Measuring Angles S: physical distance along the arc between 2 - PDF document

Angles � Angle θ is the ratio of two lengths: � R: physical distance between observer and objects [km] Measuring Angles � S: physical distance along the arc between 2 objects � Lengths are measured in same “units” (e.g., kilometers) and Angular � θ is “dimensionless” (no units), and measured in “radians” or “degrees” Resolution R S θ R Trigonometry “Angular Size” and “Resolution” + 2 2 R Y � Astronomers usually measure sizes in terms of angles instead of lengths R Y � because the distances are seldom well known S 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 � 2 π ( ≈ 6.28) radians in a circle + 2 2 R Y � 1 radian = 360 ˚ ÷ 2 π ≈ 57 ˚ R � ⇒ ≈ 206,265 seconds of arc per radian Y S θ � Angular degree ( ˚ ) is too large to be a useful R angular measure of astronomical objects S θ ≡ � 1º = 60 arc minutes R � 1 arc minute = 60 arc seconds [arcsec] [ ] opposite side Y θ ≡ = tan � 1º = 3600 arcsec adjacent side R � 1 arcsec ≈ (206,265) -1 ≈ 5 × 10 -6 radians = 5 µ radians opposite side Y 1 [ ] θ ≡ = = sin hypotenuse + 2 2 2 R Y R + 1 2 Y 1

Number of Degrees per Radian Trigonometry in Astronomy θ S Y π 2 radians per circle R ° Usually R >> S (particularly in astronomy), so Y ≈ S 360 ≈ ° 1 radian = 57.296 π 2 S Y Y 1 θ ≡ ≈ ≈ ≈ ≈ ° R R + 57 17'45" 2 2 2 R Y R + 1 2 Y [ ] [ ] θ ≈ θ ≈ θ tan sin sin [ θ ] ≈ tan [ θ ] ≈ θ θ ≈ 0 for Relationship of Trigonometric Functions for Small Angles 1 sin( π x) Check it! tan( π x) π x 18 ˚ = 18 ˚ × (2 π radians per circle) ÷ (360 ˚ per 0.5 circle) = 0.1 π radians ≈ 0.314 radians 0 Calculated Results -0.5 tan(18 ˚ ) ≈ 0.32 sin (18 ˚ ) ≈ 0.31 -1 0.314 ≈ 0.32 ≈ 0.31 -0 .5 -0.25 0 0.25 0 .5 x θ ≈ tan[ θ ] ≈ sin[ θ ] for | θ |< 0.1 π Three curves nearly match for x ≤ 0.1 ⇒ π | x | < 0.1 π ≈ 0.314 radians Astronomical Angular “Resolution” of Imaging System “Yardsticks” � Real systems cannot “resolve” objects that � Easy yardstick: your hand held at arms’ length are closer together than some limiting � fist subtends angle of ≈ 5 ˚ � spread between extended index finger and thumb ≈ 15˚ angle � “Resolution” = “Ability to Resolve” � Easy yardstick: the Moon � Reason: “Heisenberg Uncertainty Relation” � diameter of disk of Moon AND of Sun ≈ 0.5 ˚ = ½ ˚ � Fundamental limitation due to physics ½ ˚ ≈ ½ · 1/60 radian ≈ 1/100 radian ≈ 30 arcmin = 1800 arcsec 2

Image of Point Source With Smaller Lens Lens “collects” a smaller part of sphere. 1. Source emits “spherical waves” 2. Lens “collects” only part of the sphere Can’t locate the equivalent position (the “image”) as well and “flips” its curvature Creates a “fuzzier” image 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: λ λ = wavelength of light ∆ θ ≈ � collects more of the spherical wave D = diameter of lens � better able to “localize” the point source D � makes “smaller” images � smaller ∆θ between distinguished sources means � Better resolution with: BETTER resolution � larger lenses λ ∆ θ ≈ λ = wavelength of light � shorter wavelengths D D = diameter of lens � Need HUGE “lenses” at radio wavelengths to get same angular resolution 3

