OPTICS OF THE SINGLET REFRACTOR: GALILEO vs. KEPLER
This page explains with diagrams the basic optical properties of the Galilean telescope;
in particular its magnification and why the field of view is so small and varies with the position of the eye.
The diagrams also show why the field of view is so much larger when a
positive (Keplerian) eyepiece
is used.
This is followed by a brief discussion of
limitations on the resolution
of the telescope,
Galileo's understanding of optics,
and finally a series of photos of terrestrial objects
comparing the field of view and resolution in the two modes (Galilean vs. Keplerian),
and the performance of the telescope when viewing a standard test target.
The optical diagrams shown on this page, although highly schematic, are all to the same scale.
Diagrams of this sort can be (and were) generated using the two
simple rules of thin lens ray-tracing well known to all students of high school physics:
(1) a ray striking the center of a thin lens at any angle will leave the lens at the same angle
(because the front and rear surfaces are nearly parallel at the center);
and (2) the lens has a well-defined focal plane, and all rays entering the lens
parallel to a particular central ray will exit in such a fashion as to appear to radiate
from the point at which that central ray crosses the focal plane.
Although it seems to the modern mind that both these principals can be easily deduced
by reasoning about the readily observable properties of a simple lens bringing
sunlight to a focus in a plane (apparently known in Galileo's day as the "burning point" or "combustion point" or
"point of inversion"), it is unclear that anyone of Galileo's era, other than Kepler,
understood how to create accurate optical ray diagrams.
You may use the following links (or those given above) to go directly to a specific topic:
A Telescope without an Eyepiece
The first optical diagram shows the paths of light rays passing through the plano-convex
objective lens as they would appear in the absence of an eyepiece.
It is imagined that the telescope is pointed towards some object, such as the Moon,
very far away to the right.
The light striking the objective from any one point on the object is very nearly parallel,
but the bundles from different points enter at different angles (represented here
by red, green and blue).
The objective lens does not alter these basic angles (the rays through the center of the
lens are not bent), but converts the parallel bundles of light into cones with apexes
falling roughly in a single plane called the "prime focus".
The distance from the lens to its prime focus is the focal length, F, of the objective.
The diagram is only schematic, but to make the example concrete, let us suppose the object is
indeed the Moon with its nominal diameter of 32 arc-minutes (0.533 degrees),
and that the objective lens has the dimensions of that used in the present Galilean replica
(23 mm diameter and F = 1 meter).
The cones will taper from 23 mm to 0 mm over the distance of 1 meter, giving a cone angle of
1.32 degrees.
The vertical dimension of the image at the prime focus is determined by the 0.533 degree
angle of the target projected over the 1 meter focal length.
The diameter of the prime focus image is hence 18.6 mm.
Note that the prime focus image is inverted in the sense that points above the axis correspond to light
entering the telescope at an angle below the horizontal and vice versa.
Many people placing their eye to the left of the prime focus can actually focus on this aerial
image and see the (inverted) Moon much as if it was printed on a page, with an apparent angular size
depending on how close they can get their eye to the 18.6 mm diameter image and still keep it in focus.
The possibility of seeing a sharp image without an eyepiece was known, for example,
to at least one of Galileo's contemporaries, Antonio Magini,
who noted at the start of a
September 28, 1610 letter
to Galileo that if one removed the eyepiece from a Galilean telescopes and extended the tube
"everything is seen inverted and very clearly, even if quite small" (translation
by Silvio Bedini). The sequence
of images visible to the unaided eye looking, at various distances, though a convex
objective lens had also been well described some 25 years before by William Bourne
in Chapter VII of his Treatise on the Properties and Qualities of Glasses
(reproduced in Van Helden, 1977).
In the present example, if the aerial image of the Moon is viewed from a distance of 1 meter it
will have the same apparent size as the actual Moon,
and hence no magnification will be achieved.
If the observer can get his eye closer without losing focus, the Moon will appear larger than its
actual size and previously undetectable detail will become visible since it will be spread out
over a proportionately larger spacing on the retina.
The junior author of this website, for example, is quite nearsighted with little accommodation and
can normally see objects sharply only if they are in the range 10 to 20 inches from the eye.
Looking down the tube of the Galilean telescope (with no eyepiece) and getting the eye as close
as possible to the inverted image (about 10 inches) he can achieve an apparent magnification of
around three power, and is able to read 1/4-inch high letters on a standard eye chart placed 88
feet from the front of the telescope (letters slightly under 1 arc minute high).
Wearing glasses that correct his eyes for viewing objects at infinity he cannot see the aerial
image sharply at any distance.
A ten year old child with normal vision but a much larger range of accommodation could place his eye
much closer to the aerial image and as a result was able to read letters one-half to one-third
this size.
Since the diameter of the aerial image is proportional to the focal length of the objective lens,
the magnifications achievable in this way depend entirely on that focal length.
With a very long focal length objective, quite high magnification can be achieved.
As will be explained in the following section, a person with far-sightedness (hyperopia)
or a large amount of accommodation can also see a magnified erect image
by placing their eye inside the prime focus.
At least some of the references in the pre-telescopic literature to seeing distant objects
through a "perspective glass" may refer to viewing in this fashion either the erect or inverted
image formed by a simple convex lens or concave mirror.
An alternative, and more comfortable, way to see the aerial image is to let it fall on a diffusing
device, such as a ground glass or piece of paper.
This is the principle of the camera obscura, and it makes the image visible over a wider angle,
but at the expense of brightness and detail.
In addition, magnification can still only be achieved by placing the eye closer to the image than
the focal length of the objective, and achieving significant magnification usually involves
placing the eye uncomfortably close to the image.
To overcome these difficulties, the eyepiece or eye lens is introduced.
The effect of a positive eyepiece is essentially the same as wearing strong reading glasses,
making the normal eye extremely nearsighted and allowing it to be comfortably placed very close
to the aerial image.
Galileo's negative eyepiece works in a slightly different fashion, turning the
observer into an extreme hyperope.
The Same Telescope with a Negative Eyepiece (Galilean Mode)
The normal eye is most comfortable viewing light originating from a great distance.
Parallel bundles striking the eye are converted by the visual mechanism into points on the retina
(much in the manner shown above), their position on the retina giving the brain a clue as to the
angle from which they originated.
The eyepiece of a telescope has two primary functions: (1) to convert the converging bundles of
light produced by the objective into parallel bundles that can be more easily focused by the eye;
and (2) to provide magnification.
Either positive (convex) or negative (concave) eyepieces can be used.
All the telescopes used by Galileo are believed to have used plano-concave eyepieces.
When such a lens is placed inside the prime focus by its own focal length, f,
the effect is as shown:
The eye lens not only converts the cones of light into parallel pencils, it also bends the angles
of the central rays so they diverge more rapidly than when they entered the lens.
The factor by which the angle of divergence increases is the ratio F/f, which is the
magnification of the telescope.
Continuing the numerical example, by adding a 23 mm diameter eye lens of -50 mm focal length to
the 1000 mm focal length objective, a magnification of 1000/50 = 20X is obtained.
If the cones leaving the objective diverge at an angle of 0.533 degrees (32 arc-minutes),
then the parallel pencils leaving the eye lens will diverge at 20x0.533 = 10.66 degrees.
This will be the apparent angle between the Moon's upper and lower limbs as perceived by an eye
placed anywhere to the left of the negative lens, and it is 20 times the actual size of the Moon.
