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Stellar Radii - parts of Chapters 7.3, 6.1 The Sun - resolved as a disk of diameter 0.53 Calculation of the Sun’s radius Earth d Sun D (rad) = d/D = d/1AU = 0.53(2/360) = 0.0093, hence d = 0.0093AU = 0.0093 x 1.496 x 108 km = 1.392 x 106 km or Rsun = 6.96 x 105 km Can we apply same principles to the stars? e.g. Cen (like Sun) D = 1.3 pc = 2.7 x 105 AU; if Rcen = Rsun = 2Rsun/D = 0.0093 AU/2.7 x 105 AU = 3.3 x 10-8 rad = 0.007 arcsec (angular diameter of a dime 150 km away!) Can we resolve this small angle with a telescope? Stellar radii First Minimum Airy disk - diffraction pattern produced by a circular aperture (telescope mirror). First minimum usually taken as resolution limit of telescope Angular distance from centre to first ring = 1.22 /D is wavelength of observation, D diameter of telescope, in radians (see figs 6.8 - 6.10) Usually say 2 objects resolved if angular separation > /D Resolved Unresolved min Stellar Radii Example: resolution of human eye Pupil aperture = 1 cm; visible light = 500 nm = 5 x 10-5 cm (same units) diffraction = /d = 5 x 10-5 cm / 1 cm = 5 x 10-5 rad = 0.003 = 10 arcsec If aperture > 10 cm, then diffraction = 1 arcsec This is about the limit imposed by the atmosphere To resolve Cen: = 0.007 arcsec = 3.3 x 10-8 rad > min = 1.22/D D > 1.22/ In visible light, = 500 nm = 5 x 10-7 m Therefore need D > 1.22 (5 x 10-7 m)/(3.3 x 10-8) = 18.6 m Compare with world’s largest telescope, Keck = 10 m Ways to Measure Stellar Radii 44 22 LL TT RR LLT T 4R 4R LL TTRR 44 Method 1: Using Blackbody Laws: Lum. (L) = flux (F - for a BB) x surface area (A - spherical star radius R) L = T4 x 4R2 L is known if distance is known, T is obtained from spectrum or colour, solve for radius R +1/2 not -1/2 Examples: There is a tremendous range of stellar radii spanning 7-8 orders of magnitude. 22 11 2 2 RR LL 22TT RR LLTT 4.8 TTeffeff5800K 5800K MMVV4.8 Betelgeuse Betelgeuse: : M MVV5.5 5.5 TTeffeff3500K 3500K LLBetel Betel M 2.5 log 2.5 log M 2.5 log MMVVBetel VV Betel LL L Betel LLBetel Betel 5.5 2.5 log 5.5 4.8 4.8 1.5 1.5 log log LL L Betel LLBetel Betel13,200 13,200 LL 2 2 3500K RRBetel 3500K 1 1 13,200 Betel 320 320 13,200 2 2 5800K RR 5800K Sirius Sirius BB: : LL 0.03 0.03 TTeffeff27,000K 27,000K LL faint companion companion to to Sirius Sirius AA faint 2 2 1 1 27,000K RRSirius 27,000K Sirius BB 22 0.03 0.03 0.008 0.008 5800K RR 5800K approx approx. . size size ofof Earth Earth Ways to Measure Stellar Radii Method 2: Using Eclipsing Binaries: Explanation If orbital inclination i ~ 90, orbital plane is close to line of sight and stars will partially or totally eclipse each other. Duration of eclipses, combined with orbital speeds, gives stellar radii. To Sun vA Orbit seen from above vB v = vA + v B vA dA Light curve of eclipsing binary YY sagittarii Light from both stars visible V~10.7 b m 1 m 2 2.5 log 2 b1 0.7 dB 1 2 3 1 2 2 P=2.628 d Primary… Secondary minimum 4 3 4 brightness apparent brightness halved V~10.0 If we could see stellar disks during eclipse vB v Light curve v x t B = dB tB tA time v x tA = dA Ways to Measure Stellar Radii Method 3: Using Eclipsing Binaries: A Real Example Period Eclipsing double-line spectroscopic binary AR Lacertae P ~2d Radial Velcoty (km sec-1) vB~118 km/s vB vA Difference of magnitude ∆m Light Curve v = vA + vB ~ 230 km/s tA = t3 - t1 ~ 0.24 d tB = t2 - t1 ~ 0.13 d dA = v x tA ~ 4.8 x 106 km RA ~ 3.4 R Radial Velocity Curve vA~112 km/s t1 Time (days) t2 t3 dB = v x tB ~ 2.6 x 106 km RB ~ 1.9 R State of Stellar Matter With masses and radii known, the densities and state of matter of stars can be determined. Sun: Msun = 2 x 1030 kg = 2 x 1033 g Rsun = 7 x 105 km = 7 x 1010 cm Mean density sun = Msun/(4/3Rsun3) = 1.4 g/cm3 (a bit more than water) Giant Stars: Mstar ~ few x Msun in some cases Rstar ~ 500 Rsun giant typically < 10-7 g/cm3! 3000K < Teff < 50,000 K stars are gaseous