any of that says, but, for the record, I'll be sticking a 32mm Plössl in first tonight 

 

[John]

That's a very thorough and analytical approach 

I have a spreadsheet set up with my scope specs loaded and can then plug in any other eyepieces that I am thinking about to get the basic figures on what they deliver in terms of magnification and true field of view. One snag with that approach is that the specs supplied by the manufacturers are sometimes wide of the mark, occasionally well wide

What they actually deliver in terms of image sharpness across the field of view, field flatness or otherwise, eye relief accessibility, ergonomics, etc, etc seems to be a more qualitative business for which there seems to be no alternative but to try them out and see how they fare

I was very lucky that FLO kindly provided me with a regular stream of eyepieces on loan to try out and report back on

I hope you reach a satisfying conclusion in the not to distant future

By the way, I have recently owned one of the Extra Flat 27mm eyepieces. Mine was branded "Orion" but it was the same as the other clones of this eyepiece. The AFoV did seem to be around 53 degrees as stated. The eyepiece was nice to use in a 90mm mak-cassegrain and 70mm F/6.8 refractor but even in the latter the stars in the outer part of the field were looking astigmatic and I was not keen on the eyepiece in my F/5.3 12 inch dobsonian at all really. But for what it cost me (used) the eyepiece was well made and good to use in the slower scopes.

   

Edited January 21, 2021 by John

[Zermelo]

1 hour ago, John said:

By the way, I have recently owned one of the Extra Flat 27mm eyepieces. Mine was branded "Orion" but it was the same as the other clones of this eyepiece. The AFoV did seem to be around 53 degrees as stated. The eyepiece was nice to use in a 90mm mak-cassegrain and 70mm F/6.8 refractor but even in the latter the stars in the outer part of the field were looking astigmatic and I was not keen on the eyepiece in my F/5.3 12 inch dobsonian at all really. But for what it cost me (used) the eyepiece was well made and good to use in the slower scopes.

Thanks for that John. My F/5 would probably be pushing it beyond its comfort zone.

Edited January 21, 2021 by Zermelo

[Zermelo]

59 minutes ago, Mr Spock said:

Fascinating.

I have no clue what any of that says, but, for the record, I'll be sticking a 32mm Plössl in first tonight 

 

I'm just relieved that Mr. Spock says "fascinating" and not "highly illogical" 😀

 

[vlaiv]

Two remarks:

- Apparent field of view is often not accurately given and also - it can vary depending on what sort of correction was applied. There are two extremes - one of which you used for AFOV calculation based on magnification and true field of view.

One depends on relation:

y = f * tan(beta) 

and represents zero rectilinear distortion (lines remain straight)

while other on

y = f * beta

which represents zero angular magnification distortion (angles remain the same across FOV).

In both y represents linear distance in focal plane (in your case field stop radius), f represents focal length of eyepiece and beta is angle of exiting beam.

For example, with 27.3mm field stop diameter and let's say 26mm FL eyepiece, max AFOV in first case will be:

atan(13.65 / 26) = 27.7° so whole AFOV will be 56.4° (twice the angle)

in second case, it will be: 27.3 / 26 = 1.05 in radians or 60.16°

Depending what designers of eyepiece chose as their priority (how much of each distortion) AFOV will deviate from these two edge cases. You are using second formula - more often used for astronomical eyepieces - but keep in mind - wider the FOV larger the difference between two formulae

- Don't worry too much about exit pupil. Why not use 32mm Plossl eyepiece? With it, exit pupil will be 6.4mm. If your eyes have 5.5mm max pupil size - your scope will operate as if it has 130mm - or 5.1" scope. That is not major loss against 6" scope. You also have Vixen 30mm NPL if you are afraid to loose too much of light (remember, you don't know for sure if your pupil is indeed max 5.5mm). That will make your scope operate at 137.5mm of aperture that is 5.4" scope equivalent.

32mm Plossl will give you 27mm field stop in very affordable package, it is also very nice eyepiece to use. I've used it in F/5 scopes and never felt that it is somehow holding the scope back.

 

[Louis D]

Well, at f/6, this is what I got with my ~24mm eyepieces:

If you want to maximize your true field of view without using a 32mm Plossl and have decent correction and eye relief at 24mm, the APM UFF and its kin would be your best bet at 1.25".

This image shows the coverage of a couple of 32mm Plossls in the same setup:

Which is just about identical in true field of view and similar correction to the 24mm APM UFF for a lot less money.  It's your money. 😁

[Dantooine]

Wow.. that’s some detailed work and must have taken time. I congratulate you on the effort.