Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/~kjbg1/lectures/lect6.pdf Äàòà èçìåíåíèÿ: Mon Oct 6 23:36:27 2003 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 03:15:39 2012 Êîäèðîâêà: koi8-r Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï ð ï

Line Emission
Observations of spectral lines offer the possibility of determining a wealth of information about an astronomical source. A usual plan of attack is :Identify the line.

Obtain from theory/experiment a value for the absorbtion cross-section. Calculate the number of atoms/ions/molecules giving rise to the line.

The position of the lines gives information on the velocity of the emitters along the line of sight or the cosmological redshift.



Infer information about the population of states. Determine temperature and density.


Einstein Coefficients

2 E=h12 1
Spontaneous emission rate =

n B12 B21 A
21

2

n
¥¤ ¸ ¢

1

If only radiative transitions link the two states then

)

(

' & %! $ #

© § ¸ ¤

¦

)

¤0

D C8 6 @ 6 A B6 8 @ 8 A 98 6 @ 98 76 5

(

© § ¨¤ ¸

¦

(

¸0

©§

32 ¥¤ ¸ 1¢ ¸ 0

4

"!

Line profile

normalised so that

¦

¦

¦

(NB sometimes and ).

and

defined so that rates are

© ¸ ¤

¨¤ ¸

¦

Absorbtion rate =

¦

Stimulated emission rate =

© ¨§ ¥¤ ¸

© § ¸ ¤

¸ ¤

© ¥¤ ¸

¦

¡ ¡ ¡ ¡ ¡


are proper ties of the emitters, so if we can determine them in any specific case then we can apply them in any general case. Consider the case where the emitters and the radiation are in thermal equilibrium (Planck function). Also occupancy of the two states given by Boltzmann function

So if we can find one Einstein coefficient then we can calculate the other two.

R S U

Can use Fermi's Golden Rule to find

.

F

c Y ¥S R U e yx R w ¨S R Q E p p c Y ¥S R U R R S U S 't ¨s q 7 C u h f ' % yx w d 't ¨s q F u h 3f ' % yx w d B

a



R

h f ' % yx w vu d 't ¨s rq S p

c

h gf ed a

IE F U

c ba `

H 'F E C C 7 R p Y

cSi

7H GF E Ri

Finding

R S XU WT ¥S R VU T ¥S R Q P c a ` P P P P

Y


1 ¥

¥ l ¥ ¥

¦

l



¬ ¨ ¨« ¨ ¨§ C " ª© ¨ k z 1¤ C %

k k i m

ig

{ 'j f i hg f



i g

¥ ¤

l

ni g

¥¤

%



¨ k

k 'j f i hg f

s ~} r| W{ s u ¸ z r %{ ¥y ¢ 1t s n s gv n { s w t ¡ } ~ %{ y w 1v s t s

, the spontaneous emission coefficient is the energy emitted per time per volume per solid angle per unit frequency.

Radiative transfer:

is the absobtion coefficient and includes the effect of stimulated emission. It is the fractional loss of intensity per unit length.

i

m

e

"





¨

¨ k

k ri q iq e

¥y " 1t s b xs v n W{ s uw t } W{ s z r W{ s ~} r| %{ zy xw 1v us 1t s

ri q

po ni g i

ml

k

'j f i hg f e

and



d

The equation for radiative transfer is




Excitation Temperature
°¯ ® ´
Define the excitation temperature, for the two levels through

If the emitters are in local thermodynamic equilibrium then . The level is said to be thermalised.

The ratio of how much stimulated emission there is compared to absorbtion is

Stimulated emission is very impor tant at radio frequencies where so that

á à ® 'ó ¿ %¾ y½ ¼ v» ò¹ ñ¸ · ¶

®

ä ¯®

If not in thermal equilibrium then in general

½ à®ó¿ ² À Þ y½ ¼ á ½¹ ¥¨ ² ú ¨÷ ý ø ¨ ± ç ªù %ý ²

á 7à ® hþ ¿ %¾ y½ ¼ v» º¹ ¸ · G¶ ¨ µ ¨ ² µ ¨ ± µ ² ± ´ µ
.

² â

ï î ~ã xå ¡å ã 1õ ë ní ìë ô ê é è õ ~ã ~ô å

´ ¨ ± ää ï î ~ã ô ÿ ð ~î "å ÿ ê ö û ü ® ó ¿ "¾ y½ ¼ ® ö à® æ

² ±




Collisions

2 E=h12 1
Á

n C12 C
21

2

n

1

Atoms can become excited or de-excited though collisions.

Consider a system where the par ticles are in thermal equilibrium and where radiative transitions are negligable.

The 's can be related to a collisional cross-section and calculated.

¥Ä Õ Ã

and always true.

are proper ties of the par ticles so the above is

¥Ä ÈÕ Ã Ô ¥Ö ºÕ Ö

The rate of collisional de-excitation is

.

Ô Ö

Õ Ä bÃ Ô xÖ ÅÄ Ö

The rate of collisional excitation is density of colliders).

