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Angular size and resolution Astronomers usually measure sizes in terms of angles (not lengths) –This is because distances are seldom well known For small angles “theta”: –tan(theta) = sin(theta) = theta –theta = S/D where S is the distance between 2 objects and D is the distance from observer to the objects theta S D

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Angles: units of measure theta = S/D will yield angle in radians –there are 2*pi (or roughly 2*3.1416) radians in a circle –so 1 radian = 57 degrees degrees are often too big a unit to be useful –1 degree = 60 arc minutes; 1 arc minute = 60 arc seconds –1 degree = 3600 arcsec –1 radian = 2x10 5 arcsec

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Angular yardsticks Easy yardstick: your fist –fist held at arms’ length subtends angle of about 5 degrees Easy yardstick: the Moon –Moon’s disk: 1/2 degree in diameter (same for Sun) –Moon’s disk is about 1/100 of a radian –Moon’s disk is 30 arcmin or 1800 arcsec

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Telescopes and magnification Telescopes serve to magnify distant scenes Magnification = increase in angular size Simple refractor telescope (such as was used by Galileo and Kepler and contemporaries) involves use of 2 lenses –objective lens: performs light collecting and forms intermediate image –eyepiece: acts as magnifying glass to form magnified image that appears to be infinitely far away

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Telescopes and magnification Ray trace for refractor telescope demonstrates how the increase in magnification is achieved –Seeing the Light, pp , 422 From similar triangles in ray trace, can show that magnification = -f(obj)/f(ep) –f(obj) = focal length of objective lens –f(ep) = focal length of eyepiece –note that magnification is negative: image is inverted

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Magnification: requirements Unaided eye can distinguish shapes/shading on Moon’s surface (angular sizes of a few arc minutes) To increase Moon from “actual size” to “fist size” requires magnification of 10 (typical of binoculars) –with binoculars, can easily see shapes/shading on Moon’s surface (angular sizes of 10’s of arcseconds) To see further detail you can use a small telescope w/ magnification of –w/ small telescope can distinguish large craters (angular sizes of a few arc seconds)

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Aside: parallax and distance The only direct measure of distance astronomers have for objects beyond the solar system is parallax –Parallax: apparent motion of nearby stars (against a background of very distant stars) as Earth orbits the Sun –Requires taking images of the same star at two different times of the year Foreground star Background star CAUTION: NOT TO SCALE

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Parallax as a distance measure Apparent motion of 1 arcsec is defined as a distance of 1 parsec (parallax second) –1 parsec (pc) = 3.26 light years –1 light year = distance light travels in 1 year 1 parsec = 3.26 * 60sec * 60min * 24hrs * 365days * 3x10 5 km/sec so, 1 parsec (pc) is roughly 3x10 13 km (about 20 trillion miles) D = 1/P where D is distance in pc, P is parallax in arcsec Image 1 Image 2 (6 months later) Reference star Parallax (P)

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Magnification: limitations Can you use a small telescope (or a large one for that matter) to increase the angular size of the nearest star to the angular size of the Sun? –nearest star, alpha Cen, has physical diameter similar to Sun but a distance of 1.3 pc (4.3 light years), or about 1.5x10 13 km from Earth –Sun is 1.5x10 8 km from Earth –=> required magnification is 100,000

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Magnification: limitations Can one magnify images by arbitrarily large factors? Increasing magnification involves “spreading light out” over a larger imaging (detector) surface –necessitates ever-larger light-gathering power Before this become problematic, most telescope hit their diffraction limit –limiting angle roughly equal to lambda/D radians, where lambda is wavelength and D is telescope diameter Typically, before diffraction becomes a problem, the atmosphere becomes a nuisance –most telescopes limited by “seeing”: image smearing due to atmospheric turbulence

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