Astronomical Telescopes

J. Kielkopf
September 20, 2021

1 Telescope Overview
An astronomical telescope is an optical tool to capture light and deliver it to instruments
that measure the amount of light coming from different directions, its spectral content, and
its polarization. There are usually several limiting factors determing the performance of the
telescope at these tasks
• Earth’s atmosphere

• Telescope aperture

• Optical quality

• Diffraction

• Detector design

• Spectrograph and polarimeter design

• Photon statistics
to mention the most signficant. Let’s take these one by one.

Earth’s atmosphere
At any altitude a human astronomer could work, the air we breath absorbs light that would
otherwise be interesting for astronomy. This absorption includes all wavelengths below about
350 nm (3500 Å), wavelengths in the infrared above 1000 nm (1 µm or 10,000 Å) excepting
specfic bands between molecular absorption in the infrared, and longer wavelengths that are
in the radio frequency regime. Ground based optical telescopes are therefore designed for
optimal work in the visible and near infrared in these standard filter bands
Johnson-Cousins U B V R I from 365 nm to 787 nm

Sloan u g r i z from 354 nm to 905 nm

1
1 TELESCOPE OVERVIEW 2

Figure 1: Transmission of Earth’s atmosphere with the response of silicon and InGaAs
sensors.

Infrared I J H K L from 787 nm to 3450 nm

The atmosphere also scatters light, introducing stray light from urban lighting and the Moon,
emits light of its own that we term airglow, and distorts imaging. Flowing air with density
variations is often turbulent, and the variations in gas density along the optical path, both
at altitude and within the optical system, cause wavefront deviations that usually exceed
the errors in the optics and diffraction. Turbulence, what astronomers call “seeing”, limits
the image quality severely.

Telescope aperture
Today’s “small” telescope is last century’s giant, and even instruments in the class of 4
meters diameter are dwarfed by those under development. This is not to say that apertures
measured in centimeters rather than meters are not useful, but that the science that can
be done with them is in a different realm. The very largest telescopes gather light enabling
1 TELESCOPE OVERVIEW 3

Figure 2: Standard astronomical filters for the visible and near-infrared as used by the Monet
telescope. Credit: Frederick Hessmann.
1 TELESCOPE OVERVIEW 4

Figure 3: Sloan filter set.

1 TELESCOPE OVERVIEW 5

Figure 4: JHK near-infrared filter set.

1 TELESCOPE OVERVIEW 6

studies of the faintest and most distant objects of interest. Smaller telescopes offer wide
fields of view, and are appropriate for brighter closer objects. They are also naturally less
subject to turbulence, and capable of high resolution imaging without adaptive controls to
correct for the atmosphere, and their affordability means they can dedicate time to long
duration studies. Thus telescopes under 35 cm might be used by amateur astronomers for
their work, and for education. Telescopes in the 0.5 to 2.5 m class are often dedicated to
specific areas of research that require rapidly cadenced data on objects brigher than 18th
magnitude, or on spectroscopy of bright stars. Larger telescopes would primarily be directed
to fainter and transient events – distant galaxies, supernovae, faint star spectroscopy, and
small solar system objects. Currently the pair of Keck telescopes in Hawaii each with 10 m
mosaic mirrors lead the aperture race. However ESO has 4 very large telescopes (VLT)
with 8.4 m diameter mirrors in Chile, the Gemini telescopes at 8.1 m cover the northern
and southern sky from Mauna Kea and Chile, and many others have apertures in the 3 to
6 m range. The European Extremely Large Telescope (ELT) will be operating with a 39 m
aperture by 2024, and before it the Vera Rubin formerly known as the Large Synoptic Survey
Telescope or LSST, at 8.4 m, will have first light by 2022. Others in this class are planned,
or under construction. Because of the very high cost, most of these are operated by national
governments or private consortia, and all have very limited access based on membership and
competitive proposals. By contrast, Vera Rubin when it is running will produce terabytes
of data per night that will be available to astronomers in the U.S.
Telescopes in space range from the James Webb Space Telescope (JWST) to be launched
in 2021 with a 6.5 m aperture, the Hubble Space Telescope (HST) at 2.4 m, Gaia at 1.45 m,
the now-decomissioned Kepler spacecraft at 0.95 m, and the Transiting Exoplanet Survey
Satellite (TESS) launched in 2018 at a tiny 10 cm for each of its four cameras.
Typically larger apertures must be made with reflecting optics, supplemented in the
optical path by refracting elements to correct the images. The technical reason for this
is that reflecting optics have one surface that can be supported from the back, and that
segmenting the optics into multiple components to achieve very large diameters is feasible.
Reflecting optics are also achromatic and will focus equally light from the ultraviolet to the
infrared. Refracting optics in smaller apertures are valuable for their favorable focal length to
aperture ratio (the f-ratio), and for their ability to image very wide fields with good quality.
Almost all large optical telescopes incorporate refracting elements to correct for aberrations,
though some designs work well with mirrors alone for small fields of view.

