literature - fau
TRANSCRIPT
I
EF
CO
DRI
L
AI
N
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DN
XAEA
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M LMVA
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LGE
Erlangen−Nürnberg
Friedrich−Alexander−Universität
I
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Ast
roph
ysic
alR
adia
tion
Pro
cess
es
Jörn
Wilm
shttp://pulsar.sternwarte.uni-erlangen.de/wilms/teach/radproc/
Som
mer
sem
este
r20
08
Bür
o:D
r.K
arlR
emei
s-S
tern
war
te,B
ambe
rg
Tel.:
(095
1)95
222-
13
Em
I
EF
CO
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L
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AI
AD
R
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LGE
1–1
Intr
oduc
tion
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
1–2
Intr
oduc
tion
1
Sch
edul
e
Intr
oduc
tion
14.0
4.In
trod
uctio
n,M
ultiw
avel
engt
hA
stro
phys
ics
21.0
4.R
adia
tion
and
Rad
iativ
eTr
ansf
er
28.0
4.B
lack
Bod
yR
adia
tion
Cla
ssic
alR
adia
tion
The
ory
05.0
5.R
adia
tion
from
Mov
ing
Cha
rges
12.0
5.N
ole
ctur
e–
Pen
taco
st
19.0
5.B
rem
sstr
ahlu
ng
26.0
5.S
ynch
rotr
onR
adia
tion
02.0
6.C
ompt
oniz
atio
n
09.0
6.P
air
Pro
duct
ion
16.0
6.R
adia
tion
from
Nuc
lei
Ato
mic
(Qua
ntum
-Mec
hani
cal)
Pro
cess
es
23.0
6.A
tom
icS
truc
ture
30.0
6.Li
neD
iagn
ostic
s
07.0
7.M
olec
ular
Rad
iatio
n
14.0
7.N
ole
ctur
e(c
onfe
renc
e)
I
EF
CO
DRI
L
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XAEA
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C
M LMVA
AI
AD
R
E
LGE
1–3
Intr
oduc
tion
2
Lite
ratu
re
RY
BIC
KI,
G.B
.&
LIG
HT
MA
N,
A.P
.,19
79,R
adia
tive
Pro
cess
esin
Ast
roph
ysic
s,
New
York
:W
iley,
$116
A“m
ustb
uy”,
alth
ough
now
very
expe
nsiv
e(I
goti
tfor
$50)
.S
tand
ard
text
ofth
efie
ld,i
nso
me
area
sge
tting
outd
ated
,tho
ugh
–ge
titf
rom
amaz
on.c
om,n
otam
azon
.de
PA
DM
AN
AB
HA
N,
T.,2
000,
The
oret
ical
Ast
roph
ysic
s:V
olum
e1:
Ast
roph
ysic
al
Pro
cess
es,C
ambr
idge
:C
ambr
idge
Uni
v.P
ress
,65.
00C
Intr
oduc
tion
toth
eph
ysic
sof
astr
ophy
sics
.S
hort
,con
cise
,gre
at.
PA
DM
AN
AB
HA
N,
T.,2
006,
An
Invi
tatio
nto
Ast
roph
ysic
s,N
ewJe
rsey
:W
orld
Sci
entifi
c,$3
6.00
Abe
autif
ully
writ
ten
over
view
ofth
em
ajor
phys
ical
proc
esse
sre
leva
ntfo
ras
trop
hysi
cs(n
ot
only
grav
iatio
n).
I
EF
CO
DRI
L
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AD
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LGE
1–4
Intr
oduc
tion
3
Lite
ratu
re
LO
NG
AIR
,M
.S.,
1992
,Hig
hE
nerg
yA
stro
phys
ics,
Vol
.1:
Par
ticle
s,P
hoto
ns,
and
thei
rD
etec
tion,
Cam
brid
ge:
Cam
brid
geU
niv.
Pre
ss,∼
50C
Goo
din
trod
uctio
nto
high
ener
gyas
trop
hysi
cs,t
he1st
volu
me
deal
sex
tens
ivel
yw
ithhi
gh
ener
gypr
ocss
es.
Rec
omm
ende
d.U
nfor
tuna
tely
,eve
ryth
ing
isin
SIu
nits
.
SH
U,
F.H
.,19
91,T
heP
hysi
csof
Ast
roph
ysic
s,I.
Rad
iatio
n,M
illV
alle
y:
Uni
vers
ityS
cien
ceB
ooks
,70.
00C
Goo
din
trod
uctio
nto
radi
atio
npr
oces
ses,
som
eim
port
anta
reas
are
mis
sing
,tho
ugh.
Not
as
unde
rsta
ndab
leas
Ryb
icki
&Li
ghtm
an.
LA
NG
,K
.R.,
1999
,Ast
roph
ysic
alF
orm
ulae
,3rd
editi
on,2
Vol
s,H
eide
lber
g:
Spr
inge
r,2×
107
CC
olle
ctio
nof
1000
sof
form
ulae
nece
ssar
yfo
ras
trop
hysi
calr
esea
rch,
with
exha
ustiv
e
refe
renc
esto
the
orig
inal
liter
atur
e.
CO
WL
EY,
C.R
.,19
95,A
nIn
trod
uctio
nto
Cos
moc
hem
istr
y,C
ambr
idge
:
Cam
brid
geU
niv.
Pre
ss,$
37P
ract
ical
sum
mar
yof
atom
ican
dm
olec
ular
proc
esse
s.
I
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1–5
Intr
oduc
tion
4
Lite
ratu
re
Goo
dre
fere
nces
onth
eW
WW
onra
diat
ion
proc
esse
sin
clud
e
Ern
ieS
eaqu
ist,
Uni
vers
ityof
Toro
nto,
AS
T14
40F
:Rad
iatio
nP
roce
sses
Lect
ure
note
son
anad
vanc
edle
ctur
eon
astr
ophy
sica
lrad
iatio
npr
oces
ses,
appr
oxim
atel
yat
the
leve
loft
his
lect
ure,
avai
labl
eat
http://www.astro.utoronto.ca/~seaquist/radiation/notes.html.