Effect of Differences in Observer's Eyes
As with all telescopes, the exact power achieved will depend somewhat on the
properties of the observer's eye.
The figure of 20X in the present example was calculated assuming that the
observer's eye, when relaxed, focuses parallel light (equivalent to a point source at infinity).
This is not actually the case for all observers: a person with nearsightedness
(myopia) is most comfortable focusing slightly diverging rays; while a person with farsightedness
(hyperopia) can most easily focus converging light.
Opticians find it useful to describe the focal length of lenses (and the
correction necessary to restore an eye to the condition of focusing parallel light)
in terms of diopters, which are defined as 1000 mm divided by the focal
length of the lens.
In the present example, the 1000 mm focal length objective has a power of +1 diopter
and the 50 mm focal length eyepiece has a power of -20 diopter.
The magnification of the telescope, when focused at infinity, is the ratio
of the powers of its eyepiece and objective with the eyepiece power on top
(the opposite of when focal lengths are divided).
Diopters are convenient because when two lenses are sandwiched together, their
combined power in diopters is the sum of their individual powers in diopters.
The hyperope, requiring say +1 diopter of correction to achieve "normal" vision,
has, in effect, a -1 diopter concave lens added to their "normal" eye (the one that
focuses at infinity when relaxed).
When they apply their eye to the website telescope, they will be
adding -1 diopter of power to the -20 diopter eyepiece, giving it a total power
of -21 diopters, which will also be the magnification they see.
They will also require a slight adjustment in the focal position to accommodate the
effectively higher power of the eyepiece.
Similarly a myope requiring -1 diopter of correction has an eye that is the equivalent
of a normal eye plus a +1 diopter positive lens.
When looking through the website telescope, their eye will effectively reduce the
power of the concave eyepiece to -19 diopters, requiring a slight adjustment in
focal position, and reducing the power they see by that amount.
This same reasoning can be used as an alternative explanation how some people are
able to see a magnified image with no eyepiece at all: the hyperope has in
effect a weak negative eyepiece built into their natural eye; while the myope
has a weak positive one.
For 1 diopter of correction, the focal length of this natural eyepiece is 1000 mm;
for 2 diopters of correction it is 500 mm, and so on.
If a person with 1 diopter of hyperopia (farsightedness) places their eye at
the position appropriate for a 1000 mm concave eyepiece, they will see an erect Galilean
image using the objective alone.
In the present example, with a 1000 mm focal length objective, that position
(1000 mm inside the prime focus) would be right up against the objective lens,
and the power would be 1X (i.e., their eye would cancel the power of the objective).
For 2 diopters of hyperopia, the position of comfortable focus would be 500 mm
from the objective, and the power seen would be 2X.
A person with myopia (nearsightedness) of the same amounts will find it most comfortable
to place their eye outside the prime focus by the same amounts, where the natural
positive power of their eye will allow them to see an inverted
Keplerian image
with the same magnifications as just indicated.
When persons with normal vision (eyes focused at infinity when relaxed) "accommodate"
their eye to focus on near or far objects, they are effectively making their eye
myopic or hyperopic.
The accessible range of accommodation (which gets much less as we age) is also expressed in diopters.
As a result of accommodation, a person with normal vision may also be able to
see, with no eyepiece, the Galilean and/or Keplerian images described above, but
it will probably require some strain to get the eye to focus sharply at those positions.
Accommodation also allows the eye to adjust for slight errors in focal position when an
eyepiece is used.
Calculation of Field of View
The optical diagram explains not only the power of the telescope, but also how
much of the Moon will be seen at any one time, and why
moving the eye up and down or to the side changes the part that is seen.
The angular field that can be seen at any one time is determined by the range of cones from the
objective that fully or partially enter the observer's pupil.
In the situation shown schematically in the diagram, the green pencil from the center of the Moon
fully enters the dark pupil of the eye, but the red and blue pencils (from the lower and upper
limbs) completely miss it, so only an area around the center of the Moon is visible.
To see the limbs properly, the eye would have to be moved up, to intercept the red pencil, or down
to see the blue pencil.
Of course, when these limbs are visible, the rest of the Moon would not be.
Alternatively, the schematic eye could be moved forward until its pupil is pressed against the
eye lens.
In that case, not only the green pencil, but also a little bit of the red and blue pencils would
pass through the eye's pupil.
Hence the Moon's lower and upper limbs would be visible, but dimly.
This demonstrates that the field of view visible at any one time depends not only on the diameter
of the telescope objective and observer's pupil, but also on the power of the eyepiece (which
affects how close it is to the prime focus and hence the diameter of the transmitted pencils) and
how close the observer's eye is to it.
It also demonstrates that the field of view does not have a sharply defined edge.
The present telescope does not permit so wide a field of view as would be suggested by the diagram.
For a 32 arc-min moon, the spacing between the centers of the red and blue cones falling on the
eyepiece would be 32 arc-min * 950 mm = 8.8 mm.
The diameter of the parallel pencils exiting the lens is the same as the diameter of those cones
at the point where they intercept the lens.
Since the cones taper from 23 mm diameter on the objective to 0 mm diameter (at the prime focus)
in 1000 mm, and since the eyepiece is 50 mm from the prime focus, the diameter of the
cones/pencils at the eyepiece is (50/1000)*23 mm = 1.2 mm.
The spacing between the centers of the most widely spaced pencils that can be (marginally)
perceived by the observer is the diameter of his pupil plus the diameter of one of the pencils.
For a 4 mm diameter pupil, this would be 4 + 1.2 = 5.2 mm.
Comparing this to the known scale of 32 arc-min = 8.8 mm, we see that the extreme limit of the
observer's field is to see light from targets separated by (5.2/8.8)*32 = 19 arc-min.
Likewise the spacing between pencils that fully enter the observer's pupil without attenuation is
equal to the pupil's diameter minus the diameter of one pencil.
In the present case, this is 4 - 1.2 = 2.8 mm, corresponding to a target spacing of (2.8/8.8)*32 =
10 arc-min.
In summary, an observer with a 4 mm diameter pupil pressed against the eyepiece would be expected
to see objects without attenuation over a field of 10 arc-min, and with vignetting out to
19 arc-min.
The mid-point of this range, 14.5 arc-min, is determined solely by the size of the observer's
pupil divided by the distance from eyepiece to objective and is independent of the diameter of
the pencils (and hence of the diameter of the objective).
It agrees well with the observed field of about 14 arc-min, although the agreement is partly
accidental.
In reality the eye's pupil is held further back from the eyepiece, giving a smaller field,
but one compensates by measuring out closer to the limit of visibility.
The field of view and amount of vignetting at other eye positions can be calculated by
observing that the vertical spacing between the centers of the red and blue pencils always
corresponds to the angular spacing between the low and high targets -- in this case, the Moon's
lower and upper limbs.
Since these pencils diverge at a known angle, their spacing at any eye position can be calculated,
giving the vertical spacing corresponding to 32 arc-min.
It is evident from the diagram that if the eye is held close to the eyepiece, increasing the
power of the eyepiece (i.e., decreasing its focal length) would only slightly decrease the
angular field (a higher power eyepiece is placed closer to the prime focus where the vertical
spacing between the centers of the red and blue pencils is slightly larger compared to the fixed
pupil diameter).
However increasing the power of the eyepiece dramatically increases the rate at
which the field falls off as the eye is moved back from the eyepiece, since the angle at which the
red and blue pencils diverge is directly proportional to the power.
