Download Lecture 18/9 Telescopes Ulf Torkelsson 1 Optics

Lecture 18/9
Telescopes
Ulf Torkelsson
1
Optics
When light is passing through the interface between two media with different refractive indeces it
is refracted according to Snell’s law
n1 sin θ1 = n2 sin θ2 .
(1)
This effect is exploited in lenses. The focal length of a thin lens consisting of a material with
refractive index n and with spheroidal surfaces with radii of curvatures R1 and R2 is
1
1
1
= (n − 1)
+
.
(2)
f
R1
R2
Note that we here count R as positive for a convex surface and negative for a concave surface.
The focal lengt is the distance between the centre of the lens to the focal plane. The focal plane
is the plane at which parallel infalling rays will be focused to a point. This is where an image of
an astronomical object will be formed. If the incoming light rays form an angle θ with the optical
axis then the distance between the optical axis and the point at which they are focused will be
y = f θ,
(3)
which is a simple consequence of that light that goes through the center of a thin lens is never
refracted. From this we can calculate the plate scale
1
dθ
= ,
dy
f
(4)
which gives us the conversion factor between the size of the image and the angular scale on the
sky.
An important consideration for an optical system is what is the smallest angle that can be
resolved. The resolution is limited by the diffraction of the light. Start by considering a single slit
of width D. We compare a light ray at one end of the slit with a light ray at the centre of the slit.
The difference in path length between the two rays is
D
sin θ,
2
(5)
if the rays form an angle θ with the normal to the slit. If the difference corresponds to an integer
m number of half wavelengths the rays will interfere destructively either with each other or with
other rays that are located halfway between the centre of the slit and the edge of the slit. Thus
diffraction will create minima at angles that are given by
D
λ
sin θ = m ,
2
2
(6)
which gives us
λ
.
(7)
D
The Rayleigh criterion says that for two light sources to be resolved the maximum of one light
source should fall on the first minimum of the other source, so that the smallest angle that can be
resolved by the slit is
λ
= sin θmin ≈ θmin .
(8)
D
sin θ =
1
The diffraction pattern looks slightly different for a circular aperture, so that the criterion becomes
θmin = 1.22
λ
D
(9)
For a big optical telescope with an 8 m mirror operating at 500 nm, this means that the theoretical
resolution is
5 × 10−7
θmin = 1.22
= 8 × 10−8 rad = 0.016 arc seconds.
(10)
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In practice this resolution is never achieved and the real resolution is determined by the motion of
the air. Some modern telescopes can achieve a resolution down to 0.3 arc seconds on a very good
night, and adaptive optics techniques have been developed to partly correct for the influence of
the motion of the air, which can then lead to some further improvement in the image quality.
The total amount of light that is collected by the telescope is described by the illumination J,
which measures the amount of light per unit image area. The illumination is proportional to the
surface area of the telescope πD2 /4. Since the size of an image is proportional to the focal length
f the image area is proportional to f 2 and the illumination is proportional to D2 /f 2 . Often one
talks of the focal ratio
f
F = ,
(11)
D
so we have
1
J ∝ 2.
(12)
F
So far we have talked about lenses, but lenses have a serious disadvantage as optical elements.
Their refractive index n depends on the wave length of the light, which means that light of different
wave lengths are focused in different focal planes, an effect that is known as chromatic aberration.
In this sense mirrors are better optical elements since the angle under which the light is reflected
is the same for all wave lengths. For a spheroidal mirror the focal length f = R/2, where R is the
radius of curvature.
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Optical telescopes
The simplest optical telescope consists of two lenses. The large lense, the objective, with a focal
length fobj collects the light and forms an image in the focal plane. This image can then be studied
through the eyepiece with the focal length feye . The angular magnification of the system is then
m=
fobj
.
feye
(13)
In practice it is not possible to build lens telescopes, refractors, that are larger than one meter, and
they do also suffer from chromatic aberration. For these reasons all large modern telescopes are
reflectors. The problem with a reflector is that the focal plane is in front of the mirror, and there
is limited space to put detectors at this prime focus. There are then different constructions that
use auxiliary mirrors to move the focus to a more convenient place. Many amateur instruments
use Newton focus, where a plane mirror is reflecting the light out to the side. Both amateur and
professional instruments use the Cassegrain focus, where a secondary mirror is reflecting the light
back through a hole in the primary mirror, and then the focal plane is behind the primary mirror,
where there is more space for heavy equipment. The largest detectors are sometimes put at a
coudé or Nasmyth focus at the side of the telescope though.
