Determining Distances: The Baade-Wesselink Method Kerry Hensley – AS 712 – December 2, 2016

What is the Baade-Wesselink method? A type of primary distance indicator Primary: require no calibration against another method; also called an absolute distance indicator Parallax (<1 kpc) Moving-cluster method (<1 kpc) Baade-Wesselink method (Local Group galaxies; few Mpc) Secondary: require calibration against a primary distance indicator Cepheid variables (~50 Mpc) RR Lyrae variables Used for stellar-type objects for which the approximation of a spherically expanding photosphere is valid

Applications Distances to pulsating stars Supernovae Cepheid variables RR Lyrae variables Supernovae (To a certain extent) Eclipsing binaries Image Credit: From top: NASA/ESA; Chandra/Harvard; ESO/L. Calcada

“Über eine Möglichkeit, die Pulsationstheorie der δ Cephei-Veränderlichen zu prüten” The mean radius of a Cepheid variable can be determined absolutely from the brightness, color, and radial velocity of the star Method of determining distances to stellar objects Walter Baade (1926) An investigation of some of the assumptions made in the use of Baade’s method “The observations of brightness, colour and radial velocity of δ Cephei and the pulsation hypothesis” Adriaan Jan Wesselink (1946) Image Credit: allposters.com (left) and AIP (right)

Basic Method ∆𝑅 𝑑 = ∆𝜃 2 𝑅 𝑑 = tan 𝜃 2 ≈ 𝜃 2 Determine the angular and physical radius of a stellar object These two quantities are related by ∆𝑅 𝑑 = ∆𝜃 2 𝑅 𝑑 = tan 𝜃 2 ≈ 𝜃 2 𝑅 𝑑 𝜃

Basic Method Part I: Photometry Assume the star is a perfect blackbody Surface brightness is a function of the color (temperature) Observed flux/surface brightness = surface area (in arbitrary units) Image Credit: University of Georgia

Basic Method Part II: Spectroscopy Determine radial velocity curve from Doppler shift of spectral lines Determine the displacement in absolute units (km) by integrating the radial velocity curve: Δ𝑅= 𝑡 1 𝑡 2 𝑣 𝑡 𝑑𝑡 Image Credit: RIT

First Attempts Bottlinger – 1928 Becker – 1940 NOAO First Attempts Bottlinger – 1928 Unsuccessful: found the radius and the displacement were out of phase Concluded that the perfect blackbody assumption was wrong Becker – 1940 Blackbody assumption  single-valued relation between color and surface brightness Radius and displacement found to be in phase!

However… The radial velocity measured spectroscopically is not necessarily equal to the expansion velocity 𝑉 𝑝𝑢𝑙𝑠 𝑉 𝑟𝑎𝑑 𝑝= 𝑉 𝑝𝑢𝑙𝑠 𝑉 𝑟𝑎𝑑 𝑉 𝑝𝑢𝑙𝑠

Limb Darkening 𝑝= 𝑉 𝑝𝑢𝑙𝑠 𝑉 𝑟𝑎𝑑 = 2 (3−𝛽) (4−𝛽)

Extension: The Geometric B-W Method Image Credit: ESO

Summary The Baade-Wesselink method is an absolute distance indicator Relies on a combination of photometry and spectroscopy (or interferometry and spectroscopy) Used as a test of and calibration for the period-luminosity relation for Cepheid variables Image Credit: From top: NASA/ESA