Resolution of Unaided Eye Telescopes and magnification � Telescopes magnify distant scenes � Can distinguish shapes and shading of light of objects with angular sizes of a few arcminutes � Magnification = increase in angular size � (makes ∆θ appear larger) � Rule of Thumb: angular resolution of unaided eye is 1 arcminute Simple Telescopes 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 f objective � acts as “magnifying glass” applied to image from objective lens Ray incident “above” the optical axis � forms magnified image that appears to be infinitely far emerges “above” the axis away image is “upright” Galilean Telescope Keplerian Telescope θ θ′ f objective f eyelens Ray entering at angle θ emerges at angle θ′ > θ Ray incident “above” the optical axis Larger ray angle ⇒ angular magnification 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 � Seeing the Light, pp. 169-170, p. 422 � From similar triangles in ray trace, can show θ′ θ that f = − objective magnification f eyelens � f objective = focal length of objective lens Ray entering at angle θ emerges at angle θ′ � f eyelens = focal length of eyelens where | θ′ | > θ � magnification is negative ⇒ image is inverted Larger ray angle ⇒ angular magnification Ways to Specify Astronomical Magnification: Requirements Distances � To increase apparent angular size of Moon from “actual” to angular size of “fist” requires magnification � light year = distance light travels in 1 year of: ° 5 ° = × 10 0.5 1 light year = 60 sec/min × 60 min/hr × 24 hrs/day × 365.25 days/year × (3 × 10 5 ) km/sec � Typical Binocular Magnification ≈ 9.5 × 10 12 km ≈ 5.9 × 10 12 miles ≈ 6 trillion miles � 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 Parallax as Measure of Distance � The only direct measure of distance astronomers have for objects beyond the solar system is parallax � Triangulation P � Parallax: apparent motion of nearby stars (against a Background star background of very distant stars) as Earth orbits the Sun � Requires taking images of the same star at two different times of the year Image from “A” Image from “B” 6 months later Caution: NOT to scale Apparent Position of Foreground Star as seen from Location “B” A � P is the “parallax” � typically measured in arcseconds “Background” star Foreground star B (6 months later) Apparent Position of Foreground Earth’s Orbit Star as seen from Location “A” 5

Limitations to Magnification Parallax as Measure of Distance � Can you use a telescope (even a large one) to increase angular size of nearest star to match that of the Sun? � Apparent motion of 1 arcsec in 6 months defines the � nearest star is α Cen (alpha Centauri) distance of 1 parsec ( parallax of 1 second ) � Brightest star in constellation Centaurus � 1 parsec = 3.26 light years ≈ 3 × 10 13 km ≈ 20 × 10 12 miles = 20 trillion � Diameter is similar to Sun’s miles � D = P -1 � D is the distance (measured in pc) and P is parallax (in arcsec) Limitations to Magnification α Centaurus � Distance to α Cen is 1.3 pc � 1.3 pc ≈ 4.3 light years ≈ 1.5 × 10 13 km from Earth � Near South Celestial � Sun is 1.5 × 10 8 km from Earth Pole � ⇒ would require angular magnification of � Not visible from 100,000 = 10 5 Southern Cross Rochester! � ⇒ To obtain that magnification using telescope: f objective = 10 5 × f eyelens α Centaurus Magnification: Limitations Limitations to Magnification � However, atmospheric effects typically � Can one magnify images by arbitrarily large factors? dominate effects from diffraction � Increasing magnification involves “spreading light � most telescopes are limited by “seeing”: image out” over a larger imaging (detector) surface “smearing” due to atmospheric turbulence � necessitates ever-larger light-gathering power, larger � Rule of Thumb: telescopes � limiting resolution for visible light through the � BUT: Remember diffraction atmosphere is equivalent to that obtained by a � Wave nature of light, Heisenberg “uncertainty principle” telescope with D ≈ 3.5" ( ≈ 90 mm) � Diffraction is the unavoidable propensity of light to change λ direction of propagation, i.e., to “bend” ∆ θ ≈ λ = at 500nm (Green light) � Cannot focus light from a point source to an arbitrarily small D “spot” × − 9 λ 500 10 m − � Diffraction Limit of telescope = ≈ × 6 = µ 5.6 10 radians 5.6 rad ∆ θ ≈ 0.09 m ≈ ≈ D 1.2 arcsec 1/50 of eye's limit 6