Tetsuo Sasao and Andr´eB. Fletcher Introduction to VLBI Systems Chapter 1 Lecture Notes for KVN Students Based on Ajou University Lecture Notes (to be further edited) Version 2. Issued on February 28, 2005. Revised on March 18, 2005. Revised on February 4, 2006. Revised on March 5, 2008. Revised on February 24, 2008. Basic Knowledge of Radio Astronomy This chapter describes basic cooncepts on radio sources and their radi- ation mechanisms to the extent meaningful, or desirable, for understanding VLBI systems. Contents 1 ‘Windows’ for Ground–based Astronomy 3 2 Various Classifications of Astronomical Radio Sources 4 3 Spectra of Typical Continuum Radio Sources 4 4 Spectralline Radio Sources 5 5 Designations of Astronomical Radio Sources 8 6 Designations of Frequency Bands 8 1 7 Basic Quantities in Radio Astronomy 9 7.1 Intensity (Specific/Monochromatic Intensity) Iν or Brightness Bν ................................. 9 7.2 Constancy of the Monochromatic Intensity Iν ......... 10 7.3 ‘Spectral Flux Density’ or ‘Flux Density’ or ‘Flux’ Sν ..... 11 7.4 Power/Energy/Radiation Flux Density S ............ 12 7.5 Spectral Energy Density per Unit Solid Angle uν ....... 12 7.6 Spectral Energy Density Uν ................... 13 8 Emission and Absorption of Electromagnetic Radiation 14 8.1 Elementary Quantum Theory of Radiation (A. Einstein, 1916) 14 8.2 EinsteinCoefficients . 17 8.3 NumberDensityofPhotons . 18 9 Blackbody Radiation 19 9.1 Two Extreme Cases of the Planck Spectrum . 21 9.2 Wien’sLaw ............................ 22 9.3 Spectral Indices of Thermal Continuum Radio Sources . 22 9.4 Stars are Faint and Gas Clouds are Bright in the Radio Sky . 22 9.5 Stefan–BoltzmannLaw . 23 9.6 Total Blackbody Radiation from a Star or a Gas Cloud . 24 9.7 Universality of the Relationship among Einstein’s Coefficients 25 9.8 Another Important Quantity in Radio Astronomy – Brightness Temperature............................ 26 10 Radiative Transfer 26 10.1 Phenomenological Derivation of the Radiative Transfer Equation 26 10.2 Derivation of the Radiative Transfer Equation from Einstein’s Elementary Quantum Theory of Radiation . 27 10.3 The Simplest Solutions of Radiative Transfer Equation .... 28 10.4WhatisLTE?........................... 32 10.5 SpectrumoftheOrionNebula . 33 11 Synchrotron Radiation 34 11.1 Non–RelativisticCase . 35 11.2 RelativisticCase . 36 2 1 ‘Windows’ for Ground–based Astronomy Figure 1 shows how the Earth’s atmosphere transmits electromagnetic waves from the Universe. The curve over the hatched area in this figure corresponds to the height in the atmosphere at which the radiation is attenuated by a factor of 1/2. Figure 1: ‘Windows’ for ground-based astronomy (from Rohlfs & Wilson, 2001). This figure shows that the Earth’s atmosphere is largely transparent in the radio (15 MHz – 300 GHz) and visible light (360 THz – 830 THz) regions of the electromagnetic spectrum. These two regions are the main ‘windows’ for ground-based astronomical observations. VLBI (Very Long Baseline In- terferometry) is one of the techniques used to observe astronomical radio sources through the ‘radio window’. 3 2 Various Classifications of Astronomical Ra- dio Sources There are several ways to classify astronomical radio sources. Typical clas- sifications are: • – Thermal source Source emitting via a thermal mechanism (e.g. blackbody radia- tion), – Non–thermal source Source emitting via a non–thermal mechanism (e.g. synchrotron radiation, inverse Compton scattering, annihilation radiation, maser emission, etc.), • – Continuum source Source emitting over a broad range of frequencies, – Spectralline source Source emitting in narrow lines at specific frequencies, and • – Galactic source Source inside our Milky Way Galaxy, – Extragalactic source Source outside our Galaxy. In astrophysical applications of VLBI, we try to observe everything, but in geodetic VLBI, we usually observe non–thermal continuum extragalactic sources. 3 Spectra of Typical Continuum Radio Sources Figure 2 shows typical spectra of continuum radio sources. These spectra are characterized by a quantity called ‘spectral index’ α, de- −α fined by the formula: Sν ∝ ν , where Sν is a quantity called the ‘flux density’ [unit: Jy (Jansky) ≡ 10−26W m−2Hz−1], which is a measure of the strength of the radiation from a source. Thermal and non–thermal sources usually show different spectral indices, namely: α ∼= −2 for thermal sources, and α ≥ 0 for non-thermal sources. 4 Figure 2: Spectra of galactic and extragalactic continuum radio sources (from Kraus, 1986). 4 Spectralline Radio Sources An example of a thermal spectralline source showing many molecular lines is given in Figure 3. Examples of spectra of non-thermal maser line sources obtained in test observations for the VERA (VLBI Exploration of Radio Astrometry) array in Japan are shown in Figures 4, 5, and 6. 