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射电天文基础 姜碧沩 北京师范大学天文系 2009/08/24-28日,贵州大学 大纲 1. 2. 3. 4. 5. 射电天文基础 射电望远镜 连续谱辐射机制 谱线辐射机制 星际分子 参考书:《射电天文工具》 2009/08/24-28日 射电天文暑期学校 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 2009/08/24-28日 射电天文暑期学校 The waves used by optical astronomers • • • • • • • Electromagnetic Spectrum 4000 to 8000 angstroms 7.51014Hz to 3.751014Hz The Sun The solar system Stars Galaxies 2009/08/24-28日 射电天文暑期学校 The 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 2009/08/24-28日 射电天文暑期学校 Some 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 2009/08/24-28日 射电天文暑期学校 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 2009/08/24-28日 射电天文暑期学校 The 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 Telescope Type: Fully steerable Diameter: 100-m Working wavelength: up to 3mm, mainly cm 2009/08/24-28日 射电天文暑期学校 Fundamentals of Radio Astronomy • Some basic definitions • Radiative transfer • Blackbody radiation and brightness temperature • Nyquist theory and noise temperature 2009/08/24-28日 射电天文暑期学校 I: specific intensity • dW I cosddd • 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。 2009/08/24-28日 射电天文暑期学校 The total flux of a source • Total flux of a source: integration over the total solid angle of the source Ωs S • Unit I ( , ) cosd s – 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 2009/08/24-28日 射电天文暑期学校 Brightness is independent of the distance I 1 (r1) I 2 (r 2) 2009/08/24-28日 射电天文暑期学校 The total flux density depends on distance as r-2 • Total flux received at an point P from an uniformly bright sphere c S I ( , )cos d I sin cos d d s 0 c 2 R R sin c S I 2 I r r 2 2009/08/24-28日 射电天文暑期学校 Radiation energy density • Energy density per solid angle: erg cm-3Hz-1 1 u () I c • Total energy density 1 u u ( )d I d c ( 4 ) ( 4 ) 2009/08/24-28日 射电天文暑期学校 Radiative transfer • For radiation in free space the specific intensity is independent of distance. But I changes if radiation is absorbed or emitted. dI I ds, dI ds, 2009/08/24-28日 射电天文暑期学校 I (s ds) I (s) dsd dd I dsd dd dI I ds 2009/08/24-28日 射电天文暑期学校 Limiting cases • Emission only: 0 dI , ds s I ( s ) I ( s0 ) ( s )ds s0 • Absorption only: 0 s dI I , I ( s) I ( s0 ) exp ( s)ds ds s0 2009/08/24-28日 射电天文暑期学校 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 dI 0, I B (T ) ds 2h 3 1 B (T ) 2 c e h / kT 1 2009/08/24-28日 射电天文暑期学校 ( s ) cases (cont’d) Limiting I ( s) I (0)e B (T ( ))e d (s) 0 • Local Thermodynamic Equilibrium (LTE) – Kirchhoff’s Law B (T ) s ds – Optical depth d ds 0 – Equation of transfer – Solution 2009/08/24-28日 dI 1 dI I B (T ) ds d 射电天文暑期学校 LTE (cont’d) • The medium is isothermal – T(τ)=T(s)=T=const. I ( s) I (0)e ( s ) B (T )(1 e ( s ) ) • Optical depth is very large – τ(0) I B (T ) – Difference with the intensity in the absence of an intervening medium I ( s) I ( s) I (0) ( B (T ) I (0))(1 e ) 2009/08/24-28日 射电天文暑期学校 Blackbody radiation • Planck law 2h 3 1 B (T ) 2 c e h / kT 1 B (T ) 2hc 2 1 5 e hc / kT 1 • Total brightness of a blackbody 2 k 5 2 1 4 B(T ) T , 1 . 8047 10 erg cm s K 15c 2 h 3 4 4 4 2009/08/24-28日 射电天文暑期学校 Wien’s displacement law • Maxima of B (T) and Bλ(T) – Bν/ν=0 and Bλ/λ=0 – νmax max T – λmax 2009/08/24-28日 58.789 GHz K max cm T 0.28978 K 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 Rayleigh-Jeans Law • Rayleigen-Jeans Law h kT T 20.84 GHz K e c h 1 J (T ) I 2 h / kT k 2k e 1 2 h / kT • Radiation temperature h 1 kT 2 BRJ ( , T ) 2 kT c 2009/08/24-28日 2 射电天文暑期学校 Wien’s Law h kT e h kT 1 2h h / kT BW ( , T ) 2 e c 3 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 Brightness temperature Tb • 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 c 1 Tb B B 2 2k 2k 2 2009/08/24-28日 2 射电天文暑期学校 Transfer equation of Tb • Transfer equation dTb (s) Tb (s) T (s) d • General solution Tb ( s) Tb (0)e ( s ) ( s ) T ( s ) e d 0 • Two limiting cases when Tb(0)=0 – Optically thin, τ<<1 Tb T – Optically thick, τ>>1 Tb T 2009/08/24-28日 射电天文暑期学校 The Nyquist Theorem • Johnson noise – The thermal motion of the electrons in a resistor will produce 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 2009/08/24-28日 射电天文暑期学校 Electromagnetic wave propagation 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 2009/08/24-28日 射电天文暑期学校 Maxwell’s equations • Material equations J E D E B H • Maxwell’s equations D 4 B 0 1 E B c 4 1 H J D c c • Continuity equation of charge density and current J 0 2009/08/24-28日 射电天文暑期学校 Energy conservation and the Poynting vector • Energy density of an electromagnetic field 1 1 E D B H E 2 H 2 u 8 8 • Poynting vector c S E H 4π • Equation of continuity for S u S E J射电天文暑期学校 2009/08/24-28日 t Complex field vectors • Complex field vectors E E1 iE 2 e it H H 1 iH 2 e it • The Poynting vector c S ReE ReH 4 c * S Re E H 4 2009/08/24-28日 射电天文暑期学校 The wave equation 4 H 2 H 2 H 2 c c 4 E 2 E 2 E 2 c 2009/08/24-28日 c 射电天文暑期学校 Plane waves in nonconducting media • Nonconducting media – σ=0 – The wave equation 1 u 2 u 0 v 2 – Velocity of the wave v 2009/08/24-28日 c 射电天文暑期学校 Plane waves (cont’d) • Harmonic wave solution of the wave equation u u 0 e i kx t • Wave number • Phase velocity k 2 v c2 k 2 c • Index of refraction 2009/08/24-28日 c c n k v 射电天文暑期学校 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. E z 0, H z 0 E H 0 H E 2009/08/24-28日 射电天文暑期学校 c S 4 2 E Group velocity • Dispersion equation (k ) d (k ) 0 (k k 0 ) dk • Group velocity 0 d vg dk • Energy and information are usually propagated with the group velocity 2009/08/24-28日 射电天文暑期学校 Plane waves in dissipative media • Dissipative media 0 • Harmonic waves propagating in the direction of increasing x E( x, t ) E 0 e i ( kx t ) • Wave equations 2 2 4 E k c 2 i c 2 H=0 • Dispersion equation k 2 2009/08/24-28日 2 c 2 4 1 i 射电天文暑期学校 Cont’d • Wave number k a ib 2 1 4 a 1 1 c 2 2 1 4 b 1 1 c 2 • Field E( x, t ) E 0 e bx i ( axt ) e n E( x, t ) E 0 exp nx exp i x t c c 2009/08/24-28日 射电天文暑期学校 Cont’d • Index of refraction and absorption coefficient 2 1 4 n 1 1 2 2 1 4 n 1 1 2 2009/08/24-28日 射电天文暑期学校 Dispersion measure of a tenuous 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 me v mer -eE 0 e it – Solution 2009/08/24-28日 e e it v E0e i E im e me 射电天文暑期学校 Cont’d • Conductivity of the plasma Ne 2 i me ω • Wave number for a thin medium with ε≈1 andμ≈1 2 p 2 k 2 1 2 c 2 2009/08/24-28日 2 4 Ne p2 me 射电天文暑期学校 Cont’d • Phase velocity and group velocity – For ω>ωp, k is real, v>c, vg<c c v 1 vvg c 2009/08/24-28日 2 2 p 2 vg c 1 2 2 p n 1 射电天文暑期学校 2 p 2 Dispersion measure of pulsars • A pulse emitted by a pulsar at a distance L will be received after a delay 2 dl 1 1 p D 1 2 v c 2 0 g 0 L L L 2 1 e 1 dl 1 N (l ) dl 2 c 0 2me • The difference between the pulse arrival time measured at two frequencies 2 e D 2cme 2009/08/24-28日 1 1 2 2 N (l )dl 1 2 0 L 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 Cont’d • Dispersion Measure N l DM 3 d pc 0 cm DM 1 1 4 D 2 . 410 10 3 2 2 cm pc μs 1 2 MHz MHz 2009/08/24-28日 射电天文暑期学校 1 Dispersion Measure, DM, for pulsars at different Galactic latitudes 2009/08/24-28日 射电天文暑期学校 Faraday 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. 