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1 Size of Astronomical istances ecture 2 Astronomical istances istance to the Moon (1 sec) istance to the Sun (8 min) istance to other stars (years) istance to centre of our Galaxy ( 30,000 yr to centre) istances to other Galaxies ( 2 million years to Andromeda) Size of the Universe (13 billion years) istance Methods Cosmic istance adder Standard ulers > Angular Size istances Hubble aw l ( for small angles << 1 radian ) Standard Candles > uminosity istances F energy/time area ight Travel Time t distance velocity 2 c l A l 4 π 2 4 π F (e.g. within solar system) t c 2 t 1/ 2 Galaxies Type Ia Supernovae HST Cepheids Hipparcos Parallax istance Size of Size of and istance to the Moon radius Two poles, height: H North-South separation: S Shadow length at noon: 2 S 1 S ( 2 1 ) H tan H unar Eclipse gives moon ~ / 3.5 Angular iameter istance 0.25 π 180 Moon Moon Moon To find, measure H and at 2 latitudes separated by S. Ancient Greeks used Athens to Alexandria, finding ~ 6300 km aser ranging 1.2 light seconds cm accuracy

2 s Orbit size from Jupiter s Moons Size of and istance to the Sun Same angular diameter as Moon. Moon closer -- by what factor? ae 1 AU 8 light minutes. 1.5 x 1013 cm moon sun moon sun Jupiter + orbiting moons Sun Moon ue to light travel time across s orbit, Jupiter s moons appear to orbit up to 8 minutes ahead or behind schedule. Sun s Size and istance from Transits of Venus Stellar Parallaxes time cos S S a sin(44 ) Venus 0.69 a 1 radian 1 AU Method by Halley 1716 Observed 1761 and 1769 (e.g. by Capt. Cook in Tahiti) or, from Kepler s law: av 3 PV 2 ae PE elative size of orbits: 1 arcsec pc α Sun Venus 1 parsec AU 1 pc x 8 lightminutes S 1 parsec 1 parallax arcsec 1 radian arcsec E Motion in the sky combines Proper motion + Parallax. ae 1 AU 3.3 light years 3.1x1018 cm E sin E sin S α S av ae av Imaging from ground: 0.02 arcsec > 50 pc Hipparcos satellite arcsec > 300 pc GAIA satellite (2012?): 10-4 arcsec > 10 kpc uminosity istances How Far are the Stars? Use the inverse-square law: Sun : energy /time 4π 2 area F mv < +1 mag Faintest (naked-eye) stars (about 6000) : uminosity istance: Apparent magnitude: mv < + 6 mag m 2.5log(F /FVega ) elative distances : 5 mag 100 x fainter 10 x farther away e.g. 5 mags 100x fainter 10x farther away 0.1 mag 10% fainter 5% farther away Absolute magnitude M apparent magnitude m at standard distance 10 pc m M + 5log( /10 pc) istance Modulus (ignoring dust extinction): 25 mag 1010 x fainter 105 x farther away 2 (1+ x) 1+ 2x istance to a sun-like mv +1 mag star: (1.05) x 105 light minutes 1.5 light years 2 since F m M 5log( /pc) 5 mv -24 mag Brightest stars (about 10) :