Alternative Ways of Calculating the Field of View
Although this method of estimating the field of view by considering which pencils of light are
intercepted, fully or partially, by the observer's pupil is correct, the diagram given above might
suggest the results depend on the eyepiece being at focus.
In reality, the eyepiece can be moved
far in and out of focus without noticeably altering the apparent angular size of the patch of
light visible to the eye.
The following two diagrams offer an alternative explanation of the
field of view which may help to explain why.
They are based on the principle, mentioned earlier, that rays passing
through the center of lens are undeviated in direction (because the faces of the lens are
essentially parallel at that point).
They also indicate how the field of view is made up of two
parts: one due to the size of the telescope's objective, which would be present even if the
observer's pupil were reduced to a pinhole; and the other due to the finite size of the observer's
pupil, which would be present even if the objective were reduced to a pinhole.
In the present system, the observer's pupil dominates, while in a faster system the size of the
objective would become dominant.
First we consider two rays (red and blue) drawn from the center of the eyepiece to the extremes
of the objective lens.
The angle between them is the diameter of the objective divided by the
distance from eyepiece to object.
In the present case this is 23mm/950mm or 1.4 degrees.
Since these rays pass through the eyepiece undeviated, we know that rays over at least this range
of apparent angle will be perceived by the eye.
A field of 1.4 degrees apparent diameter would be visible even if the diameter of the observer's
pupil were reduced to zero.
Next we consider two rays (red and blue) drawn from the extremes of the observer's pupil to the
center of the objective. The angle between these rays is the diameter of the observer's pupil
divided by the distance from eyepiece to objective. In the present case this angle is 4mm/950mm
= 0.24 degrees. Since these rays pass through the objective undeviated we know the observer can
at least partially see targets separated by that angle. In addition, we know that if the eyepiece
is near focus the telescope will magnify the apparent angle between targets by its magnification
factor. In the present case, the apparent angle between the red
and blue rays will be 20*0.24 = 4.8 degrees.
A field of 4.8 degrees apparent diameter would be visible even if the objective lens were reduced
to a pinhole.
Examination of the overall ray diagram for the Galilean telescopes reveals that the red and blue
lines in the last diagram are actually the centers of cones from the objective which would be
approximately 50% attenuated by the observer's pupil. But we can see somewhat beyond this range,
because we know from the earlier diagram that from every tiny part of the pupil we can see a cone
1.4 degrees in diameter. The total range visible is obtained by adding or subtracting the 1.4
degree cone from the 4.8 degree one. In other words we should be able to see from 4.8 - 1.4 =
3.4 degrees, without vignetting, to an extreme of 4.8 + 1.4 = 6.2 degrees with full vignetting.
The true angular field (on the sky) would be 1/20th of these amounts or 10 to 19 arc-min.
This is identical to the previous result.
A detailed mathematical investigation of the field of view of Galilean telescopes (as applied to
the commercial binoculars of the day) was given in a 1920 article by
Hughes and Everitt.
For those familiar with geometrical optics, they make the interesting observation that the eye
lens forms a virtual image of the objective lens.
The diameter of this virtual image (the exit pupil) is D/m and its distance
to the right of the eye lens is
(1 - 1/m)*f = L/m where D is the diameter of the objective, f is the
focal length of the eye lens and m = F/f is the magnification of the telescope and
L = F - f is its length.
Since all the light must pass through this virtual image, looking through the telescope is like
looking at the target through a porthole of this diameter and at this distance.
In the present case, the "porthole" is (1 - 1/20)*50 = 48 mm inside the eyepiece, and if our eye
is focused at infinity, the edge of the porthole will appear very fuzzy.
Their result differs slightly from ours in that they assume that as one shifts one's attention
from the center of the target to the edge of the field the eye rotates about its center.
This produces a lateral motion of the pupil, which actually reduces the field, as can be seen by
consulting the optical diagram at the top of this section: if the eye rotates upward in an effort
to look in the direction of the blue rays from the Moon's upper limb, the pupil will move in such
a direction that the blue pencil will be further away from the pupil than it was before moving.
This effect is indeed real and can be demonstrated by staring at the center of the field and then
shifting one's gaze so as to look directly at the extreme edge of the field.
An object that was originally visible in peripheral vision at the edge of the field will disappear from view.
We reject this reasoning as a definition of the field
of view, however, for to our minds the field of view refers to what can be taken in by the eye in
a single view -- staring at a fixed point and noting how far we can see to the side with
peripheral vision.
This is the only definition that is consistent with what we see with the
camera, where the position of the pupil is definitely fixed.
Also, as explained above, if the pupil is allowed to move, the field of view can
be greatly increased by moving the observer's eye around the eyepiece so as to
intercept the pencils of light from different parts of the target.
Another thorough mathematical analysis of the field of view of the Galilean telescope is given,
somewhat oddly, as an appendix to the otherwise rather non-technical article by
John North in
a book about Thomas Harriot, an Elizabethan scientist and adventurer who observed the Moon and Sun
through a telescope somewhat before Galileo, but whose results were not known until his private
journals were read many years later.
North gives the formula for the field of view visible on the target as:
Field of View [arc-min] = (10800/π)*(p ± D/m)/(L + em)
where p is diameter of the observer's pupil, e is its distance from the eye lens
and the other symbols are as defined above.
The "+" condition is used to define the outer edge of the zone of vignetting,
while the "-" sign gives its inner edge.
North's result is mathematically identical to the argument given in words above,
as can be verified by substituting the given data with e = 0.
North's formula is for the true field visible.
The apparent field presented to the eye is, of course, m times larger.
The formula given by Hughes and Everitt (which is for the apparent field) is equivalent to
North's, but for e they use the distance from the eyepiece to the center of rotation of
the observer's eye, rather than the distance from the eyepiece to the front of the observer's eye.
As noted above, this results in a smaller estimate of the field of view.
Also they are dealing with a situation in which D/m > p, so the exit pencils completely
fill the observer's pupil and hence they take the difference in the opposite order.
Galileo's Early Efforts to Measure Angular Distances with Stops
Near the beginning of Sidereus Nuncius
Galileo suggested it would be possible to measure the distances between stars by putting
a series of graduated stops over the objective, calibrating the exact field provided
by each with terrestrial observations during the day:
comparing small circles viewed through the telescope with one eye to larger circles
viewed through the other eye (unaided by the telescope).
This calibration would be much more difficult than the estimation of the magnification,
since the field of view has a poorly-defined edge.
Even if a repeatable daytime calibration could be achieved, it would give
erroneous results when applied to stellar observations since the field of view
depends strongly on the diameter of the observer's pupil, which would be larger at night.
Galileo may have discovered this, for he seems to have abandoned the stops and
made his later, highly accurate, measurements of the positions of Jupiter's moons
using an adaptation of his method for making daytime estimates of magnification:
comparing the spacing between the stars seen through the telescope to the spacing
of known marks on a dimly lit target viewed through the other eye.
The observer can, with surprising ease, mentally merge the view seen through the
telescope with one eye with the image of the reticle as seen by the other.
In that way, quite accurate measurements can indeed be made.
The impossibility of accurately measuring angular distances using stops placed over the entrance aperture,
and the actual dependence of the field of view of a Galilean telescope on the diameters of
both the objective lens and the observer's pupil, as well as the fuzziness of that zone,
was the subject of a 1986 article by
Hideo Nishimura,
who expanded greatly on North's work, and confirmed his predictions experimentally.
