A short review• The basic equation of transfer for

radiation passing through gas: the change in specific intensity I is equal to:

-dI /d = I - j/ = I - S

• Assumptions– LTE… (all physical processes in balance)

S= B

– Flux constant with depth– Plane parallel atmosphere

Black Body Radiation• walls heated• emit and reabsorb

radiation• interior of

chamber in thermodynamic equilibrium

• leakage in or out of the whole is small

• Stellar photospheres approximate blackbodies• Most photons are reabsorbed near where they are emitted• Higher in the atmosphere, a star departs from a black body

See: http://homepages.wmich.edu/~korista/phys325.html

See: http://www.jb.man.ac.uk/distance/life/sample/stars/index.html

See: http://astro1.panet.utoledo.edu/~ndm/4810.html

Black Bodies - Observations

• spectrum continuous, isotropic, unpolarized• continuum intensity depends on frequency

and temperature• observed relation:

• From this observational result can be derived Wien’s law (peak intensity) and the Stefan-Boltzman law (luminosity)

• Also Rayleigh-Jeans approx. and Wien approx. of flux above and below BB peak

T

fI

3

Wien’s Law – Peak Intensity

Iis maxat max = 0.29/T ( in cm)

or’max = 0.51/T (where ’max is the wavelength at which I is max)

Thought Problem: Calculate the wavelengths at which I and I are maximum in the Sun. Think about why these are different.

Luminosity – Stefan Boltzman Law

• F = T4 or L = 4 R2 T4

• Class Problem: What is the approximate absolute magnitude of a DA white dwarf with an effective temperature of 12,000, remembering that its radius is about the same as that of the Earth?– what is the simplest approach?

Deriving the Planck Function• Several methods (2 level atom,

atomic oscillators, thermodynamics)• Use 2-level atom: Einstein

Coefficients– Spontaneous emission proportional to

Nn x Einstein probability coefficient

j = NuAulh

– Induced (stimulated) emission proportional to intensity

= NlBluh – NuBulh

Steps to the Planck Function

• Energy level populations given by the Boltzman equation:

• Include spontaneous and stimulated emission

• Solve for I, substitute Nu/Nl

• Note that

IBNIBNAN lululuulu

kTh

l

n

l

u eg

g

N

N /

u

lluul g

gBB ulul B

c

hA

2

32

Planck’s Law

• Rayleigh-Jeans Approximation (at long wavelength, h/kT is small, ex=x+1)

• Wien Approximation – (at short wavelength, h/kT is large)

1

12)(

/5

2

kThce

hcTB

1

12)(

/2

3

kThec

hTB

kThec

hI /

2

32

22

2

22 kt

ckTI

Class Problem

• The flux of M3’s IV-101 at the K-band is approximately 4.53 x 105 photons s–1 m–2 m-1. What would you expect the flux to be at 18 m? The star has a temperature of 4250K.

Using Planck’s Law

Computational form:

For cgs units with wavelength in Angstroms

1

1019.1)(

/1044.1

527

8

Txe

xTB

Class Problems

• You are studying a binary star comprised of an B8V star at Teff = 12,000 K and a K2III giant at Teff = 4500 K. The two stars are of nearly equal V magnitude. What is the ratio of their fluxes at 2 microns?

• In an eclipsing binary system, comprised of a B5V star at Teff = 16,000K and an F0III star at Teff = 7000K, the two stars are known to have nearly equal diameters. How deep will the primary and secondary eclipses be at 1.6 microns?

Class Problems

• Calculate the radius of an M dwarf having a luminosity L=10-2LSun and an effective temperature Teff=3,200 K. What is the approximate density of this M dwarf?

 • Calculate the effective temperature of a proto-

stellar object with a luminosity 50 times greater than the Sun and a diameter of 3” at a distance of 200 pc.

Class Problems

• You want to detect the faint star of an unresolved binary system comprising a B5V star and an M0V companion. What wavelength regime would you choose to try to detect the M0V star? What is the ratio of the flux from the B star to the flux from the M star at that wavelength?

• You want to detect the faint star of an an unresolved binary system comprising a K0III giant and a DA white dwarf with a temperature of 12,000 K (and MV=10.7). What wavelength regime would you choose to try to detect the white dwarf? What is the ratio of the flux from the white dwarf to the flux from the K giant at that wavelength?