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Lecture 4
By Tom Wilson
Review page 1
Interferometers on next page
Rayleigh-Jeans:
S  2.65

T  0 (')

2
 (cm)
2
 0.074
T  0 2 ('')
2 mm
True if h  << kT
S = measured: if s < B, T=TMB
s = b, T = TS
In mm / sub mm usually calibrations give
TA* = “corrected antenna temperature” corrected for atmosphere,
and telescope efficiency for very extended source
TMB: corrected for atmosphere and beam efficiency
Review page 2
Grading Across the Aperture and Far E Field
as limit of interferometer
Above: the 2
antennas on
the earth’s
surface have a
different orientation
as a function
of time.
Below: the
ordering
of
correlated
data in
(u,v) plane.
Review page 3
Gridding and sampling in (u,v)
plane
Review page 5
RECEIVERS (MM)
Mix from sky frequency to IF frequency (4 GHz) and amplify signal
TRX  TM  LM  T1  LM 
T2
G1
2 LIMITS:
 TRMS 
k
TSYS
 

D
Review page 5
Analog Coherent Receiver Block Diagram
Time
Total Amplification=1016
Frequency, f
Review page 6
In centimeters, the first stage of a receiver is a cooled transistor ampifier, a
HEMT (InP, GaAs…). For HEMTs, TRX = 2 (nGHz ) with a minimum of 4 K,
perhaps. The minimum noise for a coherent receiver is h n / k or about 5 K
at 100 GHz.
With mixers, the Rx noise is usually Double Sideband (DSB). For spectral
lines want Single Sideband (SSB) , where TSSB =2 TDSB.
Bolometers- non coherent receivers. Noise quoted in NEP (watts Hz –1/2).
For a given system on a telescope, performance is frequently given as
“detectable source in Jy in 1 sec”. For HHT and 19 channel bolometer, 1 Jy
in 1 sec. SCUBA/JCM this is 4 x better (2 x collecting area, 2 x rx
efficiency). SCUBA has 37 beams at 0.87nm.
INTERPRETATION
Continuum dust thermal emission
S (mJy)
 4 mm
N H  193
.  10
 2 (")  z 

  b  TDUST
 zSun 
24
Review page 7
Free-Free (Bremstrahlung)
T  Te     8.235  102  Te 
 0.35
  GHz  EM    1
2
Synchrotron emission
if B  10G  B 
17.6

Hz 
G

Radiation frequency is increased by beaming 1/ and doppler 1/2, so critical
Frequency is
C 
3 2
   G  sin 
4
Find  
where  
: energy spectrum of cosmic rays
1
  1
2
=104 for 10 GHz
Lecture4 page 1
RELATE ATOMIC PHYSICS TO RADIO ASTRONOMY
•
Einstein A & B coefficients and their role in Equation of radiative transfer
A+B coefficients in a 2 level system. Start with:
Nu Aul+Bul Nu U=Blu Nl U
U=4/c I
After some manipulation, get
8  3
Aul 
 Bul
3
c
gl  Blu  gu  Bul
Inserting numerical values of physical constants:


gl 
1
N l  93.5  

4.8102  GHz
g u Aul (sec) 

Tex
1 e
3
GHz
3
Aul  1.165 10 11  GHz
 ul
h
N u g u  k Tex
 e
Nl gl


   TB dV


2
TB  T  (1  e  )
Lecture4 page 2
In terms of column density, Nl, get
N l  1.94 10 
3
3
 GHz
gl
Aul g u
  T  V dV
if
h   k  T
h
N u g u k Tex
T is defined by

e
Nl gl
GROUND STATE OF HYDOGEN
HI line from overlap of proton and electron wave functions—see next slide
In Q. M., allowed transitions only between wave functions of opposite parity:


  e   f r i 
S
P
D
F
N=1
(From H. E. White, ‘ Introduction to Atomic Physics’ )
Lecture4 page 4
Apply all this to HI: “21 cm line” hyperfine transition
0 = 1.420 405…GHz
Aul = 2.87 · 10-15 sec-1
A very non classical system!!
N t  N u  N l  3N l  N l
 1.4202   1 
N l  1.94 10  

