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Angles Angle θ is the ratio of two lengths: R: Measuring Angles and Angular Resolution physical distance between observer and objects [km] physical distance along the arc between 2 objects Lengths are measured in same “units” (e.g., kilometers) θ is “dimensionless” (no units), and measured in “radians” or “degrees” S: R S θ R Trigonometry “Angular Size” and “Resolution” R2 + Y 2 Astronomers usually measure sizes in terms of angles instead of lengths because R the distances are seldom well known S Y θ θ S R R S = physical length of the arc, measured in m Y = physical length of the vertical side [m] Trigonometric Definitions Angles: units of measure R2 + Y 2 2π (≈ 6.28) radians in a circle 1 R S radian = 360˚ ÷ 2π ≈ 57 ˚ ≈ 206,265 seconds of arc per radian ⇒ Y θ R S θ≡ R opposite side Y = tan [θ ] ≡ adjacent side R opposite side Y = = sin [θ ] ≡ hypotenuse R2 + Y 2 Angular degree (˚) is too large to be a useful angular measure of astronomical objects 1º = 60 arc minutes arc minute = 60 arc seconds [arcsec] 1º = 3600 arcsec 1 arcsec ≈ (206,265)-1 ≈ 5 × 10-6 radians = 5 µradians 1 1 1+ R2 Y2 1 Number of Degrees per Radian Trigonometry in Astronomy θ 2π radians per circle S 360° ≈ 57.296° 2π ≈ 57°17 '45" 1 radian = θ≡ S Y ≈ ≈ R R Y R +Y 2 2 1 ≈ 1+ θ ≈ tan [θ ] ≈ sin [θ ] sin[θ ] ≈ tan[θ ] ≈ θ Relationship of Trigonometric Functions for Small Angles Astronomical Angular “Yardsticks” Easy yardstick: your hand held at arms’ length 0 -0 .5 -1 -0 .5 -0 .2 5 of disk of Moon AND of Sun ≈ 0.5˚ = ½˚ ½˚ ≈ ½ · 1/60 radian ≈ 1/100 radian ≈ 30 arcmin = 1800 arcsec 0 .2 5 0 .5 Three curves nearly match for x ≤ 0.1⇒ π|x| < 0.1π ≈ 0.314 radians “Resolution” of Imaging System Real systems cannot “resolve” objects that are closer together than some limiting angle Reason: “Heisenberg Uncertainty Relation” “Resolution” diameter 0 x subtends angle of ≈ 5˚ spread between extended index finger and thumb ≈ 15˚ Easy yardstick: the Moon for θ ≈ 0 sin (π x) ta n (π x) πx 0 .5 fist R2 Y2 1 Check it! 18˚ = 18˚ × (2π radians per circle) ÷ (360˚ per circle) = 0.1π radians ≈ 0.314 radians Calculated Results tan(18˚) ≈ 0.32 sin (18˚) ≈ 0.31 0.314 ≈ 0.32 ≈ 0.31 θ ≈ tan[θ ] ≈ sin[θ ] for |θ |<0.1π Y R Usually R >> S (particularly in astronomy), so Y ≈ S = “Ability to Resolve” Fundamental limitation due to physics 2 Image of Point Source 1. Source emits “spherical waves” With Smaller Lens Lens “collects” a smaller part of sphere. Can’t locate the equivalent position (the “image”) as well Creates a “fuzzier” image 2. Lens “collects” only part of the sphere and “flips” its curvature λ D 3. “piece” of sphere converges to form image Image of Two Point Sources Image of Two Point Sources Fuzzy Images “Overlap” and are difficult to distinguish (this is called “DIFFRACTION”) Apparent angular separation of the stars is ∆θ Resolution and Lens Diameter Equation for Angular Resolution Larger lens: more of the spherical wave able to “localize” the point source makes “smaller” images smaller ∆θ between distinguished sources means BETTER resolution ∆θ ≈ collects better ∆θ ≈ λ D λ = wavelength of light D = diameter of lens λ D λ = wavelength of light D = diameter of lens Better resolution with: larger lenses wavelengths shorter Need HUGE “lenses” at radio wavelengths to get same angular resolution 3 Resolution of Unaided Eye Can distinguish shapes and shading of light of objects with angular sizes of a few arcminutes Telescopes and magnification Telescopes magnify distant scenes Magnification = increase in angular size (makes Simple Telescopes ∆θ appear larger) Rule of Thumb: angular resolution of unaided eye is 1 arcminute Galilean Telescope Simple refractor telescope (as used by Galileo, Kepler, and their contemporaries) has two lenses objective lens collects light and forms intermediate image “positive power” Diameter D determines