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