lecture 4 by tom wilson. review page 1 interferometers on next page rayleigh-jeans: true if h

Lecture 4Lecture 4

By Tom Wilson

Review page 1Review page 1

Interferometers on next page Interferometers on next page

S

T

cm

T

m m

2 6 5 0 0 7 40

2

2

02

2.( ' )

( ).

( ' ' )Rayleigh-Jeans:

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

Grading Across the Aperture and Far E FieldGrading Across the Aperture and Far E Field

as limit of interferometeras limit of interferometer


Above: the 2antennas onthe earth’s surface have a different orientation as a function of time.

Below: the ordering of correlated data in (u,v) plane.


Gridding and sampling in (Gridding and sampling in (u,vu,v) ) planeplane


RECEIVERS (MM)

Mix from sky frequency to IF frequency (4 GHz) and amplify signal

T T L T LT

GR X M M M 12

1

2 LIMITS:

TT

R M SSY S

kD

Analog Coherent Receiver Block DiagramAnalog Coherent Receiver Block Diagram


Time Frequency f

Total Amplification=1016


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 NS m Jy

zz

b TH

m m

SunD U ST

1 9 3 1 0 2 42

4

.( )

(" )


Free-Free (Bremstrahlung)

T T T E Me e G H z 8 2 3 5 1 0 12 0 3 5 2

..

Synchrotron emission

if B G H zBG 1 0

1 7 6

.

Radiation frequency is increased by beaming 1/ and doppler 1/2, so critical

Frequency is

C G

3

42 sin =104 for 10 GHz

F ind w here 1

21

: energy spectrum of cosmic rays

ululul BgBg

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:

Lecture4 page 1 Lecture4 page 1

ulul Bc

A

3

38

After somemanipulation, get

Inserting numerical values of physical constants:

ex

ex

GHz

Tk

h

l

u

l

u

ulGHzul

B

Tul

GHz

u

ll

eg

g

N

N

A

dVT

eAg

gN

2311

108.4

3

10165.1

1

1

(sec)5.93 2

)1( eTTB

NNuu A Aulul+B+Bulul N Nuu U=B U=Blu lu NNll U U

UU=4=4/c I/c I

Lecture4 page 2Lecture4 page 2

In terms of column density, Nl, get

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 rf i

exTk

h

l

u

l

u

Vu

l

ul

GHzl

eg

g

N

NbydefinedisT

TkhifdVTg

g

AN

331094.1

N=1N=1

SS PP DD FF

(From H. E. White, ‘ Introduction to Atomic Physics’ )(From H. E. White, ‘ Introduction to Atomic Physics’ )

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!!

Show N T dVt 1 8 2 4 1 0 1 8.

For HI, h · / k = 0.06 K

dVTN

NNNNN

Bl

lllut

3

1

1089.2

420.11094.1

3

15

23


Lecture4 page 5 Lecture4 page 5

Frequency (Frequency (0)) AAulul (sec (sec-1-1)) gguu ggll

DI (deuterium)DI (deuterium) 327 MHz 4.65 ·10-17 4 2

3He+ 8.665 GHz8.665 GHz 1.95 ·10-12 11 33

Similar transition are for D and 3He+

EXCITATION OF 2 LEVEL SYSTEM

Competition of radiation and collisions

Answer:

TT T y

y

w here yh

k

C

Av n

exB k

1

2 1

2 1

For HI, Tex = Tk , if n >1 cm-3

hh

kk


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=2.65 10A=2.65 10-7-7 sec sec-1-1

A=7.93 10A=7.93 10-8-8 sec sec-1-1

A=2.4 10A=2.4 10-6-6 sec sec-1-1

(Use two level excitation with collision rate of 10(Use two level excitation with collision rate of 10-10-10 cm cm33 sec sec-1-1 to get n*) to get n*)


RADIO RECOMBINATION LINES

These are “Rydberg Atoms” with Principal Quantum Numbers > 20

B ohr O rb it ah

z mn

F requency z Ri k

i k

RR

mM

ne

ik M

M

2

2 22

22 2

1 1

1

m = electron mass and M = nucleon mass

Set z2 = 1i k k i

R

iR H zi i

HH

1 1

23 2 8 8 1 01 3

1 5 , .


n nn

e

n n

nn

n

G H z

m mB

ne

x

k Tp e

x

k T

e a h

mn

An

n A

n A

Ng

g AT dV

N nh

m k Te N N e

n

e

n

e

,

,

,

.sec

, sec

, . sec

.

1

22

1

9

51

9 1

1

3

1

2

1

22

3

2

2 8

5 3 6 1 0 1 5 1 0

1 0 0 0 5

1 9 4 1 0

21

Then, line intensity is proportional to NpNe

From Bremsstrahlung, so is Tc for HII regions

Find

T

T km s a

T

N H eN H

L

CG H z

e

12

1

31 1

1 3 56 9 8 5 1 0

1

.

( )( )

..

e

Saha Eq’n

Lecture4 page 9Lecture4 page 9NON-LTE EFFECTS

Nn: Actual population Nn* : LTE population

gu and ge are nearly equal for u, l > 20, so

N b Nm m m *

N

Ne

b

b

N

Nu

e

h

k T u

e

u

e

ex

*

*

Then Tex < 0 population inversion typically Tex = -300K

In Lecture 1, if || < 0 , get

T T T e T T so very w eak m aserB C E B G C E B G 1 1 " "


If lines are optically thin, what is amplified?

On ‘Tools’, p. 342 is:

T T bk T

h

b

nL L c

* 11

2

Often lots of algebra !

< 0 and depends on density, and could be large!


APPLY TO AN ACTUAL HII REGION (ORION A)

History 1965-1967

From Radio Recombination Lines (RL), From Radio Recombination Lines (RL),

TTee = 5800 K (6 cm, 5 GHz) optical is ~104 K

19681968 Theory of RL broadening theory (Griem)Theory of RL broadening theory (Griem)

19691969 2 components models with dense clumps2 components models with dense clumps

19701970 Measured TMeasured Tee rises to 7000 K (6 cm) rises to 7000 K (6 cm)

19721972 Brocklehurst & Seaton give complete theoryBrocklehurst & 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!


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”


21

2

21

2

21

2

3

1 0 2 13

1

3

2

1

29 2

4

2

0 1 3 4 61

3

S F H I cm line hyperfineg

g

S F D I cm line hyperfineg

g

S F H e cm line hyperfineg

g

n

e

n

e

n

e

.

31

30

32

30

33

2

31

2

6 0 9

3 7 0

1 5 7

P P C I m fine structure

P P C I m

P P C I m

Quantum description: Quantum description: 2S+1 2S+1LLJJ

L orb ita l angu lar m om entum

J L S to ta l angu lar m om entum

and F I J I nuclear sp in

:

:

lecture 4 by tom wilson. review page 1 interferometers on next page rayleigh-jeans: true if h

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