Measuring Radii and Temperatures of Stars Definitions (again…) Direct measurement of radii – Speckle – Interferometry – Occultations – Eclipsing binaries Photometric determinations of radii – Bolometric flux – Surface brightness – Absolute flux Determining temperatures – Absolute flux – Model photospheres – Colors – Balmer jump – Hydrogen lines – Metal lines R = radius r = distance R/r=angular diameter

Stellar Diameters Angular diameters typically measured in milli- arcseconds (mas) Angular diameter (in radians) given by physical diameter divided by distance The diameter of Aldebaran is ~40 R SUN. 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 4000A, 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 semi- major 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 P V as a function of B-V P V (B-V)=logT eff – 0.1BC P V (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 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  ) 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 T eff 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