Telescopes (3.9.1)

Lenses refract light, causing it to change direction. Convex (or converging) lenses refract parallel light rays towards each other, causing them to converge at the principle focus. Converging lenses are thicker in the middle than at the edges.

Light rays from an object that pass through a convex lens are refracted and form an image at the point where the rays converge.

Shown below are three symbols used to denote a convex lens:

To locate an image formed by a converging lens, draw three standard rays from a point on the object. Any two rays are sufficient to locate the image; the third acts as a check. Light rays travel in straight lines and are refracted only at the lens, not before it, after it, or at focal point .

1. A ray parallel to the principal axis refracts at the lens and passes through the far focal point .
2. A ray passing through the optical centre, passes straight through the lens, undeviated.
3. A ray passing through the near focal point refracts at the lens and emerges parallel to the principal axis.

The image of that point sits where the rays cross.

When ray diagrams are constructed using at least two of the three principal rays from a point on the object, the location of the corresponding image can be determined.

The image of the top of the object forms where the refracted rays from the top of the object intersect. For a real inverted image, this point lies below the principal axis.

If the base of the object lies on the principal axis, then the base of the image will also be on the principal axis. Therefore, only two rays from the top of the object are needed to find where the image will form. A third ray can be used as a check if necessary.

If the base of the object does not lie on the principal axis, then two additional rays must be drawn from the base of the object to determine the position of the base of the image.

It is important to note that ray diagrams for an astronomical telescope assume that each converging lens is a thin lens. This means that light is drawn as refracting only once, at the vertical line through the lens’s centre, rather than refracting twice at its two curved surfaces.

A ray passing through the optical centre is drawn as a straight, undeviated line. Therefore, in every telescope ray diagram, the ray from an object point passing through the optical centre of each lens can be extended as a straight line without bending. This simplifies the construction of the final image.

Lenses can form two types of images: A real image is one that can be projected onto a screen. They are formed by a convex lens where the distance from the object to the lens is greater than the focal length:

• The image formed is always real and inverted (upside down).
• Images can be magnified, diminished or remain the same size depending on the distance between the object and the lens.

Lenses can form two types of images: A virtual image is formed by a convex (converging) lens when the object is placed closer to the lens than its focal length. Virtual images cannot be projected onto a screen because the light rays only appear to diverge from a point in space rather than actually converging there.

An example of a virtual image is the reflection in a mirror. The light rays appear to be coming from behind the mirror, but the object is not actually there. A virtual image will form on the same side of the lens as the object and will be upright.

It is important to note that dashed lines represent the backward extensions of light rays and are used to locate virtual images. They do not represent real light rays.

An astronomical refracting telescope consists of two convex (converging) lenses with different focal lengths. The objective lens faces the object and has the longer focal length. The other lens is called the eyepiece lens.

Light from the distant object enters the telescope through the objective lens, then passes through the eyepiece lens and enters the eye. The distance between the two lenses can be adjusted to produce a clear, focused image.

The objective lens produces a real, inverted image inside the telescope, which is then magnified by the eyepiece lens. The image seen by the eye is magnified, virtual and inverted.

Distant objects in space can be assumed to be at infinity, meaning that the incoming rays are effectively parallel to each other.

When a telescope is in normal adjustment, the focal point of the objective lens is at the same point as the focal point of the eyepiece lens This results in parallel rays entering the eye, producing a virtual image that appears to be at infinity.

The length of the telescope is the sum of the focal length of the objective lens and the focal length of the eyepiece lens

The angular magnification of the telescope in normal adjustment can be calculated in terms of angles:

Where:

• The angle subtended by the object is the angle at the observer’s eye between rays coming from the top and bottom of the real object when viewed directly (i.e. without the telescope).

• The angle subtended by the image is the angle at the eye between rays leaving the eyepiece and determines the apparent angular size of the image seen through the telescope.

It is useful to note that it does not matter whether the angles are measured in degrees or radians, provided and use the same units.

Concave mirrors can be used to focus axial rays. As with lenses:

• The point where the rays converge is called the principal focus or focal point.
• The focal length is the distance from the principal focus to the mirror along the principal axis.

Reflecting telescopes use a parabolic concave mirror to form a real image at the mirror’s focal point. This image is then magnified through the eyepiece lens, as in a refracting telescope.

The angle of incidence of a light ray with the normal of a concave mirror’s surface is equal to the angle of reflection. Therefore, a concave mirror will focus all incoming light to a common focal point . The focal length is half the distance to the centre of curvature.

It is important to note the centre of curvature is the centre of the imaginary circle that the mirror’s curved surface is part of.

The principal focus of a concave primary mirror lies in front of the mirror. If an observer or detector were placed at this point, it would obstruct some of the incoming light.

A common solution to this problem is to use a convex secondary mirror to form a Cassegrain arrangement.

The secondary mirror is positioned just before the primary mirror’s focal point. It reflects the converging light back through a small hole in the centre of the primary mirror, where it passes to the eyepiece. In normal adjustment, the eyepiece forms a virtual image at infinity.

The Cassegrain arrangement increases the effective focal length of the telescope, thereby increasing its angular magnification while keeping the telescope physically compact.

The primary mirrors in reflecting telescopes are parabolic rather than spherical to avoid unwanted spherical aberration.

