Lecture 18 - Telescopes (2/23/96)

Seeds: Chapter 5

1. Some details ...
   • New office hours: Mondays 3:30-4:30 and Thursdays 2:00-3:00 PM in my office (DRL 2N3C).
   • Homework #3 is now due on Monday 2/26/96 instead of today. Note that the web version has some clarifications and hints.
   • Homework #4, handed out today, is due on Monday 3/4/96.
   • The updated course schedule reflects the new weekly homework scheduling, and the new reading schedule.
   • Next week we will cover Chapter 6 and hopefully start Chapter 7.
   • I noticed on the midterm that many of you were confusing the factors of 2.5 and 2.512 that appear in the intensity - magnitude relation. Note that the 2.512 is actually 10^(1/2.5) or 2.5 = 1/log(2.512) approximately (the 2.512 is rounded). The 2.5 is exact, in the relations I_A/I_B = 10^( m_B - m_A / 2.5 ) or (m_B - m_A) = 2.5 log(I_A/I_B) (see below).
2. Images from Lenses and Mirrors
   • A lens can be made from a refracting material such as glass that is shaped with oppositely curving surfaces such that parallel light rays passing through from one side are focused to a point, called the focus, on the other side of the lens.
   • The distance from the center of the lens to the focus is naturally enough called the focal distance, which we will abbreviate as f.
   • Note that the focus is on the line running through the center of the lens perpendicular to the plane that bisects the lens - this is called the optical axis. You can make a good lens out of two spherical surfaces, where the radius of curvature determines the strength of bending and thus the magnification. The useful lenses for our purpose are converging lenses, with convex surfaces so that the rays pass through and converge. If you make a lens with concave surfaces, the rays will diverge, which isnt too useful here!
   • You can focus rays also with a curved mirror. It turns out that a parabolic mirror (shaped like a parabola) will focus incoming parallel rays also to a point along the optical axis. Note that the focus is of course in front of the mirror.
   • The reason we consider parallel incoming rays is that we are most interested in looking a very far away objects like planets and stars. These objects are so far away that the light rays from them are essentially parallel by the time they reach the Earth.
   • It is interesting to look at the image formation with a lens or mirror. Just by drawing an upright object, like an arrow, with the bottom on the optical axis some distance in front of the lens, then drawing the rays, we can see where an image is formed. You only need to know that rays parallel to the optical axis are bent to pass through the focus, and that rays through the center of the lens are not deflected.
   • You find that the image is inverted (upside down!) and located some distance beyond the focus which depends upon the how far in front of the lens you drew the source, and how strong a lens you made it (what the focal distance was).
   • There is a thin-lens equation that states: 1/f = 1/s_o + 1/s_i where s_o is the distance from the source object to the lens, and s_i is the distance from the lens to the image (and f is the focal distance).
   • Note that if you move the object out to infinity in front of the lens, then the image moves closer to the focus. Thus, for astronomical purposes, the image of a distant star-field is made in the focal plane, the plane perpendicular to the optical axis and containing the focus.
   • You can make the same exercise for the parabolic mirror. You also find an inverted image that moves closer to the focal plane as you move the object to infinity.
   • In both cases for a single lens or mirror the image is inverted. Note that our eye is a single lens, and thus we see the world upside down (at least on our retina). Our brain has been trained from birth to ignore this and reprocess the information so that we perceive the world as right side up! Part of the fumbling about by little babies is learning to deal with inverted images in the eye. The optical processing by our brains is trainable - scientists have done experiments where special goggles that invert our view are worn by subjects, who of course are disoriented at first. After a few days, they find that they see the world not upside down but right side up again! This reverses when they take off the goggles, and their vision is returned to normal.
3. Refracting & Reflecting Telescopes
   • You can make a simple refracting telescope by placing a large objective lens at the top of a tube, and then a second smaller lens called an eyepiece at the bottom end of the tube. The eyepiece is place just beyond the focus and its purpose is to focus the diverging rays into parallel rays again, which are then focused by our eyes just as if we were looking at the sky without the telescope.
   • The focusing of the eyepiece controls the size of the image that appears on the eye, and thus the magnification of the telescope (see below).
   • Note that a refracting telescope, or refractor is unwieldy to make, since you need a long tube for big lenses (since its hard to make large strongly curved lenses). If you've picked up a large magnifying glass, you also know big lenses are very heavy!
   • You can make a reflecting telescope, or reflector, by placing a parabolic mirror at the bottom of a tube. The focus is then inside the tube, unless you put a second mirror in front of the focus to direct the light somewhere else.
   • The large main parabolic mirror in a reflector is called the primary mirror, and its focus inside the tube is called the prime focus. Many large optical telescopes are big enough so that there is actually room inside the tube for a person, who can observe at the prime focus (with an eyepiece)! Many radio telescopes put their receivers at prime focus also.
   • If a second mirror, or secondary mirror is placed in front of the prime focus, and is oriented at 45 degrees to the optical axis so as to reflect the light through a hole in the side of the tube, then we have what is called a Newtonian telescope, with a focus on the side and outside the telescope called the Newtonian focus. This is the first sort of reflecting telescope built 1668 by Isaac Newton (designed in 1663 by James Gregory).
   • If you orient the secondary mirror perpendicular to the optical axis, then the converging rays are pointed back down toward the primary. If a small hole is made in the center of the primary, then the focus can be place at the bottom end of the telescope where it is very convenient for observing. This design is known as a Cassegrain telescope, and the focus is of course the cassegrain focus.
   • With extra mirrors, there are an number of different configurations where the focus is directed to convenient places outside the tube. See the textbook for some of these.
