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Download Measuring Radii and Temperatures of Stars
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Measuring Radii and Temperatures of Stars 4r F 4R F 2 0 0 2 F d Teff • • Definitions (again…) Direct measurement of radii • Photometric determinations of radii 4 F d ( R / r ) 2 Teff R = radius r = distance R/r=angular diameter • 4 – Speckle – Interferometry – Occultations – Eclipsing binaries – Bolometric flux – Surface brightness – Absolute flux Determining temperatures – Absolute flux – Model photospheres – Colors – Balmer jump – Hydrogen lines – Metal lines Stellar Diameters • Angular diameters typically measured in milliarcseconds (mas) • Angular diameter (in radians) given by physical diameter divided by distance The diameter of Aldebaran is ~40 RSUN. Its distance is about 19 pc. The angular diameter of Aldebaran is … (work in cgs or MKS units or work in AU and use the definition of a parsec) What would the angular diameter of the Sun be at 10 pc? Speckle Diameters • The diffraction limit of 4-m class telescopes is ~0.02” at 4000A, comparable to the diameter of some stars • The seeing disk of a large telescope is made up of the rapid combination of multiple, diffraction-limited images • 2-d Fourier transform of short exposures will recover the intrinsic image diameter • Only a few stars have large enough angular diameters. • Speckle mostly used for binary separations Interferometry • 7.3-m interferometer originally developed by Michelson • Measured diameters for only 7 K & M giants • Until recently, only a few dozen stars had interferometric diameters CHARA Interferometer on Mt. Wilson CHARA Delay Compensator Other Methods • Occultations – Moon used as knife-edge – Diffraction pattern recorded as flux vs. time – Precision ~ 0.5 mas – A few hundred determined • Eclipsing binaries – Photometry gives ratio of radii to semimajor axes – Velocities give semi-major axes (i=90) Photometric Methods – Bolometric Flux • Must know bolometric flux of star – Distance – Temperature – Bolometric correction R L LSun RSun 2 T TSun 4 • Calibrated with – Stellar models – Nearby stars with direct measurements log R log r 2 log Teff 0.2 BC 0.2mV 7.460 Surface Brightness • To avoid uncertainties in Teff and BC • Determine PV as a function of B-V PV(B-V)=logTeff – 0.1BC PV ( B V ) a b( B V ) c( B V ) 2 d ( B V )3 log R log r 2PV ( B V ) 0.2mV 7.460 • PV(B-V) is known as the “surface brightness function” • Calibrate with directly measured diameters Absolute Flux • Determine the apparent monochromatic flux at some wavelength, F • From a model that fits the spectral energy distribution, compute the flux at the star’s surface, F • From the ratio of F/F, compute the radius F R r F 1 2 • The infrared flux method is just this method applied in the infrared. Hipparcos! • The European Hipparcos satellite determined milli-arcsec parallaxes for more than 100,000 stars. • Distances are no longer the major source of uncertainty in radius determinations for many stars • Zillions of stars within range of the Keck interferometer (3 mas at 2m) • USNO & CHARA interferometers < 1 mas – Surface structure – Pulsations – Circumstellar material