Measuring Angles and Angular Resolution 1 Angles Angle is the - - PDF document
Measuring Angles and Angular Resolution 1 Angles Angle is the ratio of two lengths: R: physical distance between observer and objects [km] S: physical distance along the arc between 2 objects Lengths are measured in same
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Angle θ is the ratio of two lengths:
R: physical distance between observer and objects [km] S: 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” R S R
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Astronomers usually measure sizes in terms
because the distances are seldom well known
S R θ
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R S R
Y
2 2
S = physical length of the arc, measured in m Y = physical length of the vertical side [m]
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2 2 2 2
tan adjacent side
1 sin hypotenuse 1 S R Y R Y R Y R Y θ θ θ ≡ ≡ = ≡ = = + +
R S R θ Y
2 2
R Y +
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2π (≈ 6.28) radians in a circle
1 radian = 360˚ ÷ 2π ≈ 57 ˚ ⇒ ≈ 206,265 seconds of arc per radian
Angular degree (˚) is too large to be a useful
1º = 60 arc minutes 1 arc minute = 60 arc seconds [arcsec] 1º = 3600 arcsec 1 arcsec ≈ (206,265)-1 ≈ 5 × 10-6 radians = 5 µradians
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° ° °
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S
R Usually R >> S (particularly in astronomy), so Y ≈ S Y
2 2 2 2
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0.5 1
0.25 0.5
sin(πx) tan(πx) πx x
Three curves nearly match for x ≤ 0.1⇒ π|x| < 0.1π ≈ 0.314 radians
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Easy yardstick: your hand held at arms’ length
fist subtends angle of ≈ 5˚ spread between extended index finger and thumb ≈ 15˚
Easy yardstick: the Moon
diameter of disk of Moon AND of Sun ≈ 0.5˚ = ½˚
½˚ ≈ ½ · 1/60 radian ≈ 1/100 radian ≈ 30 arcmin = 1800 arcsec In the DRAWING: Point A: The sky appears blue due to scattering. The scattered light from the
Point B: When this person looks toward the sun the sky appears reddish because the most of the shorter wavelength light has already been scattered away.
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Real systems cannot “resolve” objects that
“Resolution” = “Ability to Resolve”
Reason: “Heisenberg Uncertainty Relation”
Fundamental limitation due to physics
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and “flips” its curvature
form image
D λ
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Lens “collects” a smaller part of sphere. Can’t locate the equivalent position (the “image”) as well Creates a “fuzzier” image
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Fuzzy Images “Overlap” and are difficult to distinguish (this is called “DIFFRACTION”)
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Apparent angular separation of the stars is ∆θ
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Larger lens:
collects more of the spherical wave better able to “localize” the point source makes “smaller” images smaller ∆θ between distinguished sources means
BETTER resolution
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Better resolution with:
larger lenses shorter wavelengths
Need HUGE “lenses” at radio wavelengths
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Can distinguish shapes and shading of light
Rule of Thumb: angular resolution of unaided
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Telescopes magnify distant scenes Magnification = increase in angular size
(makes ∆θ appear larger)
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Simple refractor telescope (as used by Galileo,
collects light and forms intermediate image “positive power” Diameter D determines the resolution
eyepiece
acts as “magnifying glass” applied to image from
forms magnified image that appears to be infinitely far
away
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Ray incident “above” the optical axis emerges “above” the axis image is “upright” fobjective
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Ray entering at angle θ emerges at angle θ′ > θ Larger ray angle ⇒ angular magnification θ′ θ
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Ray incident “above” the optical axis emerges “below” the axis image is “inverted” fobjective feyelens
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Ray entering at angle θ emerges at angle θ′ where |θ′ | > θ Larger ray angle ⇒ angular magnification θ′ θ
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Ray trace for refractor telescope demonstrates
Seeing the Light, pp. 169-170, p. 422
From similar triangles in ray trace, can show
fobjective = focal length of objective lens feyelens = focal length of eyelens
magnification is negative ⇒ image is inverted
eyelens
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To increase apparent angular size of Moon from
“actual” to angular size of “fist” requires magnification
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
° ° =
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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
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The only direct measure of distance astronomers
have for objects beyond the solar system is parallax
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
Foreground star “Background” star
Caution: NOT to scale A B (6 months later)
Apparent Position of Foreground Star as seen from Location “B” Apparent Position of Foreground Star as seen from Location “A”
Earth’s Orbit
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P is the “parallax” typically measured in arcseconds
Image from “A” Image from “B” 6 months later
Background star
P
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Apparent motion of 1 arcsec in 6 months defines the
1 parsec = 3.26 light years ≈ 3 × 1013 km ≈ 20 × 1012 miles = 20 trillion
miles
D = P-1
D is the distance (measured in pc) and P is parallax (in arcsec)
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Can you use a telescope (even a large one) to
nearest star is α Cen (alpha Centauri)
Brightest star in constellation Centaurus
Diameter is similar to Sun’s
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Near South Celestial
Not visible from
Rochester!
Southern Cross α Centaurus
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Distance to α Cen is 1.3 pc
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
⇒ To obtain that magnification using telescope:
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Can one magnify images by arbitrarily large factors? Increasing magnification involves “spreading light
necessitates ever-larger light-gathering power, larger
telescopes
BUT: Remember diffraction
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
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However, atmospheric effects typically
most telescopes are limited by “seeing”: image
“smearing” due to atmospheric turbulence
Rule of Thumb:
limiting resolution for visible light through the
atmosphere is equivalent to that obtained by a telescope with D ≈ 3.5" (≈ 90 mm)
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