Download 06-angles and resolution

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
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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
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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”
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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”
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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
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