2009/08/24-28 2009/08/24-28. 2009/08/24-28 1. 2. 3. 4. 5

射电天文基础射电天文基础姜碧沩姜碧沩

北京师范大学天文系北京师范大学天文系

2009/08/24-282009/08/24-28 日，贵州大学日，贵州大学

2009/08/24-282009/08/24-28 日日射电天文暑期学校射电天文暑期学校

大纲大纲

1. 射电天文基础2. 射电望远镜3. 连续谱辐射机制4. 谱线辐射机制5. 星际分子

参考书：《射电天文工具》


射电天文射电天文• Radio

– 大气窗口– 地面射电天文的频率上限和下限– 空间

• Astronomy– Astro-: star

• Radio astronomy– 与其它波段的区别


The waves used by optical The waves used by optical astronomersastronomers

• Electromagnetic Spectrum

• 4000 to 8000 angstroms

• 7.51014Hz to 3.751014Hz

• The Sun

• The solar system

• Stars

• Galaxies


The radio windowThe radio window

• Atmospheric Transmission• From about 0.5mm to 20m• 600GHz to 15MHz

– Troposphere （对流层） to ionosphere– FM radio (and TV)– AM radio– Mobile phone…

• The solar system, stars, ISM, galaxies, cosmic microwave background……..– The Sun


Some advantages of radio astronomySome advantages of radio astronomy

• Transparent to terrestrial clouds: visible in cloudy time

• The Sun is quiet: visible in day time

• Transparent to the vast clouds of interstellar dust: able to see distant objects

• Different origin of radiation


射电天文的辉煌射电天文的辉煌• 获得诺贝尔奖的发现

– 宇宙微波背景辐射的发现– 脉冲星的发现：快速旋转的中子星– 双星脉冲星的发现与引力波理论的验证– 宇宙微波背景辐射的黑体形式以及非各向同性

• 其他重要贡献– 星际分子– 氢原子谱线– 恒星形成区– 磁场


The world’s largest radio telescopesThe world’s largest radio telescopes

• The Arecibo Telescope Type: Fixed reflector, movable feeds Diameter of reflector: 1000 ft (304.8 m) Surface accuracy: 2.2 mm rms Working wavelength: from cm to dm

• The Effelsberg TelescopeType: Fully steerableDiameter: 100-mWorking wavelength: up to 3mm, mainly cm


Fundamentals of Radio AstronomyFundamentals of Radio Astronomy

• Some basic definitions

• Radiative transfer

• Blackbody radiation and brightness temperature

• Nyquist theory and noise temperature


II: specific intensity: specific intensity

•

• dW ＝ infinitesimal power, in watts ，

• dσ ＝ infinitesimal area surface, in cm2 ，

• dν ＝ infinitesimal bandwidth ， in Hz ，

• θ ＝ angle between the normal to dσand the direction to dΩ

• Iν ＝ brightness or specific intensity, in Wm-2Hz-1sr-1 。

dddIdW cos


The total flux of a sourceThe total flux of a source

• Total flux of a source: integration over the total solid angle of the source Ωs

• Unit– W m-2Hz-1

– Jy• 1Jy=10-26 W m-2Hz-1= 10-23 erg s-1 cm-2Hz-1

• A 1Jy source induces an signal of only 10-15W. • Few sources are as bright as 1Jy

s

dIS cos),(


Brightness is independent of the Brightness is independent of the distancedistance

)2()1( 21 rIrI


The total flux density depends on The total flux density depends on distance as rdistance as r-2-2

• Total flux received at an point P from an uniformly bright sphere

2

0

( , ) cos sin cosc

s c

S I d I d d

r

Rc sin

I

r

RIS

2

2


Radiation energy densityRadiation energy density

• Energy density per solid angle: erg cm-3Hz-1

• Total energy density

Ic

u1

)(

)4()4(

1)(

dI

cduu


Radiative transferRadiative transfer

• For radiation in free space the specific intensity is independent of distance. But I

changes if radiation is absorbed or emitted.

