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Sp.-V/AQuan/1999/10/07:17:14 Page 121 Chapter 6 Radio Astronomy Robert M. Hjellming 6.1 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2 Atmospheric Window and Sky Brightness . . . . . . . 123 6.3 Radio Wave Propagation . . . . . . . . . . . . . . . . . 125 6.4 Radio Telescopes and Arrays . . . . . . . . . . . . . . 128 6.5 Radio Emission and Absorption Processes . . . . . . 131 6.6 Radio Astronomy References . . . . . . . . . . . . . . 140 INTRODUCTION Radio astronomy is defined by three things. First is the range of frequencies that constitute the radio windows of the Earth’s atmosphere, ranging roughly from 20 MHz to 1000 GHz. Second are the astronomical objects that emit radio waves with sufficient strength to be detectable at the Earth; some emit by thermal processes, but most are seen because of the emission from relativistic electrons (cosmic rays) interacting with local magnetic fields. Third, radio radiation behaves more like waves than particles (photons), allowing the measurement of both amplitude and phase of the radiation field. The capability to measure phase gives radio interferometry the capability to do the highest resolution imaging of astronomical objects currently possible in astronomy. The variety and number of radio sources is so large that in Table 6.1 we merely summarize a number of source catalogs devoted to these topics. 121 Sp.-V/AQuan/1999/10/07:17:14 122 / 6 Page 122 R ADIO A STRONOMY Table 6.1. List of radio source catalogs. Type of Source Contents 4C Radio Survey [1, 2] Bologna B2 Sky Survey [3–5] Bologna B3 Sky Survey [6, 7] 20 cm N. Sky Catalog [8] 6 cm Gal. Plane Survey [9] 6 cm South & Tropical Surveys [10, 11] 11 cm All Sky Catalog [12] 11 cm N. Sky Catalog [13] Bright Galaxies [14] 20 cm Gal. Plane Survey [15] 21 cm Gal. Plane Survey [16] Galaxy HI [17, 18] Atlas of Galactic HI [19] Molecular lines [20, 21] Pulsars [22] Opt. Pos. Radio Stars [23] 20 cm Radio Sources δ > −5◦ [24, 25] 4844 56 cm sources > 2 Jy, −7◦ ≤ δ ≤ 80◦ 408 MHz, 21.7◦ ≤ δ ≤ 40◦ 408 MHz, 37◦ ≤ δ ≤ 47◦ 365, 1400, 4850 MHz, 0 ≤ δ ≤ 75◦ Parkes 64 m, |b| ≤ 2◦ , l = 190◦ − 360◦ − 40◦ Parkes 64 m, −78◦ ≤ δ ≤ −10◦ Bright sources 6483 sources, |b| ≤ 5◦ , 240◦ ≤ l ≤ 357◦ All radio data pre-1975, normal galaxies VLA B Conf. Survey Eff. 100m, |b| ≤ 4◦ , 95◦ ≤ l ≤ 357◦ Galaxies, vradial ≤ 3000 km/s 21 cm H emission line profiles Frequencies, other molecular data Catalog of known pulsars 221 radio stars 1400 MHz sky atlas References 1. Pilkington, J.D.H., & Scott, P.F. 1965, Mem. R.A.S., 69, 183 2. Gower, J.F.R., Scott, P.F., & Willis D. 1967, Mem. R.A.S., 71, 144 3. Colla, G. et al. 1970, A&AS, 1, 281 4. Colla, G. et al. 1972, A&AS, 7, 1 5. Colla, G. et al. 1973, A&AS, 11, 291 6. Fanti, C. et al. 1974, A&AS, 18, 147 7. Ficcara, A., Grueff, G., & Tomesetti, G. 1985, A&AS, 59, 255 8. White, R.L., & Becker, R.H. 1992, ApJS, 79, 331 9. Haynes, R.F., Caswell, J.L., & Simons, L.W.J. 1978, Aust. J. Phys. Suppl., 48, 1 10. Griffith, M.R. et al. 1994, ApJS, 90, 179 11. Wright, A.E. et al. 1994, ApJS, 91, 111 12. Wall, J.V., & Peacock, J.A. 1985, MNRAS, 216, 173 13. Fürst, E. et al. 1990, A&AS, 85, 805 14. Haynes, R.F., Caswell, J.L., & Simons, L.W.J. 1978, Aust. J. Phys. Suppl., 45, 1 15. Zoonematkermani, S. et al. 1990, ApJS, 74, 181 16. Reich, W., Reich, P., & Fürst, E. 1990, A&AS, 83, 539 17. Tully, R.B. 1988, Nearby Galaxies Catalog (Cambridge University Press, Cambridge) 18. Huchtmeier, W.K., & Richter, O.