Contents:

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Aperture, Focal Length & Focal Ratio

Magnification

Telescope Field of View

Minimum Magnification

Maximum Magnification

Resolution

Aperture, Focal Length & Focal Ratio

First of all some definitions:

Aperture        The diameter of the main lens in a refractor or the main mirror in a reflecting telescope.  Measured in millimetres, centimetres, metres, inches etc. 

Use the letter “D” to represent the aperture.

Focal Length        This is the distance from the front surface of the mirror (or from the plane of the lens for a refractor) at which an imaginary parallel beam of light would be brought to a focus. 

Use the symbol “F” to represent the focal length.

Focal Ratio   Use the symbol “f” to represent focal ratio. 

The focal ratio is calculated by dividing the focal length (F) by the aperture (D).

                     So              f        =       F / D                     (Equation 1)

                     and             F       =       f x D                     (Equation 2)

                     A few examples :

                     A Mirror has an aperture (D) of 200 mm. 

It has a focal ratio (f) of 6. 

What is the focal length (F)? 

From Equation (2) above;

                                       F       =       f x D

                                                 =       6 x 200

                                                =       1,200 mm is the focal length. 

                     A Mirror has an aperture (D) of 250 mm. 

It has a focal length (F) of 1,250 mm. 

What is the focal ratio (f)? 

From Equation (1) above :

                                       f        =       F / D

                                                 =       1,250 / 250

                                                =       5.0 is the focal ratio.

You may have heard the terms short and long focal ratio.  The former means that “f” is a small number and the light is focussed in a relatively short distance compared with the aperture of the mirror.  This makes for a compact telescope.  But it is more difficult to manufacture the mirror precisely, and the short focal ratio causes increased off-axis aberrations (distortions) to the images, particularly coma.  Such aberrations mean that while objects at the centre of the field of view will be focussed, nearer the perimeter of the field of view, point sources (star images) will be distorted.

The alternative long focal ratio telescope brings the light gradually to a focus over a distance which is many times the aperture of the main mirror.  This type of telescope is relatively long and bulky, but the mirror is easier to manufacture and the aberrations such as coma, relatively smaller.

Magnification

So how do we calculate the magnification of a telescope? 

Another piece of information is required, the eyepiece focal length which we will call “e”. 

When you inspect an eyepiece, the eyepiece focal length will always be stamped somewhere usually upon the bezel.  The focal length is usually in millimetres, but some may be in inches.

The magnification of the telescope (M) is simply given by :

                             M      =       F / e            (F and e in the same units)

 So for our 200 mm aperture telescope of f/6 focal ratio with a focal length of 1,200 mm, an eyepiece with a 25 mm eyepiece focal length (e) would give a magnification of :

                                       M      =       1,200 / 25

                                                =       x48 (meaning times 48 magnifications).

 Similarly the same telescope with an eyepiece with a 9 mm eyepiece focal length (e) would give a magnification of :

                                                M      =       1,200 / 9

                                                          =       x133.3

Telescope Field of View

How is the telescope field of view calculated?  Some more information is required. 

Eyepieces are not created equal.  They are given names such as Kellner, Nagler, Plossl, Erfle or more generic names such as Super Wide Angle. 

There are three basic facts to know about any eyepiece, the eyepiece focal length (e), the diameter of the barrel (so it fits the diameter of your focusser), and the eyepiece field of view sometimes known as the effective field of view.

The eyepiece field of view is the theoretical field of view in degrees the eyepiece would provide at a magnification of one.  Of course such a low magnification can neither be achieved or used in practice.  Alternatively, the eyepiece field of view can be thought of as the magnification required to give a telescope field of view of one degree (about twice the angular diameter of the full Moon).

The eyepiece field of view varies with the type of eyepiece.  The actual telescope field of view is calculated by dividing the eyepiece field of view by the magnification.  So Kellners have an eyepiece field of view of about 40 degrees.  This means that at 40 magnifications a Kellner eyepiece will give a 1 degree telescope field of view no matter what the aperture of the mirror or the focal length or focal ratio of the telescope. 

Alternatively most Plossl eyepieces will have an eyepiece field of view of about 52 degrees.  So at 52 magnifications, this eyepiece would also give a 1 degree telescope field of view and at 40 magnifications would give a telescope field of view of 52/40 = 1.3 degrees.

Consider our example telescope with a 200 mm f/6 mirror with a focal length of 1,200 mm.  Consider using various eyepieces of 25 mm focal length.  Whatever the eyepiece type, you now know the magnification will always be 1,200 / 25 or x48. 

