ASTR 3130, Majewski [SPRING 2015]. Lecture Notes
ASTR 3130 (Majewski) Lecture Notes
TELESCOPES: MAGNIFICATION AND FIELD OF VIEW
See pp.116121 of Birney, Gonzalez & Oesper.
MAGNIFICATION
 We start our discussion here to coincide with Lab 1, but the least
important property of a telescope is its magnification.
In fact, for general astronomical applications it only makes sense to discuss this
property when doing eyepiece viewing (not typical for professional
astronomers), i.e. when "looking through" a telescope (rather than using a
detector to record an image, or sending light to an instrument).
(In some applications involving detailed designing of astronomical instrumentation,
the f/ratio [or exit pupil  see below]
of the telescope beam must be modified. In this case an additional
piece of optics is often introduced to effectively magnify or demagnify the image in the
focal plane. But this is something you don't need to worry about right now.)
 The optics of eyepiece viewing:
 Recall that all that is needed to form an image is a single optical element
 i.e. the telescope objective = light collector = mirror or lens.
 But we also know from experience that a lens can be used as a magnifier.
 The function of an eyepiece and eyepiece viewing is to combine these
two concepts: the eyepiece is a magnifying lens acting on
the primary (virtual) image created by the objective of a telescope.
The principle of magnification deals with the changing of the apparent angular sizes of
features in the sky when seen through an optical system.
In general, for a lens of focal length F we have the following optical
arrangement:
The smaller F is, the bigger φ is for constant S  that is,
a shorter focal length yields a larger image.
 When light rays are for an object not at infinity (light rays
not parallel), the image is at a distance given by the thin lens formula:
1 / F = 1 / S + 1 / S'
(Note, if S = infinity, back to image at focal length distance).
 Magnification occurs when the apparent angular scale of an image
is made larger than the angular scale of the actual object.
From Roy & Clarke, Astronomy: Principles and Practice. (Note: this
figure has a slight techncial flaw in that it says "Object at infinity" but were that the case,
then the ideal distance between the "collector" and eyepiece would be exactly F_{c}
+ F_{e}. Thus, the picture, as drawn, corresponds to the image formation
for an object not at infinity. Compare to the cartoon drawing of the same setup below for
the situation of imaging an object at infinity, or compare to the Roy & Clarke
"Figure 17.2" below for
the same.)
As one can see from this figure, the angular size of the object seen at infinity is
α_{c}, but after magnification by the eyepiece of the telescope, the apparent
angular size of the image of the object is the larger angle α_{e}.
The figure below shows that one can connect the magnification, given by
θ_{i} / θ_{o} shown
in the figure, to the relative focal lengths of the objective and the eyepiece,
F_{o} / F_{e} through use
of similar triangles attached at the vector d.
Optical arrangement for viewing an object at infinity with telescope and eyepiece.
Figure by Mark Whittle.
Thus, one finds that the magnification, θ_{i} / θ_{o} = F_{o} / F_{e}.
Notice also the inversion of the image in this configuration.
So, there is no single "magnification" for a telescope  it depends on the
focal length of the eyepiece selected.
Smaller focal length eyepiece > larger magnification on same telescope.
For example, for a Meade 8" telescope (F_{o} = 2000 mm):
F_{e}

mag

32 mm 
62.5 
20 mm 
100 
9.5 mm 
210 
Note how the magnification increases as the eyepiece focal length decreases.
EXIT PUPIL
Note the figure of the magnification process below.
From Roy & Clarke, Astronomy: Principles and Practice.
LIMITS TO MAGNIFICATION
There are practical limits to magnification.
 On the low magnification end, there is a practical limit in that we
would like the eye to have access to all light that the objective
of the telescope collects.
 This means we would like the exit pupil of the system to be
smaller than the pupil of the eye.
 Obviously, it makes no sense to have an exit pupil
that is larger than the dilated eye (or some light will miss your eye).
Recall that this is a variable with people's age, but for most students
in this class, d ~ 7 mm is about the size of your dilated pupil.
Eyepieces with exit pupils larger than this waste the telescope's
primary advantage  light gathering power  by putting some of the
light onto your iris instead of into your eye. This is called
vignetting, where your iris is blocking some of the light from entering your eye.
Thus, we find that m > D / (7 mm) is a useful lower limit to the
magnification for a telescope.
(Note that if the eyepiece makes a 7 mm exit pupil, older people
like me with smaller dilated pupils will not see as bright an image!)
 But an even more stringent lower limit on the magnification comes
about if one considers the resolution (finest perceptible detail) of the human eye.
 Obviously, we lose information and detail if the telescope
can deliver better resolution than we can perceive.
 Recall that the resolution of the eye (i.e., the smallest angular
scale the eye can discriminate) is at best
1 arcminute = 60 arcsecs.
 (As we will see) the finest resolution delivered by a telescope
of diameter D in mm
is given roughly by the Rayleigh formula:
( 140 mm / D ) arcseconds
 This implies that we want a magnification, m sufficient to
convert the ( 140 mm / D ) arcsec resolution of the telescope
to more than the 60 arcsecond resolution of the eye, so that
(m) ( 140 mm / D ) arcseconds > ( 60 arcseconds )
Thus
m > 3 D / (7 mm) ~ D / (2 mm)
 Note that this calculation is based on the theoretical
resolution of the telescope, which is normally not achieved
for larger telescopes because of the limitations of
atmospheric seeing.
This limit is generally about 1 arcsecond, which is about the
theoretical limit delivered by a D = 6inch = 152 mm
telescopes.
Thus, for telescopes with D > 6inches, we can simply state a lower limit
of about m > 60.
 There are also upper limits to practical magnification, because:
 It is very hard to make quality eyepieces with very short
focal lengths.