Optical quality
Optical elements work by modifying the plane wavefront of light from a distant object so
that it arrives coherently at a focus where a sensor, each pixel of the camera, can detect
it and provide a measurable signal for analysis. If the optical element is not perfect, the
wavefront that leaves it will not converge to a single point, and the image of a distant point
source will spread over pixels and be blurry. The conventional standard is that the surfaces
must preserve the wavefront to within λ/4, so-called “quarter-wave” optics, to avoid this
1 TELESCOPE OVERVIEW 7

Figure 5: University of Louisville’s “Shared Skies” CDK20-North telescope at Moore Obser-

vatory.
1 TELESCOPE OVERVIEW 8

Figure 6: University of Louisville’s “Shared Skies” CDK700 telescope at the University of

Southern Queensland’s Mt. Kent Observatory in Australia.
1 TELESCOPE OVERVIEW 9

Figure 7: University of Louisville’s “Manner Telescope”, an 0.6-meter Ritchie-Chrétein lo-

cated at the University of Arizona’s Stewart Observatory on Mt. Lemmon near Tucson.
1 TELESCOPE OVERVIEW 10

outcome. However for the best performance and minimizing the spread of the light outside
the diffraction pattern, λ/20 is a more desirable constraint. Such precision is a challenge
to achieve in any optical element, and especially a very large one. For the extremely large
telescopes in use today, some adaptive control of the optical surface is required to meet this
high standard. Since monolithic mirrors in apertures of several meters are being made that
are of this quality, they may be combined in a mosaic to make larger instruments when
supported by actuators that respond to measurements of wavefront error.

Diffraction
Ultimately the ability of a telescope to resolve detail is limited by diffraction. For at telescope
of diameter d in meters, the light spreads over an angle of approximately 0.1/d arcseconds.
Thus a telescope with an aperture of 10 cm, that is 0.1 meter, resolves about 1 second of arc.
In principle, a 10 m diameter telescope will resolve 0.01 arcseconds or 10 milli-arcseconds.
This is not achieved in practice from the ground because of optical defects and atmospheric
turbulence, but both can be mitigated by actively adapting the optics to correct for image
errors.

Detector design
For astronomical imaging, the telescope focuses light onto a detector that is typically a
charge coupled device with elements that are several microns (1 µm is 10−6 m) across. Each
imaging sensor may contain 4096 × 4096 pixels or more, and cover an area of 50 mm on a
side. At the detector, the focal plane scale is set by simple geometry. If x is a distance in
the focal plane and f is the focal length, the angle imaged on x is

θ = arcsin(x/f ) ≈ x/f (1)

From this it follows that for a fixed distance set by the detectors structure, the angle on the
sky is smaller the larger the focal length. The area of the sky covered by the detector is also
smaller the larger the focal length. Telescope designs constrain the ratio of focal length to
aperture to values usually greater than 2:1 (an f/2 optical system), and more typically in
the largest telescopes to 8:1 or more. Auxilliary refracting optics may speed up the system.
However, the largest telescopes still require very large detector arrays to cover a reasonable
field of view. For more modest optics, the telescope’s imaging field is covered by a single
detector. As an example, a 0.5 meter telescope with a focal ratio of 7 would have a focal
length of 3.5 meters. The focal plane scale is 59 arcsecond/mm, and a 10 µm pixel will
resolve 0.59 arcsecond. That is, the atmospheric blurring will spread light over an area
about 2 pixels wide, while 4096 pixels will span 40 arcminutes. That’s larger than the full
Moon.
2 TELESCOPE OPTICS 11

