Transcript Document

Physics 681: Solar Physics and
Instrumentation – Lecture 4
Carsten Denker
NJIT Physics Department
Center for Solar–Terrestrial Research
Solar Spectrum
The spectrum of the Sun: The dashed line is the
curve of an ideal blackbody having the Sun’s
effective temperature.
September 13, 2005
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Radiation Field
I 
E d 
d  dtdA cos  d 
J


(average intensity) 
3

 s m sr 
 E  d   I  d  dtdA cos  d   I  d  dtdA cos  sin  d 
I 
1
4
 I d  
1
4
2

 
0
0
sin  d  d  (m ean intensity)
iosotropic radiation field

I  I
isotropic blackbody radiation

I   B
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Energy Transport




Radiation and convection
2
Energy flux (luminosity per unit area): F  F R  FC  L / 4  r
Mean free path of photons is small in the solar interior  local
thermal equilibrium (LTE)
Radiative transfer equation
cos 

      I  B
dr
Total energy flux
F 

dI


0
F d w ith F 

w ith B  T  
 cos  I d
Temperature gradient
T
m

3 L
256   r T
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2
4
3
w ith
1





4


0
2
3   dr
  dT
0
c
dB
1 dB
dB

2h
e
h / kT

d
d
dT
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3
1
4
dB dT
3   dT dr
Temperature Definitions
excitation tem perature N b
(Boltzm an equation)

Na
ionization tem perature N i  1
(S aha equation)
Ni
gbe
gae
 E b / kT
 E a / kT

gb
e
 ( E b  E a / kT )
ga
2 kT Z i  1  2  m e kT 



2
Pe Z i 
h

 m 
n v dv  n 

(B oltzm an distribution)
2

kT


kinetic tem perature
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and Pe  n e kT
3/2
2
e
 m v / 2 kT
2
2 hc 
1

B  (T ) 
 hc /  kT

5
(P lanck function)
 e
1
color tem perature
3/2
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4  v dv
2
Opacity
Any process that removes photons from a beam of light
will be called absorption.
dI       I  ds (   [m
d        ds
2
1
kg ] absorption coefficient or opacity)
(optical depth)
s
      , f    ,i       d s
0
s
0    ,0       d s    
0
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
s
0
   ds
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The optical depth may be thought of as the number of
mean free paths from the original position to the surface,
as measured along the ray’s path.
I   I  ,0 e
 
(p u re ab so rp tio n )

1
(gas optically thick)

1
(gas optically thin)
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Radiative Transfer Equation
dI   j  ds
( j  [m s
3
1
sr ] em ission coefficient)
dI       I  ds  j  ds
 
S 
 
1
dI 
   ds
j
([J s

1
 I 
dI 
   ds
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1
m
j

3
1
sr ] source function)
 I  S
(transfer equation)
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The intensity of the light tends to
become equal to the local value of
the source function.
For the case of thermodynamic
equilibrium, the source function is
equal to the Planck function S = B.
dI 
ds
dI 
 0  I  S
 0  intensity decreases w ith distance
ds
dI 
ds
 0  I  approaches S 
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Photon Interactions
Bound–bound transitions
 Bound–free absorption
 Free–free absorption
(bremsstrahlung)
 Electron scattering

 bf  1.31  10
 19
3
m
3
2
8  e 
 29
2


6.65

10
m

2 
3  mec 
2
T
1 



5 
n  500 nm 
(T hom pson scattering)
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Continuum Opacity
Photoionization of H– ions
 
hc


1240 eV nm
 1640 nm
0.754 eV
     , bb    ,bf    , ff    , es
(total opacity)
The total opacity depends not only on
the wavelength of the light being
absorbed but also on the composition,
density, and temperature of the stellar
material.  Rosseland mean opacity
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Structure of Spectral Lines
W 

Fc  F
d
(equivalent w idth)
Fc
natural broadening
D oppler broadening
pressure broadening
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 1
1
 



2 c   t i  t f

 
 
2
2
2 kT
c
m

2
1
c   t




 n
2

c
2 kT

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m
Nuclear Reactions
http://nobelprize.org/physics/articles/fusion/sun_pp-chain.html
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CNO Cycle
http://outreach.atnf.csiro.au/education/senior/astrophysics/stellarevolution_mainsequence.html
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