Physics 320: Astronomy and Astrophysics – Lecture IX

Carsten Denker

Physics Department Center for Solar–Terrestrial Research

NJIT

Problem 9.1

E bb F

  4 3  3

r aT

eye 4 

L

/ 4 

r

2 with

L

 11 J with

r

eye  2 10 J/s  

E



t



FA

eye and

t

2

r

eye /

c

 1.5 cm and

T

 310 K 

E

bulb 

LA

eye 4  However:  max 2

r

eye 2

r c

    15 J 

E bb E

bulb otsensitive to IR)

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Center for Solar-Terrestrial Research October 29th, 2003

Problem 9.2

  

hc

   4 

e hc

8  / 

kT

 1 

d



n

 

N

 0  

nV

   8  3

k T

3 3

h c

8  3

k T

3 3  0 

e x x

2  1

dx

with

x

 2.404114



V

 

hc

 15 / 

kT

with

T

 477 K and

V

 1 m 3

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Center for Solar-Terrestrial Research October 29th, 2003

Problem 9.7

 500 nm  2 0.0264 m /kg and   1.2 kg/m 3   1  500 nm  31.6 m However:    2 3 

d

 2 3  21.0 m

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Center for Solar-Terrestrial Research October 29th, 2003

Stellar Atmospheres

 The Description of the Radiation Field  Stellar Opacity  Radiative Transfer  The Structure of Spectral Lines

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Center for Solar-Terrestrial Research October 29th, 2003

Opacity

Any process that removes photons from a beam of light will be called absorption.

dI

     (   2  1 [m kg ] absorption coefficient or opacity)

d

     

ds

(optical depth)    ,

f

   ,

i

   0

s ds

0    ,0    0

s

 

ds

     0

s

 

ds

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Center for Solar-Terrestrial Research October 29th, 2003

Opacity (cont.)

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

          (pure absorption) 1 (gas optically thick) 1 (gas opticall hin)

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Center for Solar-Terrestrial Research October 29th, 2003

Photon Interactions

 Bound–bound transitions  Bound–free absorption  Free–free absorption (bremsstrahlung)  Electron scattering 

bf

  19 1

n

5  500 nm  3 m 3 

T

 8  3  

e

2

m c e

2   2   29 m 2 (Thompson scatt eri n ) free–free absorption of a photon

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Center for Solar-Terrestrial Research October 29th, 2003

Continuum Opacity

Photoionization of H – ions.

 

hc

  1240 eV nm 0.754 eV  1640 nm      ,

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 NJIT

Center for Solar-Terrestrial Research October 29th, 2003

Emission

Any process that adds photons to a beam of light will be called emission.

d l N

(random walk)

l l

1 1

l l

1 2

l l

1

N l l

2 1

l l

2 2

l l

1

N l N l

1

l N l

2

l N l N d

2 

Nl

2  cos  21   [cos  12  cos cos 

N

1   23  cos  13   cos 

N

2   cos  2

N

cos  1

N

   cos    1) ]   

NJIT

Center for Solar-Terrestrial Research l N

  

l October 29th, 2003

Limb Darkening

Looking into a star at any angle, we always look back to an optical depth of about   = 2/3, as measured by a straight line back along the line–of–sight.     0

s ds dP

rad

dr

  

c F

rad (radiative pressure and flux)

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Center for Solar-Terrestrial Research October 29th, 2003

Radiative Transfer Equation

d I

 

j

 

ds dI

      

j

 

ds

 3  1   1

dI



ds



I

  

j

 

S

  

j

   1  3  1 ([J s m sr ] source function)   1

dI



ds



I

 

S

 (transfer equa tion)

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Center for Solar-Terrestrial Research October 29th, 2003

Radiative Transfer Equation (cont.)

dI



ds dI



ds dI



ds I

 

S

 intensity decreases with distance

I

 approaches

S

 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  .

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Center for Solar-Terrestrial Research October 29th, 2003

Solving RTE

dI d

   

I

 

S

   ,

v

    

z

0   ,

v

cos 

dz

and

dz



ds

cos     ,

v

sec   cos 

dI



d

  ,

v



I

 

S

 plane parallel atmos phere

I

  0   and

S

  0 

d d



v



I

cos 

d

  

Id

 

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  cos 

dI d



S

 

dF

rad

d



v

 4  

I



S



October 29th, 2003

Solving RTE (cont.)

d d



v



I

cos 2 

d

   cos 

d

   2     0    0 

I

cos 

d

 

S

 cos 

d

  0 

dP

rad

d



v

 1

F

rad

c



dP

rad

dr

  

c F

rad (speherical coordinates)

F

rad  const.



F

surf  

T e

4

dF

rad

d



v I



S P

rad  1

c F

rad 

v



C

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Center for Solar-Terrestrial Research October 29th, 2003

Eddington Approximation

I

in  0 at 

v

      

I F

rad

P

rad    1 2   

I

out

I

out 2  3

c



I

out  

I

in

I

in  

I

in    4  3

c I

 4   3

c

4  3

I I

 1

F

rad 

v c



C

and

C



F

rad 

v

 2 3 

I

  4 2 3

c

3  

F

rad

T e

4 

v

 2 3 LTE 

I B



T

 4 

T

4  3 4

T e

4 

v

 2 3

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Center for Solar-Terrestrial Research October 29th, 2003

Structure of Spectral Lines

W

 

F c



F



F c d

 (equivalent width) natural broadenin g   2 2 

c

  1 

t i

 1 

t f

  Doppler broadening  2 

c

2

kT m

pressure broadening

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Center for Solar-Terrestrial Research

 2

c

1  

t

 

c n

 2

kT m October 29th, 2003

Homework Class Project

 Continue improving the PPT presentation.

 Use the abstract from the previous assignment as a starting point for a PowerPoint presentation.

 The PPT presentation should have between 5 and 10 slides.

 Bring a print-out of the draft version to the next class as a discussion template for group work  Homework is due Wednesday November 5 th , 2003 at the beginning of the lecture!



Exhibition name competition!

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Center for Solar-Terrestrial Research October 29th, 2003

Homework

 Homework is due Wednesday November 5 th , 2003 at the beginning of the lecture!

 Homework assignment:

Problems 9.9, 9.12, and 9.13

!

 Late homework receives only half the credit!

 The homework is group homework!

 Homework should be handed in as a text document!

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Center for Solar-Terrestrial Research October 29th, 2003