Juri
Pou
tane
n,U
nive
rsity
ofO
ulu,
Rad
iativ
eP
roce
sses
inA
stro
phys
ics
2006
Mor
efo
rmal
than
this
cour
se,g
ood
intr
oduc
tion
onre
lativ
istic
mec
hani
sms,
whi
chw
ew
ill
igno
refo
rtim
ere
ason
s,av
aila
ble
athttp://cc.oulu.fi/~jpoutane/teaching/rad06.html.
Jelle
Kaa
stra
etal
.,T
herm
alR
adia
tion
Pro
cess
es,S
pace
Sci
.Rev
.,su
bmitt
edW
ellw
ritte
nov
ervi
ewof
the
atom
icpr
oces
ses
whi
char
ere
leva
ntfo
rth
eco
mpu
tatio
nof
phot
oion
ized
plas
mas
(an
area
whi
chw
ew
illno
tbe
able
todi
scus
sin
deta
ildu
eto
time
reas
ons)
;ava
ilabl
eat
http://arxiv.org/abs/0801.1011.
shor
ter
high
erlo
wer
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eter
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ton
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aves
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Visual
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
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C
M LMVA
AI
AD
R
E
LGE
1–7
Mul
tiwav
elen
gth
Ast
roph
ysic
s2
Ele
ctro
mag
netic
Spe
ctru
m,I
I
As
we
allk
now
,lig
htca
nbe
char
acte
rized
by
Wav
elen
gth
:λ,m
easu
red
inm
,mm
,cm
,nm
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qu
ency
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easu
red
inH
z,M
Hz.
En
erg
y:E
,mea
sure
din
J,er
g,R
ydbe
rgs,
eV,k
eV,M
eV,G
eV.
Tem
per
atu
re:T
,mea
sure
din
K.
The
sequ
antit
ies
are
rela
ted:
λν
=c
E=
hν
T=
E/k
(1.1
)
whe
rec
=29
9792
458
ms−
1∼
2.99
8×
1010
cms−
1(1
.2)
h=
6.62
6069
3(11
)×
10−
34J
s∼
6.62
6×
10−
27er
gs
(1.3
)
k=
1.38
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5(24
)×
10−
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K−
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×10
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erg
K−
1(1
.4)
Con
stan
tsar
e20
02C
OD
ATA
valu
es,http://physics.nist.gov/cuu/Constants/index.html
unce
rtai
nty
is1σ
inun
itsof
last
digi
tsho
wn.
1–7
Con
vers
ion
tabl
e(c
ourt
esy
Eur
eka
Sci
entifi
c,www.eurekasci.com
):
From
⇓To
=⇒
λ[Å
]λ[µ
m]
λ[c
m]
ν[H
z]E
[keV
]E
[erg
]
λ[Å
]1
10−
4λ
10−
8λ
3×
1018
/λ12
.4/λ
2×
10−
8/λ
λ[µ
m]
104λ
110
−4λ
3×
1014
/λ1.
24×
10−
3/λ
2×
10−
12/λ
λ[c
m]
108λ
104λ
13×
1010
/λ1.
24×
10−
7/λ
2×
10−
16/λ
ν[H
z]3×
1018
/ν3×
1014
/ν3×
1010
/ν1
4.14
×10
−18
ν6.
63×
10−
27ν
E[k
eV]
12.4
/E1.
24×
10−
3/E
1.24
×10
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2.42
×10
17E
11.
60×
10−
9E
E[e
rg]
2×
10−
8/E
2×
10−
12/E
2×
10−
16/E
1.51
×10
26E
6.24
×10
8E
1
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
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LGE
1–8
Mul
tiwav
elen
gth
Ast
roph
ysic
s3
Why
Mul
tiwav
elen
gth
Ast
rono
my?
,I
Str
uctu
reof
Act
ive
Gal
actic
Nuc
lei(
AG
N):
•su
perm
assi
vebl
ack
hole
(107
M⊙
)
•ac
cret
ion
disk
(M∼
1..
.2M
⊙yr
−1)
•la
rge
lum
inos
ity(L
∼10
10L⊙
)
•S
chw
arzs
child
radi
us2G
M/c
2∼
1A
U
I
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CO
DRI
L
AI
N
R
DN
XAEA
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C
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AI
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1–8
Mul
tiwav
elen
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Ast
roph
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s4
Why
Mul
tiwav
elen
gth
Ast
rono
my?
,II
Str
uctu
reof
Act
ive
Gal
actic
Nuc
lei(
AG
N):
•su
perm
assi
vebl
ack
hole
(107
M⊙
)
•ac
cret
ion
disk
(M∼
1..
.2M
⊙yr
−1)
•la
rge
lum
inos
ity(L
∼10
10L⊙
)
•S
chw
arzs
child
radi
us2G
M/c
2∼
1A
U
•of
ten
rela
tivis
ticje
ts,w
here
mat
eria
lis
acce
lera
ted
toth
esp
eed
oflig
ht
AG
Nw
ithje
ts:
quas
ars,
blaz
ars.
..
AG
Nw
ithou
tje
ts:
Sey
fert
gala
xies
Inth
efo
llow
ing
asan
exam
ple:
Cen
taur
usA
(NG
C51
28)
•on
eof
the
brig
htes
trad
io
sour
ces
inth
esk
y
•di
stan
ce:
11m
illio
nlig
htye
ars
•gi
ante
llipt
ical
gala
xy(m
ore
prop
erly
:S
0),m
erge
dw
ith
spira
lgal
axy
abou
t100
mill
ion
year
sag
o,re
mna
ntof
the
spira
lsee
nas
dust
lane
.
AG
Nar
eex
cept
iona
llygo
odex
ampl
esfo
rth
eim
port
ance
ofm
ulti-
wav
elen
gth
astr
onom
y.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
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C
M LMVA
AI
AD
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LGE
1–10
Mul
tiwav
elen
gth
Ast
roph
ysic
s6
Cen
taur
usA
Cen
A:V
LTK
ueye
n+F
OR
S2,
cour
tesy
ES
O
Opt
ical
:
The
rmal
emis
sion
from
star
s
and
gas,
i.e.,
brem
sstr
ahlu
ng
(fre
e-fr
eera
diat
ion)
,lin
e
emis
sion
,dus
tsca
tterin
g,..