3 Main-Sequence Fitting Cepheid Variable Stars H ionisation instability drives pulsations. Pulsation period ~ sound travel time Period-uminosity relationship ~ P1.3 Bright Cluster 1 Calibrate using parallax, main-sequence fitting. Also from Supernova 1987A, light travel time to circumstellar ring --> MC 51 kpc +/- 6%. Cluster 2 Faint Blue (hot) Hubble used Cepheids in ocal Group < 2 Mpc. Magnitude shift gives ratio of star cluster distances: Stars with Hipparcos parallaxes calibrate the Colour-Magnitude (Hertzsprung-ussel) diagram. ed (cool) HST sees Cepheids in Virgo Cluster < 20 Mpc. m1 m2 5log(1 /2 ) Cepheid Period-uminosity istance to the Galactic Centre ( Galactic Centre ) 8.5 kpc MC: m - M 18.5 mag ust in Galactic isk log (m-m+5)/5 4.7 AV 1 mag / kpc 50 kpc yr variables in Galactic Bulge MV( yr ) ~ +0.5 mag Globular Clusters in Galactic Halo 47 Tuc Cepheids Main Sequence Μ87 istance to arge Magellenic Cloud istances to Galaxies SN 1987a explosion illuminates circumstellar gas ring. ight travel time gives linear size. Observed angular size then gives distance. Standard Candles? Cepheids ( to 20 Mpc ) Brightest stars Planetary nebulae Globular Clusters / 1 i 0 Gives (MC) 51 kpc 2 the Cepheid distances Checks Supernovae ( e.g. Type 1a Mpc ) c t0 c t1 + sini c t sini Galaxies (e.g. using uminosity-otation Velocity correlations) c ( t 2 t1 ) 2 sini Giant Elliptical in Virgo Cluster 400 light days ~104 globular clusters

4 M100 HST designed to find Cepheids in Virgo Cluster Galaxies Cepheid istance to M100 MC: 50 kpc M100 Cepheid variables m - M 18.5 mag Match periods to get same luminosity. ifference in apparent mag gives ratio of distances. 50 kpc x 10(25-13)/5 Hubble s iagram (~1924) 13 Mpc m M 5log( /pc) 5 P 50d > MV mag mv 25 mag m - M 30.5 mag log(/pc) (m-m+5)/ Mpc Velocity > Cepheid istances H070 istance > Slope H0 500 km/s/mpc (!) Cepheid distance calibration was wrong (dust in Milky Way was not yet recognised). Freedman et al. Hubble was wrong ( but his idea was good ). Why go beyond Cepheids? CMB dipole --> Milky Way velocity HST sees Cepheids to Mpc. H0 x 70 x 15 ~ 1000 km/s. not really far enough galaxy pecular velocities ~500 km/s. galaxies falling toward Virgo cluster. ΔT V V 600 km s 1 T c argely due to Milky Way (ocal Group) falling toward Virgo Cluster.

5 SN Type Ia in Virgo Galaxy NGC 4526 Supernova outshines the entire galaxy, but only for a month or so. Type II -- massive stars ( M > 8 M SUN ) explode at end of life. Type Ia -- white dwarf in a binary system accretes mass, collapses when M W 1.4 M SUN. Good standard bombs. Calibrate SN distances using HST to see Cepheids in Virgo galaxies. Calibrating Standard Bombs 1. Brighter ones decline more slowly. 2. Time runs slower by factor (1+z). AFTE correcting: Constant peak brightness M B Observed peak magnitude: m M + 5 log (d/mpc) + 25 gives the distance! Time > Absolute magnitude M B > SN Ia distances: accuracy ~0.15 mag ~8% in distance Freedman et al. ust corrections ---> improve accuracy Galaxy uminosity Calibrations 4π 2 F K V 4 V 2 K 4π F Tully - Fisher relation spirals : V rotation velocity ( HI 21 cm emission line width ) Faber - Jackson relation ellipticals : V stellar velocity dispersion ( optical absorption line widths ) etermine K using galaxies with Type Ia Supernovae. Measure flux F and velocity V to determine distance. The Fundamental Plane of Ellipticals improves the F-J relation by including a surface brightness correction. H Freedman, et al ± 3 ± 7 km s Mpc Frailty of the istance adder Parallax pc ( GAIA kpc ) Cepheids ~100 pc - 20 Mpc ( HST ) Type Ia SNe Mpc ( 8m ) z ~ 1.5 ( HST ) ittle overlap between Cepheids and SN Ia. Only 3 galaxies with both Cepheids and SN Ia

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