Image formation

Telescopes and antennas are the light collectors of astronomy. They come in varying shapes and sizes that depend in part on the frequency of radiation they are designed to detect. Most systems concentrate the incoming radiation by means of focusing. Optical telescopes gather light with a lens or a reflecting surface (a mirror). Radio telescopes make use of reflecting metal surfaces. X-ray telescopes make use of the reflecting character of a smooth metal surface for x rays impinging on it at a low glancing ("grazing") angle, like a stone skipping on water. Some radio and x-ray detection systems and all gamma-ray systems do not focus the radiation. These are non-focusing systems.

Focal Range Ray

Figure 5.1. The focusing characteristics of an ideal thin lens. The focal length fL and the aperture d are each measured in meters. (a) Parallel beam arriving along the lens axis, and focusing at distance fL. (b) Off-axis parallel beam (small angle a) also converging at distance fL, but displaced a distance 5 = fL tan a from the lens axis, (c) Extended source subtending angle a and depositing, in a fixed time, energy Ep a (d/fL)2 onto a single pixel of the image plane.

Figure 5.1. The focusing characteristics of an ideal thin lens. The focal length fL and the aperture d are each measured in meters. (a) Parallel beam arriving along the lens axis, and focusing at distance fL. (b) Off-axis parallel beam (small angle a) also converging at distance fL, but displaced a distance 5 = fL tan a from the lens axis, (c) Extended source subtending angle a and depositing, in a fixed time, energy Ep a (d/fL)2 onto a single pixel of the image plane.

Focusing systems

Focal length and plate scale

The radiation from a very distant point-like star arrives at earth as a parallel beam of light. If the light impinges normally onto a thin (ideal) convex lens (Fig. 1a), a parallel bundle of rays will focus to an on-axis point image in the focal plane, a distance fL (focal length) beyond the lens. If the parallel rays arrive at an angle a from the lens axis, they will also focus to a point in the focal plane (Fig. 1b), but at a distance s removed from the optical axis. The vertical position of the focus is defined by the ray that passes through the center of the lens; it will transit the (thin) lens without being deviated. The relation between these quantities is s = fLtana —> fL a (m) (5.1)

a small

This geometry can also be applied to a properly figured concave mirror (or system of mirrors) that brings a parallel beam of light to a focus.

If there were two stars, one on axis and the other off axis as shown in Figs. 1a,b, they would be separated in the sky by the angle a and on the plate by the distance s. The relation (1) gives the star separation in the focal plane, or on a photographic plate exposed in the focal plane. A nebula with angular diameter a (Fig. 1c) would have this same diameter s on the photograph. A large focal length yields a large star separation or nebular image, and a small focal length yields a small image. A small focal length requires the lens or mirror to refract the rays more strongly (Fig. 1a), and this leads to difficulties of design and limitations in performance (e.g., depth and breadth of the well-focused region). On the other hand, such a telescope can fit inside a smaller, and hence cheaper, telescope building.

The plate scale describes the angle that is imaged onto unit length of the plate; it is simply the ratio of a and s which is the inverse of the focal length, a1

s fL

The units are m-1 or, equivalently, radians/meter; radians have no dimension. A large plate scale means the image size s is small and vice versa. In practice, the plate scale is usually given in "arcsec per mm" (''/mm). The focal length of the Lick 3-m (diameter) telescope is fL = 15.2 m giving a plate scale of 14''/mm at the prime focus. A 1° piece of the sky would occupy a full 1/4 m in the focal plane. A given telescope may offer a choice of several focal lengths, for example by changing the secondary mirror (see below).

Aperture and deposited energy The rate of energy deposited on a single grain of film, or on the single pixel of a modern electronic imaging device, determines whether a given incident energy flux can be detected in a given time. A large telescope aperture (diameter d) will increase the energy flow onto the detector because a larger part of the incoming wavefront is intercepted and focused. For a perfect point source, perfect atmosphere, and perfect lens, all the collected photons from a source will be deposited onto the same grain of the film or the same pixel of an electronic detector. In this ideal case, the aperture alone determines the needed exposure; the focal length does not enter.

However, if the celestial source has a significant finite angular size, like the moon or a nebula, the energy will be deposited over a number of grains or pixels (Fig. 1c). If the image is spread over a large number of pixels because of a large focal length, a longer exposure is required to obtain a detectable signal in a given pixel. If, in contrast, the image is concentrated in a small region because of a short focal length, there is more energy deposited in a given pixel in a given time. In this case, the source image appears smaller but is more quickly detected.

The area of a circular image of diameter s is proportional to s2. Thus, for a nebula of (fixed) angular size a, the energy deposited onto a single pixel for a fixed telescope mirror aperture is Ep a s-2 a f- where we used s a fL from (1). Also, a larger aperture will allow more photons to be collected proportionally to the collecting area a of the mirror where a a d2. Thus, for diffuse sources, the energy deposited per unit time onto a single pixel depends on both the aperture and the focal length,

+ Ep a ( — ) (Energy per unit time onto single pixel) (5.3)

The ratio fL/d is called the focal ratio,

From (3) and (4) it is apparent that the focal ratio is an inverse measure ofhow fast energy is deposited on an element of the image plane. One refers to the "speed" of the optical system; it is proportional to the energy Ep deposited in a given time. Thus, speed a Ep a R-2. A greater speed means a photograph or other measurement may be carried out in less time.

The focal ratio is usually indicated with the notation " f/ R"; i.e., a focal ratio of 6 is written " f /6", when, in fact, it is the aperture d which is equal to fL/R; see (4). When you see " f /6", think "R = 6, the focal length fL is 6 times the aperture d". Amateur cameras usually have a focal ratio adjustment which can be varied from about f /2 to about f /16. This is accomplished by changing the aperture with an adjustable diaphragm; the focal length does not change. A "50-mm lens" refers to the focal length fL. Zoom lenses change the focal length.

The speed of an optical system depends only on the ratio R; it is independent of the specific camera used. A 1-m telescope with f /6 optics will be just as fast as a 4-m system with f/6 optics. Less light is collected by the smaller 1-m aperture, but the focal length is also shorter. This causes the energy to be concentrated onto a smaller area in the image plane. Thus the energy deposited per pixel remains the same for the two systems. Slower optics (greater R) are sometimes desirable; the image is more spread out (greater magnification) and the angular resolution is improved. But it takes longer to get a good exposure.

A special kind of telescope design, the Schmidt telescope uses a refracting corrector plate in front of the principal (primary) mirror to produce high quality images over a large 5° x 5° angular field. It features a short focal length. The focal ratio for the large Palomar Schmidt telescope is R = 2.5. The short focal length yields low magnification causing a lot of energy to be focused onto each pixel. The Schmidt design is thus very fast. It was the ideal instrument to make the Palomar Observatory Sky Survey described in Section 3.4.

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