Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/~kjbg1/lectures/lect_2.pdf Äàòà èçìåíåíèÿ: Fri Oct 3 18:09:38 2003 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 03:38:43 2012 Êîäèðîâêà: koi8-r Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï

The Transfer of Radiation
When observing an astronomical source we may be looking through a cloud of matter which lies along the line of sight. This matter may absorb the radiation from the source, scatter it or in fact emit fur ther radiation. Each of these will change the source's apparent intensity.

Absorption
ds dA

I



I+ dI



Consider light shining into a sparse cloud of perfectly absorbing spheres of cross section and number density . As the beam of area propagates a distance into the cloud it encounters a total absorbing cross-section of , so of the beam to be absorbed. we expect a fraction

Thus the absorption coefficient is defined as the fractional loss of intensity per unit length, with dimensions . It follows that the photon mean free path .

"!

¦¥ ¤ © ¨§ ¤

¸

¡ ' &% $ ¡ #

¡ ¸ ¡ ¸ ¨§ ¤ ¡ ¡ ¨§ ¤ ¡ ¡ ¡ ¦ ¤

¸¡

¨§ ¤

¢¡

¨§ ¤

¸¡

¦¥ ¤




Emission
An excited atom can return to its ground state through two distinct mechanisms: (i) the atom emits energy spontaneously; (ii) it is stimulated into emission by the presence of electromagnetic radiation. The amount of stimulated emission is propor tional to , as was the amount of absorption. Therefore for simplicity it can be considered to be negative absorption and its effect included . in
ds dA

dI



We define a spontaneous emission coefficient which is the energy emitted per unit time per unit volume per unit solid angle per unit frequency.

Thus on crossing a length a beam's specific intensity is increased by spontaneous emission by

¨V G 03 ¦U TG ¨S G ¨R G QP G I ¨H G "C 6 F ED "C 6 BA @9 87 6 5

¨V G

03

I

W0 )

G

4

This has units

10 )

03

0 2

( (


The optical depth
A medium is said to be opaque or optically thick if on average a photon cannot pass through the medium without absorption. Conversely, a transparent medium is said to be optically thin. Both these proper ties are functions of wavelength; for example, a pane of glass is optically thin in the optical, but optically thick in the infrared.

.

The mean optical depth travelled by a photon before absorption is 1. In an optically thick homogenous medium the number of steps taken for a photon to diffuse out .

Example

r

h

through 200m of water bottom of the sea.

t

Qs r

g

h

through 2m of water bottom of a swimming pool.

t

g

h

through 20m of water

. You can see to the You cannot see to the

f 'Y X

A medium is optically thick at a par ticular frequency if

pi X h

¨d cb Y a g

1Y X

We define the optical depth

` YX X X Xe g e e q q e


The Radiative Transfer Equation
Combining emission and absorbtion gives the Radiative Transfer Equation

This can be rewritten as

where we have defined the source function . This is the value approached by given sufficient optical depth. For the case of a homogeneous cloud with a constant source function and optical depth with initial incident intensity , the emergent intensity is

This makes a lot of sense: the emergent radiation is the sum of the incident intensity attenuated by the total optical depth plus the sum of each section of cloud emission attenuated by the optical depth from that point to the receiver.

v

y w w w

w f 8

y ¨x u w w 'v w Ww ¦v u y w u w wv w ¦v u ed 8 v w v &w y wv


I

0

I'

I' = I0 e - + S 0 (1- e - )


Kirchhoff 's law calculates the source function for a body in thermal equilibrium. depends only upon the material and the temperature, so we can find its general form with a par ticular example. Consider a blackbody cavity at temperature T with a hole which is plugged by material also at temperature T (i.e. it is in thermal equilibrium with the cavity).
Plug of material Temperature T B (T) I


Blackbody Cavity Temperature T

The emergent intensity otherwise the material has gained energy from something at the same temperature.

Hence in general the source function for a body in thermal equilibrium is equal to the Planck function.

m p r mq r m l

mp

The incident intensity from the blackbody is

nm l

vr

m p r m q

m u t s m l ix m q w r Wm q t o

hk
.

m u t mqt

j ih g s ml

Kirchhoff 's Law,

o o


Example
From Teide Observatory, Tenerife, the optical depth of the sky is approximately at 30 GHz (mostly due to absorbtion by water vapour). Assume that the entire atmosphere is in thermal equilibrium at a temperature of 300K.