,

,

dsdI

dsIdI


( ) ( )I s ds I s dsd d d I dsd d d

I

ds

dI


Limiting casesLimiting cases

• Emission only:

• Absorption only:

0

dsssIsIds

dI s

s

)()()(,0

0

0

dsssIsII

ds

dI s

s

)(exp)()(,0

0


Limiting cases (cont’d)Limiting cases (cont’d)

• Thermodynamic Equilibrium (TE): radiation is in complete equilibrium with its surroundings, the brightness distribution is described by the Planck function, which depends only on the thermodynamic temperature T of the surroundings

)(,0 TBIds

dI

1

12)(

/2

3

kThec

hTB


Limiting cases (cont’d)Limiting cases (cont’d)

• Local Thermodynamic Equilibrium (LTE)– Kirchhoff’s Law

– Optical depth

– Equation of transfer

– Solution

)(TB

dsd 0

s

ds

)(1

TBId

dI

ds

dI

)(

0

)( ))(()0()(s

s deTBeIsI


LTE (cont’d)LTE (cont’d)

• The medium is isothermal– T(τ) ＝ T(s) ＝ T=const.

• Optical depth is very large– τ(0)

– Difference with the intensity in the absence of an intervening medium

)1)(()0()( )()( ss eTBeIsI

)(TBI

)1))(0()(()0()()(

eITBIsIsI


Blackbody radiationBlackbody radiation

• Planck law

• Total brightness of a blackbody

1

12)(

/2

3

kThec

hTB

1

12)(

/5

2

kThce

hcTB

412532

444 Kscmerg108047.1

15

2,)(

hc

kTTB


Wien’s displacement lawWien’s displacement law

• Maxima of B (T) and Bλ(T) Bν/ν=0 and Bλ/λ ＝ 0

– νmax

– λmax

K789.58

GHzmax T

28978.0Kcm

max

T


Rayleigh-Jeans LawRayleigh-Jeans Law

• Rayleigen-Jeans Law • Radiation temperature

kTh

kT

he kTh 1/

kTc

TBRJ 2

22),(

1

1

2)(

/2

2

kThek

hI

k

cTJ

K84.20

GHz

T


Wien’s LawWien’s Law

1kT

h

e

kTh

kThW e

c

hTB /

2

32),(


Brightness temperature TBrightness temperature Tbb

• One of the important features of the Rayleigh-Jeans law is the implication that the brightness and the thermodynamic temperature of the blackbody that emits the radiation is strictly proportional.

• In radio astronomy, the brightness of the extended source is measured by its brightness temperature which would result in the given brightness if inserted into the Rayleigh-Jeans law

B

kB

k

cT

2

1

2

2

2

2

b


Transfer equation of TTransfer equation of Tbb

• Transfer equation

• General solution

• Two limiting cases when Tb(0)=0– Optically thin, τ<<1

– Optically thick, τ>>1

)()()(

bb sTsTd

sdT

desTeTsTs

s )(

0

)(bb )()0()(

TT b

TT b


The Nyquist TheoremThe Nyquist Theorem

• Johnson noise– The thermal motion of the electrons in a resistor will p

roduce a noise power which is the noise determined by the temperature of the resistor

• The average noise power per unit bandwidth produced by a resistor R is proportional to the its temperature, i.e. the noise temperature, and independent of its resistance

P=kTN


Electromagnetic wave propagation Electromagnetic wave propagation fundamentals fundamentals

• Maxwell’s equations• Energy conservation and the Poynting vector• Complex field vectors• The wave equation• Plane waves in nonconducting media• Wave packets and the group velocity• Plane waves in dissipative media• The dispersion measure of a tenuous plasma


Maxwell’s equationsMaxwell’s equations

• Material equations

• Maxwell’s equations

• Continuity equation of charge density and current

HB

ED

EJ

Dc

Jc

H

Bc

E

B

D

14

1

0

4

0 J


Energy conservation and the PoyEnergy conservation and the Poynting vectornting vector

• Energy density of an electromagnetic field

• Poynting vector

• Equation of continuity for S

22

8

1

8

1HEHBDE

u

HES π

c

4

JES t

u


Complex field vectorsComplex field vectors

• Complex field vectors

• The Poynting vector

tiei 21 EEE tiei 21 HHH

HES ReRe4

c

*HES Re4c


The wave equationThe wave equation

EEE 22

2 4

cc

HHH 22

2 4

cc


Plane waves in nonconducting mediaPlane waves in nonconducting media

• Nonconducting media– σ ＝ 0 – The wave equation

– Velocity of the wave

01

22 u

vu

c

v


Plane waves (cont’d)Plane waves (cont’d)

• Harmonic wave solution of the wave equation

• Wave number

• Phase velocity

• Index of refraction

tkxieuu 0

22

2 c

k

c

kv

kc

v

cn


Plane waves (cont’d)Plane waves (cont’d)

• A wave that propagates in the positive z direction is considered to be plane if the surfaces of constant phase forms planes z=const.