• If the mirror is spherical, the outermost rays are brought to a focus closer to the mirror than the inner rays. This produces a blurred image. 

 A *parabolic mirror is shaped so that all rays parallel to the principal axis are reflected to the same principal focus, producing a sharp image.

Chromatic aberration is an optical defect in refracting lens telescopes where different wavelengths of light are focused at different points.

Different colours of light have different wavelengths and, therefore, are refracted by different amounts when passed through a lens. This causes image blurring and colour fringing.

It is important to note that mirrors do not suffer from chromatic aberration as they do not refract light.

Optical reflecting telescopes and optical refracting telescopes are similar, but differ in several ways. It is important to know their relative merits:

Similar to an optical reflecting telescope, a radio telescope uses a parabolic dish to reflect and focus radio waves. Because radio waves have much longer wavelengths than visible light, this dish is constructed from wire mesh rather than a mirror.

The waves are reflected by the mesh and focused onto an antenna placed at the focal point. A preamplifier is also placed at the focal point to boost the signal. This signal is further amplified before passing through a tuner to remove unwanted frequencies and noise, then it is sent into a computer for analysis.

A mesh dish is easier and cheaper to manufacture than a mirror for an optical telescope, as small imperfections do not affect the focus. The longer the detected wavelength, the less it is affected by imperfections.

Because X-rays have much shorter wavelengths, they do not reflect off surfaces as readily as longer-wavelength electromagnetic waves do when they strike directly. However, they can be reflected when they hit the surface at a very shallow, grazing angle.

By using a series of carefully arranged mirrors at shallow angles, X-rays can be focused onto a detector. As a result, X-ray telescopes have a very different design compared to conventional telescopes.

Infrared (IR) and ultraviolet (UV) telescopes are very similar to optical reflecting telescopes in that they use parabolic mirrors and charge-coupled devices (CCDs).

• UV telescopes are more sensitive to imperfections. Therefore, the mirrors must be even more precisely constructed than optical telescopes.
• IR telescopes are less affected by imperfections in the mirror due to the longer wavelength of IR light. However, these telescopes require additional cooling using liquid helium due to the heating effect of IR waves.

The Earth’s atmosphere only transmits certain wavelengths of electromagnetic radiation. Radio and optical wavelengths can be observed from the ground. For other wavelengths, such as IR or UV, the best solution is often to use telescopes on balloons, on high mountains, or in orbit around Earth.

The level of detail observed through a telescope is known as the resolving power and is dependent on the width of the objective lens.

The angular separation between two distant stars is the angle between two straight imaginary lines between the observer and the two stars. The smallest angular separation at which a telescope can detect two objects is called its minimum angular resolution.

Resolving power is inversely proportional to the minimum angular resolution ; the smaller the minimum angular resolution, the better the resolving power.

The resolution of a telescope is limited by the amount of diffraction. When light is diffracted through a circular aperture, it produces a circular diffraction pattern.

The diffraction pattern from a circular aperture is called an Airy pattern, characterised by a bright central Airy disc and concentric dark/bright rings.

Two stars are just resolvable if the centre of one star’s Airy disc coincides with the first minimum (dark ring) of the other’s Airy pattern.

The airy pattern shows a bright central maximum with fainter maxima on either side. The accompanying graph is a one-dimensional slice through the centre of this pattern:

In a telescope, the objective lens or mirror has a finite diameter, acting as an aperture that diffracts light from distant objects. This can be modelled as a single slit. The following equations give the angular position of the dark fringes, which occur when:

​
Where:

• is the order of the minimum,
• is the wavelength , and
• is the slit width

The central maximum is at , and the first minima are found on either side when . For small angles, , the minima can be written as:

Diffraction always occurs when light passes through an aperture, causing the image to spread out slightly. The narrower the aperture, the greater the diffraction.

For telescopes, the diameter of the objective lens determines the extent of diffraction. The smaller the objective lens, the greater the diffraction.

An example of this is when two close stars are just resolved and then observed with a telescope equipped with a narrower objective lens; an observer would see them as a single blurred image. Using a telescope with a wider objective lens would resolve this issue, allowing the two stars to be seen separately.

The image above shows two stars observed through three different telescopes.

A charged-coupled device (CCD) is an array of light-sensitive pixels, within a silicon chip comprised of several silicon layers, that become charged when exposed to incident photons. CCDs can be used as a detection method in large-diameter telescopes, but they are also employed in mobile phones and cameras.

The silicon chip is made out of approximately 16 million pixels aligned in a 2D array. Electrodes connected to the device form potential wells that trap any emitted photoelectrons, which are then moved to process the information and form a high-resolution digital image.

CCDs can detect more than just visible light; they can also respond to other regions of the electromagnetic spectrum. Their linear response means the signal produced is proportional to the intensity of radiation received, so faint details can still be detected within brighter images.

Quantum efficiency is the percentage of incident photons that are detected by a light detector. For a pixel, this is the percentage of incident photons that release an electron.

The high quantum efficiency of charged-coupled devices (CCDs) gives them an advantage over the naked eye and photographic film:

• Capable of detecting fainter images and resolving much finer details.
• Can detect minor changes in images, particularly in fast-changing ones.
• Wider range of wavelength detection than the human eye compared to meaning they can also detect infrared and ultraviolet light.
• Maximum quantum efficiency can be obtained across a wide range of wavelengths ().