 The eye tends not to form good images when the exit pupil
becomes smaller than 0.8 mm.
 These two considerations lead to a rough empirical rule of
thumb, Whittaker's rule, which is that
m < D / (0.8 mm), or
m <~ D / (mm)
that is, the magnification should not exceed the diameter of
the objective in mm units.
(This is merely a statement that we don't want to scale down the size
of the entrance pupil [objective lens] by a factor [given by m] more
than will make the exit pupil smaller than about 1 mm.)
Thus, for the 6inch refractor, the highest useful magnification is about 150X.
 Higher power does not always mean a better image.
 Atmospheric turbulence (seeing), optical aberrations,
and telescope vibrations are all magnified.
 As an image is magnified, its surface brightness 
the amount of light falling on any fixed area of the retina 
is reduced, making it harder to see.
 Thus, it makes no sense to magnify beyond the
limit where the resolution of the eye
is much better than the resolution of the image
delivered by either the telescopic or atmospheric
limiting resolutions.
This type of consideration leads to the "Lewis equation"
given in Roy & Clarke:
m < 27.8 D ^{0.5}
where D is in mm.
 Keep in mind that these calculations are not hard and fast rules, but
rather general guides to useful magnification.
 As may be seen, (ignoring the Lewis Equation and using the other limits calculated above)
a very rough useful range of magnification is then (with
D in mm):
D > m > D / 2
FIELD OF VIEW
Field of view is the maximum angular size visible through an optical system.
The actual field of view can
depend on several factors relating to the internal design of the telescope
tube and (when using an eyepiece rather than a detector at the focal plane)
the internal structure of the eyepiece optics  either which can introduce a
field stop that limits the greatest angle for which a light ray can enter the
system with respect to the optical axis.
But generally, as the magnification goes up, the FOV goes down (generally
linearly).
Eyepieces generally are characterized to have an apparent field of view (FOV),
which is the maximum possible angle as viewed through the eyepiece alone.
 This roughly corresponds to the angles α_{e} = θ_{i}
in the various figures above.
 As you might suspect, the actual field of view on the
sky delivered by the eyepiece is inversely proportional to the degree of magnification,
and so typically will be given by the
(eyepiece apparent FOV / m ).
For example, a 25mm focal length eyepiece with a 40 degree FOV, if used
on a telescope with a focal length of 2000 mm, will give:
a magnification of ( 2000 mm ) / ( 25 mm ) = 80,
and a FOV of (40 degrees) / ( 80 ) = 0.5 degrees.
EYEPIECES
A few comments about eyepieces.
Recall optically what the eyepiece is doing generally:
 An eyepiece takes the (virtual) image formed by the objective and magnifies it so that the human eye can
perceive more details.
From http://www.televue.com/engine/TV3b_page.asp?id=212&Tab=#.VMhjQlr92M4 .
And a nice summary of the equations (some given before, some not, but derivable from what was said above):
From http://www.televue.com/engine/TV3b_page.asp?id=212&Tab=#.VMhjQlr92M4 .
Recall that the exit pupil of the system is that place where all rays from all parts of the field that
can be viewed by the system can be intercepted. It is ideally where you would put your eye.
The eye relief is how close your eye has to be to the end of the eyepiece to be at the exit pupil
(and therefore, be able to see the entire "picture").
 An eyepiece with good eye relief is an especially important consideration for people who wear glasses, because
those glasses have to be in between the eye and the eyepiece.
There are a large range of eyepiece designs that have been created to address various desires for the
properties of the delivered image.
From http://www.weasner.com/etx/techtips/eyepiece_designs.html.
The design of an eyepiece must balance a number of considerations. Among the considerations and things we would
like the eyepiece to do are:
 Aberrations: Produce an image with minimal or no aberrations
within the range of objective focal ratios it is designed to magnify.
 Wide field: Display a wide field of view without perceptible curvature or distortion.
 Throughput and Contrast: Transmit almost all light delivered by the telescope and
without ghosts, glare or scatter.
 Eye Relief: Provide enough eye relief for comfortable viewing.
 Durability: Because eyepieces are constantly handled, we want them to be able to
withstand years of normal use, variable environmental exposure and the occasional accident
without compromising any of the previous qualities.
 Affordable: Combined with durability, this drives the commercial marketability of the eyepiece.
In summary, and to paraphrase http://www.handprint.com/ASTRO/ae5.html, the ideal eyepiece should deliver images
that are crisp, wide, flat, bright, and dark, while being comfortable to use, durable and affordable.
Obviously, there is no magic prescription that can always achieve all of this,
and the variety of eyepiece designs is partly because of:
 the evolution in optical design sophistication, especially with the advent of computer aided design
capability.
 different eyepieces are built for an optimization in some properties (often at the expense of others).
From http://www.weasner.com/etx/techtips/eyepiece_designs.html.
Barlow Lens
A particularly relevant type of eyepiece is the Barlow Lens, which you will use in Lab 2.
The Barlow is basically a negative powered lens, which, when employed in the telescope, acts to delay
convergence of the light from the obkective.
From http://starizona.com/acb/basics/equip_optics101_optical.aspx.
Because the Barlow Lens does NOT make the rays parallel for entrance into the eye, it cannot be the last
part of the system.
Typically we insert the Barlow in front of another eyepiece (we screw another eypiece into the Barlow).
The net effect of the Barlow is to lengthen the effective focal length of the objective, which then means
that the utilized eyepiece will deliver even more magnification.
Typical Barlow lenses increase magnification by factors of either 2X or 3X.
From wikipedia.
All material copyright © 2002,2006,2008,2012,2015 Steven R. Majewski. All rights
reserved. These notes are intended for the private,
noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the
University of Virginia.