For a Kellner eyepiece

          Eyepiece field of view      =       40°

          Telescope field of view    =       Eyepiece field of view / Magnification

          =       40/48

                                                =       0.83 degrees

                                                =       50.0 minutes of arc

For a Plossl eyepiece

          Eyepiece field of view      =       52°

          Telescope field of view    =       Eyepiece field of view / Magnification

          =       52/48

                                                =       1.08 degrees

                                                =       65.0 minutes of arc

So for our 200 mm f/6 telescope, a 25 mm Kellner eyepiece would show a patch of sky about 1.66 times the diameter of the full Moon at a magnification of x48. 

The 25 mm Plossl eyepiece would show a patch of sky about 2.17 times the diameter of the full Moon at the same magnification!

Minimum Magnification

There is a minimum recommended magnification for each telescope, because of the size of the image formed by the eyepiece (known as the exit pupil) and its relation to the diameter of the pupil of the eye. 

The iris is only fully open when the eye is dark adapted.  In young people a fully dark adapted eye has a pupil diameter of about 7mm.  This value decreases if there is any stray light about, or with age.  In the latter circumstances, the maximum pupil diameter may only be 5 or 6 mm (or lower).

If the exit pupil is greater than 7mm, the telescope is effectively wasting light because it cannot enter the eye.  This is critical if you are observing faint objects or faint detail.

How can the diameter of the exit pupil be calculated?  Again (luckily) it is a simple calculation.

Exit Pupil of Telescopic Image           = Aperture / Magnification

                                                          <7

So for each telescope aperture, there is a minimum magnification based on the exit pupil diameter formed.  Note that the exit pupil diameter is independent of the eyepiece field of view discussed above.

Maximum Magnification

Beware of telescopes that are advertised with an “up to 600 magnifications” type of approach.  Any telescope can achieve almost unlimited magnification (in theory) – it is just that the results will be less than memorable when the optics are pushed too hard.

The maximum magnification achievable is only limited by the eyepiece focal length.  With a 4mm eyepiece focal length, our example 200mm f/6 telescope of 1200mm focal length would achieve a magnification of 1200/4 or x300.  A 2mm eyepiece focal length would provide 1200/2 or x600, and so on.

But using higher magnifications will not necessarily show any more detail.  This is because we exist at the bottom of an ocean of air, our atmosphere, which is always turbulent to some degree.  Telescopes not only magnify the image of distant astronomical objects, they effectively magnify the atmospheric turbulence as well.

So eyepiece (magnification) choice for achieving best detail in observing the planets,  the Moon or close binary stars will often be a matter of trial and error.  More often than not you will get the best results with moderate magnification.  Only occasionally, on nights of better “seeing” will higher powers reveal more detail.  On typical nights, excessive magnification just results in excessive blurring, and less observable detail.

Resolution

The resolution of a telescope is the amount of detail theoretically detectable.  For point sources of light (stars), resolution is the angular separation between two objects which is just at the limit of detectability, where the two images almost overlap.  Note that telescopes cannot form a point source image of a point object due to a process called diffraction.  Perfect optics actually create a tiny circular disk-like image surrounded by fainter concentric circles, known as the Airey Disk).

The theoretical resolution is related solely to the aperture (D), and the wavelength of the electromagnetic radiation.  In visual wavelengths, the theoretical resolution of a telescope in seconds of arc is given by :

Resolution = 122 / D         (seconds of arc)

          Where D is the aperture in millimetres.

          Note  1 second of arc     = 1/60^th of a minute of arc

                    1 minute of arc      = 1/60^th of a degree

          So     1 second of arc     = 1/3600^th of a degree

The theoretical resolution of the ASTRONZ range of telescopes at visual wavelengths is as follows:

In practice, the actual resolution of a telescope is limited by the “seeing”, or atmospheric turbulence.  In most parts of New Zealand, average seeing would be 2 seconds of arc (2”) or worse, sometimes much worse.  So on average nights, high aperture telescopes will be unable to resolve any more detail than those with more modest apertures.

The best seeing conditions in the world on Mauna Kea, Cerro Tololo or Sierra Paranal provide approximately 0.5 second of arc seeing.  A view through the Keck telescopes in Hawaii or one of the VLT’s in Chile would provide no greater resolution than that of a 250mm aperture telescope, although you would be able to see fainter objects.  It is to escape the seeing limitations of ground based telescopes that the Hubble Space Telescope has been so important.  Even though it has only a relatively small 2 metre diameter mirror, that mirror is able to perform to its theoretical resolution, unlike the much larger ground based scopes.