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
Fr 22 44 RF
42
0)/( effTrRd
F
R = radius r = distanceR/r=angular diameter
4
0 effTdF
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 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 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
42
SunSunSun T
T
R
R
L
L
5.72.0)(2.0log2loglog VSuneff mBCBCTrR
(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)=logTeff – 0.1BC
• PV(B-V) is known as the “surface brightness function”
• Calibrate with directly measured diameters
32 )()()()( VBdVBcVBbaVBPV
460.72.0)(2loglog VV mVBPrR
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.
2
1
F
FrR
Fr 22 44 RF
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 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
