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Lecture 3Lecture 3
By Tom Wilson
Summary of Lecture 1Summary of Lecture 1
Noise in a ReceiverNoise in a Receiver
timetime
on sourceon source
TT
R M SSY S
Receiver itself,Receiver itself,
atmosphere,atmosphere,
ground ground
and sourceand source
Analying bandwidth (for lines, need 3 resolution Analying bandwidth (for lines, need 3 resolution
elements on the line above the ½ power point)elements on the line above the ½ power point)
Temperatures from thermal hot and cold Temperatures from thermal hot and cold load measurements using the receiver. load measurements using the receiver.
Hot-cold load measurementsHot-cold load measurements
Absorber at a given temperature
Input to receiver
(to determine receiver noise contribution)
OK for heterodyne receivers, but OK for heterodyne receivers, but not for Bolometers for Bolometers
Current Receiver Noise TemperaturesCurrent Receiver Noise Temperatures
Tmin=h/kfor coherentreceivers
TTSYSSYS=T=TRXRX e e Atmospheric optical depthAtmospheric optical depth
ReceiversReceivers
Heterodyne for spectral linesHeterodyne for spectral lines High velocity resolutionHigh velocity resolution flexibility, but not multi-pixel receivers in the mm/sub-mmflexibility, but not multi-pixel receivers in the mm/sub-mm
Bolometers for continuumBolometers for continuum Multi-pixel camerasMulti-pixel cameras
k
NEPATAT SYS
RMS2
NoiseNoise
Equivalent Equivalent
powerpower(about (about 1)1)
Types of Receivers
Fractional R
esolution
Lecture 3 page 7Lecture 3 page 7
BOLOMETERS VS COHERENT RECEIVERS
JCMT: 15 m sub mm dish,A = 0.5 at = 0.87 mm, = 345 GHz= 345 GHz
With SCUBA, can detect a source with 0.16 Jy in With SCUBA, can detect a source with 0.16 Jy in 11 secondsecond. So RMS is ¼ . So RMS is ¼ of this peak value or of this peak value or SS0.04 Jy in 1 sec.0.04 Jy in 1 sec.
Compare to a coherent receiver:Compare to a coherent receiver:
TTSYSSYS = 50 K, = 50 K, = 2 GHz, integration time= 2 GHz, integration time= 1 sec
In antenna temperature.In antenna temperature.
From Lecture 2: From Lecture 2:
For JCMT, For JCMT,
Or Or
So comparable for 1 beam, but SCUBA has 37 beams & MAMBO has 117 So comparable for 1 beam, but SCUBA has 37 beams & MAMBO has 117 beams. beams.
TK
KR M S
5 0
2 1 01 2 1 0
9
3.
ST
DA
a m 3 5 2 0 2
S Ta
A
1 6
S JyR M S 4 1 0 2
(arc sec)=(arc sec)=kkmm)mm)/D(m)/D(m)
of order 1.2 for single dishesof order 1.2 for single dishes
SS=3520 T=3520 TAA /( /(AA D D22))
ST
DP A S k T A AA
ae A e a g
3 5 2 01
22
Rayleigh-Jeans
ST
cm
TM B M B
cm
2 6 5 2 6 50
2
20
2
2.( ' )
( ).
( ' )
Or
S
T
m m
M B
0 0 7 4 0
2
2.( ' ' )
( )
Gaussian beams:
Summary of Lecture 2Summary of Lecture 2
(Show that these(Show that these
are consistent)are consistent)
In JyIn Jy
22
2
222
22
Beams
ssMB
sBeamobs
ssobsMB
TT
TT
True source size True source size
and temperatureand temperature
apparent source size apparent source size
and temperatureand temperature
Can make a relation for flux density similar to that for Can make a relation for flux density similar to that for
Main Beam Brightness temperature:Main Beam Brightness temperature:
S(total)=S(peak) S(total)=S(peak) .. ( (SS 22 + + BB 22 )/ )/ BB 22
Example: Orion A is an HII region with a total flux Example: Orion A is an HII region with a total flux
density of 380 Jy at 1.3 cm. The size is 2.5’ (FWHP). If the radiodensity of 380 Jy at 1.3 cm. The size is 2.5’ (FWHP). If the radio
telescope beam size is 40” (FWHP), what is the peak flux density?telescope beam size is 40” (FWHP), what is the peak flux density?