Unfortunately this work is available only in Japanese.
Like North, Nishimura does not seem to have considered the small effect, important to Hughes and Everitt,
of the rotation of the observer's eye as the observer's attention shifts from one
part of the field to another.
In addition to Nishimura's paper, there are some nice optical diagrams showing the start, middle, and end of the
vignetted zone at the edge of the field of view, and its dependence on the size of the
observer's pupil, on the French language website by
Joseph Hormière.
Hormière appears to have worked out the theory independent of any of the previous
authors.
The other pages on his site contain many interesting observations about the geometric optics
of Galileo's telescopes.
Hormière suggests that in Sidereus Nuncius
Galileo may have been referring to placing a diaphragm next to the
objective lens, as he later did, rather than over it.
However, in a later book (the opening pages of his
Discourse on Bodies Floating in Water)
Galileo says he had not yet developed an accurate method
for judging angular distances at the time he wrote Sidereus Nuncius
(see our Jupiter page).
The small size of the exit pencils and the fact that the pencils from different parts of the
target are physically separated from one another in the focused Galilean system has an interesting
and little-noted consequence: the telescope becomes a sensitive indicator of imperfections in
the observer's (or camera's) eye.
As noted above, in the present 20X telescope the pencils are already only 1.2 mm in diameter.
An imperfection over a millimeter-sized area of the observer's cornea or lens will appear as a
ripple or other non-uniformity (e.g., in brightness) in the image.
If a higher power (shorter focal length) eyepiece is used the diameter of the exit pencils will
become still smaller in proportion and imperfections over even smaller areas will become apparent.
To distinguish the imperfections that are due to his own eye from those in the telescope or in
the target itself, the experienced observer learns to turn his head, rotate the telescope and
sweep its field over the target.
Each kind of imperfection will behave in a different way.
The Same Telescope with a Positive Eyepiece (Keplerian Mode)
The great German mathematician-astronomer Johannes Kepler (1571-1630) suggested the idea of using
a positive eyepiece with a telescope in his classic work on the formation of images by lenses,
Dioptrice, which appeared less than a year after Galileo's initial announcement of
his astronomical discoveries.
The following diagram illustrates the effect of using a positive eye lens with identical focal
length and diameter to that in the previous example:
To convert the cones of light from the objective into parallel pencils, the positive lens must be
placed its own focal length BEHIND (rather than in front of) the prime focus.
Just as with the negative lens, it not only renders the converging light parallel but bends the
central axis of the cones.
The direction of the bending is, however, reversed.
The red pencil, for example, which exited the top of the negative eye lens going UP at a certain
angle, now exits the top of the positive eye lens going DOWN at the same angle.
Hence an eye placed to the left of the positive eye lens sees an inverted image: light from the
target's lower limb appears to be coming from above, while light from the upper limb appears to
come from below (and similarly for left and right).
The diameter of the parallel pencils and the magnification in their angles of divergence are
exactly the same as for the negative lens of the same focal length.
That is, in the present numerical example, the pencils are 1.2 mm in diameter and the angle
between the red and blue pencils is 10.66 degrees (20 times the actual diameter of the Moon).
If the eye were pressed right against the positive eye lens, the field of view would be determined
by exactly the same considerations as in the Galilean case.
The result would actually be slightly smaller because the eyepiece is farther from the objective
(1050 mm vs. 950 mm).
The diameter of the 32 arc-minute Moon is larger, relative to the fixed pupil diameter, at the
point where it strikes the positive lens: 9.8 mm on the positive lens vs. 8.8 mm on the negative
lens; so the field visible with a 4 mm diameter eye pressed against the negative lens is about
(4/8.8)*32 = 14.5 arc-minutes vs. (4/9.8)*32 = 13.1 arc-minutes with the positive lens.
However, the positive lens has a magical property: all the pencils exiting the lens cross at a
particular point on the telescope's axis called the "exit pupil".
If the observer places his eye so that his own pupil is roughly centered on the exit pupil all of
the pencils exiting from the eye lens will be seen equally well, and the apparent field of view
will be limited only by the diameter of the eye lens.
For pencils starting at a spacing of 9.8 mm and converging at an angle of 10.66 degrees, this
point is expected to fall 53 mm beyond the eye lens, and the diameter of the point of convergence
is 1.2 mm, the same as the diameter of any one pencil. In fact a 23 mm diameter lens with a focal
length of 50 mm is so steeply curved (radius of curved surface = 25 mm), that the actual
refractions at the edge are a little different than would be expected from this simple logic.
The pencils from the periphery of the eyepiece converge closer to the lens than those from the
center.
A more careful calculation indicates the point of maximum convergence is nearly twice the expected
diameter and about 42 mm behind the last surface of the eyepiece.
From 42 mm away, the apparent angular diameter of a 23 mm diameter eyepiece is about 30 degrees,
corresponding, in a 20X telescope, to a true field of view of 30/20X = 1.5 degrees = 90
arc-minutes; however, the magnification for the peripheral rays is also a little different and
the maximum true field is more like 78 arc-minutes.
This is some four or five times the diameter of the largest field that can be seen at any one
time through a negative eyepiece of the same power.
As with the negative lens, a certain amount of vignetting is expected at the edge of the field,
but its origin is different: all the pencils exiting the eye lens pass fully through the
observer's pupil, but the ones originating near the edge of the eyepiece are only partially
illuminated because the cone from the objective will be partially blocked by the edge of the
eyepiece.
The lateral motion corresponding to the transition from full passage to full blockage is the
diameter of the cone at the eye lens (= diameter of exit pencil).
For the numerical example, a 1.2 mm motion at 1050 mm from the objective corresponds to 3.9
arc-minutes, about half the width of the ring of vignetting expected in the Galilean image with
eye pressed against eyepiece.
The previous section noted the sensitivity of the Galilean telescope to imperfections in the
observer's eye.
The 1.2 mm diameter for the Keplerian exit pencils is the same as in the Galilean telescope; however, if the
observer places his eye in the exit pupil of the Keplerian telescope, all the exit pencils will pass
through the same 1.2 mm area of his pupil.
Hence all parts of the target will be equally affected by imperfections in the observer's eye and
one is not aware of the ripples that can be seen in the Galilean image. In a sense, the Galilean
telescope has a similar exit pupil, but it is apparent from consulting the diagrams given above that
when a negative eyepiece is used the plane in which the exit pencils would theoretically converge is
on the objective side of the eyepiece, and the observer cannot place his eye's pupil there.
Although Kepler and Galileo corresponded, and Kepler's publication of the positive eyepiece idea
was made while Galileo was still actively pursuing his astronomical observations, we can find no
suggestion that Galileo, or even Kepler, ever tried actually using one.
As pointed out by Albert Van Helden
in 1974, Kepler seems to have presented the possibility
of producing a magnified, but inverted visual image (and also the possibility of correcting
the inversion by adding a third lens) more as a theoretical curiosity than as a practical
suggestion.
According to Van Helden, both Kepler and the readers of Dioptrice were
probably completely unaware of either of the great advantages
of his design: the immensely larger field of view and the possibility of inserting a
measuring device into the prime focus image.
Since Kepler correctly pointed out the obvious disadvantage of the inverted visual
image, there may have been little impetus for exploring his designs.
The positive eyepiece found early use (e.g., by Scheiner) in forming projected images,
where it reversed an inversion of the projected image inherent to the Galilean design.