.)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
1–11
Mul
tiwav
elen
gth
Ast
roph
ysic
s7
Cen
taur
usA
2MA
SS
,cou
rtes
yIP
AC
,Uni
v.M
assa
chus
etts
Nea
rIn
frar
ed:
The
rmal
emis
sion
,mai
nly
from
star
s,si
mila
rto
optic
al,b
utdu
stle
ss
appa
rent
=⇒
Opa
city
ofdu
stin
IRis
smal
ler.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
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LGE
1–12
Mul
tiwav
elen
gth
Ast
roph
ysic
s8
Cen
taur
usA
Spi
tzer
Spa
ceTe
lesc
ope,
cour
tesy
Cal
tech
/NA
SA
Mid
Infr
ared
(3.6
–8µ
m):
The
rmal
emis
sion
from
dust
star
tsto
dom
inat
e,
cont
ribut
ion
of
ther
mal
emis
sion
from
star
sst
ill
sign
ifica
nt.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
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LGE
1–13
Mul
tiwav
elen
gth
Ast
roph
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s9
Cen
taur
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ISO
,cou
rtes
yE
SA
-ES
TE
C
Far
Infr
ared
(7µ
m):
The
rmal
emis
sion
from
dust
Res
olut
ion
ofth
isim
age
isw
orse
than
the
prev
ious
Spi
tzer
tele
scop
eim
age.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
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LGE
1–14
Mul
tiwav
elen
gth
Ast
roph
ysic
s10
Cen
taur
usA
VLA
,cou
rtes
yN
RA
O
Rad
io(6
cm):
Syn
chro
tron
radi
atio
nfr
omje
ts
and
blac
kho
le.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
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LGE
1–14
Mul
tiwav
elen
gth
Ast
roph
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s11
Cen
taur
usA
VLA
/opt
ical
,cou
rtes
yS
TS
cI
Rad
io(6
cm):
Syn
chro
tron
radi
atio
nfr
omje
ts
and
blac
kho
le.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
1–15
Mul
tiwav
elen
gth
Ast
roph
ysic
s12
Cen
taur
usA
GA
LEX
,cou
rtes
yN
AS
A/C
alte
ch
UV
(30–
300
nm):
The
rmal
UV
emis
sion
from
youn
g
star
s(in
NE
corn
er)
Pho
toab
sorp
tion
and
abso
rptio
n
bydu
stby
dust
lane
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
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LGE
1–16
Mul
tiwav
elen
gth
Ast
roph
ysic
s13
Cen
taur
usA
Cha
ndra
,cou
rtes
yC
XC
X-r
ays
(2–1
0ke
V):
•S
ynch
rotr
onra
diat
ion
from
jet,
•C
ompt
oniz
edph
oton
sfr
om
blac
kho
le,
•ot
her
emis
sion
from
X-r
ay
bina
ries
and
back
grou
ndA
GN
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
1–16
Mul
tiwav
elen
gth
Ast
roph
ysic
s14
Cen
taur
usA
Cha
ndra
,cou
rtes
yC
XC
X-r
ays
(2–1
0ke
V):
•S
ynch
rotr
onra
diat
ion
from
jet,
•C
ompt
oniz
edph
oton
sfr
om
blac
kho
le,
•ot
her
emis
sion
from
X-r
ay
bina
ries
and
back
grou
ndA
GN
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
1–17
Mul
tiwav
elen
gth
Ast
roph
ysic
s15
Cen
taur
usA
CG
RO
-CO
MP
TE
L,co
urte
syM
PE
/H.S
tein
le
γ-r
ays
(1–3
0M
eV):
Com
pton
ized
sync
hrot
ron
radi
atio
n
from
jeta
nd/o
rbl
ack
hole
.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
1–18
Mul
tiwav
elen
gth
Ast
roph
ysic
s16
Cen
taur
usA
Ste
inle
(200
6,C
hin.
J.A
stro
n.A
stro
phys
.6(S
uppl
.1),
106)
Bro
ad-b
and
spec
trum
ofC
enA
:Spe
ctra
l
Ene
rgy
Dis
trib
utio
n
(SE
D)νf ν
isfla
t
=⇒
sim
ilar
ener
gy
outp
utat
all
wav
eban
ds!
Sho
wn
isa
νf ν
-plo
t,w
here
ν:
freq
uenc
y,f ν
:flu
xde
nsity
atfr
eque
ncy
ν(u
nits
off ν
are
erg
s−1
cm−
2H
z−1).
Sin
ce∫
ν 2 ν 1νf ν
dν
=∫
lnν 2
lnν 1
f νd
lnν
plot
ting
νf ν
ina
log-
log-
plot
give
sa
mea
sure
ofth
een
ergy
emitt
edpe
rfr
eque
ncy
deca
de.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–1
Rad
iatio
nan
dR
adia
tive
Tran
sfer
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–2
Ele
ctro
mag
netic
Wav
es1
Max
wel
l’sE
quat
ions
Incl
assi
cale
lect
rody
nam
ics,
elec
trom
agne
tism
isde
scrib
edby
Max
wel
l’seq
uatio
ns:
Cou
lom
b’s
law
:
∇·E
=
[
1
4πǫ 0
]
4πρ
(2.1
)
Law
ofIn
duct
ion
(Far
aday
):
∇×
E=−
[c]1 c
∂B ∂t
(2.2
)
Non
exis
tenc
eof
Mag
netic
Mon
opol
es(G
ilber
t):
∇·B
=0
(2.3
)
Am
père
’sla
w:
∇×
B=
[
cµ0
4π
]
4π cj
+
[
1 c
]
1 c
∂E ∂t
(2.4
)
Whe
recu
rren
t,j
,and
char
gede
nsity
,ρ,a
reco
uple
dby
the
equa
tion
ofco
ntin
uity
,
∂ρ
∂t
+∇
·j
=0
(2.5
)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–3
Ele
ctro
mag
netic
Wav
es2
EM
Wav
es,I
One
ofth
em
osti
mpo
rtan
tsol
utio
nsto
Max
wel
l’seq
uatio
nsis
the
elec
trom
agne
ticw
ave
in
vacu
um.
