Astrophysics Tutorial 2 - The Reflecting Telescope

Contents

Reflecting Telescopes

Resolving Power

Diffraction Effects

Rayleigh's Criterion

Derivation of the Equation

Light Detectors

 

Reflecting Telescopes

Lenses suffer a major drawbacks:

 

Concave mirrors can be used to project a real image:

 

 

 

The advantages of a concave mirror that is front-silvered are:

The focal length of a concave mirror is half the radius of curvature.  Spherical mirrors are easy to produce, but the image can be distorted by spherical aberration, so a parabolic shape is used to give perfect focusing.   You may wish to review curved mirrors in Waves Tutorial 6.

 

 

Many other receivers of electromagnetic radiation use parabolic mirrors

 

The diagrams show the two kinds of reflecting telescope:

 

 

This is called the Newtonian system.  Light is reflected to an eyepiece at the side of the telescope.

 

 

This telescope uses the Cassegrain system.  The eyepiece is at the back of the telescope.  The hole in the centre of the mirror does not affect the viewing ability.  Both kinds are found in observatories.  The mirror shown in this diagram is a plane mirror.  However many instruments use a convex mirror as the secondary mirror.  Some radio telescopes also use the Cassegrain system of mirrors.

 

 

All large telescopes use the reflecting system.   The largest telescope in the world has a 5 metre diameter concave mirror which requires many tonnes of glass, a considerable cooling time, and many hundred of hours of grinding to get it to a perfect shape.  It was silvered with a few grams of aluminium.

 

 

Question 1

Suggest reasons for the following:

(a) The silvering on a telescope mirror is on the top surface.

(b) The hole in the centre of the mirror of the Cassegrain system does not affect the viewing ability of the instrument. 

Answer

 

 

Resolving Power

The resolving power of any optical instrument is an indication of how good it is at distinguishing two objects close to one another.  For example, at a long distance two car headlights appear as a single blob of light.  At about 5 km, we can tell that they are two separate lights.  This is because the eye can resolve down to an angle about 3 × 10-4 radians.

 

Astronomers use angles in radians or degrees:

 

Radians have the advantage that for small angles:

 

sin q = tan q = q

 

This makes trigonometrical functions easier.  However astronomers tend to use arc-seconds which are useful for describing very small areas of sky.

 

Question 2

The Moon has a diameter of about 3500 km and is about 400 000 km from the Earth.  What is the angle in radians that the Moon subtends to an observer on the Earth? What is this in degrees? 

Answer

Question 3

Entirely coincidentally the angle subtended by the Sun is exactly the same as the angle subtended by the Moon.  The distance between the Earth and the Sun is 150 x 106 km.  What is the diameter of the Sun?

Answer

 

 

The observation of objects in space is made difficult because the atmosphere is turbulent.  This results in the twinkling or scintillation of stars.  Light pollution from street lights does not help either.  Dust in the atmosphere causes scattering of light. Major observatories have moved as far away as possible from cities and are situated on high mountains. 

 

 

Photograph from NASA

 

The best images of them all come from the Hubble Telescope, a Cassegrain instrument which is in orbit above the Earth.  There are no problems with distortion of the atmosphere up in space, but doing routine maintenance is not very easy.  The quality of pictures produced has been very high.

 

 

Photograph from NASA

 

 

 

Diffraction Effects

When light enters a telescope, it is passing through a gap.  It spreads out by the process of diffraction.  You will remember from AS Waves how when light passes through a single slit, dark and bright fringes are made.  (You might want to break off and revise that bit.  Go to Waves Tutorial 8.)  The resulting pattern is called a Fraunhofer Diffraction pattern:

 

 

 

 

 

Fraunhofer diffraction also occurs with circular openings.  If we use a circular aperture we get an effect like this:

 

 

 

The central bright spot is called an Airy Disc.

 

The telescope can be modelled as a single circular hole.  Diffraction is inevitable, but the larger the hole, the narrower the diffraction pattern becomes.  (Conversely a narrow aperture leads to a greater diffraction pattern.

 

Rayleigh's Criterion

Consider two stars A and B.  They are separated by an angle of q radians.

 

 

If the angle q gets smaller, the two stars seem to merge into one.  Therefore we need a more powerful telescope to observe them.  The physicist John William Strutt, Lord Rayleigh (1842 - 1919) studied the effect of overlapping of fringes and came up with the Rayleigh's Criterion.   Rayleigh’s criterion says:
 

The resolution of the images of two point objects is not possible if any part of the central spot of either image lies inside the first dark ring of the other image.

 

 
 

 

The angular separation at which the two objects are resolved is given by the formula:

 

 

[q - angular separation (rad); l - wavelength (m); D - aperture width (m)]

 

The angle term q must be in radians.  This is because sin q q in radians.  If the angle is in degrees, the relationship becomes:

 

 

Where:

 

the two stars cannot be resolved.  To improve the resolution of a telescope, we need to have a large aperture and a short wavelength.

 

In the AQA syllabus, the aperture is regarded as a single slit.  In other syllabuses, the aperture is regarded as a circular disc.  In which case, the relationship is modified to:

 

 

 

In practice, although telescopes have much better resolution than the eye, this is limited by the atmosphere.  Telescopes have large apertures to allow as much light to get in as possible.

 

 

Derivation of the Equation

We know that the diffraction equation for a diffraction grating is:

Where:

 

In this case, n = 1 and d = diameter of the telescope.  Also for small angle in radians, sin q = q.

 

So we can now write:

 

And we can rearrange to give:


 

 

Angle q must be in radians. Make sure that your calculator is set to radians.
 

 

Question 4

What is the resolving power of a telescope of diameter 15 cm at a wavelength of 600 nm? 

Answer

 

Light Detectors

Once we have got a good image down the telescope, we need to have a way of recording it.  In early astronomy the human eye was used, and the results depended on the artistic ability of the astronomer.  Photographic techniques were used from the middle of the Nineteenth Century.

 

The resolution depends not just on the Rayleigh Criterion, but also on the emulsion of the film.  Very fine grain films are used for astronomical observation.  The quality of the picture needed to be high and precision mechanisms were essential for tracking individual stars across the sky.  If the grains of film are larger than the resolution of the telescope, than that is the limiting factor.

 

For many years, photographic film was the only way to record images of telescopes.  The quality of the images using wide format film could be outstanding.  Getting good quality images requires:

This is because the film needs photons to deposit the silver grains.  The film has to be very fine-grained, to give it the quality of image. This makes the film slow.  Therefore the telescope needs to track across the sky in order to follow the object under study.

 

Another problem with photographic film is that it has to be chemically processed.  While it's a straight forward process, it has been known to go wrong.  If the film consists of a number of exposures that have taken hours on cold nights to expose...
 

 

More recently a charged coupled device is used and is connected to a computer.  The computer can quickly do comparisons of images which would take a skilled astronomer several days.  The CCD works like this, as shown in this animation:

Animation by Michael Schmid, Wikimedia Commons

 

The picture below shows a CCD.

 

Kaspar Metz, Wikimedia Commons

 

 

The CCD is about the size of a postage stamp and can have many millions of pixels on it.  They work on the principles of quantum physics.  They have these advantages over film:

 

The graph below shows the quantum efficiency of the CCD:

 

 

The eye has a quantum efficiency of only 1 %.

 

With a CCD, the image is uploaded to a computer, where software is used to combine the images to give a colour picture.  The pictures can be of stunning quality.