However the difference in field of view would not have been apparent in this application,
because the full field is projected (as shown below) in both cases.
According to King's highly-respected History of the Telescope it was
apparently a number of years before anyone (possibly Fontana) actually
looked through the positive eyepiece and discovered its larger field.
Despite this conventional wisdom, Keplerian telescopes evidently existed from
a very early time.
Silvio Bedini
reproduces a photograph of one made by or for the bookseller Augustus Cracaw.
It is listed in a catalog of the Kunstkammer of Dresden compiled in January, 1613,
and was presumably constructed before that date, although it is, of course, not known
how this telescope was used.
It is frequently claimed that despite its vastly superior field of view the positive eyepiece was
not adopted by the early astronomers because it introduced additional aberrations.
For example, a 1944 Encyclopaedia Britannica article on the telescope by Sirs David Gill
and Arthur Eddington states that "The sharpness of image in Kepler's telescope is very inferior
to that of the Galilean instrument."
We are unable to discover any reason, theoretical or practical, for thinking that a positive
eyepiece, with the same curve and diameter, would offer any different axial resolution than a
negative eyepiece in a telescope of the present aperture and focal length.
The following photographs demonstrate this.
With a positive eyepiece one will naturally tend to look farther off axis, where aberrations do
indeed become apparent; however, if one attempted to look equally far off axis with the negative
eyepiece (which requires physically moving the eye's pupil to the side), the aberrations would be
the same. In other words, aside from the erect image and more compact tube, we can see no optical
advantage to Galileo's negative eyepiece.
The intrinsic limits to the resolution of the telescope, whether of Galilean or
Keplerian design, are discussed more fully below.
Viewing the Sun in Projection
For those still not convinced that the limited field of view of the Galilean telescope
is primarily due to the finite size of the observer's pupil and not something intrinsic
to the design of the telescope, the fact that the entire Sun can be seen in projection
(whereas only a portion can be seen by direct viewing) should provide ample proof.
In his 1613 The Sunspot Letters,
Galileo attributes the discovery of the possibility of observing the Sun by eyepiece
projection to his friend and protege Benedetto Castelli.
It is also mentioned (and illustrated by a nice little sketch) in a
July 14, 1612 letter
to Galileo from his artist friend Cigoli.
Cigoli credits a Gismondi Coccapani with having shown the method to him.
It was later used extensively by Galileo's great German rival Scheiner, who initially
attempted direct observation of the Sun with colored glass filters placed over the
telescope's objective.
The possibility of eyepiece projection had presumably been independently discovered
by David and Johannes Fabricius, for
they apparently mention using it in their publication on sunspots which preceded the
announcements by Scheiner and Galileo.
Pierre Humbert
indicates that eyepiece projection was also used by the French
artist Claude Mellan
in preparing his remarkable engravings of the full disk of the Moon with a Galilean telescope
(although a large projected lunar image would be extremely dim,
and it seems unlikely much detail could have been seen).
As shown in the second optical diagram near the top of this page,
when the negative eyepiece of the Galilean telescope is in its normal position for viewing with an eye
focused at infinity, the rays from a particular point on the Sun (or Moon) leave
the eyepiece in parallel bundles.
If these are allowed to fall on a paper held
far enough to the left so that the bundles from different points are separated
one will see a large but very dim image of the object at which the telescope is pointed.
If the eye lens is moved out slightly (away from the objective), as shown in the diagram
immediately above, it will still make the bundles more parallel than they
were approaching the prime focus but not enough so as to completely overcome their
convergence.
Hence they will come to a distinct focus at a point somewhere to the left of the prime focus, as shown.
Since the same amount of light will be concentrated
into a smaller image it will be correspondingly brighter.
Note that the paper will receive all the bundles passing through the eyepiece,
not just those that would normally
pass through the observer's pupil, and hence the whole field of view of the telescope
is seen in a single image.
In the extreme case where the eyepiece is pulled out all the way to the prime
focus, the lens will have no effect on the convergence of the bundles and a paper held at that position
will simply receive the direct image projected by the objective.
This is the principle of the camera obscura with its fixed size image.
The eyepiece provides the possibility of easily modifying the size (and brightness)
of the image.
Since the quantity of light in the projected circle is exactly the same as that falling on the telescope's
objective, when the eyepiece is adjusted to give a projected image the same diameter
as the clear aperture of the objective, the intensity of the image will be the
same as that of the ambient sunlight falling on the telescope, and hence will be
hard to distinguish unless the paper is shaded.
If the projected image is made larger, even more shading will be needed to view it.
The ideal situation is when the telescope is used to project the image through a
small aperture into a completely darkened room.
Eyepiece projection can be achieved with both Galilean and Keplerian telescopes.
In the Keplerian design the eyepiece is also pulled out slightly (away from the objective)
to give convergence to the exiting bundles of light and form a projected image.
The Galilean design is slightly superior for solar observation because the
light never comes to a focus except in the projected image.
In a Keplerian telescope there are potentially two hot spots where the beam of solar light
is channeled into a small circle: one at the prime focus and
one at the exit pupil, with the attendant possibility of igniting a cardboard telescope.
For the small diameters and very long focal lengths of typical 17th century objective lenses,
the concentration of light at the prime focus was not very strong and never posed
any real threat of fire inside the tube.
However, the concentration of light at the exit pupil of a Keplerian telescope can
be intense and not only ignite a piece of paper when the Sun is bright but also damage
an unwitting observer's eye if one attempts, for example, to view the Sun directly through haze
(at the expense of seeing only a small field, like that of a Galilean telescope, a wise observer
will hold the eye well back away from the exit pupil).
In return for these problems there is no advantage in the size or quality of the projected
image: in both cases the whole Sun will be visible basically if and only if the full
diameter of the prime focus image can fit through the eyepiece.
In the numerical example given above, with a 1000 mm focal length objective the
diameter of the prime focus image of
the Sun (or Moon, which subtends about the same angle in the sky) is about 18.6 mm.
For use in projection, the negative eyepiece of a 20X Galilean telescope would be
placed somewhere between the prime focus and its normal position 50 mm closer
to the objective.
At this latter point, the entire cone of rays from the Sun will be only slightly
larger than its diameter at the prime focus.
Hence any eye lens more than about 19 or 20 mm in diameter will pass all the
rays received from the Sun and form a complete image.
Any smaller diameter eye lens will vignette the projected image.
For projecting images of terrestrial objects (as for use by artists),
the positive eyepiece is preferred, since it produces an upright projected image.
The projected image produced with a negative eyepiece is upside down.
This is the opposite of the case for visual observation through the lens, in which
the Galilean image is upright and the Keplerian image is inverted.
Although it contains a complicated objective with many elements, the modern
telephoto lens is a not very distant cousin of the Galilean design shown
above.
A final negative element permits projection onto the film or sensor, in a short distance,
of an image much larger than could be achieved by the front lens alone.
The Barlow lens of the amateur astronomer, a negative element introduced just before
the light reaches its focus, behaves in a very similar way, creating an
enlarged image slightly beyond the original focal point, which is then
examined with the modern derivative of a Keplerian eyepiece.
Resolution and Aberration
The resolution of a telescope refers to the true (as opposed to magnified) angle
between the most closely spaced features it can separate.
The maximum resolution achievable by any telescope is limited by a phenomenon called diffraction --
the spreading out of light due to its wave nature.
