Mathematics of Telescopes

Mathematics of Telescopes
Although telescopes has been around for several hundreds of years, there has been great discrepancy as to who invented it first. Here is one authors opinion. Lippershey was a Dutch spectacle marker during the early 17th century (approximately 1600). He was one of the first who created the ?looker? (now called telescope) by placing two pieces of lenses together. The discovery that placing lenses together can magnify images were made by children who took Lippershey?s spectacles and looked at a distant church tower.
One of the most influential scientist associated with the telescope has to be Galileo. He took the design and reinvented the telescope into one of the first refractive telescopes we use to this day.

Galileo used this great invention to report astronomical facts such as the moon is cover with craters instead of being smooth, the Milky Way is composed of millions of stars, and Jupiter have four moons. Perhaps the most famous discovery is the Earth revolves around the Sun and the Earth is not the center of the universe (even though he was discredited at the time).

Newton was also involved with telescopes. After his growing interest with light bending, he applied his knowledge of the reflecting and refracting properties of light and invented the first reflective telescopes. Newton?s reflective telescopes vastly improve the clarity of images as well as escaping from chromatic abberration. In order to fully understand the concepts and ideas of how a telescopes work, some knowledge of simple optics are required.

Some Simple Optics

Refraction
Refraction occurs when light travels from one medium crosses a boundary and enters another medium of different properties. For example, light traveling from air to water. The amount of refraction (or bending) can be calculated using Snell?s Law.

Refractor

How Refractor Telescopes Work
The principle behind the refractive telescopes is the use of two glass lenses (objective lens and eyepiece lens) to gather and bend parallel light rays in a certain way so that the image fits the size of the eye?s pupil. Light rays is gather through the opening of the telescope called the aperture and passes through the objective lens and refracts onto a single point called the focal point. From there the light rays continue the same direction until it hits the eyepiece lens which also refract the light back into parallel rays. During the process, the image that enters our eyes is actually reverse of the original image and magnified because the size in which we preceive the image. The following diagram will show the components and the process of light rays traveling through a refractor telescope.

Magnification of a refractor telescope mainly depends on the distance of the eyepiece lens with regard to the focal point because the focal length determines how large the image get scale by. Typically a long focal length of the objective lens combine with a short focal length of the eyepiece lens will yield greater magnification. However, magnification is not the most important optical property for viewing astronomical phenomenon. The aperture size is more important because it accounts for the amount of light the telescope receives and the more light, the brighter the image. The size of aperture also accounts for the resolution power of the telescope. Larger aperture size will yield high quality of image, more crisp and detail image, and increase clarity of the image.

Advantages/Disadvantages
Advantages of a refractor telescopes

After initial setup, refractor telescopes are less likely to be misalign than reflector telescopes.
No cleaning is necessary inside the telescope because the tube is sealed off from contact with air.
Temperature and air current cannot effect the refractor as much as a reflector because of the isolation from outside atmosphere.
Disadvantages of a refractor telescopes

Refractor suffers from chromatic aberration due to the different index of refraction for different wavelengths.
Quality of transmission of light through the lens depends on the wavelength.
Quality of transmission of light decreases as the thickness of lens increase.
Difficult to construct a perfect lens in terms of curvature and purity inside the lens.
The objective lens will sag due to its own weight because the lens is only supported at the edge.
The cost of a objective lens increases as the diameter of the lens increase, so cost efficiency verse quality of image, reflectors is better than refractors.
The limitation of diameter size of the objective lens because the larger the lens, more problems such as chromatic aberration will occur.
Ray Tracing Through Refractor Telescope
The following is a sample trace of a ray through the refractor telescope lens system. Several assumption is made such as the radius of curvature for the lenses and index of refraction.

Reflector

How Reflector Telescopes Work
The main difference between a refractor and a reflector is the latter telescope involves both refraction and reflection of light rays. Reflector telescopes uses two mirrors (primary and secondary mirror) as well as a glass lens (eyepiece lens) in their lens system. Similar to refractors, light rays are gather through the aperture. However instead of bending the rays through a glass lens immediately, rays are to travel to the back of the telescope where the primary mirror is located. The primary mirror is shaped parabolically so that all incoming parallel rays will reflect off the mirror at their own specific angle and hit the surface of the secondary mirror. Once again a reflection will occur and all the rays will converge to a single point call the focal point. It is from that point where optics of refractors and reflectors are common. Light rays will continue to travel and refracts at the eyepiece lens and forms an image that is magnified to people?s perception. However this image is also in reverse to the original image. Refer to the following diagram to understand the components and processes in which light rays travel through a reflector telescope.

Magnification of a reflector telescope mainly depends on the distance of the eyepiece lens with regard to the focal point because the focal length determines how large the image get scale by. Typically a long objective focal length combine with a short focal length of the eyepiece lens will yield greater magnification. However, magnification is not the most important optical property for viewing astronomical phenomenon. The aperture size is more important because it accounts for the amount of light the telescope receives and the more light, the brighter the image. The size of aperture also accounts for the resolution power of the telescope. Larger aperture size will yield high quality of image, more crisp and detail image, and increase clarity of the image.

Advantages/Disadvantages
Advantages of a reflector telescopes

The main advantage is reflector telescope can escape from chromatic aberration because wavelength does not effect reflection.
The primary mirror is very stable because it is located at the back of the telescope and can be support in the back.
More cost effective than refractor of similar size.
Easier to make a high quality mirror than lens because mirror need to only concern with one side of the curvature.
Disadvantages of a reflector telescopes

Optical misalignment can occur quite easily.
Require frequent cleaning because the inside is expose to the atmosphere.
Secondary mirror can cause diffraction of original incoming light rays causing the ?christmas star effect? where a bright object have spikes.

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