Vac
uum
mea
ns:
ρ=
0an
dj
=0
(2.6
)
Incg
s(“
Gau
ssia
n”)
units
,Max
wel
l’seq
uatio
nsth
enbe
com
e
∇·E
=0
∇×
E=−
1 c
∂B ∂t
∇·B
=0
∇×
B=
1 c
∂E ∂t
(2.7
)
As
show
non
the
hand
out,
thes
ear
eeq
uiva
lent
toth
eva
cuum
wav
eeq
uatio
ns:
∂2E
∂t2
−c2∇
2E
=0
and
∂2B
∂t2
−c2∇
2B
=0
(2.8
)
The
equa
tions
are
hom
ogen
eous
wav
eeq
uatio
ns.
1.N
ote
that
Max
wel
lfor
∂E
/∂t
and
∂B
/∂t
impl
yE
⊥B
.
2.G
ener
also
lutio
nsus
ually
obta
ined
usin
gF
ourie
rtr
ansf
orm
s.
2–3
Tode
rive
the
wav
eeq
uatio
ns,
look
atth
eM
axw
elle
quat
ions
inva
cuum
:
∇·E
=0
∇×
E=
−1 c
∂B ∂t
∇·B
=0
∇×
B=
1 c
∂E ∂t
(2.7
)
The
refo
re
∇×
(∇×
E)=
−1 c
∂ ∂t(∇
×B
)=
−1 c2
∂2E
∂t2
(2.9
)
But
∇×
(∇×
E)
=∇
(∇·E
)−
∇2E
(2.1
0)
and
furt
herm
ore∇
·E
=0
inva
cuum
.T
here
fore
∂2E
∂t2
−c2∇
2E
=0
(2.8
)
The
wav
eeq
uatio
nfo
rB
isob
tain
edin
asi
mila
rm
anne
r(s
ince
Max
wel
l’seq
uatio
nsar
ein
varia
ntw
ithre
spec
tto
the
chan
geE
→B
and
B→
−E
).
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–4
Ele
ctro
mag
netic
Wav
es3
EM
Wav
es,I
I
One
poss
ible
solu
tion
toth
ew
ave
equa
tions
∇2E
=1 c2
∂2E
∂t2
and
∇2B
=1 c2
∂2B
∂t2
(2.8
)
isgi
ven
by E(r
,t)
=a
1E
0ei(
ωt−
k·r
)an
dB
(r,t
)=
a2B
0ei(
ωt−
k·r
)(2
.11)
whe
rea
1,2
are
unit
vect
ors
spec
ifyin
gth
edi
rect
ions
ofE
and
B,a
ndth
ew
ave
vect
or,k
isre
late
dto
the
freq
uenc
y,ν
,and
wav
elen
gth,
λ,v
ia
|k|=
2π λ=
2πν c
=ω c
(2.1
2)
The
solu
tions
toE
q.2.
8ar
epl
ane
wav
estr
avel
ing
inth
edi
rect
ion
ofk
.
See
hand
outf
orpr
oof.
2–4
Plu
ggin
gth
ew
ave
equa
tions
into
Max
wel
l’seq
uatio
nsal
low
sus
toun
ders
tand
afe
wfu
rthe
rpr
oper
ties
ofel
ectr
omag
netic
radi
atio
n,w
hich
will
beof
grea
tus
ela
ter
on.
Inor
der
tom
ake
prog
ress
,w
ene
edth
efo
llow
ing
vect
orid
entit
ies:
∇·(
fa)=
f(∇
·a)+
(∇f)·a
=(∇
f)·a
(2.1
3)
∇×
(fa)=
f(∇
×a)+
(∇f)×
a=
(∇f)×
a(2
.14)
whe
reth
e2nd
equa
tion
hold
sfo
ra
=co
nst..
For
apl
ane
wav
e,th
eab
ove
equa
tions
give
(not
eth
at∇
wor
kson
ron
ly!)
∇·E
=∇
(
E0ei(
ωt−
k·r))
·a1
=−
iE0ei(
ωt−
k·r) (
k·a
1)
(2.1
5)
But
inva
cuum
,ρ=
0an
dth
eref
ore
Cou
lom
b’s
law
(Eq.
(2.1
))gi
ves
∇·E
=0
(2.1
6)
and
ther
efor
e
k·a
1=
0(2
.17)
Asi
mila
rca
lcul
atio
nfo
rB
show
sth
at
k·a
2=
0(2
.18)
Fur
ther
mor
e,
∇×
E=
∇(
E0ei(
ωt−
k·r))
×a
1=
−iE
0ei(
ωt−
k×
r)(k
×a
1)
(2.1
9)
But
beca
use
ofFa
rada
y’s
law
ofin
duct
ion
(Eq.
2.2)
,
∇×
E=
−1 c
∂B ∂t
=−
1 c
∂ ∂t
(
a2B
0ei(
ωt−
k·r))
=−
iω cB
0ei(
ωt−
k·r) a
2(2
.20)
and
ther
efor
e
E0(k
×a
1)
=ω c
B0a
2(2
.21)
and
anal
oguo
usly
B0(k
×a
2)
=−
ω cB
0a
1(2
.22)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–5
Ele
ctro
mag
netic
Wav
es4
EM
Wav
es,I
II
As
show
non
the
hand
out:
•a
1,a
2,a
ndk
are
orth
ogon
alto
each
othe
r
B
k
E
•E
0=
ωB
0/k
can
d
B0
=ωE
0/k
c,su
chth
at
E0
=(
ω kc)
2
E0
(2.2
3)
or
ω2
=c2
k2
(2.2
4)
But
sinc
ek
=2π
/λ=
2πν/c
λν
=c
(2.2
5)
•B
ecau
seof
the
abov
e,in
Gau
ssia
nun
its E0
=B
0(2
.26)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–6
Ele
ctro
mag
netic
Wav
es5
Con
tinui
tyan
dLo
rent
zF
orce
Ass
ume
allc
harg
esar
epo
intc
harg
es,t
hen
ρ(x
,t)
=∑
i
q iδ(
x−
xi(t)
)(2
.27)
j(x
,t)
=∑
i
q iv
iδ(x
−x
i(t)
)(2
.28)
Pos
ition
sx
i(t)
are
com
pute
dfr
om
dx
i
dt
=v
ian
ddp
i
dt
=F
i(2
.29)
whe
re
pi=
γim
ivi
and
γi=
1√
1−
(vi/
c)2
(2.3
0)
and
the
forc
eF
isty
pica
llydo
min
ated
byth
eLo
rent
zfo
rce:
F=
q
(
E+
v c×
B
)
(2.3
1)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–7
Ele
ctro
mag
netic
Wav
es6
Poy
ntin
g’s
theo
rem
The
EM
field
and
part
icle
sca
ndo
wor
kon
each
othe
r.