2009/08/24-28日 射电天文暑期学校 Equation of motion for an electron in the presence of a magnetic field mv mr -eE r B If the magnetic field B is oriented in the z direction rx ry e e Bry E x m m e e Brx E y m m 2009/08/24-28日 e e r i Br E m m r rx ir y E E x iE y 射电天文暑期学校 Solution Linearly polarized wave can be regarded as the superposition of circularly polarized waves 1 1 E x ( E E ), E y ( E E ) 2 2i Solution in the form of harmonic waves E Aei ( k x t ) r r0 e 2009/08/24-28日 i ( k x t ) 射电天文暑期学校 Parameters of the material 2 Ne Conductivity: purely imaginary i e m B m e c B Cyclotron frequency which is in resonance m with the gyration frequency of the electrons in e the magnetic field c B 2m 2 2 p 2 k 1 Wave number c 2 ( c ) 2009/08/24-28日 射电天文暑期学校 Phase propagation velocity Index of refraction 2 p 2 n 1 ( c ) Phase propagation velocity 2009/08/24-28日 v c / n 射电天文暑期学校 Relative phase difference Two circularly polarized waves will have a relative phase difference after a propagation distance due to the slightly different phase velocity 2 (k k )z 3 p2c 2 Ne3 B z 2 2 z 2 2c m c L e 1 B N (z )dz 2 2 // 2 m c 0 5 8.1 10 rad m 2009/08/24-28日 2 L / pc 0 B// N z 3 d Gauss cm pc 射电天文暑期学校 Rotation Measure RM 5 8.110 -2 rad m L / pc 0 B// N z 3 d Gauss cm pc 1 2 rad rad 2 2 1 2 m m B// 6 RM 1.23 10 Gauss DM 2009/08/24-28日 Magnetic field parallel to the line of sight 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 Example • 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 2009/08/24-28日 射电天文暑期学校 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? 2009/08/24-28日 射电天文暑期学校 Example • 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. 2009/08/24-28日 射电天文暑期学校 examples • 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? 2009/08/24-28日 射电天文暑期学校 Homework • 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 2 E (4 / c 2 ) E The solution to this equation is an exponentially decaying wave. Use this to estimate the 1/e penetration depth δ. Estimate the value of c / 4 for copper, which has (in CGS units) σ=1017s-1 and μ≈1 for =1010Hz. 2009/08/24-28日 射电天文暑期学校 Cont’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 v 1.2 104 ( DM ) 1[ / MHz ]3 • 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 t 8.3 10 DM [ / MHz ] B s 3 2009/08/24-28日 射电天文暑期学校 3 Examples • In the near future there may be an anticollision 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? 2009/08/24-28日 射电天文暑期学校 Examples • 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? 2009/08/24-28日 射电天文暑期学校 Examples • 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? 2009/08/24-28日 射电天文暑期学校 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 2009/08/24-28日 射电天文暑期学校 bandwidth. Homework • 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? 2009/08/24-28日 射电天文暑期学校 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? 2009/08/24-28日 射电天文暑期学校 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? 2009/08/24-28日 射电天文暑期学校 Telescopes • The Green Bank Telescope Type: off-axis, fully steerable Diameter: 100 by 110 meters Surface accuracy: 1.2mm--0.3mm Working wavelengths: cm to mm • The Parkes Telescope Diameter: 64-m, in the southern sky Working wavelength: cm • The Nobeyama 45-m • JCMT JCMT with no membrane 15-m, sub-mm(surface accuracy 14-18 m), Mauna Kea 2009/08/24-28日 射电天文暑期学校 Telescopes • Interferometers VLBA: 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 2009/08/24-28日 射电天文暑期学校 Radio 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 Telescope 30 elements, Guizhou Large radio telescope: 50-m MSRT FAST DLH Urumqi Sheshan 2009/08/24-28日 射电天文暑期学校 The 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 • Research Fainter objects, finer structure 2009/08/24-28日 射电天文暑期学校 Homework • 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 ~105cm3. Use νp N kHz 8.97 e 3 cm Where p is the plasma cutoff frequency. 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 JCMT 2009/08/24-28日 射电天文暑期学校 JCMT without membrane 2009/08/24-28日 射电天文暑期学校 Parkes 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 佘山 2009/08/24-28日 射电天文暑期学校 乌鲁木齐 2009/08/24-28日 射电天文暑期学校 VLA 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 FAST 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 Nobeyama 2009/08/24-28日 射电天文暑期学校 White light, radio and X-ray Sun 2009/08/24-28日 射电天文暑期学校 德令哈 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校 2009/08/24-28日 射电天文暑期学校