A point of light is actually focused into a central spot surrounded by a faint
pattern of rings, as can be seen in our photo of Mizar;
or perhaps more clearly in the picture at left: a full color simulation of the diffraction spot
expected from a point source of white light as viewed near the focus of the
website telescope (BK7 glass, 23 mm aperture/1000 mm focal length with eyepiece 945 mm
from objective; the simulation is 40 arc-seconds on a side).
For light of a single color, the width of the diffraction pattern is directly proportional to the wavelength
and inversely proportional to the aperture (diameter of the front opening) of the telescope.
Although various definitions can be proposed depending on the nature of the target
and experience of the observer, the usual figure quoted for the ultimate resolution is based on
a standard proposed by Lord Rayleigh in 1879 and, as applied to a telescope,
refers roughly to the minimum spacing at which two equal points of light can be distinguished.
For a wavelength in the middle of the visible band (mercury green line = 5461 Å)
and an aperture of 1 inch (25.4 mm) the Rayleigh limit is 5.41 arc-seconds.
The maximum possible resolution of the present telescope is slightly less since
its aperture is a little smaller.
Projected over a distance of 1000 mm, 5.4 arc-seconds corresponds to a spot of
light a little over 0.05 mm in diameter at the prime focus.
For a star image, the resolution limit (5.4 arc-seconds in the present example)
refers to the diameter of first dark ring; the diameter of the bright central spot
of light will appear slightly smaller.
As indicated above, the diameter of this spot will decrease as the diameter of
the objective lens is increased.
Images of extended sources (such as a planet or a two-dimensional test target)
suffer from diffractive blurring just as much
as do those from point sources, because in truth the image of an extended source
is nothing more than the sum of the diffraction patterns from each individual
point in that source.
A telescope will achieve the Rayleigh limit only if it is optically perfect: that is, the surfaces must be such that in the absence of diffraction the
parallel rays of light from a distant source would be brought to a perfect point focus.
Most failures to do this are called aberrations.
A Galilean telescope is expected to suffer from two principal aberrations: spherical and chromatic.
Spherical aberration arises because the spherical shapes of the lens surfaces are not the ideal ones for bringing parallel light to a point focus (Kepler, whose understanding of the
theory of optics was apparently much more profound than Galileo's, already knew that the ideal singlet lens would have a hyperbolic surface).
Rays striking the periphery of a spherical objective will focus at a closer distance than ones striking the center.
This imparts an additional fuzziness to the image.
For the present 23 mm diameter objective lens this effect is very tiny: if the front surface is a perfect sphere and the central rays for a particular color come to a focus at
a point 1000 mm from the lens, the rays from around the edge of the lens will focus at a point about 0.1 mm closer to the objective.
From the cone angle (expanding by 23 mm in 1000 mm) it is easy to see that at the point where the central rays focus, the peripheral rays will be spread over a circle about 0.002 mm in diameter.
Since this is much smaller than the 0.05 mm of intrinsic spreading due to diffraction, the effects of spherical aberration in the objective are quite negligible.
The eyepiece has a much more steeply curved surface, and although used only over a small area (limited by the diameter of the observer's pupil) might have a greater effect.
Its exact effect can be explored by tracing a bundle of parallel rays backwards from the observer's pupil through the eyelens to their virtual focus.
If the spot they form is smaller than the size of the spot formed by the objective, then the eyepiece can be expected to do an accurate job of "resolving" the astronomical image;
if the spot formed by the eyepiece is larger, it will not be able to resolve all the detail present at the prime focus. For the maximal 8 mm diameter bundle that can be used by the eye,
the diffractional blurring (by comparison with the result for a 1 inch aperture) will be 17 arc-seconds.
Projected over the focal length of 50 mm this gives a spot 0.004 mm in diameter.
However, the peripheral rays will focus about 0.19 mm closer than the central ones (the spherical aberration).
In the middle of this range, the minimum spot diameter that can be achieved by cutting through the overlapping cones of light is about 0.015 mm.
Since this is about one-third the size of the smallest detail expected in the prime focus image produced by the objective, the eyepiece should, at most, add a very slight
additional degradation even when the observer's pupil is at its maximum size.
Chromatic aberration ("color") arises because, just like a prism, a lens made of
glass bends blue light more strongly than red -- hence the blue rays come to a
focus closer to the lens than the red ones, with the other colors in between.
Even if it is able to form a perfect point for light of one color, the other
colors will not come to a focus at the same point.
The individual images, which we could see by using a filter to pick out the individual wavelengths,
suffer only from diffraction and spherical aberration, but when we look without
a filter we will inevitably see a combination of a focused image in one color
superimposed on unfocused images in other colors.
Sliding the eyepiece in and out we can focus on the various spectral colors in succession.
When focused on a bright light optimized for the green or yellow, the combination
of out of focus red and blue light will typically produce a violet-tinged halo as shown in
our over-exposed pictures of Jupiter and
Saturn.
During his 1923 study,
Ronchi measured the properties of the three objectives preserved in Florence at a variety of wavelengths.
He found that the focal length increased
by amounts between 2.3 and 2.9% as the wavelength was increased from 4500 Å (blue) to 6500 Å (red).
This compares to 2.1% expected for the common modern optical glass
BK-7, meaning that
telescopes made from a glass of Galileo's day probably had slightly more
chromatic aberration than a modern replica.
Projected over a 1000 mm focal length 2.6% of dispersion translates into differences
of 2-3 cm in the locations of the prime foci for the various colors.
In the middle of this range, where green light comes to a point focus, the rays
from the extreme blue and red light will be spread over a diameter of about 0.3 mm.
This is much larger than the intrinsic fuzziness imposed by diffraction, suggesting
that chromatic aberration will be the most serious limitation on the resolution of the telescope.
Both practical experience and simulation
demonstrate that chromatic aberration produces a somewhat unexpected effect:
the telescope itself acts as a kind of chromatic filter.
At each focal position one color is sharply in focus while the others are
sufficiently out of focus that they are seen as a kind of very diffuse background
haze which decreases the contrast of the image but affects the resolution less
than might have been imagined.
In other words, if one is viewing a point source of white light, such as a star,
as the eyepiece is moved in and out the star's apparent color will change, but
it can be seen sharply, in one color or another, over a considerable range.
The actual resolution of the present telescope, which is remarkably close to the Rayleigh limit,
is demonstrated photographically below.
Galileo's Understanding of his Telescope
In addition to the present discussion and the references cited on our
Additional Information page,
Yaakov Zik has written extensively
about the optics of Galileo's telescopes.
Zik's discussion tends more towards philosophical implications than towards
optical engineering insights.
There is also, what seems to us, a marked tendency towards drawing sweeping conclusions
from extremely limited evidence.
For example, Zik claims that Galileo's off-hand comment that to verify his astronomical
observations one needs a telescope that produces sharp images to indicate that
Galileo had a profound understanding of optical aberrations, lacking only the
modern terminology.
To us, the fact that some telescopes produce good images and others bad
ones seems a very commonplace observation, and does not necessarily imply any
understanding of the cause.
Likewise Zik says that Galileo's mention in his book about comets of the comet-like
flare of light seen surrounding the reflection of a candle from a smudge on a glass wine carafe
indicates that Galileo understood not only spherical and chromatic aberration, but also
the aberration now called coma, produced by tipped optics.
This, again, seems a sweeping conclusion, since the origin of this effect is quite
different from that of the flared smudge of light one sees trying to image the
sun with a tilted burning glass (which is much more closely related to coma, and
which Galileo does not chose to mention).