0,0 zz HE

0HE

H

E2

4ES

c


Group velocityGroup velocity

• Dispersion equation

• Group velocity

• Energy and information are usually propagated with the group velocity

)(k

)()( 00

0 kkdk

dk

dk

dvg


Plane waves in dissipative mediaPlane waves in dissipative media

• Dissipative media • Harmonic waves propagating in the direction of

increasing x• Wave equations

• Dispersion equation

0

)(),( tkxietx 0EE

04

22

22 ＝

H

E

ci

ck

4

12

22 i

ck


Cont’dCont’d• Wave number

• Field )(),( taxibxeetx 0EE

1

41

2

12

ca

1

41

2

12

cb

ibak

tx

c

nixn

ctx

expexp),( 0EE


Cont’dCont’d

• Index of refraction and absorption coefficient

1

41

2

12

n

1

41

2

12

n


Dispersion measure of a tenuous Dispersion measure of a tenuous plasma plasma

• Plasma: free electrons and ions are uniformly distributed so that the total space charge density is zero

• Tenuous plasma– Interstellar medium– dissipative medium

• Equation of motion of free electrons

– Solution

tieemm 0ee E-rv

EEv 0

ee m

eie

im

e ti


Cont’d Cont’d

• Conductivity of the plasma

• Wave number for a thin medium with ε≈1 andμ≈1

ωm

Nei

e

2

2

2p

2

22 1

ck

e

22p

4

m

Ne


Cont’d Cont’d

• Phase velocity and group velocity– For ω>ωp, k is real, v>c, vg<c

2

2p1

c

v2

2p

g 1

cv

2cvvg 2

2p1

n


Dispersion measure of pulsarsDispersion measure of pulsars• A pulse emitted by a pulsar at a distance L

will be received after a delay

• The difference between the pulse arrival time measured at two frequencies

L

e

Lp

L

gD dllN

m

e

cdl

cv

dl

02

2

02

2

0

)(1

21

1

2

11

1

L

eD dllN

cm

e

022

21

2

)(11

2


Cont’d Cont’d • Dispersion Measure

pccmDM

03

ld

N

1

2

2

2

1

D43

MHz

1

MHz

1

μs10410.2

pccm

DM


Dispersion Measure, DM, for pulsars at different Galactic latitudes


Faraday rotationFaraday rotation

• In 1845, Faraday detected that the polarization angle of dielectric material will rotate if a magnetic field is applied to the material in the direction of the light propagation

• The rotation of the plane of polarization of an EM wave as it passes through a region containing free electrons and a magnetic field, also known as Faraday effect. The amount of rotation, in radians, is given by RMλ2, where RM is the rotation measure of the source and λ is the wavelength. Observation of the Faraday rotation in pulsars is the most important means of determining the magnetic field of the Galaxy. It is named after the English physicist Michael Faraday.


Equation of motion for an electron Equation of motion for an electron in the presence of a magnetic fieldin the presence of a magnetic field

BrE-rv emm

If the magnetic field B is oriented in the z direction

yxy

xyx

Em

erB

m

er

Em

erB

m

er

yx

yx

iEEE

irrr

Em

erB

m

eir


SolutionSolution

)(2

1),(

2

1 EE

iEEEE yx

Linearly polarized wave can be regarded as the superposition of circularly polarized waves

Solution in the form of harmonic waves)( txkiAeE

)(0

txkierr


Parameters of the materialParameters of the material

Conductivity: purely imaginary

Bm

em

Nei

2

Bm

e

Bm

e

2c

c

Cyclotron frequency which is in resonance with the gyration frequency of the electrons

in the magnetic field

)(

1c

2p

2

22

ck

Wave number


Phase propagation velocityPhase propagation velocity

Index of refraction

)(1

c

2p2

n

Phase propagation velocity ncv /


Relative phase differenceRelative phase differenceTwo circularly polarized waves will have a relative phase difference after a propagation distance due to the slightly different phase velocity

zkk )(2 2 3p c

2 2 2

2

2

Ne Bz z

c m c

3

//2 20

1( )