The
flow
ofen
ergy
due
toth
isw
ork
isgi
ven
byP
oynt
ing’
sth
eore
m,
∂ ∂t
(
E2
8π+
B2
8π
)
=∇
·S−
j·E
(2.3
2)
orin
wor
ds:
The
chan
geof
the
ener
gyde
nsity
(E2+
B2)/
8πat
ace
rtai
npo
sitio
n
equa
lsth
ew
ork
dW
=j·E
dt
done
onm
atte
rby
the
EM
field
atth
at
posi
tion
plus
the
dive
rgen
ceof
the
Poy
ntin
gve
ctor
.
whe
reth
eP
oynt
ing
vect
or,S
,is
defin
edby
S=
c 4π(E
×B
)(2
.33)
The
units
ofS
are
erg
cm−
2s−
1,i
.e.,
pow
ertr
ansp
orte
dth
roug
han
area
,i.e
.,S
=dW
/dA
),an
d(E
2+
B2)/
8πis
anen
ergy
dens
ity,i
.e.,
has
units
erg
cm−
3.
2–7
Inor
der
tode
rive
Poy
ntin
g’s
theo
rem
,le
t’slo
okat
the
mec
hani
calw
ork
done
ona
part
icle
iby
the
elec
tric
and
the
mag
netic
field
s.T
his
wor
kis
give
nby
vi·F
i=
q iv
i·(
E+
vi c×
B)
=q i
vi·E
(2.3
4)
sinc
ev
i⊥
(vi×
B).
Sum
min
gov
eral
lpar
ticle
sat
ace
rtai
npo
sitio
nth
engi
ves
P=
∑
i
δ(x−
xi(
t))q
ivi·E
=j·E
(2.3
5)
Bec
ause
ofA
mpè
re’s
law
,
∇×
B=
4π cj
+1 c
∂E ∂t
⇒j
=c 4π
(∇×
B)−
1 4π
∂E ∂t
(2.4
)
we
find
j·E
=c 4π
(
E·(∇
×B
)−
1 cE
·∂E ∂t
)
(2.3
6)
=c 4π
E·(∇
×B
)−
1 8π
∂(E
2)
∂t
(2.3
7)
Ana
logo
usly
to(a
×b)
·c=
(b×
c)·a
one
has
E·(∇
×B
)=
∇B·(B
×E
)=
−∇
B·(
E×
B)
(2.3
8)
whe
re∇
Bop
erat
eson
lyon
B.
The
refo
re
j·E
=−
c 4π∇
B·(E
×B
)−
1 8π
∂E
2
∂t
(2.3
9)
Not
eth
at
∇·(E
×B
)=
∇B·(E
×B
)+
∇E·(
E×
B)
(2.4
0)
The
refo
re,
add
(c/4
π)∇
E(E
×B
)to
Eq.
2.39
: j·E
=−
c 4π
(
∇B·(
E×
B)+
∇E(E
×B
)−
∇E(E
×B
))
−1 8π
∂E
2
∂t
(2.4
1)
=−
c 4π∇
·(E
×B
)+
c 4π∇
E(E
×B
)−
1 8π
∂E
2
∂t
(2.4
2)
2–7
but∇
E(E
×B
)=
(∇×
E)·B
(sim
ilar
toE
q.2.
38) =−
c 4π∇
·(E
×B
)+
c 4π(∇
×E
)·B
−1 8π
∂E
2
∂t
(2.4
3)
Cal
cula
te∇
×E
usin
gFa
rada
y’s
law
(Eq.
2.2)
,i.e
.,
c 4π(∇
×E
)·B
=−
1 4π
∂B ∂t·B
=−
1 8π
∂ ∂t(B
·B)
=−
1 8π
∂B
2
∂t
(2.4
4)
such
that
we
final
lyob
tain
Poy
ntin
g’s
theo
rem
−j·E
=c 4π∇
·(E
×B
)+
∂ ∂t
(
E2
8π+
B2
8π
)
(2.3
2)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–8
Rad
iatio
nQ
uant
ities
1
Flu
xD
ensi
ty,I
dAW
eno
wre
late
the
elec
trod
ynam
ic
quan
titie
ssu
chas
E,B
,orS
to
mea
sura
bles
.B
efor
ew
eca
ndo
this
,
need
toin
trod
uce
som
ede
finiti
ons.
Defi
nitio
n.E
nerg
yflu
x,F
,is
defin
ed
asth
een
ergy
dE
pass
ing
thro
ugh
area
dA
intim
ein
terv
aldt:
dE
=F
dA
dt
(2.4
5)
Uni
tsof
Far
eer
gcm
−2
s−1.
Fde
pend
son
the
orie
ntat
ion
ofdA
,and
can
also
depe
ndon
the
freq
uenc
y.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–9
Rad
iatio
nQ
uant
ities
2
Flu
xD
ensi
ty,I
I
Flu
xfr
oman
isot
ropi
cra
diat
ion
sour
ce,i
.e.,
aso
urce
emitt
ing
equa
lam
ount
sof
ener
gyin
alld
irect
ions
.
Sph
eric
ally
sym
met
ricst
ars
are
isot
ropi
cra
diat
ion
sour
ces,
othe
ras
tron
omic
alob
ject
ssu
chas
,e.g
.,A
ctiv
eG
alac
ticN
ucle
i,ar
eno
t.
r1 r 2
Bec
ause
ofen
ergy
cons
erva
tion,
flux
thro
ugh
two
shel
lsar
ound
sour
ceis
iden
tical
:
4πr2 1
F(r
1)
=4π
r2 2F
(r2)
(2.4
6)
and
ther
efor
ew
eob
tain
the
inve
rse
squa
rela
w,
F(r
)=
cons
t.
r2(2
.47)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–10
Rad
iatio
nQ
uant
ities
3
Spe
cific
Inte
nsity
,I
Bet
ter
desc
riptio
nof
radi
atio
n:en
ergy
carr
ied
alon
gin
divi
dual
ray.
dΩ
n
Pro
blem
:R
ays
are
infin
itely
thin
=⇒
No
ener
gyca
rrie
dby
them
...
