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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


See pp.116-121 of Birney, Gonzalez & Oesper.


  • 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:

    1. 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.
    2. But we also know from experience that a lens can be used as a magnifier.
    3. 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 Fc + Fe. 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 αc, but after magnification by the eyepiece of the telescope, the apparent angular size of the image of the object is the larger angle αe.

The figure below shows that one can connect the magnification, given by θi / θo shown in the figure, to the relative focal lengths of the objective and the eyepiece, Fo / Fe 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, θi / θo = Fo / Fe.

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 (Fo = 2000 mm):
    Fe mag
    32 mm 62.5
    20 mm 100
    9.5 mm 210

    Note how the magnification increases as the eyepiece focal length decreases.


    Note the figure of the magnification process below.

    • The telescope objective acts as the entrance pupil of the optical system.

    • The exit pupil of the system is the image of the objective formed by the eyepiece.

      All rays from all parts of the field that can be viewed by the system pass through the exit pupil.

      The combined objective-eyepiece system is effectively "translating", mapping, scaling or "downsizing" the entrance pupil to the exit pupil (albeit with an inversion of the image).

      • Compare the rays coming into the entrance pupil to those rays coming into the exit pupil (they resemble one another, only inverted in the figure shown below).

        Follow the path of the sets of rays of different colors from the entrance to the exit pupil in this image. As you can see, the rays from each colored set are parallel going into the entrance pupil (objective) and parallel coming through the exit pupil. Hence, the exit pupil is a map (or image or translation) of the entrance pupil, just over a smaller diameter. From .

        Here is a photo of an exit pupil seen from the eyepiece end of the telescope. The white disk in the middle is an image of the objective, or exit pupil. Note how it will fit into your eye pupil. From
      • For example, rays coming into the entrance pupil parallel to the optical axis (i.e. horizontal) are coming out of the exit pupil also parallel and horizontal (though more densely packed).

        (Note also that rays coming into the entrance pupil parallel and at angle αc are coming out of the exit pupil also parallel to one another, albeit at the different angle αe.)

        From Roy & Clarke, Astronomy: Principles and Practice.
      • Note that for this objective-eyepiece configuration, because the light is coming out of the exit pupil in parallel beams, it is left to your eye (cornea + crystalline lens) to form the ultimate image that is detected by your retina.

        This is something that is worth driving home -- unlike an optical system with a single powered element (e.g., a telescope with only an objective lens or a single curved mirror) that converts parallel rays into light rays converging to a focus, by inserting an eyepiece into the system we are actually reconverting the converging light rays produced by the objective lens back into parallel rays, like those that came into the optical system to begin with!

      • By acting as a translation from the (usually much larger) entrance pupil to the (usually smaller) exit pupil (typically matched to the size of your eye), the combined optical system is acting (if you think about things in reverse) as if to transform the collecting area of your eye to a collecting area equal in size to the objective lens.

      From Roy & Clarke, Astronomy: Principles and Practice.
      It is this "translation of the entrance pupil to the exit pupil" that gives rise to the above (often-made) statement that the exit pupil is an "image" of the objective lens.

    • By the way, the distance from the last lens of the eyepiece to the position of the exit pupil is called the eye relief.

      This distance is important in the case where you must wear glasses -- e.g. because of astigmatism -- because you need enough eye relief to accommodate your glasses (about 15 mm).

    • The geometry in the figure shows that if there is any magnification (i.e. αe > αc), the size of the exit pupil is necessarily smaller than that of the entrance pupil.

    • In fact, if one considers rays coming in parallel to the optical axis on the left, the triangle formed by the lens D extending to an apex at the position of the primary (virtual) image at F is similar to a triangle formed starting at this same point F and extending to a vertical line segment in the eyepiece lens that is the diameter of the exit pupil, d.

      Then one sees that magnification is given by a variety of ratios (using the word "collector" for "objective"):

      m = αe / αc = Fc / Fe

      (as already shown above), and, in addition, therefore that

      m = Fc / Fe = D / d

      We see that magnification is not only given by the ratio of objective to eyepiece focal lengths, but also by the ratio of the entrance to exit pupil diameters.

    From Roy & Clarke, Astronomy: Principles and Practice.


    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 )


        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 = 6-inch = 152 mm telescopes.

        Thus, for telescopes with D > 6-inches, 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 6-inch 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 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 αe = θi 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 25-mm 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.


    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 .

    And a nice summary of the equations (some given before, some not, but derivable from what was said above):

    From .

    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.


    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, 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).


    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.


    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.

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    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.