.(

is the

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¨Ä ÈÕ bÃ Õ Ö

É Õ Ä Ã Ä Ö

Õ

Ë

Ë

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Õ Ä Ã

Ê

Ã

Õ Ä Ã

Á Á Á Á Á


Critical Density for Collisional De-excitation
nÝ Ø ×
Assume negligable. is low so absorbtion and stimulated emission are

Substituting for

collisional de-excitation much faster than spontaneous emission; approaches its equilibrium value and the line is thermalised. spontaneous emission much faster than collisional de-excitation; each time a collision excites the high enequiergy state it decays through emitting a photon librium value ­ sub-thermal excitation.

rÝ Ø

If we relax the assumption that modify .

is low the effect is to

W W X¤

×

Ù

GF C ED C B ¡ × A @9 87 65 43 21 0 )( ' &% $ #" ! ¦ ¤ × ¤ ¢ ¤ ¢ ¤ ©¨ §¤ × ¦ ¥¤ ¸¢ ¡
and is the critical density for collisional de-excitation.

gives

¤ ¢

Ú ¥×

¤

×

T! a T! S Q `Y IH

!C C

V IH × U U Ú T! R! SQ P¦ IH

×

PV IH

V ( IH

WW

×

¥× Ú × Ú × × × Ç Ç Ç Ç Ç

Ç Ù


Example: Cooling rate of gaseous nebulae
Cooling occurs when collisions excite gas into high energy state which then radiates a photon; this leaves the nebula and the cloud cools. For high density most collisions which excites a gas par ticle are de-excited by collisions too. Number of exrate of emission rate of cited gas par ticles cooling . For low density most collisions which excites a gas par ticle results in emission of a photon (because collisional de-excitation negligable). Rate of collisions rate of cooling .

Below this density the number of CO molecules in the level is lower than one would expect if the system was in thermal equilibrium weak emission ­ the line is subthermally excited. So generally only see CO from regions where

dc

v i

vu

Gy y v e c g & ©y w v i ©y w y xw vu b

For the CO

rotational line at 115 GHz, .

t

sr qp

Example: CO

rotational line

g Xi c f

gcf

Xe c d d c

ec hh c

gcf

g

ic f

cf

Gy y

w

b b b b




Line Shape
Spectral lines always have a width ­ 3 contributions to this width. Natural line shape:gives the mean occupation time in the upper state; Heisenberg's uncer tainty relationship gives the corresponding energy width of the line. Lorentzian line shape.

Pressure broadening:- collisions while emitting disrupt the radiation train line width. Lorentzian line shape. Doppler broadening:- velocity of emitter along the line of sight results in a Doppler shift. Many emitters with Maxwellian random velocities result in Gaussian line shape, width dependant on temperature or bulk motion of the gas. In most cases the natural line shape and pressure broadening can be ignored and we just consider Doppler broadening. The shape will also depend on the optical depth. The brightness temperature is a function of frequency given by

xw qh p `v h

d § z T } { z n n @t k & t k o

n 6u l t Tl k Iq p

i h g fe d sr

T ~} |{ z

o 4n ml k j

y







Consider an emitter moving along the line of sight at speed . The emitted light at frequency in the emitteds rest frame is Doppler shifted to freqency given by

If the velocity distribution is Maxwellian then the velocity distribution is given by

Equating

and

gives



Note that since includes a depth of a line is also propor tional to

term, then the optical .





¥



¡ & ¥ ¥ 6

¡ & §¦ 2¥









¨ ¨ &¤ ¸ ¢ © & I f 6 f &¤ ¸ ¢ ¡ & I f



G

§



6

f

Calculating

« ª ¨ ¨ f






Figure 1: Low optical depth cloud

Figure 2: High optical depth cloud


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&à ¶
along the line of sight.

&· ¶

½ ¾½ ¨

¼ )± I¸ &» º ¹ ± ° ¸ &· ¶ µ´ ² ¨ ² ± ° ¯

The actual profile of an absobtion line may be difficult to measure precisely.

Equivalent Width

Measure the area in the line and calculate the equivalent . width,

Can then estimate

þ¿

¯ ® ® ¬

I

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å åå åå äåäå
I

0

I



Equal Areas

W



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¬


Example: H 21 cm line

pancy time

For a cloud at temperature 100 K and density the levels are set time between H atom collisions is 100 years by collisions. collisional de-excitation much faster than spontaneous emission and the line is thermalised. The spin temperature,

An optically thin line will give an estimate of

&Ý Ø Å Ç ÉÈ

An optically thick line will give an estimate of

ÃÀ XÔ Å Þ Ä ù Õ ©ý ÿ Þ Ä ù ¡Ö ã ì ©ý ÿ â Þ ì ù ú ¥÷ ì &ù ©ý ÿ ø Gç ûü &ð æ ï î ì
GHz; years.

ò Å µÛ É È Í 6Ì Å ïî ÞÐ ï ßÏ ûý Îð ý ï &Í 6Ì Ë î ì 2ë Ê

ï |É È

Ë 2ë Ê )Ü Â &Æ ÓÒ Ñ Ð ï

¾Á ©ý ÿ À 8æ óò ñð ÿ ï ¥î íì ë ê é èõ ã îÅ ìÅ

ô ô ô ô ô ô

Magnetic dipole transition between nuclear hyperfine levels F = 0, 1; Degeneracies 1,3 occu-

K.

.

.


Summary

Fermi's Golden Rule can be used to find ; the other two coefficients can be calculated from this.

Collisions can also excite/de-excite atomic transitions with and . rates governed by There is a critical density above which the rate of cooling of a gas varies linearly with density and below which it varies quadratically. Emission line widths are determined by their natural line shapes (uncer tainty principle), pressure broadening and Doppler broadening.

Ú

¢

Ú



§¦ ¥

¤

¸

Ú

¤

¥Ú

¢

¸

Ú

¢

¡

Ú×

Ù Ù Ù Ù Ù

The equation of radiative transfer for emission lines can be written in terms of the transitions Einstein coefficients , and .