To us, it seems likely that Galileo's understanding of his telescopes was much more
practical and experimental than theoretical.
One of Zik's main contentions is that Galileo's success, relative to his contemporaries,
was due to his addition of aperture stops over the objective.
In his 2001 and 2002 papers, Zik provides an illustration which appears to represent a
simulation of the point image produced by a perfect
spherical objective (i.e., a lens with perfect spherical surfaces) of 1.33 m focal
length stopped to 26 mm and versus the same lens opened to 32 mm.
Zik says that although (presumably) Galileo's historic discoveries could have been
achieved with the "reasonable image" produced by such a lens at 26 mm aperture,
when the same lens is opened to the 32 mm aperture, the effects of chromatic aberration
"actually prevent one from using the telescope for celestial observation."
We find this conclusion hard to understand, since the increase in spot diameter
shown in the illustration is at most 10 or 15%.
Moreover, our own
simulations
do not corroborate even this modest increase in
spot size: on the contrary, at the long focal lengths used in Galileo's telescopes,
we find a steady increase in resolution with aperture.
As proof of this, one need look only at the image of Saturn obtained by
Thierry Lepine
using a 70 mm singlet objective of 1000 mm focal length.
Its resolution is certainly as good or better than we have been able to achieve with a 23 mm objective,
although at a given focal length, the contrast should suffer if the aperture is increased excessively.
Historically, there is evidence from Galileo's letters that he believed some of his
larger diameter objectives were poorly figured near the edge, and benefited from
being stopped down.
This seems to be corroborated by Ronchi's
1923 tests of surviving lenses attributed to Galileo;
although Ronchi's own tests with an artificial star do not show any dramatic difference
in the size of the focused image with and without the existing stops.
In any event, the motivation for stopping down the objectives is to avoid areas of
poor figure.
It is not, as Zik suggests, to reduce chromatic aberration.
Even more mysterious is the practice, not mentioned in Zik's articles but evident
in the IMSS telescopes, of placing stops,
similar in diameter to those over the objective, between the observer's eye and the eyepiece lens.
These stops have no obvious optical advantage, yet they clearly restrict the
ability of the observer to redirect the field of view by moving his eye around the eyepiece,
making the telescope more difficult to point and operate.
Galileo himself appears to mention only placing stops over the objective.
Sven Dupré,
another recent PhD in the history of science, has also weighed in
on the extent of Galileo's theoretical understanding of the optics of his telescope in the second
issue of the IMSS's new e-journal Galilæana.
Dupré's article deals with Galileo's bold assertion in Sidereus Nuncius
that, upon hearing of the possibility of a telescope, he hit upon his working
design by thought, rather than by experiment.
It has never been clear whether the ray diagram
Galileo published in Sidereus Nuncius
(which faithfully reproduces the drawing
he made in his manuscript) demonstrates a clear understanding
of how the telescope works or not.
Galileo's diagram shows the light bending only at the objective, and the
accompanying text assigns no obvious role to the eyepiece; yet we know that the
focal length of the eyepiece is crucial to determining the magnification.
Galileo's diagram also indicates, according to his reasoning, that the
telescope would provide infinite power if one were to place one's eye
at the focal point of the objective (since the rays "leaving" the telescope
to the right would be parallel to the axis -- that is, the telescope would,
in that situation, make a true angle of zero appear to the eye as a finite
angle equal to that subtended by the objective lens).
Dupré seems to be making two main points in his effort to elucidate
Galileo's reasoning:
- That Galileo's explanation of how the telescope worked would have been
understandable to a Florentine gentlemen of his day, schooled in
mathematical perspective, which included the idea of a point of inversion
for concave mirrors and convex lenses (the "point of inversion" referring to
the parallel rays from a distant source crossing over at the focal point of the mirror or lens).
- That there was no concept of a point of inversion for a concave lens (many people,
even today, have trouble understanding the idea of the focal length of a
negative lens or how it can be measured). Therefore, the concave eyepiece
played no role in Galileo's thinking about the magnification; other than it
served to somehow sharpen the otherwise fuzzy magnified image produced by the
objective.
Dupré is himself a little vague about what these two assumptions would imply.
He says at one point that Galileo's understanding of Florentine optics would have
led him conclude that the magnification would be determined by the focal length
of the objective, and at another point, by its diameter.
In fact, Galileo's letters clearly demonstrate that he understood that for a
given objective lens using a less steeply curved eyepiece gave lower magnification,
and more steeply curved ones gave higher magnification.
This can perhaps be reconciled with his ray diagram if we take what he seems
to be indicating as the pupil of the observer's eye to represent instead the
center of the eyepiece lens.
Since the rays that pass through the center of the lens are undeviated in angle,
Galileo's diagram would then be exactly equivalent the modern optical diagrams
shown earlier on this page, and predict exactly the same power as we do today
for a given eyepiece position.
The reason, then, that Galileo might expect the more steeply curved eyepiece
to give higher magnification is that it allows the observer (for probably unknown
reasons) to see a clear image with his eye (and accompanying eyepiece) closer to
the objective's focal (or "inversion") point at which infinite magnification would be obtained.
What Galileo's diagram fails to explain is where an eyepiece of a given steepness
needs to be placed, and how the refraction taking place in the eyepiece manages
to make an otherwise fuzzy image appear sharp to the observer's eye.
In this, we feel Kepler's understanding was far deeper.
Galileo also clearly (and incorrectly) thought his ray diagram implied that the field
of view would be proportional to the diameter
of the objective lens, a belief he later had to admit was wrong, or at least inaccurate in practice
since it neglects the very sizeable effect from the observer's pupil.
Although Kepler is generally credited with discovering a way to vastly increase the field of view,
it is not clear that he had any better idea of how to predict it, or even knew
that a positive eye lens would have this effect.
Photographic Comparisons -- Field of View
The performance of the present Galilean replica, when used with a positive eye lens, differs
somewhat from the model described above because, as explained on the main page, baffles were
added to prevent glancing reflections from the interior wall of the small-diameter white PVC tube
from reaching the eye.
Consulting the optical diagrams given above it can be seen that when the telescope is being used
in the Galilean mode and the observer has his eye near the optical axis, the baffles do not
restrict access of rays from the objective to his pupil since only the central part of the
eyepiece is used. In the Keplerian mode, however, one ideally tries to make use of the entire
surface of the eyepiece.
Since the baffles block the rays from the objective that would normally strike the periphery of
the eyepiece, they effectively stop the eyepiece down to a smaller size, restricting the Keplerian
field and also limiting the number of fields that can be seen by moving the observer's eye to the
side in the Galilean mode.
In the present replica it is the baffles, rather than the eyepiece, that determine the limit of
the field visible in the Keplerian mode.
Indeed, when the telescope is used in the Keplerian mode, one baffle in the sliding eyepiece tube
falls quite close to the prime focus and can be seen sharply focused at the edge of the field.
The subject here is a chimney 340 feet from the telescope.
The large image was photographed with the positive eyepiece.
A 16 mm diameter baffle close to the prime focus limits the field to 55 arc minutes.
Without the baffle (i.e., if the 23 mm diameter of the eyepiece were completely illuminated), one
would see a field nearly half again as wide.
The small image shows, to the same scale, the small portion of the Keplerian view visible at any
one time with a negative eyepiece of the same focal length.
The full diameter of the Galilean field, which is unaffected by the baffle, is 14 arc minutes.