Use the R-J relation to find the peak main beam brightnessUse the R-J relation to find the peak main beam brightness
temperature. temperature. Solution: peak Jy/beam=9.5; TSolution: peak Jy/beam=9.5; TAA=8.8K, T=8.8K, TBB=24 K=24 K
Far FieldFar Field Diffraction and Fourier Transforms Diffraction and Fourier Transforms
(Exact calculations require programs such as GRASP)(Exact calculations require programs such as GRASP)
(radiation passing through an opening)(radiation passing through an opening)
Lecture 3 page 1Lecture 3 page 1
Grading Across the Aperture and Far E FieldGrading Across the Aperture and Far E FieldLecture 3 page 2Lecture 3 page 2
ALMA ALMA
Front-End
DigitizerClock
Local Oscillator
ANTENNA
Data Encoder12*10Gb/s
12 Optical Transmitters
12->1 DWD Optical Mux
Digitizer8* 4Gs/s -3bit ADC
8* 250 MHz, 48bit out
IF-Processing(8 * 2-4GHz sub-bands)
Fiber Patch-PanelFrom 270 stations to 64 DTS Inputs
Optical De-Mux& Amplifier
Digital De-Formatter
Correlator
Technical Building
Tunable Filter Bank
Fiber
Lecture 3, page 3Lecture 3, page 3
Sketch of 2 element Sketch of 2 element interferometerinterferometer
((u,vu,v) plane and image plane ) plane and image plane
These are related by Fourier transformsThese are related by Fourier transformsThe distance between antennas varies, so The distance between antennas varies, so we sample different source structureswe sample different source structuresOn the next overheads, we indicate how On the next overheads, we indicate how structures are sampled. Following structures are sampled. Following tradition, tradition, u u represents one dimension represents one dimension distributions, with x as the separtion in distributions, with x as the separtion in wavelenghts wavelenghts uu=2=2xxandandv=2v=2yy
Earth Rotation Aperture SynthesisEarth Rotation Aperture Synthesis
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
Sensitivity: http://www.eso.org/projects/alma/science/bin/sensitivity.htmlSensitivity: http://www.eso.org/projects/alma/science/bin/sensitivity.html
VLA VLA uvuv plane response plane response
Dataas taken
Datawith MEM
with MEMand Self-Calibration
The radiogalaxy Cygnus Aas measured with all configurations of the VLA
From W. D. Cotton (in ‘The Role of VLBI in Astrophysics, Astrometry, From W. D. Cotton (in ‘The Role of VLBI in Astrophysics, Astrometry,
And Geodesy, ed Mantovani & Kus, Kluwer 2004)And Geodesy, ed Mantovani & Kus, Kluwer 2004)
Lecture 3 page 16Lecture 3 page 16
BROADBAND RADIATION
• Black body (Moon, planets, 3K background)
• Dust thermal emission
• Bremsstrahlung (free-free)
• Synchrotron (relativistic electrons in magnetic fields)
• Inverse Compton Scattering (S-Z)
Dust: Mostly carbon, silicon with ice mantles “ground up planets”
From Hildebrand (1983)
D U STSun
H
z
zb N
7 1 0 2 1 2
22
2
:
:
:
:
min
cmindensitycolumnN
datathefittoaccounttofactorb
mmin
H
DUST
Lecture 3 page 17Lecture 3 page 17
For warm grains
Use R-J
get
T TD U ST D U ST
S k TD U ST D U ST 2 2
NS m Jy
zz
b TH
m m
SunD U ST
1 9 3 1 0 2 42
4
.( )
(" )
EXAMPLE: Dust emission from Orion KL
The Orion “hot core” has the following properties:
1 0
1 6 0
1 0
1 5 1 0 5 0 0
27 3
2 1
"
( )
. ( )
T K
n H cm
D ist cm pc
D U ST
Lecture 3 page 18Lecture 3 page 18
Calculate the column density N(H2) = n L and the 1.3 mm dust flux density, S, ,
if z = zif z = zSunSun, b = 1.9 , b = 1.9
If the value of L = diameter, use L (diameter in cm) x (size in radians) =
= 7.5 1016 cm
Them N(cm-2) = 7.5 1016 cm x 107 cm-3 = 7.5 1023 cm-2 for H2
N(H) = 2 N(H N(H22) = 1.5 ) = 1.5 1024 cm-2
So
1 5 1 0 1 9 3 1 01 3 0 0
1 0 1 9 1 6 0
1 0
1 0
2 4 2 44
2
4
. .( ) ( )
( ) ( . ) ( )
( )
S m Jy
S m Jy
S Jy
At 4 mm, S S is 81 times smaller or 120 mJy. At 0.39 mm, Sis 81 times smaller or 120 mJy. At 0.39 mm, S is 81 times is 81 times
larger or 810 Jylarger or 810 Jy
Lecture 3 page 19Lecture 3 page 19
BREMSSTRAHLUNG (FREE-FREE)
Hydrogen is ionized by O, B, electron and protons interact, electrons radiate. Classically:
P t ev t
cL arm or( )
( )( )
2
32
2
3
Power radiated during encounter:W
e
cv t d t
2
3
2
32 ( )
Find
4
3
22 6
3 22
1
z e
c
N N
m
m
k T
p
pi e
e
ln
From the Kirchhoff relation, get
B T k Tc
k Tso
N N L
Te e
i e
e( ) 2 2
2
2
2 3
22
velocityvelocity
‘ ‘ pp’ is impact parameter’ is impact parameter
frequencyfrequency
Lecture 3 page 20Lecture 3 page 20
T T eB e 1
When = 1: = 1: 0
0 6 4 3
6
0 4 7 6
0 3 0 4 5
..
