Download Homework 2 key: Radiation processes, Larmor formula

Radio astronomy 29:186 Spring 2012
Homework 2 key: Radiation processes, Larmor formula
Define useful constants, conversions (N.B. not all are needed for any given assignment)
 23 J
kB  1.38 10

5
σsb  5.67 10
K
 34
h  6.6260693 10
26
cm  s K  sr
W
 26

RE  6378 km
3
mJy  10
2
 Jy
m  Hz
 31
 27
 kg
 19
J
mp  1.67 10
3
amp
 19
 kg
qe  1.602 10
6
keV  10  eV
 7 newton

2
μ0  4 π 10
8
AU  1.5 10  km
4
pc  3.06 10  m
Jy  10
me  9.109 10
eV  1.6 10
2
16
 joule sec
Lsun  4 10  W
erg

MeV  10  eV
 coul
5
Rsun  7 10  km
 12 farad
ε0  8.85 10

m
1.[1] Estimate the number density of cosmic microwave background (CMB) photons,
assuming (correctly) that the CMB radiation originated from a nearly perfect black
body emitter at a temperature of 2.72 K. Express in cm-3. Hint: Find the energy
density, and assume each photon has exactly kT energy.
T  2.72 K
ucmb 
4
4
 σsb T
c
Ecmb  kB T
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 13 erg

3
ucmb  4.141  10
ncmb 
ucmb
Ecmb
1
cm
3
3
ncmb  1.103  10  cm
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2. [1] The Green Bank telescope (diameter = 100m) switches between a radio source
unknown flux density and a resistive load with a physical temperature T = 250K. The
power received by the radio source and the resistor are equal. What is the flux density
of the radio source (in Jansky)? Assume the telescope collects 70% of the incident flux
of the source.
T  250 K
Pres  3.45  10

Hz
π 2
Aeff  0.7  d
4
d  100 m
S 
 21 W
Pres  kB T
Pres
Aeff
S  62.753 Jy
3. [1] A ball with a total charge of 1 coulomb and a mass 1 kg is dropped from the top of a
tall building of height 100 m. What is the total power radiated as a function of height? (Ignor
air friction).
Q  1 coul
a  9.8
m
2
s
2 2
2 Q a
1
P  

3 c3 4 π ε0
P  2.137  10
 14
W
Power is constant until it hits
ground!
4. [2] This problem calculates the power radiated from thermal electrons in the lower solar
corona. The plasma has a mean temperature 105 K, and the electrons are immersed in a
magnetic field of field strength B = 1 Gauss.
(a) What is the mean velocity of the electrons?
5
T  10  K
E 
3
k T
2 B
V 
2
E
me
3 km
V  2.132  10 
sec
(b) What is the mean acceleration of each electron?
B  1 gauss
F  qe V B
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a 
F
me
13 m
2
a  3.749  10 
s
2
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(c) What is the total power radiated per electron?
2 2
2 qe  a
1
Pelec  

3 c3 4 π ε0
 27
Pelec  8.027  10
W
(d) Assume a mean density 1012 elec/cm3 and a thickness 0.1Rsun starting at the base
at the solar photosphere. What is the total radiated power? Compare with the
observed solar x-ray luminosirty (Lx = 1027 ergs/sec)
5
Rsun  7 10  km
12
3
ne  10  cm
ΔR  0.1 Rsun
27 erg
Pxray  10 
s
2
Volume  4 π Rsun  ΔR
Ne  Volume ne
44
Ne  4.31  10
25 erg
Ptotal  Pelec Ne Ptotal  3.46  10 
sec
Ptotal
Pxray
 3.46 %
5.[2] A linear accelerator of length 10 m uniformly accelerates protons to kinetic
energy 100 MeV. Ignore relativistic effects.
2
L  10 m
E  10  MeV
(a) What is the power radiated by each proton (Watts)?
vfinal 
2
E
mp
vfinal  0.462 c
2
a 
vfinal
2 L
14 m
2
a  9.581  10
s
2 2
2 qe  a
1
P  

3 c3 4 π ε0
P  5.241  10
 24
W
(b) What fraction of the energy imparted to the protons is lost to radiation?
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2 L
7
t  1.445  10 s
a
t 
Erad  P t
total energy radiated: Note that
power radiated is constant along
path since acceleration is constant
 18
Erad  4.733  10
Erad
E
 MeV
 18
 4.733  10
%
Not much!
(b) Plot the normalized (1= max) power pattern of the radiation [use a polar plot,
indicate the direction of motion of the protons]. Tip:If using MathCAD, the poalr
plot option will be useful
P ( θ)  sin ( θ)
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2
θ  0 0.001  2 π
4
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120
90
1.2
60
1
0.8
150
30
0.6
0.4
0.2
P ( θ) 180
0
0
210
Proton
velocity is
along θ = 0
330
240
300
270
θ
6. [2] A white dwarf star with surface temperature 85,000K and radius = 1 Earth radius i
surrounded by an ionized sphere shell of HII gas of radius 0.1 pc. Assuming the tempera
the gas is 10,000K.
(a) What fraction of the star's radiation is capable of ionizing neutral hydrogen?
Plot the Planck function vs. frequency; indicate the region of ionizing photons
with a vertical line (use the 'marker' option in MathCAD plots).
3
h ν 
ν 
B ( ν T)  2
  exp  h
 1

2
c 
 kB T  
νLy 
c
912 Angstrom
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1
Twd  85000 K
ν  0.1 νLy 0.11 νLy  10 νLy
5
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4
1.510
110

1
4

B ν T wd
510
5
0
0
2
4
6
8
ν
νLy




100 νLy
1 νLy






B ν Twd dν
 84.573 %
100 νLy
0.01  νLy


B ν Twd dν
(b) What is the mean electron density in the ionized sphere? Assume the
recombination coefficient is α = 3 x 10-14 cm-3 sec-1.
Rs  0.5 pc
B ( T) 
Rwd  RE
σsb 4
T
π
ELy  13.6 eV
NLy 
0.85Lwd
ELy
2
3
ne  10  cm
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 14
αH  3 10
Lwd  B ( T)  4 π Rwd
2
3
1
 cm  sec
Lwd  2.306 Lsun
RHII  0.1 pc
44 1
NLy  3.604  10
s
initial guess value
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Given
 3 NLy 

RHII = 

 4 π α  n 2 
H e 

1
3
 
3
Find ne  316.373 cm
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4 π
3
27
 ΔR  1.437  10 L
3
2
29
2 π Rsun ΔR  2.155  10
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L
8
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is
ature of
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