   • Note that both for refracting and reflecting telescopes, the image quality degrades (becomes distorted and blurry) as you look at stars that are at increasing angles from the optical axis. This is because the simple lens or parabolic mirror only has a point-like focus for rays parallel to the optical axis - rays at slight angles actually focus into a little blurry spot that gets bigger the further you go off-axis. Thus the useable field-of-view, the angle from the center of the image that is not terribly distorted, is fairly small for these telescopes.
   • For a refracting telescope, complicated lenses with different surfaces and extra lenses can be used to "correct" for these aberrations. For a reflector, the secondary mirror can be specially figured and a large thin correcting plate or lens can be placed at the top of the tube. If this is done to a Cassegrain telescope, then it becomes a Schmidt-Cassegrain telescope (the lens is called the Schmidt corrector). This is the most popular sort of serious astronomical telescope.
   • Note that if a Schmidt corrector is place on a prime-focus telescope, you get a telescope with an excellent wide field of view. If you place a photographic plate or other detector at the prime focus, then you get a Schmidt Camera. These are used to make sky surveys, with a single photographic plate that covers as much as 6 degrees!
4. The Powers of a Telescope
   • Three "powers" of a telescope:
     1. Light Gathering => brightness of images
     2. Resolving => sharpness of images
     3. Magnifying => size of images
   • Light Gathering Power (LGP) is a measure of the ability of the telescope to collect light energy and concentrate it into the image or detector. Each photon of a given frequency f carries a quanta of energy h f, so the total energy collected is proportional to the number of photons collected, which in turn is proportional to the area of the primary mirror or objective lens of the telescope. Think of a telescope as a big "light bucket" collecting a rain of photons. Since the area of a telescope of diameter D is proportional to D^2, the light gathering power LGP is proportional to D^2. Note that LGP is really only useful as a ratio between two different telescopes. If a quantitative measure is needed, the telescope area A = Pi D^2 / 4 is used.
   • The brightness B of an image of a source of intensity I for telescope area A is given by B = A I.
   • Two telescopes looking at the same source (same intensity) form images of brightnesses given by the ratio of the areas.
   • Two telescopes forming images of the same brightness are looking at sources with ratio of intesities given by the inverse ratio of areas.
   • The Resolving Power of a telescope is a measure of the sharpness of the detail that can be seen in an image made by the telescope. Strictly speaking, it is the degree of angular localisation of the light on the sky coming from a single point source - that is, the degree of "blurring" that occurs for a source of infintesimal angular size (like a star for most telescopes).
   • The resolution of a telescope is the angular diameter of the blurred disk of a point source, and is usually measured in arc-seconds. Thus, the resolution is also a measure of how close two astronomical objects such as stars can be on the sky and be seen as distinct.
   • The reason the image is blurred by the telescope is because the waves corresponding to the photons being collected by the telescope are being cut off by the finite size of the collecting mirror or lens. We need the wave nature of light to understand this effect (known as diffraction).
   • It turns out (see next lecture), that the resolution (in radians) is just the wavelength divided by the diameter of the dish (in the same units, such as meters) - see the panels below. For resolution in arcseconds, we multiply by the usual factor of 206265 arcsec/radian.
   • Note that for optical light, with a representative wavelength of 550 nm for example, a 1-meter diameter telescope has a resolution of 0.116 arcseconds.
   • Larger telescopes have better (smaller) resolution, and can distingish things that are closer together as well as show finer detail on extended sources.
   • However, the Earth's atmosphere causes its own blurring, limiting ground based telescopes, even those on high mountain tops, to a resolution of about 0.3 to 1 arcseconds for visible wavelengths of light. Thus, from the ground, the resolution of a telescope of diameter greater than about 0.3 meters is limited by the atmosphere, not its size, at optical wavelengths.
   • To obtain better resolutions than this, we must put our telescope above the atmosphere in space or on high-flying aircraft.
   • The other power of a telescope is magnifying power, or the size of the image produced by the telescope. This depends upon the focal length of the primary or objective f_o, and the focal length of the eyepiece or any re-imaging optics f_e. The magnification is given by the ratio of the two, m = f_o/f_e - this is how many times larger an image appears than if you just looked with your eye alone. Since it can be changed simply by changing the focal length of the eyepiece, it is the least important of the powers.
5. Telescopes of the World
   • There are a good number of powerful telescopes now operating around the world. Below is a quick survey of some of the best of these. You can also explore the internet on your own by using your favorite Web searching engine using "telescope" or "observatory" as a keyword (try this).
   • Although most optical telescopes operate alone as a single (often large) telescope, some radio telescopes are linked together to form what are called interferometers, or interferometer arrays.
   • An interferometer array is like one large telescope made up of smaller telescopes - it has the resolution of a single telescope of a diameter equal to the largest distance between two telescopes in the array, but has the light gathering power of the sum of the collecting areas of the individual telescopes (usually much less than the area of a single telescope the size of the array). Thus, an interferometer can give superb resolution for even modest size telescopes.
   • It is difficult to link an interferometer together. You need to adjust things to a fraction of a wavelength. This is easiest in the radio part of the spectrum, where wavelengths range from 1 mm to 1 meter. In the optical, with wavelengths of 1000 nm or less, this is extremely difficult as you might imagine! There are several groups of astronomers around the world working on this important technological problem.
   • For more on interferometers, see the next lecture.

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