.
TE M
pc cmB
For Orion A, 00 = 1 GHz, or 30 cm.
1 8 0 0 0
1
5 4 0 0
2 3 2 4
2
G H z T T K
G H z T TT
at G H z T K
at G H z T K
B e
B ee
B
B
But
SkT kT
E M constB e
2 22 2 2
8 2 3 5 1 0 2 1 3 5 2
..
T E Me G H z
--
What is the What is the
Relation forRelation for
TTBB??
Lecture 3 page 21Lecture 3 page 21
Orion A HII Region
Te = 8500 K, = 2.5’ (FWHP), so use
is much less than 1, at = 23 GHz so T = T = 23 GHz so T = Tee
From 100-m, TMB (main beam) = 24 K in a 40” beam, so TMB (main beam) == T(true)
24 = (8500) (8.235 10(8.235 10-2-2) ) (8500)(8500)-1.35-1.35 (23)(23)-2.1-2.1 EMEM
so EM = 4 so EM = 4 101066 cm cm-6-6 pc = N pc = Nee N Ni i
LL
If L = 25’ = 0.33 pc converted to radians @ 500 pc get NIf L = 25’ = 0.33 pc converted to radians @ 500 pc get Nee = N = Nii = 3.5 = 3.5 101033 cm cm-3-3
This is the RMS density. Calculate the mass of ionized gas. This is the RMS density. Calculate the mass of ionized gas.
Rough number since know Orion A is not spherical. From spectral lines know Ne = 10= 1044 cm cm-3-3, so L = 0.03 pc. Then M = 0.6 M, so L = 0.03 pc. Then M = 0.6 MSunSun in ionized gas in ionized gas
Free-Free Intensity and Flux Density as function of FrequencyFree-Free Intensity and Flux Density as function of Frequency
(Problem: Use the (Problem: Use the
exampleexample
of Orion to of Orion to
check thesecheck these
Curves)Curves)
Lecture 3 page 22Lecture 3 page 22
Lecture 3 page 4Lecture 3 page 4
Free-Free Emission from Planetary Nebulae
NGC7027 (a PNe) has SS = 5.4 Jy at 1.3 cm. What is the T= 5.4 Jy at 1.3 cm. What is the TMBMB (main beam (main beam
brightness temperature) if the 100-m FWHP beam size is 43”?brightness temperature) if the 100-m FWHP beam size is 43”?
UseUse
Where Where 0 is the telescope beam size in are min.
Suppose the “true” gaussian source size is 10”, what is TB (true brightness temperature). Could use
ST
cm
T
T K
M B
M B
M B
2 6 5
5 4 2 6 5
4 36 0
1 3
6 7
02
2
2
2
.( )
. ..
.
STM B
5 4 2 6 5
1 06 0
1 3
2
2. ..
(Problem: Repeat for the (Problem: Repeat for the
30-m, with beam 30-m, with beam
27’’, wavelength 27’’, wavelength
3.5 mm, flux density3.5 mm, flux density
4.7 Jy)4.7 Jy)
Lecture 3 page 5Lecture 3 page 5
Or
And get
We know that the electron temperature of NGC7027 is Te = 14000 K. Use equation of radiative transfer:
To get 0 0 0 9.
