Astro 405/505, fall semester 2007
Homework, 4th set, solutions.
Problem 7: Radiation spectra under LTE
The supernova remnant subtends a solid angle element
d
Ω
= 1
.
2
·
10

6
sr.
The intensity
therefore is
I
ν
=
F
ν
d
Ω
= 1
.
3
·
10

13
erg
/
cm
2
/
s
/
Hz
/
sr
which we need to compare with the Planckian, because that is the solution of the radiation
transport equation for LTE and large optical depth.
B
ν
(
T
) =
2
h ν
3
c
2
1
exp
±
hν
kT
²

1
⇒
B
100
(
T
) = (1
.
5
·
10

23
cgs)
1
exp
±
hν
kT
²

1
I
ν
=
B
100
(
T
)
requires
hν
¿
kT
or, more precisely,
kT
= (8
.
7
·
10
9
)
hν
⇒
T
’
4
·
10
7
K
The radiation maximum occurs at about
hν
’
3
kT
or about
2
·
10
18
Hz.
The emission is not necessarily optically thick, in which case the radiating material can
only be hotter than calculated here.
Problem 8: Stellar emission
This is what we did in session 7, where we derive the intensity as a function of aspect angle
(Eq.7.4)
I
ν
(
μ, τ
z
= 0) =
1
μ
Z
∞
0
dτ
0
z
S
ν
exp

τ
0
z
μ
!
Here
μ
= cos
θ
=
~e
k
·
~e
n
which is unity for normal incidence (we look at the center of the sun)
and approaching zero for grazing incidence (we look at the edge of the sun). In addition our
assumption of LTE implies
S
ν
=
B
ν
(
T
)
.
a) If the temperature
T
is constant, then the source function can be pulled out of the integral
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 Fall '08
 RABE
 Physics, Derivative, Work, Radiation, photosphere

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