2

LeB N z dz

m c

pccmGaussm

101.8rad

pc/

03

//

25 z

dNBL


Rotation MeasureRotation Measure

2

2

2

1

21

pc/

03

//52-

mm

radrad

pccmGauss101.8

m rad

RM

zd

NBL

DM

RM1023.1

Gauss6//

B Magnetic field parallel to the line of sight


ExampleExample

• Determine the upper limit of the angle through which a linearly polarized EM wave is rotated when it traverses the ionosphere. Take the following parameters: an ionospheric depth of 20km, an average electron density of 105cm-3 and a magnetic field strength (assumed to be parallel to the direction of wave propagation) of 1G. – Find RM– Carry out the calculation for the Faraday rotation,

Δψ for frequencies of 100MHz, 1GHz and 10GHz, if the rotation is Δψ/rad=(λ/m)2RM

– What is the effect if the magnetic field direction is perpendicular to the direction of propagation? What is the effect on circularly polarized EM waves?


• Repeat previous problem for the conditions which hold in the solar system: the average charged particle density in the solar system is 5 cm-3, the magnetic field 5μG, and the average path 10AU. What is the maximum amount of Faraday rotation of an EM wave of frequency 100MHz, 1GHz? Must radio astronomical results correct for this?


ExampleExample

• A source is 100% linearly polarized in the north-south direction. Express this in terms of Stokes parameters.

• Intense spectral line emission at 18cm wavelength is caused by maser action of the OH molecule. At certain frequencies, such emission shows nearly 100% circular polarization, but little or no linear polarization. Express this in terms of Stokes parameters.


examplesexamples

• If the DM for a given pulsar is 50, and the value of RM is 1.2×102, what is the value of the line-of-sight magnetic field? If the magnetic field perpendicular to the line of sight has the same strength, what is the total magnetic field?


HomeworkHomework

• A plane electromagnetic wave perpendicularly approaches a surface with conductivityσ. The wave penetrates to a depth of δ. Apply equation (2.25), taking σ>>ε/4π, so The solution to this equation is an exponentially decaying wave. Use this to estimate the 1/e penetration depth δ. Estimate the value of

for copper, which has (in CGS units) σ=1017s-1 and μ≈1 for =1010Hz.

EcE )/4( 22

4/c


Cont’dCont’d

• Assume that pulsars emit narrow periodic pulses at all frequencies simultaneously. Use eq. (2.83) to show that a narrow pulse (width of order 10-6s) will traverse the radio spectrum at a rate, in MHz s-1, of

• Show that a receiver bandwidth will lead to the smearing of a very narrow pulse which passes through the ISM with dispersion measure DM, to a width

314 ]/[)(102.1 MHzDMv

s ]/[103.8 33 BMHzDMt


ExamplesExamples• In the near future there may be an anti-collisi

on radar installed on automobiles. This will operate at ~70GHz. The bandwidth is proposed to be 100MHz, and at a distance of 3m, the power per area is 10-9Wm-2. Assume the power level is uniform over the entire bandwidth of 100MHz. What is the flux density of this radar at 1km distance? A typical radio telescope can measure to the mJy level. At what distance will such radars disturb such radio astronomy measurements?


ExamplesExamples

• A signal passes through two cables with the same optical depth τ. They have temperatures T1 and T2, with T1>T2. Which should be connected first to obtain the lowest output power from this arrangement?


ExamplesExamples

• The 2.73K microwave background is one of the most important pieces of evidence in support of the big bang theory. The expansion of the universe is characterized by the redshift z. The ratio of the observed wavelength λo to the (laboratory) rest wavelength λr is related to z by z=(λo / λr)-1. The dependence of the temperature of the 2.73K microwave background on z is T=2.73(1+z). What is the value of T at z=2.28? What is the value at z=5 and z=1000?