=⇒
Look
aten
ergy
pass
ing
thro
ugh
area
dA
(with
norm
aln
)in
all
rays
goin
gin
tosp
atia
ldire
ctio
n
dΩ
.
The
spec
ific
inte
nsity
,Iν,i
nth
eba
nd
ν,..
.,ν
+dν
isde
fined
via
dE
=I ν
dA
dtdΩ
dν
(2.4
8)
I νis
mea
sure
din
units
ofer
gs−
1cm
−2
sr−
1H
z−1
and
depe
nds
onlo
catio
n,
dire
ctio
n,an
dfr
eque
ncy.
Inan
isot
ropi
cra
diat
ion
field
,Iν
=co
nst.
for
alld
irect
ions
.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–11
Rad
iatio
nQ
uant
ities
4
Spe
cific
Inte
nsity
,II
Usi
ngth
ede
finiti
onof
I,w
eca
nca
lcul
ate
the
netfl
uxin
adi
rect
ion
n.
n
dΩ
θ
Con
trib
utio
nto
flux
indi
rect
ion
nfr
om
flux
into
dire
ctio
ndΩ
:
dF
ν=
I νco
sθ
dΩ
(2.4
9)
Inte
grat
eov
eral
lang
les
toob
tain
the
tota
lflux
:
Fν
=
∫
4πsr
I νco
sθ
dΩ
=
∫
π
θ=0
∫
2π
0
I ν(θ
,φ)co
sθ
sin
θdθ
dφ
(2.5
0)
2–11
θ
Esp
ecia
llyin
the
theo
ryof
stel
lar
atm
osph
eres
,w
here
ofte
nde
als
with
the
radi
ativ
etr
ansp
ort
thro
ugh
asl
abof
mat
eria
land
can
assu
me
cylin
dric
alsy
mm
etry
,one
writ
es
µ=
cosθ
(2.5
1)
whe
reθ
isth
ean
gle
betw
een
the
z-d
irect
ion
(“up
”-“d
own”
dire
ctio
n)an
dth
edi
rect
ion
ofth
elig
htra
y.T
hen
dµ
dθ
=−
sin
θ(2
.52)
and
Fν
=
∫
−1
µ=
+1
∫
2π
φ=
0I ν
(µ,θ
)(−
µ)si
nθ
dµ
sin
θdφ
(2.5
3)
=
∫
+1
µ=−
1
∫
2π
φ=
0I µ
(µ,θ
)µdµ
dφ
(2.5
4)
taki
ngin
toac
coun
tthe
cylin
dric
alsy
mm
etry
,
=2π
∫
+1
−1
I ν(µ
)µdµ
(2.5
5)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–12
Rad
iatio
nQ
uant
ities
5
Spe
cific
Inte
nsity
,III
The
spec
ific
inte
nsity
isco
nsta
ntal
ong
the
line
ofsi
ght.
Pro
of:
Con
side
rth
efo
llow
ing
figur
e:
R
dAdA
12
Ene
rgy
carr
ied
thro
ugh
area
dA
1an
d
dA
2is
give
nby
dE
1=
I 1dA
1dtdΩ
1dν
(2.5
6)
dE
2=
I 2dA
2dtdΩ
2dν
(2.5
7)
whe
redΩ
1:
solid
angl
esu
bten
ded
by
dA
2at
dA
1,a
ndvi
ceve
rsa:
dΩ
1=
dA
2/R
2(2
.58)
dΩ
2=
dA
1/R
2(2
.59)
Ene
rgy
cons
erva
tion
impl
ies
dE
1=
dE
2,i
.e.,
I 1dA
1dt
dA
2
R2
dν
=I 2
dA
2dt
dA
1
R2
dν
(2.6
0)
that
is:I 1
=I 2
=co
nst..
QE
D.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–13
Rad
iatio
nQ
uant
ities
6
Spe
cific
Inte
nsity
,IV
I=
cons
t.al
ong
ara
ydo
esno
tcon
trad
ictt
hein
vers
esq
uare
law
!
θR
c
θP
r
Pro
of:
Ass
ume
sphe
reof
unifo
rmbr
ight
ness
,
B,t
heflu
xm
easu
red
atpo
intP
isth
eflu
x
from
allv
isib
lepo
ints
ofsp
here
:
F=
∫
Ico
sθ
dΩ
(2.6
1)
=B
·∫
π
0
∫
θ c
0si
nθ
cosθ
dθ
dφ
(2.6
2)
whe
resi
nθ c
=R
/r.
The
refo
re
F=
2πB
∫
arcs
in(R
/r)
0si
nθ
cosθ
dθ
(2.6
3)
butb
ecau
se∫
α 0si
nx
cosx
dx
=1 2si
n2α
,
=2π
B·
1 2
(
R r
)
2
=πB
(
R r
)
2
∝1 r2
(2.6
4)
=⇒
inve
rse
squa
rela
wis
cons
eque
nce
ofde
crea
sing
solid
angl
eof
obje
cts!
2–13
One
cons
eque
nce
ofE
q.2.
64is
that
the
flux
onth
esu
rfac
eof
aso
urce
ofun
iform
brig
htne
ssis
give
nby
F=
πB
(2.6
5)
The
refo
re,
espe
cial
lyfo
rst
ella
rat
mos
pher
es,
one
som
etim
esde
fines
the
astr
ophy
sica
lflux
,
F:=
F π(2
.66)
such
that
for
star
s(w
hich
roug
hly
have
unifo
rmsu
rfac
ebr
ight
ness
),F
=B
(2.6
7)
Be
awar
eof
this
sour
ceof
poss
ible
conf
usio
n!
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–14
Rad
iatio
nQ
uant
ities
7
Ene
rgy
Den
sity
and
Mea
nIn
tens
ity
The
last
impo
rtan
trad
iatio
nqu
antit
y(f
orth
em
omen
t)is
the
ener
gyde
nsity
,uν.