Spectrograph and polarimeter design

Telescopes may also be used to feed light into other instruments, most commonly spectro-
graphs to disperse the light into components, and polarimeters to enable precise determina-
tion of polarization. These instruments have their own constraints. For example, polarime-
ters are sensitive to any element of the optics that may change the polarization state of the
light, and therefore reflections that are not normal to the surface of the optic are undesirable.
Spectrographs require very small entrance apertures to limit the spectral range, and very
long focal ratios to minimize aberrations. When coupled with fiber optics, the size of the
fiber must match on one end the characteristics of the spectrograph, and on the other the
characteristics of the telescope. As a consequence, the cost and complexity of a spectrograph
may rival that of the telescope itself, and may determine the preferred telescope design.

Photon statistics
Lastly, the measurement of light from faint sources is determined by the precision with which
the number of photons can be determined. Photon√flux is a Gaussian random process, and
√ standard deviation in counting N photons is4± N . Thus the signal-to-noise ratio is also
the
N and to determine a signal to 1% requires 10 photons detected. Larger telescopes collect
proportionally more photons, and as a practical matter with an aperture of 0.5 m stars as
faint as 20th magnitude can be measured. To go fainter it takes more light and therefore
more telescope area. A telescope 10 times larger in diameter has 100 times more area and
collects 100 times more light. A factor of 100 in light is 5 magnitudes on the astronomical
scale. Weighing in with this simple math, the airglow also contributes at the level of a 20th
magnitude star per arcsecond, so sites with less airglow and better seeing, allowing smaller
image blur, are favored for detecting the faintest, most distant objects.

2 Telescope Optics
Typically an astronomical telescope uses large reflecting optical surfaces to collect light and
bring it to a focus on a detector. For visible light imaging in affordable smaller telescopes,
this is achieved by using a primary concave mirror and a secondary convex mirror paired with
non-spherical surfaces to produce the best image quality at the focal plane for the intended
purpose. A commercial design known as a “corrected” Dall-Kirkham Cassegrain system is
shown in Figure 8.
In the Dall-Kirkham design the primary mirror is ellipsoidal and the secondary mirror
is spherical. The curvatures are chosen to minimize coma and spherical aberrations. The
transmission corrector optics shown in the figure improve the overall image quality out to
a wide field of view, and enable fast (small ratio of focal length to aperture) optics that
reduce the length of the telescope. They also flatten the field of view to match today’s
detectors. These so-called CDK instruments are now widely used in telescopes up to 2 meter
in aperture. [1] An analysis of the geometrical optics of this design is shown in Figure 9
3 DIFFRACTION 12

Figure 8: Ray tracing a cross-section of a 0.5 meter corrected Dall-Kirkham telescope.

For larger research instruments it is common to use an aspheric primary and secondary
in a Ritchie-Chrétien design, which also produces high quality images but without the trans-
mission optics. Such RC telescopes have poorer performance off-axis than the CDK, and
because of the aspheric surfaces are more expensive to manufacture. However without the
transmission optics they can reach into the near ultraviolet, or into the infrared.

3 Diffraction
Light leaves a distant source with the properties of a spherical wave. That is, the phase of
the wave is constant on the surface of any sphere surrounding the source. When the radius
of the sphere is large, in a small enough local region it would be indistinguishable from a
plane. Thus, a wave arriving from a star at a telescope on Earth is described as a plane
wave. We consider the special case of the wave incident exactly on the optical axis, which is
to say that the telescope is pointed precisely at the star.
The entrance aperture of the telescope has an area A that usually would be circular.
Because the area limits the light that can go forward into the optics, the wave that propagates
through the optical system is modified by diffraction. In the conventional description of
diffraction of a wave we neglect the effects of polarization and quantum optics, and simply
sum all of the sources of amplitude over the aperture. This description is known as Kirchhoff’s
3 DIFFRACTION 13

