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Angular size and resolution
• Astronomers usually measure sizes in terms of
angles (not lengths)
– This is because distances are seldom well known
• For small angles “theta”:
– tan(theta) = sin(theta) = theta
– theta = S/D where S is the distance between 2 objects
and D is the distance from observer to the objects
S
theta
D
Angles: units of measure
• theta = S/D will yield angle in radians
– there are 2*pi (or roughly 2*3.1416) radians in a circle
– so 1 radian = 57 degrees
• degrees are often too big a unit to be useful
– 1 degree = 60 arc minutes; 1 arc minute = 60 arc seconds
– 1 degree = 3600 arcsec
– 1 radian = 2x105 arcsec
Angular yardsticks
• Easy yardstick: your fist
– fist held at arms’ length subtends angle of about 5 degrees
• Easy yardstick: the Moon
– Moon’s disk: 1/2 degree in diameter (same for Sun)
– Moon’s disk is about 1/100 of a radian
– Moon’s disk is 30 arcmin or 1800 arcsec
Telescopes and magnification
• Telescopes serve to magnify distant scenes
• Magnification = increase in angular size
• Simple refractor telescope (such as was used by Galileo
and Kepler and contemporaries) involves use of 2 lenses
– objective lens: performs light collecting and forms
intermediate image
– eyepiece: acts as magnifying glass to form magnified
image that appears to be infinitely far away
Telescopes and magnification
• Ray trace for refractor telescope demonstrates how
the increase in magnification is achieved
– Seeing the Light, pp 169-170, 422
• From similar triangles in ray trace, can show that
magnification = -f(obj)/f(ep)
– f(obj) = focal length of objective lens
– f(ep) = focal length of eyepiece
– note that magnification is negative: image is inverted
Magnification: requirements
• Unaided eye can distinguish shapes/shading on Moon’s
surface (angular sizes of a few arc minutes)
• To increase Moon from “actual size” to “fist size” requires
magnification of 10 (typical of binoculars)
– 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 a small telescope w/
magnification of 100-300
– w/ small telescope can distinguish large craters (angular
sizes of a few arc seconds)
Aside: parallax and distance
• The only direct measure of distance astronomers have for
objects beyond the solar system is parallax
– 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
Background star
Foreground star
CAUTION: NOT TO SCALE
Parallax as a distance measure
Reference star
Image 1
Parallax (P)
Image 2 (6 months later)
• Apparent motion of 1 arcsec is defined as a
distance of 1 parsec (parallax second)
– 1 parsec (pc) = 3.26 light years
– 1 light year = distance light travels in 1 year
• 1 parsec = 3.26 * 60sec * 60min * 24hrs * 365days * 3x105 km/sec
• so, 1 parsec (pc) is roughly 3x1013 km (about 20 trillion miles)
• D = 1/P
where D is distance in pc, P is parallax in arcsec
Magnification: limitations
• Can you use a small telescope (or a large one for
that matter) to increase the angular size of the
nearest star to the angular size of the Sun?
– nearest star, alpha Cen, has physical diameter similar to
Sun but a distance of 1.3 pc (4.3 light years), or about
1.5x1013 km from Earth
– Sun is 1.5x108 km from Earth
– => required magnification is 100,000
Magnification: limitations
• 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
• Before this become problematic, most telescope hit their
diffraction limit
– limiting angle roughly equal to lambda/D radians, where lambda is
wavelength and D is telescope diameter
• Typically, before diffraction becomes a problem, the
atmosphere becomes a nuisance
– most telescopes limited by “seeing”: image smearing due to
atmospheric turbulence