Transcript The Transfer Equation

1 dI l


I

S
l
l
The Transfer Equation k l r dl
• The basic equation of transfer for radiation passing
through gas: the change in specific intensity In is
equal to:
dIl = intensity emitted – intensity absorbed
dIl = jlrdl – klrIl dl
-dIl /dtl = Il - jl/kl = Il - Sl
• This is the basic equation which must be solved to
compute the spectrum emerging from or passing
through a gas.
Thermodynamic Equilibrium
• Every process of absorption is balanced by a
process of emission; no energy is added or
subtracted from the radiation
• Then the total flux is constant with depth
Frad  Fsurface  Te4
– flux is the energy passing through a unit surface area
integrated over all directions
– mean intensity is the directional average of the specific
intensity
• When we assume LTE, we are assuming
that Sl=Bl
dI l
 I l  Bl
dt l
Simplifying Assumptions
• Plane parallel atmospheres (the depth of a
star’s atmosphere is thin compared to its radius,
and the MFP of a photon is short compared to
the depth of the atmosphere
• Opacity is independent of wavelength (a gray
atmosphere)

I   I l dl
0

S   Sl dl
0
Black Bodies - Observations
• spectrum continuous, isotropic, unpolarized
• continuum intensity depends on frequency
and temperature
3 n 
• observed relation:
In  n f  
T 
• From this can be derived Wien’s law and the
Stefan-Boltzman law
• Also Rayleigh-Jeans Approx. and Wien
Approx.
Black Bodies
• Wien’s Law – Peak intensity
• Stefan-Boltzman Law – Luminosity
• Planck’s Law – Energy Distribution
– Rayleigh-Jeans approximation
– Wien approximation
Wien’s Law – Peak Intensity
 Il is max at lmax = 0.29/T (l in cm) (or l’max = 0.51/T
where l’max is the wavelength at which In is max)
 Thought Problem: Calculate the wavelengths at which
In and Il are maximum in the Sun. Think about why
these are different.
Luminosity – Stefan Boltzman Law
• F = T4 or L = 4p 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
jnr = NuAulhn
– Induced (stimulated) emission proportional
to intensity
knrIn = NlBluInhn – NuBulInhn
Steps to the Planck Function
• Energy level populations given by the
Boltzman equation:
N u g n  hn / kT

e
Nl gl
• Include spontaneous and stimulated emission
Nu Aul  Nu Bul In  Nl Blu In
• Solve for I, substitute Nu/Nl
• Note that
g 
Bul  Blu  l 
 gu 
 2hn 3 
Aul   2  Bul
 c 
2hn 3
1
Bn (T )  2 hn / kT
c e
1
2
2hc
1
Bl (T )  5 hc / lkT
l e
1
Planck’s Law
• Rayleigh-Jeans Approximation (at long wavelength, hn/kT is
small, ex=x+1)
In  2kT
n2
c
2
 2ktl2
• Wien Approximation – (at short wavelength, hn/kT is large)
2hn 3 hn / kT
In  2 e
c
Class Problem
• The flux of M3’s IV-101 at the K-band is
approximately 4.53 x 105 photons s–1 m–2 mm-1.
What would you expect the flux to be at 18 mm?
The star has a temperature of 4250K.
Using Planck’s Law
Computational form:
Bl (T ) 
1.19 x10 27 l5
1.44 x108 / lT
e
1
For cgs units with wavelength in Angstroms
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 protostellar 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?