Figure 9: Analyis of the image quality a 0.5 meter corrected Dall-Kirkham telescope.

diffraction theory, in which the amplitude of the wave into an arbitrary direction after the
aperture is given by [2] Z Z
U (P ) = C exp(−iks)dS (2)
A

Here k = 2π/λ where λ is the wavelength of the light and s is the distance from the element
in the aperture of area dS to the point at which the light is detected. The value of ks is the
phase of the wave at the detector.
This expression applies in the special case for which the incident light is a plane wave
so that all elements of the aperture develop a new wave in phase. The wavelets from each
element arrive at the detector with different phases because the transit time from that
element to the detector varies over the aperture. If we collect all the light going into a
particular direction after the aperture by observing the sum of the wavelets at an infinite
distance away then this integral may be evaluated exactly in cases where the boundary of
the opening is a rectangle or a circle. This special case is called Fraunhofer diffraction. For
a lens forming an image of a distant star, it describes the image of the star in the focal plane
because a perfect lens collects all the light going into a direction and brings it to focus in a
single point. Both circular and rectangular apertures have exact solutions that will enable
the prediction of the angular resolution of a telescope.
4 RECTANGULAR APERTURE 14

4 Rectangular Aperture
We will do the rectangle first because the integration is more familiar. Since we are interested
only in the change in the phase of the wave coming from different elements we can use
Z +a Z +b
U (P ) = C exp(−ik(px + qy))dxdy (3)
−a −b

where x and y are within the aperture. For z along the optical axis, we use θ and φ to
represent the angles of the light propagation direction measured in the (x, z) and (y, z)
planes, and define p = sin(θ) and q = sin(φ). For the small angles encountered in astronomy,
p and q are nearly equal to the angles to the optical axis in radians.
The two integrals are independent and separable. Each one is of this form:
Z +a
1
exp(−ik(px))dx = − (exp(−ikpa) − exp(ikpa)) (4)
−a ikp
= 2 sin(kpa)/(kp) (5)

The intensity of the light at the focus is proportional to the square of the amplitude and is
given by

I(P ) = |U (P )|2 (6)

I(P ) = I0 (sin(kpa)/(kpa))2 (sin(kqb)/(kqb))2 (7)

where I0 is the intensity at the center.

The function sin(β)/β is 1 when β = 0 and is 0 when β = π. It reduces in amplitude
with increasing β, oscillating as it goes. This behavior produces the diffraction fringes with
maxima nearly uniformly spaced but of diminishing significance at large β. The zeros are at
multiples of π where the sin is zero:

kpa = ±mπ (8)

kqb = ±nπ (9)

with m and n integers 1,2,3,...

The first minimum is at n = 1 and the angle given by

p = π/ka (10)
p = λ/2a (11)
θ ≈ λ/2a (12)

which is to say that the angle of the beam to the first minimum is the wavelength of the
light divided by the aperture.
4 RECTANGULAR APERTURE 15

Figure 10: Diffraction of a 50 cm wide rectangular aperture at 500 nm.

5 CIRCULAR APERTURE 16

5 Circular Aperture
In the case of a circular aperture the symmetry allows integration in only one variable. The
equation for the amplitude is found by using

x = ρ cos(η) (13)
y = ρ sin(η) (14)
p = w cos(ψ) (15)
q = w sin(ψ) (16)

where ρ and η are the radius and azimuthal angle within the circular aperture, and where
w = sin(θ) and ψ are measured in the image space. The angle ψ, like η, is measured
azimuthally around the optical axis. The angle θ is measured from the optical axis to the
direction of propagation and is analogous to θ in the rectangular case above.
With these substitutions, the amplitude in the focus in the direction P , defined by θ and
ψ, is Z Z a 2π
U (P ) = C exp(−ikρw cos(η − ψ))ρdρdη (17)
0 0
The integral representation of the Bessel function is
i−n 2π
Z
Jn (x) = exp(ix cos(a)) exp(ina)da (18)
2π 0
leading to
Z a
U (P ) = 2πC J0 (kρw)ρdρ (19)
0
 