This is a source which is thermal, so the radiation is free-free or Bremsstrahlung
2
22
10
1043MBB TT
KTB 124
eTT eB 1
Lecture 3 page 23Lecture 3 page 23
SYNCHROTON RADIATION (NON-THERMAL)
Highly relativistic electrons spiraling in a B field with a frequency
B
e B
m
1
P: Power radiated by electron (lab)
P’: Power radiated by electron (rest frame)
so P = P’
Pd
dtP
d
d td t d t
''
'' '
Transformation of acceleration
So
a a' 2
Pe
ca a v
e B
mvB
2
3
2
34 2
Radiation patterns of an electron in a B fieldRadiation patterns of an electron in a B field
B Field,B Field,
V about 0V about 0
VelocityVelocity
B Field,B Field,
V about 0.2 cV about 0.2 c
Lecture 3 page 24Lecture 3 page 24
Lecture 3 page 25Lecture 3 page 25
Then
E: Particle energy
Is difficult to separate energy of electron from B field strength
To get spectral distribution, use
Find a synchrotron spectrum ::
dN
dEk E
1
21 0 7 2 4. .
Synchrotron radiation is found to be linearly polarized
(power law (power law
distribution of distribution of
Cosmic Rays)Cosmic Rays)
22232
222
3
2BE
cm
BveP
Lecture 3 page 26Lecture 3 page 26
SINGLE ELECTRON SYNCHROTRON EMISSION
For relativistic electrons, the emitted pulse is 1/ shorter due to relativistic beaming while the Doppler effect gives rise to a factor 1/2
B: Frequency of rotation
So for B = 10G, G, B is even lower when <1
B
B
G
1 7 6.
GB H z
1 7 6
Thus in frequency reach a critical value
So if B = 10 G, G, G = 176 Hz, to reach CC = 10 GHz, = 10 GHz, = 1.6 101044
In Synchrotron emission, we measure only the most relativistic In Synchrotron emission, we measure only the most relativistic particlesparticles
C G
3
42 sin
Lecture 3 page 27Lecture 3 page 27
SYNCHROTRON ENERGY CONSIDERATIONS
Allow one to determine the minimum or equipartion energy
Inverse Compton effect
When the radiation density is equal to magnetic energy density there can be energy losses in addition to synchrotron energy losses. R & W don’t do much, but Kellermann & Owen give:
W S R VT O Tn
4
78
7
3
7
L
L
T TC om pton
Synch
M A X M A X
1
2 1 01
1
2 1 01 2
5
2
5
m ax
This is the basis of the statement: “1012 K is the highest source temperature possible”
Lecture 3 page 6Lecture 3 page 6
Non-thermal sources
Cas A: at 100 MHz, SS= 3 10= 3 1044 Jy, Jy, s=4’ (source size), 4’ (source size), = 3 m = 300 cm
ST
cm
T
K T source
M B
2 6 5
3 1 0 2 6 51 6
9 1 0
7 5 1 0
0
2
2
44
8
.( ' )
( )
.
. ( )
Thermal sources have limit T = 2101044 K K
Assume that for Cas A, T=7.5 10Assume that for Cas A, T=7.5 1088 ( (mm3) 3) -2.8-2.8
What is the source temperature at 3 mm?What is the source temperature at 3 mm?
Source Source
Specral Specral
IndexIndex
Lecture 3 page 29Lecture 3 page 29
SUNYAEV-ZELDOVICH EFFECT
Clusters of galaxies are filled with hot diffuse gas. Photons from the 3 K background are scattered in this cluster gas.
More photons are given energy than lose energy on the low frequency side of the Planck curve.
On the high frequency side, some photons are shifted to lower energies,
but still a reduction in the 3 K background. At 160 GHz, have a cross over from absorption at longer wavelengths to emission at shorter wavelengths, so have zero absorption. The
absorption is:T
TT N L
SZe e
2 2 4 1 0 3 4.
When combined with X ray luminosity, which is Bremsstrahlung (free-free), proportional to Ne
2 L, can solve for source distance. Given L, can solve for source distance. Given
velocity of source, get velocity of source, get HUBBLE CONSTANTHUBBLE CONSTANT. However there can be . However there can be systematic effects such as clumping. systematic effects such as clumping.
Lecture 3 page 30 Lecture 3 page 30
EXAMPLE OF S-Z EFFECT
The cluster CL0016 +16 shows on S-Z absorption of –700 K at 1 cm K at 1 cm wavelengthwavelength
Z = redshift of CL0016 +16 is 0.541Z = redshift of CL0016 +16 is 0.541
X ray data: TX ray data: Tee = 1.6 10 = 1.6 1088 K K
Cluster size = 30“ to 19”Cluster size = 30“ to 19”
RMS RMS NNee = 1.2 10 = 1.2 10-2-2 cm cm-3-3
So So Kcm
T
T
SZ
70010710310101024.2 4242834