 T dV
15     B
 2.89 10   3 
3
Show
N t  1824
.  1018   TdV
For HI, h · / k = 0.06 K
Lecture4 page 5
Similar transition are for D and 3He+
Frequency (0)
Aul (sec-1)
gu
gl
DI (deuterium)
327 MHz
4.65 ·10-17
4
2
3He+
8.665 GHz
1.95 ·10-12
1
3
EXCITATION OF 2 LEVEL SYSTEM
Competition of radiation and collisions
Answer:
TB  Tk  y
1 y
h   C21
h
y

   v  n
k A21
k
Tex 
where
For HI, Tex = Tk , if n >1 cm-3
Lecture4 page 6
HI Clouds assumed to be in pressure equilibrium
HI: Used to obtain dynamics of galaxies, “HI masses” of galaxies, map
rotation curve of our galaxy, …
See absorption line if TBG > Tex (HI) = Tk , geometry and cloud size
relative to background continuum plays a role - - complex but solvable !
A=2.4 10-6 sec-1
A=2.65 10-7 sec-1
A=7.93 10-8 sec-1
(Use two level excitation with collision rate of 10-10 cm3 sec-1 to get n*)
Lecture4 page 7
RADIO RECOMBINATION LINES
These are “Rydberg Atoms” with Principal Quantum Numbers > 20
h2
Bohr Orbit an  2 2  n 2
z  me
1 1
Frequency ik  z 2  RM   2  2 
i
k 
R
RM 
m
1
M
i k
m = electron mass and M = nucleon mass
Set z2 = 1
i  k  1 k  i  1
i ,i 1 
2  RH
i3
RH  3.288  1015 Hz
Lecture4 page 7
e  an e h 2
 n ,n  1 

 n2
2
8  me
9
1
n

1
,
A

5

10
sec
5.36  109
An ,n 1 
sec 1
5
n
n  100, A  0.5 sec 1
2
gn  GHz
N n  194
.  10 

 T dV
gn 1 Am,m1  B
3
Saha Eq’n
3
2
xn


h
  e k Te  N p  N e
N n  n2  
 2  m  k  Te 
2
e
xn
k Te
1
Then, line intensity is proportional to NpNe
From Bremsstrahlung, so is Tc for HII regions
Find
TL
TC
  1 2  6.985  103

 
 GHz
1  
kms
a




1.1
Te1.35

 N ( He  ) 
1 
 
N
(
H
)

NON-LTE EFFECTS
Nn: Actual population
Nn* : LTE population
Lecture4 page 9
Nm  bm  N m*
gu and ge are nearly equal for u, l > 20, so
h
*





Nu
b
N
u
u
k Tex
e
    *
Ne
 be   N e 
Then Tex < 0 population inversion typically Tex = -300K
In Lecture 1, if |  | < 0 , get
TB  TCE  TBG    e   1  TCE  TBG   
   1 so " very weak maser"
Lecture4 page 10
If lines are optically thin, what is amplified?
On ‘Tools’, p. 342 is:
 1

TL  TL*  b   1   c   
 2


k  T   b
 
h     n
Often lots of algebra !
< 0 and depends on density, and could be large!
Lecture4 page 11
APPLY TO AN ACTUAL HII REGION (ORION A)
History 19651967
From Radio Recombination Lines (RL),
Te = 5800 K (6 cm, 5 GHz) optical is ~104 K
1968
Theory of RL broadening theory (Griem)
1969
2 components models with dense clumps
1970
Measured Te rises to 7000 K (6 cm)
1972
Brocklehurst & Seaton give complete theory
At low frequencies core radiation broadened (large n)
At high frequencies, cores dominates, but not much maser emission in
diffuse foreground gas.
Brown, Lockman & Knapp in Annual Reviews (1978) proposed a
recombination line theory with large EM, low ne and lots of line
masering. This is not matched by measurements. So unrealistic!
Lecture4 page 12
This is where MACROPHYSICS meets MICROPHYSICS
Macrophysics: structure of a source on parsec
Microphysics: cross sections, local populations of atoms
Confluences of these – i.e. masering depends on atomic physics and
source structure could be thought of as “Radiative transfer”
Lecture4 page 13
Quantum description:
2S+1L
J

L: orbital angular momentum
  
J  L  S total angular momentum
  

and F  I  J
I : nuclear spin
2
2
2
S1
S1
S1
 gn  3
21cm line hyperfine 

 ge  1 
F  1  0 HI
2
3
1
F 
DI
2
2
2
F  0  1 He
3
2

 gn  4
92cm line hyperfine 

 ge  2 
 gn  1
3.46cm line hyperfine 

 ge  3
3
P1  3P0 CI
609 m fine structure
3
P2  3P0 CI
370m
3
P3  3P1
2
CI 157 m
2