After being magnified 20 times, its apparent size (that is, the size of this image as perceived by the human
eye looking though the present telescope) is about the same as the diameter of a U.S.
quarter viewed at a comfortable reading distance of 12 inches.
In this second example, the telescope was pointed at the top of a power pole 578 feet away.
Again, the large image shows the field visible with the 50 mm positive eyepiece, while the small
image is that visible with the 50 mm negative eyepiece.
In both examples, the Keplerian view has been inverted to match the orientation of the Galilean
image.
The difference in contrast and color between the two images is due to their having been taken on
different days.
If you are interested in more detail about how the colors change with focus in the two modes, check the
separate Focus Page.
Detailed images of the right hand insulator (taken on still different days) are shown in the following section.
There are several reasons why the measured Galilean field of 14 arc minutes differs from the
estimate of 10-19 arc minutes made based on the optical diagram.
First, the pupil size of the camera used to take the photographs is different from the 4 mm size
assumed in the discussion of the optical diagrams.
In fact, according to Olympus' literature, the optical system of the Camedia C3000z camera has a
maximum focal length of 19.5 mm with an aperture ratio of f/2.8.
This means the effective entrance pupil, as used here, is 19.5/2.8 = 7.0 mm in diameter.
Second, the pupil is deep within the camera's lens and cannot be pressed directly against the
telescope's eyepiece.
We do not know the exact position of the camera's entrance pupil but it acts as if it is about
2 inches back from the last surface of the eyepiece.
Third, the 10-19 arc min estimate covers a range over which the intensity falls off due to
vignetting.
A more reasonable estimate of the visual edge of the field is probably something like the point
at which the intensity falls off to 10% or so of that in the center of the image.
As shown in the optical diagrams, the field of view diminishes rapidly as the pupil is moved back
from the eye lens.
The human eye normally has a smaller pupil than the camera but it can be placed closer to eyepiece.
Since a 7 mm pupil two inches from the eyepiece gives the same field as a 4 mm pupil one-half inch
away, the Camedia C3000z, somewhat fortuitously, gives a fairly accurate impression of what one
sees looking in the telescope in both modes.
Image Resolution
Left: A reference picture of the insulator on the phone pole taken with the Celestron C8
and reduced to the scale of the Galilean photos.
Center: Detail from the center of a Galilean photograph showing the same insulator as seen
through the 23 mm refractor with its negative eyepiece.
Right: Detail from the center of photograph taken with the positive eyepiece using the same
camera settings.
The Celestron and positive eyepiece images have been inverted and rotated to match the Galilean
view.
The intrinsic scale of the Galilean images is 0.73 arc-sec per pixel (click to see full-scale
images).
The spacing between the five loops of the small wire tightly wrapped around the multi-strand
cable just to the left of the insulator is slightly over 6 arc-seconds per turn, very close to
the expected Rayleigh limit of 5.9 arc-seconds for a 23 mm circular aperture operating in the
green.
The spacing (at their closest point) between the two wires wrapped across the front of the
insulator is actually slightly less than the spacing between the loops, even though they appear
better resolved in the Galilean photos. The Galilean telescope appears to be resolving very close
to the Rayleigh limit with either eyepiece. The differences in color fringing between the positive and negative
eyepieces are not intrinsic to the two modes: rather they are due to a
slight difference in the choice of best focal position for the two photos. For a complete focus sequence see the
separate Focus Page,
which demonstrates that essentially no difference in central color is expected when
the eyepiece is placed at comparable positions in the Galilean versus Keplerian modes.
To confirm our resolution estimate we took this photo of an NBS Circular 428 25X Photographic Test Chart
taped to a light pole 214 feet from the front of the telescope.
Please note that if you click on the preceding link you will most likely be
seeing the target much larger than its real size (the whole target is only about 70 mm wide).
This target is intended for testing photographic lenses by analyzing the images captured on film.
The numbers printed next to the patterns are the number of lines per millimeter
when that pattern is reduced 25X.
Hence, on the original target used for this test, the lines in the pattern labeled "5"
have a spacing of 5 mm from center to center.
At 214 feet, the spacing between the lines (as seen without a telescope) happens to be
79/n arc-seconds where n = lines per millimeter printed on target.
The line spacing labels are not legible in this photo, however, proceeding from
the bottom up, the first four groups
in the two center columns are n = 7 (11.3 arc-sec),
10 (7.9 arc-sec), 14 (5.6 arc-sec) and 20 (4.0 arc-sec).
The 5.6 arc-sec group appears to be well resolved, while the 4.0 arc-sec group
(which has a large "X" printed between
the horizontal and vertical patterns) is unresolved.
We conclude that the photographic resolution limit of this telescope is around 5 arc-seconds.
This resolution is substantially better than the 20 arc-seconds estimated by
Greco, Molesini and Quercioli
for a near-perfect 26 mm aperture singlet refractor degraded by chromatic aberration.
As Dr. Molesini has kindly pointed out to us, the exact figure of 20 arc-seconds
was based on a standard, but somewhat arbitrary, criterion for evaluating a
calculated response curve called the Modulation Transfer Function (MTF).
The actual cut-off is not sharply defined, and an experienced
observer may see some detail well beyond this limit.
When looking through the present telescope, both our eyes and the Olympus Camedia
C3000z resolve features much smaller than 20 arc-seconds,
as did Galileo;
and we find, if anything, that the color fringing is less disturbing to the human eye than it is to the camera.
Our own simulations
of the effect of averaging the chromatic dispersion of Galilean telescopes over
the visual spectrum, performed in a somewhat different way, also suggest an expected
resolution substantially better than 20 arc-seconds for our 23 mm aperture.
REFERENCES
Images (unless otherwise credited) © Tom Pope and Jim Mosher
Last modified: March 31, 2006
A point of light is actually focused into a central spot surrounded by a faint
pattern of rings, as can be seen in our photo of Mizar;
or perhaps more clearly in the picture at left: a full color simulation of the diffraction spot
expected from a point source of white light as viewed near the focus of the
website telescope (BK7 glass, 23 mm aperture/1000 mm focal length with eyepiece 945 mm
from objective; the simulation is 40 arc-seconds on a side).
For light of a single color, the width of the diffraction pattern is directly proportional to the wavelength
and inversely proportional to the aperture (diameter of the front opening) of the telescope.
Although various definitions can be proposed depending on the nature of the target
and experience of the observer, the usual figure quoted for the ultimate resolution is based on
a standard proposed by Lord Rayleigh in 1879 and, as applied to a telescope,
refers roughly to the minimum spacing at which two equal points of light can be distinguished.
For a wavelength in the middle of the visible band (mercury green line = 5461 Å)
and an aperture of 1 inch (25.4 mm) the Rayleigh limit is 5.41 arc-seconds.
The maximum possible resolution of the present telescope is slightly less since
its aperture is a little smaller.
Projected over a distance of 1000 mm, 5.4 arc-seconds corresponds to a spot of
light a little over 0.05 mm in diameter at the prime focus.
For a star image, the resolution limit (5.4 arc-seconds in the present example)
refers to the diameter of first dark ring; the diameter of the bright central spot
of light will appear slightly smaller.
As indicated above, the diameter of this spot will decrease as the diameter of
the objective lens is increased.
Images of extended sources (such as a planet or a two-dimensional test target)
suffer from diffractive blurring just as much
as do those from point sources, because in truth the image of an extended source
is nothing more than the sum of the diffraction patterns from each individual
point in that source.
