Angular size and resolution

1. 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

2. 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

3. 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 Angular yardsticks

4. 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

5. 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

6. 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)

7. 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

8. Parallax as a distance measure • 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 Reference star Parallax (P) Image 1 Image 2 (6 months later)

9. 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

10. 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