Examples Examples

• The pulsar in the Crab nebula has a dispersion measure DM=57 cm-3pc, and a period of 0.0333s. Staelin and Reifenstein (1969 Science 162, 1481) discovered this pulsar at ν=110MHz, using a 1MHz-wide receiver bandwidth. Someone tells you that “this pulsar would not have been found at 110MHz if the pulses all had the same amplitude.” Do you believe this? Use the following relation to support your decision: the smearing Δt of a short pulse is (202/νMHz)3DM ms per MHz of receiver bandwidth.


HomeworkHomework

• A cable has an optical depth τof 0.1 and a temperature of 300K. A signal of peak temperature 1K is connected to the input of this cable. Use equation (1.34) in the textbook with T being the temperature of the cable and T (0) the temperature of the input signal. What is the temperature of the output of the signal? Would cooling the cable help to improve the detectability of the input signal?


Homework (cont’d)Homework (cont’d)

• A signal passes through two cables with the same optical depth, t. These have temperatures T1 and T2, with T1>T2. Which cable should be connected first to obtain the lowest output power from this arrangement?


Homework (cont’d)Homework (cont’d)

• Apply the Stefan-Boltzman relation to the Sun and the planets to estimate the surface temperature if each planet is assumed to absorb all of the radiation it receives (this is an albedo of zero – this is the upper limit the planet can absorb since in reality some radiation is reflected). As a first approximation, assume that the planets have no atmosphere and no internal heating sources and that the rapid rotation equalizes the surface temperatures. The distances for assumed circular orbits (in AU) are: Mercury (0.39AU), Venus (0.72AU), Earth (1 AU) , Mars (1.5AU), Jupiter (5.2AU). At a wavelength of 68cm, Jupiter was found to have a brightness temperature of more than 500K. Could the temperature of Jupiter be caused by solar heating?


TelescopesTelescopes

• The Green Bank TelescopeType: off-axis, fully steerableDiameter: 100 by 110 metersSurface accuracy: 1.2mm--0.3mmWorking wavelengths: cm to mm

• The Parkes TelescopeDiameter: 64-m, in the southern skyWorking wavelength: cm

• The Nobeyama 45-m• JCMT JCMT with no membrane

15-m, sub-mm(surface accuracy 14-18 m), Mauna Kea


TelescopesTelescopes

• InterferometersVLBA: 10 radio telescopes across USA

VLA: 27 25-m antennas, Y-shape, largest separation of antenna 36km (0.04 arcsecond at 43GHz)

The VLA looking south

MERLIN: an array of radio telescopes in UK, with separation up to 217km (0.05 arcsecond at 5GHz)

• List of radio telescopes


Radio astronomy in ChinaRadio astronomy in China

• Telescopes• Miyun Synthesis Radio Telescope: linear array of 28

9-m antennas working at 232MHz• Shanghai: 25-m• Urumuqi: 25-m• Qinghai Delingha: 13.7-m

• Projects• FAST: Five hundred meter Aperture Spherical Teles

cope 30 elements, Guizhou• Large radio telescope: 50-m• MSRT FAST DLH Urumqi Sheshan


The future of radio astronomyThe future of radio astronomy

• Bigger telescopes– Atacama Large Millimeter Array(ALMA)

• ESO,IRAM,OSO,NFRA,NRAO,NAOJ……• 64 12-m antennas, 10mm-0.35mm, 150m-10km• Year 2012

– VSOP-2

• ResearchFainter objects, finer structure


HomeworkHomework

• If the average electron density in the interstellar medium is 0.03 cm-3, what is the lowest frequency of electromagnetic radiation which one can receive due to the plasma cutoff? Compare this to the ionospheric cutoff frequency if the electron density, Ne, in the ionosphere is ~105cm-3. Use

Where p is the plasma cutoff frequency.

3978

cm

N.

kHz

νep

2009/08/24-282009/08/24-28 日日射电天文暑期学校射电天文暑期学校JCMT


JCMT without membrane


Parkes


佘山


乌鲁木齐


VLA


FAST


Nobeyama


White light, radio and X-ray Sun


德令哈

2009/08/24-28 2009/08/24-28. 2009/08/24-28 1. 2. 3. 4. 5

Documents

jy slide

blackbody slide

distance slide

total energy density

star radio astronomy

noise temperature slide

intervening medium slide

bright sphere slide