Another important aspect of a telescope is its mechanical construction. The telescope must
follow the star or galaxy during the observation. In older telescopes this was solved by making one
of the axes that the telescope is moving around parallel to the axis of the Earth. Then it is enough
to rotate this axis one round per 23 hours and 56 minutes. However the mechanical construction
is difficult, and new telescopes today usually use an alt-azimuthal mount with one vertical and one
horizontal axis, and then a sophisticated computer is controlling the motion of the telescope.
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While mirrors are not subject to chromatic aberration there are other distortions of the images
that remain. A spherical mirror will reflect light close to and farther away from the optical axis
to different foci, an effect that is called spherical aberration. This can be avoided by using a
parabolic mirror, but then the images will develop a cometlike coma pointing towards the centre of
the image, an effect that can be avoided by using a more sophisticated construction, for instance a
Ritchey-Chretien telescope with a hyperbolic primary mirror and a non-spherical secondary mirror,
but some image defects will always remain.
Another solution to the spherical aberration is to introduce a corrector plate in front of the
spherical mirror. Such Schmidt-telescopes are often used with a photographic plate at the prime
focus for wide-angle sky surveys. In most other modern applications the photographic plates have
been replaced by CCD-detectors that collect the electrons that are released when the photons strike
the detectors. These detectors have the advantage that they are much more efficient in registering
the individual photons, but they cover much smaller areas than photographic plates do.
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Radio telescopes
It is not possible to use lenses in radio telescopes, so most of the radio telescopes today are
reflectors, though there are some telescopes that are just assemblages of very simple antennae.
The main limitation of a radio telescope is that since it is working at a much longer wave length
its resolution is much worse. In case we want a telescope at 21 cm (the wave length of atomic
hydrogen) to achieve the same resolution as an optical telescope (1 arc second = 5 × 10−6 rad) we
will need a reflector of the diameter
D=
1.22 × 0.21
1.22λ
=
= 53 km.
θmin
5 × 10−6
(14)
Obviously it is not possible to build so large telescopes. The solution is to connect smaller telescopes
to each other and let the signals from these telescopes interfere with each other. The telescopes can
then re-create the resolution of a larger telescope of the size of the distance between the telescopes.
In VLBI it is even possible to use telescopes on different continents to achieve a resolution that is
better than any optical telescope.
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Space-based astronomy
Our atmosphere is only transparent to optical light and radio waves, and even some radio waves
are absorbed by the atmosphere. In order to explore other parts of the electromagnetic spectrum
one must take the telescopes outside of the atmosphere. Such instruments have been developed to
study infrared, ultraviolet, X-ray and gamma rays.
Infrared and ultraviolet telescopes look fairly similar to optical telescopes, but there are some
important differences. A special problem for infrared telescopes is that anything which is hot emits
infrared radiation, so these instruments must be cooled. The detectors are also different from the
CCD-detectors that optical instruments use, since infrared light is not energetic enough to knock
out electrons from ordinary materials. However by using special materials it has been possible to
construct array detectors similar to CCDs for infrared astronomy. A big infrared telescope, Spitzer,
was launched a couple of years ago and the area is therefore developing quickly for the moment.
Ultraviolet astronomy on the other hand is limited by that ordinary glass absorbs ultraviolet
light. The telescopes therefore have to be built of special materials, but otherwise they are similar to
optical telescopes. One of the most successful astronomical satellites ever, IUE, was an ultraviolet
telescope. It was launched in the late 1970s and had an expected life time of a couple of years.
Eventually it was shut down in the 1990s because most of its research could then be done by the
Hubble Space Telescope, which is much bigger.
In the extreme ultraviolet and in X-rays it is not possible to use ordinary mirrors any longer
since the radiation will rather go through the mirror or be absorbed in the mirror, than reflect
off the mirror surface. It is possible though to use special mirrors in which the radiation falls in
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almost parallel to the surface of the mirror, which has the shape of a segment of a paraboloid.
Such mirror designs are used on the leading X-ray satellites today, the American Chandra-AXAF,
and the European Newton-XMM.
In gamma rays it is not possible at all to focus the radiation, but rather one uses detectors
that have more in common with the detectors that are used by particle physicists. The position of
a gamma ray source is then decided by tracing what direction the gamma-ray photon went in as
it crossed the detector. The satellite Compton-GRO was very important for developing the area
in the 1990s, and it has now been succeeded by the new satellite Integral, which however does
not cover the high-energy gamma-rays that were studied by the EGRET instrument on ComptonGRO. These energies will be covered by another satellite GLAST, which is due to be launched
within a couple of years. At even higher energies it is possible to use the atmosphere of the Earth
as the detector, and that is done by Earth-based facilities such as the Pierre Auger observatory in
Argentine.
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