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Home , Angular resolution, Radian

• θ is the ratio of two lengths: “Angular Size” and “Resolution” – R: physical distance between observer and objects [km] • Astronomers usually measure sizes in terms of – S: physical distance along the arc between 2 objects angles instead of lengths – Lengths are measured in same “units” (e.g., kilometers) – because the distances are seldom well known – θ is “dimensionless” (no units), and measured in “” or “degrees”

θ S R S R

θ

R

Trigonometry Definitions

22 22 R +Y RY+ R S Y R θ S Y R S θ θ ≡ R opposite side Y R tan[]θ ≡= adjacent side R opposite sideY 1 S = physical length of the arc, measured in m sin[]θ ≡== Y = physical length of the vertical side [m] hypotenuse RY22+ R 2 1+ Y 2

Angles: units of measure Number of Degrees per •2π (≈ 6.28) radians in a –1 radian =360º÷ 2π≈57º – ⇒≈206,265 of arc per radian 2π radians per circle ° 360 ° • Angular (º) is too large to be a useful angular 1 radian = ≈ 57.296 measure of astronomical objects 2π –1º= 60 arc ° – 1 arc = 60 arc seconds [arcsec] ≈ 57 17'45" – 1º = 3600 arcsec – 1 arcsec ≈ (206,265)-1 ≈ 5 × 10-6 radians = 5 µradians

1 sin[θ] ≈ tan[θ] ≈ θ for θ ≈ 0 in Astronomy 1

θ S Y sin(πx) tan(πx) R 0.5 πx Usually R >> S, so Y ≈ S

SY Y 1 0 θ ≡≈≈ ≈ RR R22+YR 2 1+ -0.5 Y 2

-1 θ ≈≈tan[θθ] sin[ ] -0.5 -0.25 0 0.25 0.5 x Three curves nearly match for x ≤ 0.1⇒ π|x|< 0.1π ≈ 0.314 radians

Relationship of Trigonometric Astronomical Angular “Yardsticks” Functions for Small Angles Check it! • Easy yardstick: your hand held at arms’ length 18° = 18° × (2π radians per circle) ÷ (360° per circle) – fist subtends angle of ≈ 5° = 0.1π radians ≈ 0.314 radians – spread between extended index finger and thumb ≈ 15° Calculated Results tan(18°) ≈ 0.32 • Easy yardstick: the Moon sin (18°) ≈ 0.31 – diameter of disk of Moon AND of Sun ≈ 0.5° = ½° 0.314 ≈ 0.32 ≈ 0.31 ½° ≈ ½ · 1/60 radian ≈ 1/100 radian ≈ 30 arcmin = 1800 arcsec

θ ≈ tan[θ] ≈ sin[θ] for |θ |<0.1π

“Resolution” of Imaging System Image of

1. Source emits “spherical ” 2. “collects” only part of the sphere • Real systems cannot “resolve” objects that and “flips” its curvature are closer together than some limiting angle – “Resolution” = “Ability to Resolve” λ D • Reason: “Heisenberg Uncertainty Relation” – Fundamental limitation due to

3. “piece” of sphere converges to form image

2 With Smaller Lens Image of Two Point Sources

Lens “collects” a smaller part of sphere. Fuzzy Images “Overlap” Can’t locate the equivalent (the “image”) as well and are difficult to distinguish Creates a “fuzzier” image (this is called “”)

Image of Two Point Sources Resolution and Lens Diameter

• Larger lens: – collects more of the spherical Apparent angular separation of the stars is ∆θ – better able to “localize” the point source – makes “smaller” images – smaller ∆θ between distinguished sources means BETTER resolution λ ∆θ ≈ λ = of D D = diameter of lens

Equation for Resolution of Unaided

λ • Can distinguish shapes and shading of light of ∆≈θ λ = wavelength of light objects with angular sizes of a few arcminutes D D = diameter of lens • Better resolution with: • Rule of Thumb: angular resolution of unaided eye is 1 arcminute – larger – shorter • Need HUGE “lenses” at radio wavelengths to get the same resolution

3 and Simple Telescopes • Telescopes magnify distant scenes • Simple refractor (as used by Galileo, Kepler, • Magnification = increase in angular size and their contemporaries) has two lenses – lens –(makes ∆θ appear larger) • collects light and forms intermediate image • “positive ” • Diameter D determines the resolution – eyepiece • acts as “magnifying glass” • forms magnified image that appears to be infinitely far away