For
ace
rtai
ndi
rect
ion,
Ω,a
ndvo
lum
eel
emen
tdV
,uν
isde
fined
via
dE
=u
ν(Ω
)dV
dΩ
dν
(2.6
8)
But
for
light
,the
volu
me
elem
entc
anbe
writ
ten
as
dV
=c
dt·
dA
(2.6
9)
such
that
Eq.
2.68
beco
mes
dE
=cu
ν(Ω
)dΩ
dtdν
(2.7
0)
Com
pare
this
toth
ede
finiti
onof
the
inte
nsity
:
dE
=I ν
dA
dtdΩ
dν
(2.4
8)
The
refo
re,
uν(Ω
)=
I ν/c
(2.7
1)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–15
Rad
iatio
nQ
uant
ities
8
Ene
rgy
Den
sity
and
Mea
nIn
tens
ity
The
tota
lene
rgy
dens
ityat
freq
uenc
yν
isth
en
uν
=
∫
4πsr
uν(Ω
)dΩ
=1 c
∫
4πsr
I ν(Ω
)dΩ
=:
4π cJ
ν(2
.72)
whe
reth
em
ean
inte
nsity
isde
fined
by
Jν
=1 4π
∫
4πsr
I ν(Ω
)dΩ
(2.7
3)
Not
eth
atfo
ran
isot
ropi
cra
diat
ion
field
,Iν(Ω
)=
I ν,I
ν=
Jν.
The
tota
lrad
iatio
nde
nsity
isob
tain
edby
inte
grat
ion:
u=
∫
uν
dν
=4π c
∫
∞
0
Jν
dν
(2.7
4)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–16
Rad
iativ
eTr
ansf
er1
Intr
oduc
tion
Afte
rde
finin
gra
diat
ion
quan
titie
s,w
eca
nno
wst
udy
the
tran
spor
tofr
adia
tion
(“ra
diat
ive
tran
sfer
”).
Thr
eeef
fect
sin
fluen
cera
diat
ion
1.em
issi
onof
radi
atio
n
2.ab
sorp
tion
ofra
diat
ion
3.sc
atte
ring
Sca
tterin
gca
nbe
trea
ted
asco
mbi
natio
nof
emis
sion
and
abso
rptio
n,w
illno
t
talk
abou
ttha
ther
e.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–17
Rad
iativ
eTr
ansf
er2
Em
issi
on
Rad
iatio
nca
nbe
emitt
ed,a
ddin
gen
ergy
toth
ebe
am:
dE
ν=
j νdV
dΩ
dt
(2.7
5)
whe
rej ν
isco
effic
ient
for
spon
tane
ous
emis
sion
,i.e
.,en
ergy
adde
dpe
run
it
volu
me,
unit
angl
e,an
dun
ittim
e,un
itsof
j νar
eer
gcm
−3
s−1
sr−
1H
z−1.
The
chan
gein
inte
nsity
is
dI ν
=j ν
ds
(2.7
6)
(writ
ing
dV
=dA
ds)
Not
eth
atj ν
depe
nds
ondi
rect
ion!
For
anis
otro
pic
emitt
eron
ly(e
.g.,
rand
omly
orie
nted
sour
ces)
:
j ν=
1 4πP
ν(2
.77)
whe
reP
νis
the
radi
ated
pow
erpe
run
itvo
lum
e.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–18
Rad
iativ
eTr
ansf
er3
Abs
orpt
ion
Rad
iatio
nis
also
abso
rbed
.
Med
ium
with
part
icle
num
ber
dens
ityn
(cm
−3),
each
havi
ngef
fect
ive
abso
rbin
g
area
(cro
ssse
ctio
n)σ
ν(c
m2):
•N
umbe
rof
abso
rber
s:n
dA
ds
•To
tala
bsor
bing
area
:nσ
νdA
ds
=⇒
Ene
rgy
abso
rbed
outo
fbea
m:
−dI ν
dA
dΩ
dtdν
=Inσ
νdA
ds
dΩ
dtdν
(2.7
8)
or
dI ν
=−
nσ
νI ν
ds
=:−
ανI ν
ds
(2.7
9)
whe
reα
ν:
abso
rptio
nco
effic
ient
(uni
tscm
−1).
2–18
Inth
est
udy
ofst
ella
rin
terio
rs,o
neof
ten
enco
unte
rsth
ean
gle
inte
grat
edem
issi
vity
,ǫν,d
efine
das
the
emitt
eden
ergy
per
unit
mas
s,
dE
ν=
ǫ νρ
dV
dtdt
dΩ
4π(2
.80)
whe
reth
efa
ctor
dΩ
/4π
isth
efr
actio
nof
ener
gyem
itted
into
dire
ctio
ndΩ
.
For
isot
ropi
cem
issi
on,
ǫ ν=
ρ
4πj ν
(2.8
1)
Like
wis
e,th
eab
sorp
tivity
isal
som
easu
red
per
unit
mas
sof
abso
rbin
gm
ater
ial,
thro
ugh
αν
=nρ
σν ρ
=ρ·nσ
ν
ρ=
:ρκ
ν(2
.82)
whe
reκ
νis
the
mas
sab
sorp
tion
coef
ficie
nt,o
rth
eop
acity
coef
ficie
nt,a
ndha
sun
itsof
cm2g−
1.
I
EF
CO
DRI
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AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–19
Rad
iativ
eTr
ansf
er4
Equ
atio
nof
radi
ativ
etr
ansf
er
Com
bini
ngE
qs.2
.76
and
2.79
give
s
dI ν ds
=j ν
−α
νI ν
(2.8
3)
the
equa
tion
ofra
diat
ive
tran
sfer
(RT
)
For
pure
emis
sion
,αν
=0,
such
that
dI ν ds
=j ν
(2.8
4)
Sep
arat
ion
ofva
riabl
esan
das
sum
ing
the
path
star
tsat
s=
0gi
ves:
I ν(s
)=
I ν(0
)+
∫
s
0
j ν(s
′ )ds′
(2.8
5)
For
pure
emis
sion
,the
brig
htne
ssin
crea
seis
equa
lto
the
inte
grat
ed
emis
sivi
tyal
ong
the
line
ofsi
ght.
