Angular size and resolution

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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 Easy yardstick: the Moon
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 In the DRAWING: Point A: The sky appears blue due to scattering. The scattered light from the other ray is linearly polarized. 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.

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 , 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 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 Parallax (P) Image 1 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