Aperture Stop versus Exit Pupil Stop

Ok, have to add my 2 cents:

The telescope forms an image that is 'diffraction limited' with respect
to the focal plane of the telescope.

The eye forms an image that is 'diffraction limited' with respect to its
focal plane.

Eg: a pair of 10x50 binoculars. Airy disc is about 2 Arcseconds of the
true field. given a 50 degree apparent field, the Airy disc is about 20
arcseconds of the apparent field. For the eyeball, in the daytime, with
a 2mm pupil and 25mm focal length, the eye's ariy disc is about 56
arcseconds of the apparent field, so the eye is the limiting factor. At
night, when the eye's pupil is 6mm accross, the eye's Airy disc is 18
arcseconds of the apparent field, so the telescope is now the limiting
factor.

for an 4" f/10 refractor, at 100x, the airy disc is about 113 arcseconds
of the apparent field. the eye's airy disc is 18 and 56 arcseconds
respectively at daytime and nighttime, so the telescope is always the
limiting factor.

did I get it right ?

Eric.


Alan French wrote:
"Frank Bov" wrote in message
...

[SNIP]
Now, at low power, the eyepiece does not magnifying the image enough for

the

resolution at the focal plane to matter; the eye's the limiting factor. So
at very low power, once the exit pupil exceeds the dilated eye, the
resolution in the perceived image stays the same as if the exit pupil just
filled it. [SNIP]


Frank,

Yes, that's one reason the debate I'm in elsewhere is so strange. Folks
want to believe they are utilizing the full resolution of a pair of 8x42
binoculars on a bright sunny day, yet 8 power is not enough magnification to
use the resolution of even a much smaller lens.

Clear skies, Alan

Alan French wrote:
[Frank Bov wrote:]
[SNIP]
Now, at low power, the eyepiece does not magnifying the image enough for the
resolution at the focal plane to matter; the eye's the limiting factor. So
at very low power, once the exit pupil exceeds the dilated eye, the
resolution in the perceived image stays the same as if the exit pupil just
filled it. [SNIP]

Frank,

Yes, that's one reason the debate I'm in elsewhere is so strange. Folks
want to believe they are utilizing the full resolution of a pair of 8x42
binoculars on a bright sunny day, yet 8 power is not enough magnification to
use the resolution of even a much smaller lens.

In the full light of day, visual acuity allows resolution of angular
separations as small as about 1 arc minute, on average. If we apply Dawes
limit, we can predict a resolution limit for a single 42-mm aperture objective.
(Is there a Dawes limit for binocular optics?)

Dawes limit is often stated as R (resolution in arc seconds) = 4.56/D, where D
is the aperture of the optic in inches. Converting to millimeters
(1-inch=25.4-mm), R = 115.8/D. Applying this to a 42-mm objective, we get a
theoretical resolution (angular separation) limit of 115.8/42 = 2.75, or about
3 arc seconds. The magnification needed to make that 3" angle appear 1' in size
to the eye would be 20X. Theoretically, we can say this is the magnification
needed to allow the eye to make full use of the resolution potential of that
aperture.

Under low light conditions, even more magnification is needed. Visual acuity
degrades to about 4 arc minutes, on average, if the observer is viewing objects
against a sufficiently dark background. Theoretically, we would need to magnify
the image by 80X to take full advantage of the resolution potential of that
same 42-mm objective.

Of course, visual acuity varies with the individual. So, to state the above as
hard and fast rules applying to all observers would be inaccurate. But the
above analysis does illustrate a foundation for the commonly quoted guideline
that a 0.5-mm exit pupil is the highest useful magnification for an
astronomical telescope. This may, perhaps, be better stated as the highest
"needed" magnification for a given aperture. It is not uncommon for observers
to use magnifications producing exit pupils smaller than 0.5-mm. But assuming
at least average visual acuity, magnification producing a 0.5-mm exit pupil is
enough to allow an observer to resolve to the full potential of the aperture
being used.

Getting back to Alan's original question I think it's fair to say in a general
sense that, under daylight conditions, an observer isn't taking full advantage
of the resolution potential of that aperture unless the exit pupil is ~2-mm or
smaller.

Regards,

Bill Ferris
"Cosmic Voyage: The Online Resource for Amateur Astronomers"
URL: http://www.cosmic-voyage.net
=============
Email: Remove "ic" from .comic above to respond

Consider a pair of 8x42 binoculars used at night by a person with a
5.25mm dark adapted eye:

The magnification is 8 times that of the unaided (5.25mm) eye.

The light grasp [(42/2)*(42/2)]/[(5.25/2)*(5.25/2)] = 64 times that of
the unaided (5.25mm) eye.

Resolution is 42/5.25 = 8 times that of the unaided (5.25mm) eye.

XXXXX XXXXX XXXXX XXXXX XXXXX

Consider the same pair of 8x42 binoculars used in the daytime by a
person with a daylight adapted 2mm eye pupil:

The magnification is 8 times that of the unaided (2mm) eye.

The light grasp [(2/5.25)*(42/2)*(2/5.25)*(42/2)]/[(2/2)*(2/2)] = 64
times that of the unaided (2mm) eye.

Resolution will be the effective binocular aperture (42/(5.25/2))
divided by the aperture of the unaided (2mm) eye = 8 times that of the
unaided (2mm) eye.

Conclusion: A person with a pupil too small to accept the entire exit
pupil will gain just as much as the person with a pupil just
sufficient enough to accept the entire exit pupil of the binoculars.

OTOH, the person with the 2mm eye pupil would gain just as much with a
pair of 8x16 binoculars as he would with the more expensive and
heavier pair of 8x42 binoculars.

P.S. I know I've not really addressed Alan's question. I've not
compared the resolution as seen, using the binoculars, by the two
individuals with different size eye pupils. OTOH, the magnification
is low enough (in all realistic circumstances) that for all
*practical* purposes the actual resolutions could be considered equal.

Bill Greer

"Bill Greer" wrote in message
...
[SNIP]
P.S. I know I've not really addressed Alan's question. I've not
compared the resolution as seen, using the binoculars, by the two
individuals with different size eye pupils. OTOH, the magnification
is low enough (in all realistic circumstances) that for all
*practical* purposes the actual resolutions could be considered equal.

Bill,

True. You need at least 21x to utilize the full resolution of a 42mm lens,
so at 7x the theoretical resolution is irrelevant. It's more a matter of
understanding how things work.

Clear skies, Alan