Subtracting off the contribution from the atmosphere, the source appears at of its actual surface brightness.

"¥ ± ¸° B¯ B® n¥

{

Consider observation of a source surface brightness

| | { | {

Consider emission just from the atmosphere i.e. Since the atmosphere is in thermal equilibrium .



»

n¥ ¨ ²± ° B¯ B® n¥



©

|¤} { º¡¸ ¸ z ¹ ¨ µ´ "¥ » ¬ "¥ ¨ ± ¸° B¯ B® · "¥

B 8 e T 8 @

 ǻ

©

| ¤ ¸¢ } { @¨ §¦ "¥

~| Q} | { z

¿¾ ½ ¼

{

W

y z e | ¨ª © ¶¨ µ´ "¥ y &¡ z ' ¦ '

y y

.


The Effects of Scattering
We have until now considered pure absorption and emission. There are many applications when this approach is sufficient, but also many where this is not. For example, the transfer of optical photons in stellar interiors, or the propagation of infrared photons from a protostar through a dusty cloud, both involve scattering. Scattering can be described by a new emission coefficient, but this is dependent on the incident radiation field; this makes it impossible to integrate the equation of transfer directly. In general, scattering is very complicated -- it is not isotropic and may involve energy loss of the photons.


Summary of Key Points
the absorption coefficient, is the fractional loss of intensity per unit length. It includes the effect of stimuated emission. (

the spontaneous emission coefficient, is the energy emitted per unit time per unit volume per unit solid angle per unit frequency. the source function, is the value approached by sufficient optical depth. .

Kirchhoff 's Law states that for emitters in thermal equilibrium .

the radiative transfer equation for a homogeneous cloud in thermal equilibrium is

öí

¨ê 'é ö à è õ ö ä

ö ï û Bó ¨ò ñ ð ü ô â iæ Bó ¨ò ñ ð @ÿ í õ ö í

ö à î ö ë õ ö ì

ô ã &ö ä

öï õ öì

ö áë þ ô å 1ö ä ö ä þ ö ì þ þ þ

ö áà þ

the optical depth, determines whether a medium is opaque ) or transparent ( ). .

given


Blackbody Radiation

Unpolarised

e.g.

-- Cosmic Microwave Background

É ÜÂ Æ ÓÒ ÑÐ ß É Ï ù Å cÎ ¶Í ÌË Ê ¹È Õ ÃÔ ù Å Ä

Á ù Ö Á Wù ÀÞ ç

ú

÷ ù ýø ç ç ç

ç

Emitters and radiation in thermal equilibrium


Free-Free or Bremsstrahlung Radiation

e
-

Emission as result of collisions between charged par ticles, usually electrons and ions.

Emitters in thermal equilibrium

Unpolarised

e.g.

-- HII regions -- X-ray emission from clusters of galaxies -- Ionised winds from stars

ÇÝÇÝÇÙÇÙÇÝÇÙÇÝÇÙ ÝÝÙÙÝÙÝÙ

ÇÛÇØÇÛÇØÇÛÇØ ÛØÛØÛØ × × × ×

Z

+


Sychrotron Radiation

B e

Emission as result of relativistic electrons gyrating round magnetic field lines
¢ ¢ ¢

Emitters not in thermal equilibrium Polarised e.g.
¢

-- quasars -- radio galaxies -- supernovae

ÌÚÌÚÌÚÌÚ

¡ ¡ ¡ ¡

ÚÚÚÚ


Dust

Emission as result of thermal excitation.
¸ ¸ ¸ ¸

Emitters in thermal equilibrium Polarised e.g. -- star forming regions -- optical extinction and polarisation through the galaxy -- diffuse scattered light from the Inter-Stellar Medium


Line Emission

Emission as result of electronic transitions; molecular vibrations and rotations.
¤ ¤ ¤ ¤

Emitters sometimes in thermal equilibrium. Sometimes polarised e.g. -- -- -- -- 1.4 GHz neutral hydrogen line CO emission Lyman series Masers