Galilean Telescope Galilean Telescope

θ θ′

fobjective Ray entering at angle θ emerges at angle θ′ > θ Ray incident “above” the optical axis emerges “above” the axis Larger ray angle ⇒ angular magnification image is “upright”

Keplerian Telescope Keplerian Telescope

θ θ′

Ray entering at angle θ emerges at angle θ′ fobjective feyelens where |θ′ |> θ Ray incident “above” the optical axis emerges “below” the axis Larger ray angle ⇒ angular magnification image is “inverted”

4 Telescopes and magnification Magnification: Requirements • Ray trace for refractor telescope demonstrates how • To increase apparent angular size of Moon from “actual” to the increase in magnification is achieved angular size of “fist” requires magnification of: – Seeing the Light, pp. 169-170, p. 422 5° =×10 • From similar triangles in ray trace, can show that 0.5° f • Typical Binocular Magnification magnification =− objective f – with , can easily see shapes/shading on eyelens Moon’s surface (angular sizes of 10's of arcseconds) – f = of objective lens objective • To see further detail you can use small telescope w/ – f = focal length of eyelens eyelens magnification of 100-300 – can distinguish large craters w/ small telescope • magnification is negative ⇒ image is inverted – angular sizes of a few arcseconds

Ways to Specify Astronomical Aside: and distance • Only direct measure of distance astronomers have for Distances objects beyond solar system is parallax – Parallax: apparent motion of nearby stars against background of • (AU) very distant stars as Earth orbits the Sun – distance from Earth to Sun – Requires images of the same star at two different times of year separated by 6 months 8 – 1 AU ≈ 93,000,000 miles ≈ 1.5 × 10 km Caution: NOT to scale Apparent Position of Foreground A Star as seen from Location “B”

• light year = distance light travels in 1 year “Background” star

1 light year Foreground star = 60 sec/min × 60 min/hr × 24 hrs/ × 365.25 days/year × (3 × 105) km/sec ≈ 9.5 × 1012 km ≈ 5.9 × 1012 miles ≈ 6 trillion miles B (6 months later) Apparent Position of Foreground Earth’s Orbit Star as seen from Location “A”

Definition of Astronomical Parallax as Measure of Distance Parallax • “half-angle” of triangle to foreground star is 1" – Recall that 1 radian = 206,265" Background star P – 1" = (206,265)-1 radians ≈ 5×10-6 radians = 5 µradians • R = 206,265 AU ≈ 2×105 AU ≈ 3×1013 km –1 ≈ 3×1013 km ≈ 20 trillion miles ≈ 3.26 light years Image from “A” Image from “B” 6 months later

1 AU • P is the “parallax” 1" • typically measured in arcseconds R • Gives measure of distance from Earth to nearby star Foreground star (distant stars assumed to be an “infinite” distance away)

5 Parallax as Measure of Distance Limitations to Magnification

• R = P-1 – R is the distance (measured in pc) and P is parallax (in arcsec) • Can you use a telescope to increase angular size of nearest star to match that of the Sun? – Star with parallax (half angle!) of ½" is at distance of 2 pc ≈ 6.5 light years – nearest star is α Cen (alpha Centauri) – Star with parallax of 0.1" is at distance of 10 pc ≈ 32 light years – Diameter is similar to Sun’s – Distance is 1.3 pc • SMALLER PARALLAX MEANS FURTHER AWAY •1.3 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 5 – ⇒ fobjective=10 × feyelens

Limitations to Magnification Magnification: limitations • BUT: atmospheric effects typically dominate diffraction • BUT: you can’t magnify images by arbitrarily large factors! effects • Remember diffraction! – most telescopes are limited by “seeing”: image “smearing” due to atmospheric turbulence – Diffraction is the unavoidable propensity of light to change direction of propagation, i.e., to “bend” • Rule of Thumb: – Cannot light from a point source to an arbitrarily small “spot” – limiting resolution for visible light through atmosphere is equivalent to that obtained by a telescope with D˚≈˚3.5" (≈ 90 mm) • Increasing magnification involves “spreading light out” over a larger imaging (detector) surface λ • Diffraction Limit of a telescope ∆≈θλ at = 500nm (Green light) D 500× 10−9 m λ =≈×=5.6 10−6 radians 5.6µ ∆≈θ 0.09m D ≈≈1.2 arcsec 1/ 50 of eye's limit

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