Lecture 1 PHYS Lecture 1: Angles, Parallax and Astronomical Distance Units Objectives: Understand basic astronomical terminology to describe angles Learn how to estimate angles Describe parallax and thereby define the Astronomical Unit (AU) and parsec (pc) Additional Reading: Kaufmann – chap 1 Zeilik – chap 3-3

Lecture 1 PHYS Basics: Early Astronomy all about measuring angles 360˚ = 2π rad = 1 full circle Astronomers also use arcminutes and arcseconds, where: –1 degree = 60 arcmins = 3600 arcsecs –1˚ = 60’ =3600” Also useful to know that: –1 rad = 360˚/2π = 57.3˚ = 206,265” Angular diameter of the Sun and Moon are ≈30’ This system inherited (as was time) from the ancient Babylonians who counted in a base-60 (sexagesimal) system Try estimating the angle between two points on the blackboard – it will be different for different rows! Now try it with this rule-of-thumb as a guide: – hold your hand outstretched at arm’s length – tip of your little finger to that of your thumb subtends ≈ 20˚ – similarly, the tip alone of your index finger subtends ≈ 1˚ (i.e. about twice diameter of Sun or Moon!)

Lecture 1 PHYS Small angle approximations: Astronomical angles often small  can avoid trigonometry! formally, tan(α/2)=(l/2)/d but, for θ « 1, tan θ ≈ θ and hence –α = l/d N.B. VERY IMPORTANT: tan θ ≈ θ requires θ in RADIANS! thus α = l/d gives an angle in radians! e.g. calculate angle subtended by Sun (diam. 698,000 km) at distance of 150 million km. –Exact: α = 2 tan -1 (6.98x10 5 /2x1.5x10 8 ) = radians –Approx: α = 6.98x10 5 /1.5x10 8 = radians

Lecture 1 PHYS Trigonometric Parallax: “Parallax”: apparent relative motion of objects at different distances due to motion of observer –E.g. on a train, nearby objects seem to move faster than more distant ones – sense angular speed instead of true speed Parallax can be used to measure distance! Trigonometric Parallax is THE fundamental step to deriving all astronomical distances. Basic geometry uses Earth’s orbit as a baseline. Star appears to “wobble” back and forth by angle p

Lecture 1 PHYS Can simulate this effect with “star-field” on black-board! –Hold out finger (=“nearby star”) at arm’s length –Close one eye and move head from side to side –Now try with finger closer –Aim is to judge motion relative to background star-field Small angle approx  p = r e / d Therefore distance d of star is simply d = r e / p (with p in radians) Normally p is measured in arcsecs, d = r e / p(arcsecs) Setting r e = x m  d = x m / p Important Definitions: –Mean Earth-Sun distance r e is known as the Astronomical Unit (AU) –p is called the Trigonometric Parallax, or usually just the Parallax –Distance at which a star has p = 1 arcsec is called a parsec Therefore 1 parsec (=1pc) = AU = x m And hence d (parsec) = 1 / p (arcsec)

Lecture 1 PHYS Actual parallaxes and distances: Determining stellar distances was the “holy grail” of 17 th and 18 th century astronomers! But first parallax (p = 0.31”) not measured until 1838 (Bessel) What is largest parallax of any star? –Answer: p = 0.8” (i.e. d = 1.2pc) of Proxima Centauri, the nearest star N.B. no star has p > 1”! –How does this compare with typical ground-based angular resolution? e.g. brightest star in the sky, Sirius, has p = 0.379”. How far away is it? –Answer: from d = 1 / p, Sirius is 1 / = 2.64 pc from Earth e.g. largest planet in Solar System, Jupiter, is 5.2 AU from the Sun. If we could resolve angles as small as 0.1”, up to what maximum distance could we see similar planets around nearby stars? –At d pc, 5.2 AU will subtend 5.2 / d”. Thus we require 5.2 /d > 0.1, and so d < 52 pc. Can we in fact do this, or are there other problems? –Brightness contrast prevents it as the Sun is so much brighter than Jupiter

Lecture 1 PHYS Advanced Topic: In fact, regular stellar movements of amplitude ~20” were found by James Bradley in But their maxima were displaced by 3 months compared to what was expected for parallax! What was this effect? –Answer: STELLAR ABERRATION – due to Earth’s motion around the Sun and the finite velocity of light! –θ ≈ tan θ = v / c ≈ 30 / 300,000 = rads = 20”

Lecture 1 PHYS ESA’s HIPPARCOS mission: Motions of stars in the Hyades: