## Deriving a Value for the Focal Ratio

The image scale, s, is just the reciprocal of the focal length of the telescope, F, in radians. To convert radians to seconds of arc multiply by 648,000/n. Hence the image scale in seconds of arc is

648,000

The desired image scale is one that will make the distance between two stars at the focal plane greater than the diameter of the micrometer wires. Well, what is the minimum separation between two stars? The answer depends on the resolution of the telescope.

The resolution of the telescope in seconds of arc is given by

where D is the aperture of the telescope in millimetres.

Now, let the image scale equal the resolution, i.e. equate the right hand sides of these two equations, and then multiply both sides by F/120. The result is

F 5400

As F/D is the focal ratio of the telescope this equation gives the focal ratio at which the image scale equals the resolution. That is to say, this is the focal ratio at which the image scale is the resolution of the telescope per millimetre. So, for a 20-cm telescope, which has a resolution of 0.6'', a focal ratio of 1718.87 would give an image scale of 0.6 seconds per millimetre. However, unless the wires of the micrometer are 1 mm in diameter, a focal ratio this large will not be required. The required focal ratio will be 1718.87 multiplied by the diameter of the micrometer wires, i.e.

where w is the diameter of the micrometer wires.

In the case of the RETEL micrometer, w = 0.012 mm. Hence, F/D = 20.6. This means that a focal ratio greater than 20.6 will ensure that the distance between two stars that the telescope can resolve will be greater than the diameter of micrometer wires.

There is a school of thought that holds that as the separation is measured with two wires, the separation at the focal plane should be at least twice the diameter of the micrometer wires. In the above case, this would mean that a focal ratio of 41.2 would be required.

Either way, a very long focal ratio is required. Telescopes do not usually come with focal ratios of this order. The way to effectively increase the focal ratio is to use a Barlow lens. The amplification factor of a Barlow lens is about 2-3 times. To achieve an effective focal ratio of 41.2 with a 3x Barlow would require a focal ratio of about 13.7, whereas an f/10 telescope with a 2x Barlow would achieve an effective focal ratio of 20.

These figures suggest that a telescope with a focal ratio of at least 10 would be required for double star observing, providing it is used with a Barlow lens of at least 2x amplification. If a telescope of a shorter focal ratio is used then the resolution of the telescope, for measuring double stars, is going to be limited by the size of the micrometer wires rather than by the aperture of the telescope.

Of course, if a double image micrometer were being used instead of a filar micrometer, then a shorter focal ratio, higher power eyepiece combination would be feasible. The focal ratio then would be limited by the focal length of the eyepiece being used. The focal ratio should be at least numerically equal to the focal length of the eyepiece in millimetres in order to achieve twice the resolving magnification.

So, if the eyepiece has a focal length of 9 mm the telescope should have a focal ratio of at least f/9. Conversely, if the telescope has a focal ratio of f/9 the focal length of the micrometer eyepiece should be 9 mm at the most. (This relationship between the focal length of the eyepiece and the focal ratio of the telescope holds also when using a filar micrometer. The size of the wires of the filar micrometer, however, dictate focal ratios that are numerically well in excess of the eyepiece focal length.)

There are, of course, double star observers whose instruments do not meet the above criterion, however it is something that a prospective double star observer should consider when deciding on what instruments to choose.

Observing Double Stars with an Alt-azimuth Mounted Telescope

The application of computer technology to telescope drives has enabled sidereal tracking to be automated on alt-azimuth mounted telescopes. Alt-azimuth mounted telescopes, however, turn about an axis through the zenith instead of an axis through the pole, as do equatorially mounted telescopes. This means that the fixed point on the celestial sphere for such telescopes is the zenith, instead of the pole. As a consequence of this stars in the field of the eyepiece rotate around the centre of the field as the telescope follows the stars across the sky. In the case of a double star this will cause the companion to circle the primary star in the course of the night.

An example of this field rotation, as it is called, is the belt of Orion. In northern latitudes, the three stars that form the belt stand vertically when the constellation is rising, but lie along the horizon when it is setting. In the southern hemisphere the orientation is reversed: lying when rising, standing when setting.

Figure 22.1. The Astronomical Triangle is formed by three points on the celestial sphere: the Pole (P), the zenith (Z), and a star (S). The sides of the triangle are PZ = 90° - 0 (the co-latitude), PS = 90° - 8 (the co-declination), and ZS = z (the zenith distance). The internal angles ZPS = H is the hour angle of the star, PZS = A is the azimuth of the star and PSZ = q is the parallactic angle.

Figure 22.1. The Astronomical Triangle is formed by three points on the celestial sphere: the Pole (P), the zenith (Z), and a star (S). The sides of the triangle are PZ = 90° - 0 (the co-latitude), PS = 90° - 8 (the co-declination), and ZS = z (the zenith distance). The internal angles ZPS = H is the hour angle of the star, PZS = A is the azimuth of the star and PSZ = q is the parallactic angle.