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

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Measuring Angles and Angular Resolution

Angles

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 θ

“Angular Size” and “Resolution”

Astronomers usually measure sizes in terms

• f angles instead of lengths

because the distances are seldom well known

S R θ

Trigonometry

R S R

θ

Y

2 2

R Y +

S = physical length of the arc, measured in m Y = physical length of the vertical side [m]

Trigonometric Definitions

[ ] [ ]

2 2 2 2

• pposite side

tan adjacent side

• pposite 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 +

Angles: units of measure

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

angular measure of astronomical objects

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

2

Number of Degrees per Radian

2 radians per circle 360 1 radian = 57.296 2 57 17'45" π π

° ° °

≈ ≈

Trigonometry in Astronomy

S

θ

R Usually R >> S (particularly in astronomy), so Y ≈ S Y

2 2 2 2

1 1 S Y Y R R R Y R Y θ ≡ ≈ ≈ ≈ + +

[ ] [ ]

tan sin θ θ θ ≈ ≈

Relationship of Trigonometric Functions for Small Angles

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

• 1
• 0.5

0.5 1

• 0 .5
• 0.25

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

sin[θ ] ≈ tan[θ ] ≈ θ

for θ ≈ 0

Astronomical Angular “Yardsticks”

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

“Resolution” of Imaging System

Real systems cannot “resolve” objects that

are closer together than some limiting angle

“Resolution” = “Ability to Resolve”

Reason: “Heisenberg Uncertainty Relation”

Fundamental limitation due to physics

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Image of Point Source

• 1. Source emits “spherical waves”
• 2. Lens “collects” only part of the sphere

and “flips” its curvature

• 3. “piece” of sphere converges to

form image

D λ

With Smaller Lens

Lens “collects” a smaller part of sphere. Can’t locate the equivalent position (the “image”) as well Creates a “fuzzier” image

Image of Two Point Sources

Fuzzy Images “Overlap” and are difficult to distinguish (this is called “DIFFRACTION”)

Image of Two Point Sources

Apparent angular separation of the stars is ∆θ

Resolution and Lens Diameter

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

D λ θ ∆ ≈

λ = wavelength of light D = diameter of lens

Equation for Angular Resolution

Better resolution with:

larger lenses shorter wavelengths

Need HUGE “lenses” at radio wavelengths

to get same angular resolution

D λ θ ∆ ≈

λ = wavelength of light D = diameter of lens

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Resolution of Unaided Eye

Can distinguish shapes and shading of light

• f objects with angular sizes of a few

arcminutes

Rule of Thumb: angular resolution of unaided

eye is 1 arcminute

Telescopes and magnification

Telescopes magnify distant scenes Magnification = increase in angular size

(makes ∆θ appear larger)

Simple Telescopes

Simple refractor telescope (as used by Galileo,

Kepler, and their contemporaries) has two lenses

• bjective lens

collects light and forms intermediate image “positive power” Diameter D determines the resolution

eyepiece

acts as “magnifying glass” applied to image from

• bjective lens

forms magnified image that appears to be infinitely far

away

Galilean Telescope

Ray incident “above” the optical axis emerges “above” the axis image is “upright” fobjective

Galilean Telescope

Ray entering at angle θ emerges at angle θ′ > θ Larger ray angle ⇒ angular magnification θ′ θ

Keplerian Telescope

Ray incident “above” the optical axis emerges “below” the axis image is “inverted” fobjective feyelens

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Keplerian Telescope

Ray entering at angle θ emerges at angle θ′ where |θ′ | > θ Larger ray angle ⇒ angular magnification θ′ θ

Telescopes and magnification

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

that

fobjective = focal length of objective lens feyelens = focal length of eyelens

magnification is negative ⇒ image is inverted

• bjective

eyelens

f magnification f = −

Magnification: Requirements

To increase apparent angular size of Moon from

“actual” to angular size of “fist” requires magnification

• f:

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

5 10 0.5

° ° =

×

Ways to Specify Astronomical Distances

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

Aside: parallax and distance

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

Parallax as Measure of Distance

P is the “parallax” typically measured in arcseconds

Image from “A” Image from “B” 6 months later

Background star

P

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Parallax as Measure of Distance

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 × 1013 km ≈ 20 × 1012 miles = 20 trillion

miles D = P-1

D is the distance (measured in pc) and P is parallax (in arcsec)

Limitations to Magnification

Can you use a telescope (even a large one) to

increase angular size of nearest star to match that of the Sun?

nearest star is α Cen (alpha Centauri)

Brightest star in constellation Centaurus

Diameter is similar to Sun’s

α Centaurus

Near South Celestial

Pole

Not visible from

Rochester!

Southern Cross α Centaurus

Limitations to Magnification

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

100,000 = 105

⇒ To obtain that magnification using telescope:

fobjective=105 × feyelens

Can one magnify images by arbitrarily large factors? Increasing magnification involves “spreading light

• ut” over a larger imaging (detector) surface

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

Limitations to Magnification

D λ θ ∆ ≈

Magnification: Limitations

However, atmospheric effects typically

dominate effects from diffraction

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)

9 6

at 500nm (Green light) 500 10 5.6 10 5.6 rad 0.09 1.2 arcsec 1/50 of eye's limit D m radians m λ θ λ µ

− −

∆ ≈ = × = ≈ × = ≈ ≈