Download Chapter 14 – Chemical Analysis

Measuring Radii and
Temperatures of
Stars
4r F  4R F
2


0


0
2
F d  Teff
•
•
Definitions (again…)
Direct measurement of radii
•
Photometric determinations of radii
4
F d  ( R / r ) 2 Teff
R = radius
r = distance
R/r=angular diameter
•
4
– Speckle
– Interferometry
– Occultations
– Eclipsing binaries
– Bolometric flux
– Surface brightness
– Absolute flux
Determining temperatures
– 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
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 semimajor axes
– Velocities give semi-major axes (i=90)
Photometric Methods – Bolometric Flux
• Must know bolometric flux of star
– Distance
– Temperature
– Bolometric correction
 R 
L

 
LSun  RSun 
2
 T 


 TSun 
4
• Calibrated with
– Stellar models
– Nearby stars with direct measurements
log R  log r  2 log Teff  0.2(BC  BCSun )  0.2mV  7.5
(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 )  a  b( B  V )  c( B  V ) 2  d ( B  V )3
log R  log r  2PV ( B  V )  0.2mV  7.460
• PV(B-V) is known as the “surface brightness
function”
• Calibrate with directly measured diameters
Absolute Flux
4r F  4R F
2
2
• 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
F 

R  r

F
 

1
2
• 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 2m)
• 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