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–20
Rad
iativ
eTr
ansf
er5
Opt
ical
Dep
th,I
For
pure
abso
rptio
n,E
q.(2
.83)
isdI ν ds
=−
ανI ν
(s)
(2.8
6)
Atd
epth
sof
am
ediu
mirr
adia
ted
with
I(s
=0)
=I 0
,sep
arat
ion
ofva
riabl
esgi
ves
I ν(s
)=
I 0ex
p
(
−
∫
s
0α
ν(s
′ )ds′
)
=:I 0
e−τ ν
(2.8
7)
For
pure
abso
rptio
n,th
ein
tens
ityde
crea
ses
expo
nent
ially
with
the
optic
alde
pth.
The
optic
alde
pth,
τ,i
sde
fined
by
τ ν(s
)=
∫
s
0α
ν(s
′ )ds′
=
∫
s
0n(s
′ )σ
νds′
=nσ
νs
(2.8
8)
whe
reth
ela
stst
epis
valid
for
aho
mog
eneo
usm
ediu
m.
τis
cent
ralt
ode
scrib
ing
mos
teffe
cts
inR
T.
τ>
1=⇒
“med
ium
isop
tical
lyth
ick
or“o
paqu
e”
τ<
1=⇒
“med
ium
isop
tical
lyth
inor
“tra
nspa
rent
”
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–21
Rad
iativ
eTr
ansf
er6
Opt
ical
Dep
th,I
I
Exp
onen
tiala
bsor
ptio
nla
w=⇒
Pro
babi
lity
for
phot
onto
trav
eldi
stan
ce>
τ:
exp(−
τ).
=⇒
Mea
nop
tical
dept
htr
avel
ed:
〈τν〉
=
∫
∞
0
τ νex
p(−
τ ν)dτ ν
=1
(2.8
9)
The
path
leng
thco
rres
pond
ing
toτ
=1
isca
lled
the
mea
nfr
eepa
th,
τ ν=
nσ
νl ν
=⇒
〈l〉
=1
nσ
ν(2
.90)
Inan
inho
mog
eneo
usm
ediu
m,t
hem
ean
free
path
isde
fined
loca
lly.
Exa
mpl
e:In
the
cent
erof
the
Sun
,ρ∼
150
gcm
−3,(
wro
ngly
)as
sum
ing
pure
H,t
his
mea
ns
n∼
9×
1025
part
icle
scm
−3.
The
cros
sse
ctio
nis
onth
eor
der
of10
−25
cm2
(see
late
r)=⇒
〈l〉 ⊙
∼1
mm
.C
ompa
reth
isto
aso
lar
radi
usof
7×
1010
cm..
.
I
EF
CO
DRI
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AI
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XAEA
ESII
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M LMVA
AI
AD
R
E
LGE
2–22
Rad
iativ
eTr
ansf
er7
For
mal
Sol
utio
n
Itis
usef
ulto
writ
eR
Tin
term
sof
τ.
τ ν=
∫
s 0α
ν(s
′ )ds′
impl
ies
ds
=dτ ν
/αν,s
uch
that
αν
dI ν
dτ ν
=j ν
−α
νI ν
(2.9
1)
and
ther
efor
e,dI ν
dτ ν
=j ν α
ν−
I ν=
:S
ν−
I ν(2
.92)
whe
reS
νis
calle
dth
eso
urce
func
tion.
One
can
obta
ina
form
also
lutio
nof
the
tran
sfer
equa
tion
inte
rms
ofS
(see
man
uscr
ipt)
:
I ν(τ
ν)
=I ν
(0)e
−τ ν
+
∫
τ ν
0e−
(τν−
τ′ ν) S
(τ′ ν)dτ′ ν
(2.9
3)
Inte
rpre
tatio
n:
The
inte
nsity
emer
ging
from
anab
sorb
ing
med
ium
equa
lsth
eirr
adia
ted
inte
nsity
,cor
rect
edfo
r
abso
rptio
npl
usth
e(a
bsor
bed)
inte
grat
edso
urce
cont
ribut
ions
.
Sin
ceα
νca
nco
ntai
nef
fect
sfr
omst
imul
ated
emis
sion
(∝I ν
,see
late
r),t
hefo
rmal
solu
tion
can
notb
eob
tain
edfo
ran
ym
eani
ngfu
lphy
sica
lsys
tem
.
2–22
The
deriv
atio
nof
Eq.
2.93
isst
raig
htfo
rwar
dal
gebr
a:
dI ν
dτ ν
=S
ν−
I ν(2
.94)
eτν
dI ν
dτ ν
=eτ
ν
Sν−
eτν
I ν(2
.95)
eτν
dI ν
dτ ν
+eτ
ν
I ν=
eτν
Sν
(2.9
6)
d dτ ν
(eτν
I ν)=
eτν
Sν
(2.9
7)
Sep
arat
ion
ofva
riabl
esgi
ves
eτν
I ν(τ
ν)−
I ν(0
)=
∫
τν
0eτ
′ νS
ν(τ
′ ν)dτ′ ν
(2.9
8)
such
that
final
ly
I ν(τ
ν)
=I ν
(0)e
−τν
+
∫
τν
0e−
(τν−
τ′ ν) S
(τ′ ν)dτ′ ν
(2.9
3)
I
EF
CO
DRI
L
AI
N
R
DN
XAEA
ESII
C
M LMVA
AI
AD
R
E
LGE
2–23
Rad
iativ
eTr
ansf
er8
For
mal
Sol
utio
n
Ifth
era
diat
ion
isin
com
plet
eth
erm
odyn
amic
aleq
uilib
rium
,not
hing
isal
low
edto
chan
ge,i
.e.,
Eq.
2.92
read
s
dI ν
dτ ν
=S
ν−
I ν=
0(2
.99)
such
that
Inth
erm
odyn
amic
equi
libriu
m:S
ν=
I ν=
Bν
whe
reB
νis
the
Pla
nck
func
tion,
Bν
=2h
ν3
c21
exp(h
ν/k
T)−
1(3
.31)
tobe
deriv
edne
xt.
Ifth
eas
sum
ptio
nS
ν=
Bν
ism
ade
loca
lly,o
nesp
eaks
oflo
calt
herm
odyn
amic
equi
libriu
m,L
TE
.