5 Figure 3: Thermal molecular lines observed in the Orion-KL star–forming region (Kaifu et al., 1985). Figure 4: H2O maser lines in Cep A (Left), and RT Vir (Right) (courtesy of VERA group, Japan, 2002). 6 Figure 5: SiO maser lines in Orion-KL (Left), and R Cas (Right) observed in pointing tests for a VERA antenna (courtesy of VERA group, Japan, 2002). Figure 6: H2O maser lines in W49 N and OH43.8, observed simultaneously with a VERA dual-beam antenna (courtesy of VERA group, Japan, 2002). 7 5 Designations of Astronomical Radio Sources There are many designation systems for naming radio sources. Examples are given in the following. As a result, radio sources are often identified by several different names. • IAU 1974 system of designation (‘IAU name’, see, e.g., Explanatory Supplement to American Astronomical Almanac) – identifies a radio source by a six, seven, or eight-digit number, such as 0134+329, which tells us that the right ascension of the source is 01h 34m, and its declination is +32.◦9 (usually in the B1950 equinox system). – sometimes source types or catalog acronyms are added, e.g., pul- sars are called ‘PSR0950+08’, and sources from the Parkes Sky Survey as ‘PKS1322−42’. – when the right ascension and declination values are given in the J2000 equinox system, a letter ‘J’ is added. For example, 0134+329 is otherwise called J0137+331. • 3C-name, 4C-name – identifies an extragalactic radio source by a serial number in the Third Cambridge Catalog, and by declination in the Fourth Cam- bridge Survey Catalog. – examples are 3C84, 3C273, 3C345, 4C39.25, etc. • W-name – is based on Westerhout’s (1958) catalog of HII regions. – examples are W49 N, W3(OH), W51 M, etc. 6 Designations of Frequency Bands The basic unit of radio frequency is Hz (Hertz: cycle per second). In order to describe high frequency, typically used in radio astronomical observations or communications, we frequently use the following secondary units: kHz (kilohertz: 103 Hz), MHz (megahertz: 106 Hz), GHz (gigahertz: 109 Hz), and THz (terahertz: 1012 Hz). 8 Figure 7: Frequency designations. Radio frequency bands used in astronomical observations, as well as in communications, are designated by one or a few alphabetic characters. Two systems of designations are shown in Figure 7. In radio astronomy, IEEE STD-521-1976 designations have been used. Examples are: • L–band : GPS satellite, observations of OH masers, • S– and X–bands : geodetic VLBI observations, • K–band : observations of H2O masers, • Q–band : observations of SiO masers. 7 Basic Quantities in Radio Astronomy The following quantities are commonly used in radio astronomy to charac- terize radio waves coming from celestial bodies. 7.1 Intensity (Specific/Monochromatic Intensity) Iν or Brightness Bν The intensity Iν(s) (or brightness Bν (s)) is the quantity of electromagnetic radiation energy incoming from a certain direction s in the sky, per unit solid angle, per unit time, per unit area perpendicular to this direction, and per unit frequency bandwidth with center frequency ν. The SI unit of this specific (or monochromatic) intensity is thus: W m−2 Hz−1 sr−1. 9 In terms of the monochromatic intensity Iν , the power dW of radiation coming from a direction s within a solid angle dΩ, through a cross-section of area dσ with a normal inclined by an angle θ from the direction s, within a frequency bandwidth dν centered at ν, is given by: dW = Iν(s) cos θdΩ dσ dν, (1) as shown in Figure 8. s dΩ θ dσ Figure 8: Power dW of radiation with Specific/Monochromatic Intensity Iν (s) (or Brightness Bν (s)) coming through a solid angle dΩ, a cross-section dσ, and a frequency bandwidth dν is given by dW = Iν (s) cos θ dΩ dσ dν. 7.2 Constancy of the Monochromatic Intensity Iν The monochromatic intensity Iν remains constant along a ray path, irre- spective of the distance from the emitting source, as long as no absorption, emission, or scattering occurs along the path. An explanation of this con- stancy may be given as follows. Let us assume a spherically symmetric distribution of radiation from an emitting source at the center, without any intervening absorption, emission, or scattering, as shown in Figure 9. Let a cone-like ray tube of constant opening angle intersect a sphere of radius r, forming a cross–section of area σ, and let the source be seen to subtend a solid angle Ω at this radius. The power of the radiation with bandwidth ∆ν passing through this cross–section is W = Iν Ω σ∆ν, which must be constant at any radius along the tube, as long as the radius is much larger than the source size. Now, the solid angle of the source Ω and the area of the cross–section σ vary as Ω ∝ r−2 and σ ∝ r2, 10 W2 σ 2 Ω 2 W 1 σ 1 r Ω 2 1 r 1 Figure 9: Constancy of the monochromatic intensity Iν along the ray path. respectively. Therefore, the monochromatic intensity Iν = W/(Ω σ ∆ν) must be constant along the path.