Issues with the use of telescopes

Magnification

Magnification determines how much larger the image is as compared to the size of the source of the light (the object)

Magnification =fo

fe

Where

fo is the focal length of the objective

fe is the focal length of the eyepiece

Issues with the use of telescopes

Magnification

Magnification =fo

fe

A cheap telescope has an objective focal length of 600 mm, an objective diameter of 0.05 m and an eyepiece focal length of 20 mm. What is the magnification of this telescope?

Given

fo = 600 mm

fe = 20 mm

D = 0.05 m

M = 600 mm / 20 mm = 30

Issues with the use of telescopes

Magnification

Magnification =fo

fe

A cheap telescope has an objective focal length of 600 mm , an objective diameter of 0.05 m and an eyepiece focal length of 5 mm. What is the magnification of this telescope?

Given

fo = 600 mm

fe = 5 mm

D = 0.05 m

M = 600 mm / 5 mm = 120

Issues with the use of telescopes

Magnification

Magnification =fo

fe

An expensive telescope has an objective focal length of 2400 mm , an objective diameter of 0.2 m and an eyepiece focal length of 20 mm. What is the magnification of this telescope?

Given

fo = 2400 mm

fe = 20 mm

D = 0.2 m

M = 2400 mm / 20 mm = 120

Issues with the use of telescopes

Magnification

Question: Is the cheap telescope with a 5 mm eyepiece as good as the expensive telescope with a 20 mm eyepiece?

What do you think?

Issues with the use of telescopes

Resolution

More important (possibly more important) than magnification is resolution.

Resolution – the property of an instrument to identify (resolve) small details.

The smallest angular size identifiable by an instrument is given by

min = .25

DWhere

is the wavelength of the EM waves being collected in m (1 m = 10-6 m)

D is the diameter of the aperture (the opening which collects the wave) in meters

The calculated value of will be in seconds of arc (arc seconds)

Issues with the use of telescopes

Resolution

min is called the diffraction limited resolution of the telescope

Issues with the use of telescopes

Resolution

For the naked eye,

Shortest visible wavelength 400 x 10-9 m = .4 m

Diameter of the aperture (the pupil) 3 mm = 3 x 10-3 m

θmin = (0.25) (0.4 ) / (3 x 10-3 ) ≈ 0.33 arc sec

min 33” = .55’ = .0093o

The average human eye can resolve object with an angular diameter of about a half a minute.

min (in arc sec) = .25 (in m )

D (in m)

Issues with the use of telescopes

Resolution

For the Mount Palomar 200 inch optical telescope, Shortest visible wavelength 400 x 10-9 m = .4 m Diameter of the aperture (the objective) = 200 in = 5.08 m

θmin = (0.25) (0.4 ) / (5.08 ) ≈ 1.96 x 10-2 arc sec

min 1.96 x 10-2 “ = 3.2 x 10-5 ‘ = 5.5 x 10-7 degrees

The Mount Palomar telescope can resolve objects about 1700 times smaller than the naked eye

min (in arc sec) = .25 (in m )

D (in m)

Issues with the use of telescopes

Resolution – The Hubble Space TelescopeHubble works on the same principle as the first reflecting telescope built in the 1600s by Isaac Newton. Light enters the telescope and strikes a concave primary mirror, which acts like a lens to focus the light. The bigger the mirror, the better the image.

In Hubble, light from the primary mirror is reflected to a smaller secondary mirror in front of the primary mirror, then back through a hole in the primary to instruments clustered behind the focal plane (where the image is in focus).       

Mirror sizePrimary mirror: 2.4 m – (94.5 inches) in diameter

Secondary mirror: 0.3 m - (12 inches) in diameter  

Angular resolutionHubble's angular resolution is 0.05 arcsecond. This is the "sharpness" of Hubble's vision. If you could see as well as Hubble, you could stand in New York City and distinguish two fireflies, 1 m (3.3 feet) apart, in San Francisco.

Issues with the use of telescopesResolution

If the Mount Palomar 200 inch optical telescope recorded radio waves of wavelength 1 meter, wavelength 1 m = 1 x 106 m Diameter of the aperture (the objective) = 200 in = 5.08 m

θmin = (0.25) (0.1 x 106 ) / (5.08) ≈ 4.9 x 104 arc sec

min 4.9 x 104 “ = 820’ = 13.7o

The angular diameter of the moon = 30’ The angular diameter of the Andromeda Galaxy 178’

The Mount Palomar telescope would not be able to resolve these objectsIt would not be able to “see” the moon !

min (in arc sec) = .25 (in m )

D (in m)

Issues with the use of telescopesResolution

For the National Radio Astronomical Observatory Robert C. Byrd Radio Telescope,

wavelength 1 m = 1 x 106 m

Diameter of the aperture (the objective) = 100 m

θmin = (0.25) (1 x 106 ) / (100 ) ≈ 2500 arc sec

min 2500” = 41’ = .69o

The angular diameter of the moon = 30’

The angular diameter of the Andromeda Galaxy 178’

The NRAO telescope would be able (roughly) to resolve radio sources of these angular diameters

min (in arc sec) = .25 (in m )

D (in m)

Issues with the use of telescopesResolution

For the Arecibo Radio telescope,

wavelength 1 x 106 m

Diameter of the aperture (the objective) = 305 m

θmin = (0.25) (1 x 106 ) / (305 ) ≈ 819 arc sec

min 819” = 13.7’ = .22o

The angular diameter of the moon = 30’

The angular diameter of the Andromeda Galaxy 178’

The Arecibo telescope would easily be able to resolve radio sources of these angular diameters

min (in arc sec) = .25 (in m )

D (in m)

Issues with the use of telescopes

Magnification

Question: Is the cheap telescope with a 5 mm eyepiece as good as the expensive telescope with a 20 mm eyepiece?

The magnifications in both cases are the same. However, the diffraction limited resolutions are (using 0.4 μm for the visible wavelength)

Θmin,cheap = (0.25) (0.4) / (0.05) = 2 arc sec

Θmin,expensive = (0.25) (0.4) / (0.2) = 0.5 arc sec

The expensive telescope will resolve objects 4 times smaller than the cheap telescope. In part, the expense of a larger telescope is related to resolution more that magnification.