Measuring Radii and Temperatures of Stars R radius

Measuring Radii and Temperatures of Stars R = radius r = distance R/r=angular diameter • • Definitions (again…) Direct measurement of radii • Photometric determinations of radii • Determining temperatures – Speckle – Interferometry – Occultations – Eclipsing binaries – Bolometric flux – Surface brightness – 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 ~20 mas at 4000 A, comparable to the diameter of a few 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 • But 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 • Calibrated with – Stellar models – Nearby stars with direct measurements (R is radius in solar units, r is distance in parsecs)

Surface Brightness • To avoid uncertainties in Teff and BC • Determine PV as a function of B-V PV(B-V)=log. Teff – 0. 1 BC • 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, Fn • From a model that fits the spectral energy distribution, compute the flux at the star’s surface, Fn • From the ratio of Fn/Fn, compute the radius • 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 2 m) • USNO & CHARA interferometers < 1 mas – Surface structure – Pulsations – Circumstellar material

Determining Temperatures • Recall the definition of the effective temperature • Model photospheres • Temperature calibrations – Teff vs. B-V • Slope of the Paschen continuum • Color indices – synthetic colors • Balmer Jump (in hotter stars, but also pressure sensitive) • Hydrogen lines • Metal lines and metal line ratios

Temperatures – Balmer Jump and Balmer Continuum “The determination of Teff of B, A and F main sequence stars from the continuum between 3200 A and 3600 A; ” Sokolov, N. A. ; Astronomy and Astrophysics Supplement, v. 110, p. 553

Using Line Ratios

Calibration of line depth ratios

More line ratios