the resolution eyepiece acts as “magnifying glass” applied to image from objective lens forms magnified image that appears to be infinitely far away Galilean Telescope fobjective Ray incident “above” the optical axis emerges “above” the axis image is “upright” Keplerian Telescope θ θ′ Ray entering at angle θ emerges at angle θ′ > θ Larger ray angle ⇒ angular magnification fobjective feyelens Ray incident “above” the optical axis emerges “below” the axis image is “inverted” 4 Telescopes and magnification Keplerian Telescope Ray trace for refractor telescope demonstrates how the increase in magnification is achieved From similar triangles in ray trace, can show that Seeing θ′ θ the Light, pp. 169-170, p. 422 magnification = − = focal length of objective lens feyelens = focal length of eyelens Larger ray angle ⇒ angular magnification To increase apparent angular size of Moon from “actual” to angular size of “fist” requires magnification of: ° 5 = 10 × 0.5 ° The only direct measure of distance astronomers have for objects beyond the solar system is parallax Parallax as Measure of Distance Triangulation 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 Caution: NOT to scale A light year = distance light travels in 1 year 1 light year = 60 sec/min × 60 min/hr × 24 hrs/day × 365.25 days/year × (3 × 105) km/sec ≈ 9.5 × 1012 km ≈ 5.9 × 1012 miles ≈ 6 trillion miles Typical Binocular Magnification 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 small telescope w/ magnification of 100-300 can distinguish large craters w/ small telescope angular sizes of a few arcseconds Aside: parallax and distance magnification is negative ⇒ image is inverted Ways to Specify Astronomical Distances Magnification: Requirements f eyelens fobjective Ray entering at angle θ emerges at angle θ′ where |θ′ | > θ f objective Background star Image from “A” P Image from “B” 6 months later Apparent Position of Foreground Star as seen from Location “B” “Background” star P is the “parallax” typically measured in arcseconds Foreground star B (6 months later) Earth’s Orbit Apparent Position of Foreground Star as seen from Location “A” 5 Parallax as Measure of Distance Limitations to Magnification Apparent motion of 1 arcsec in 6 months defines the distance of 1 parsec (parallax of 1 second) 1 parsec = 3.26 light years ≈ 3 × miles D 1013 km ≈ 20 × 1012 nearest miles = 20 trillion star is α Cen (alpha Centauri) Brightest star in constellation Centaurus Diameter is similar to Sun’s = P-1 D is the distance (measured in pc) and P is parallax (in arcsec) Limitations to Magnification α Centaurus Can you use a telescope (even a large one) to increase angular size of nearest star to match that of the Sun? Distance to α Cen is 1.3 pc 1.3 Near South Celestial Pole Not visible from Rochester! Southern Cross pc ≈ 4.3 light years ≈ 1.5×1013 km from Earth Sun is 1.5 × 108 km from Earth ⇒ would require angular magnification of 100,000 = 105 ⇒ To obtain that magnification using telescope: fobjective=105 × feyelens α Centaurus Limitations to Magnification 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, larger telescopes Wave nature of light, Heisenberg “uncertainty principle” Diffraction is the unavoidable propensity of light to change direction of propagation, i.e., to “bend” Cannot focus light from a point source to an arbitrarily small “spot” Diffraction Limit of telescope However, atmospheric effects typically dominate effects from diffraction most telescopes are limited by “seeing”: image “smearing” due to atmospheric turbulence Rule of Thumb: BUT: Remember diffraction Magnification: Limitations ∆θ ≈ λ D limiting resolution for visible light through the atmosphere is equivalent to that obtained by a telescope with D ≈ 3.5" (≈ 90 mm) ∆θ ≈ λ at λ = 500nm (Green light) D −9 500 × 10 m = ≈ 5.6 ×10−6 radians = 5.6 µ rad 0.09 m ≈ 1.2 arcsec ≈ 1/ 50 of eye's limit 6