Angular size and resolution Astronomers usually measure sizes

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 = 2 x 105 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 * 60 sec * 60 min * 24 hrs * 365 days * 3 x 105 km/sec • so, 1 parsec (pc) is roughly 3 x 1013 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. 5 x 1013 km from Earth – Sun is 1. 5 x 108 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