2J1 (kaw)
= Cπa2 (20)
kaw
The intensity for a circular aperture is
 2
2J1 (kaw)
I(P ) = I0 (21)
kaw
As with the rectangular aperture, this is a central peak surrounded by rings of diminishing
intensity at larger radii. The first zero of J1 (x)/x is at x = 1.220π. Thus the first dark ring
has a radius

w = 1.220π/ka (22)
= 1.220λ/2a (23)

The angle from the axis to the first dark ring is 1.22λ/D where D is the diameter of the
aperture. As a rule of thumb, for visible light, this angle is approximately 1 arcsecond for
an aperture of 10 cm
5 CIRCULAR APERTURE 17

Figure 11: Diffraction of a 50 cm diameter circular aperture at 500 nm.

6 EXAMPLE 18

Figure 12: Logarithmic view of the point spread function for a diffraction-limited 50 cm
diameter telescope at 500 nm.

6 Example
As an example consider a telescope with a diameter of 0.5 meters (50 cm, 20 inches). Its
ideal performance is independent of its focal length and determined only by the diffraction
pattern generated by its entrance aperture. There is a small difference in whether the
aperture is circular or square because the minima are at slightly different angles, with the
circular pattern about 20% larger. The first minimum is at about 220 milli-arcseconds. At
1 arcsecond the maximum diffracted light is about 10−3 of the peak.
A telescope such as this could, in principle resolve two stars approximately 220 mas apart.
If a second star were centered on the zero of the first, it is simple to show that there would
be a small discernable dip in the intensity of the sum of the two patterns that would make
the second star detectable. However, if the second star were much fainter than the first, it
would be much harder to detect unless it were farther away. This ”point spread function”
decreases by about 1000 times from the peak at 1 arcsecond.
7 EPSILON LYRAE 19

7 Epsilon Lyrae
The star system  Lyrae is seen by the unaided eye as a bright star in Lyra, and in binoculars
or a small telescope as a double star with a separation of 173 arcseconds. Each of the stars
is itself double:

1 Combined magnitude 4.7, two stars of magnitude 5.0 and 6.1 separated by 2.39” at a
position angle of 348 degrees, orbiting with a period of 1725 years.

2 Combined magnitude 4.6, two stars of magnitude 5.2 and 5.5 separated by 2.37” at a
position angle of 78 degrees, orbiting with a period of 724 years.

The star system is about 162 light years (49.7 parsecs). Note that a parsec is the distance
at which 1 astronomical unit subtends an angle of 1 arcsecond.
The individual stars are clearly resolved in any small telescope with sufficient magnifi-
cation. In a large telescope, however, atmospheric turbulence distorts the wavefront of the
light from the stars, and creates a dynamic image that may blur these details when the
“seeing” is poor.

References
[1] Richard Hedrick. Planewave Instruments. 2021. url: https://planewave.com/ (visited
on 09/19/2021).
[2] Max Born and Emil Wolf. Principles of Optics. 7th ed. Cambridge: Cambridge Univer-
sity Press, 1999. isbn: 0-521-642221.
REFERENCES 20

Figure 13: The constellation of Lyra, overhead in the northern hemisphere night sky at sunset
in the fall. This image was recorded with a Nikon digital SLR, and a 3 second exposure with
an 85 mm focal length lens at f/1.8. The bright star at the upper right is Vega, α Lyrae,
and  Lyrae is the the double star at the top right.
REFERENCES 21

Figure 14:  Lyrae in a 0.2 second exposure with the 0.5 m corrected Dall-Kirkham telescope
at Moore Observatory. Compare the stars in the figure with the magnitudes, separations,
and orientations described above. The line between the two double stars is approximately
N-S. North is on the left.
REFERENCES 22

Figure 15: 1 Lyrae in a 0.03 second exposure with the 0.6 m Ritchie-Chretien telescope at
Moore Observatory.
REFERENCES 23

Figure 16: 2 Lyrae in a 0.03 second exposure with the 0.6 m Ritchie-Chretien telescope at
Moore Observatory. You can see the resolution of the telescope in the individual sharply
defined pixels, and the atmospheric blurring which takes light from the central diffraction
maximum and spreads it around the image.