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

2. 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

3. 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

4. “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)

5. 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

6. 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

7. 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

8. 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!

9. 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

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

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

12. 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