-G. 1989, A General Catalog of HI Observations of Galaxies: The Reference Catalog (Springer-Verlag, New York) 19. Hartman, D., & Burton, W.D. 1995, Atlas of Galactic HI (Cambridge University Press, Cambridge) 20. Lovas, F.J. 1992, J. Phys. Chem. Ref. Data, 21, 181 21. Lovas, F.J., Snyder, L.E., & Johnson, D.R. 1979, ApJS, 41, 451 22. Taylor, J.H., Manchester, R.N., & Lyne, A.G. 1993, ApJS, 66, 529 23. Réquième, Y. & Mazurier, J.M. 1991, A&AS, 89, 311 24. Condon, J.J., Broderick, J.J., & Seielstad, G.A. 1989, AJ, 97, 1064 25. Condon, J.J., Broderick, J.J., & Seielstad, G.A. 1991, AJ, 102, 2041 Other information and images from surveys can be obtained from Internet pages maintained by the National Center for Super Computing Applications (NCSA) Astronomy Digital Image Library (ADIL, http://imagelib.ncsa.uiuc.edu/imagelib.html) and the National Radio Astronomy Observatory (NRAO, http://info.aoc.nrao.edu). More extensive radio (and other) catalogs can be found from Internet pages for the NASA Astrophysics Data System (http://adswww.harvard.edu/index.html). In Figure 6.1 schematic radio spectra representative of a variety of continuum radio sources are plotted in the form of flux density (in units of Jy = Jansky = 10−23 ergs cm−2 s−1 Hz−1 = 10−26 W m−2 Hz−1 ) as a function of frequency. The sources and spectra in Figure 6.1 are schematic Sp.-V/AQuan/1999/10/07:17:14 Page 123 6.2 ATMOSPHERIC W INDOW AND S KY B RIGHTNESS / 123 108 Active Sun 107 106 105 Flux Density [Jy] 104 103 Sky Background Jupiter (radiation belts) Quiet Sun Cyg A Crab Virgo A Nebula M31 102 Moon Cas A Crab Pulsar (unpulsed) 3C295 3C273 101 Orion Nebula 100 Jupiter BL Lac 10-1 Mars SS433 Crab Pulsar (pulsed) Cyg X-3 (non-flaring) 10-2 Antares 10-3 10 100 1000 10000 Figure 6.1. Radio spectra of various types of sources in the form of flux density as a function of frequency ν. indicators of typical behavior for a wide variety of objects. Major solar-system radio sources are the active Sun, the quiet Sun, the Earth’s Moon, Mars, the surface of Jupiter, and Jupiter’s radiation belts. Stellar system sources are: pulsars, with the pulsed and unpulsed (dashed line) emission from the Crab pulsar; the partially ionized coronal emission of the red supergiant Antares; and the X-ray binaries SS433 and Cyg X-3 (quiescent). The optically thick and thin portions of the radio spectrum of the ionized gas from an HII region is indicated for the Orion Nebula. Cas A is a remnant of a supernova explosion in our Galaxy; and the Crab nebula spectrum is representative of the relatively rare plerion, i.e., nebulosities energized by pulsars. The radio spectrum for M31, the Andromeda nebula, is representative of spiral galaxy behavior. Strong radio galaxies are represented by Cyg A and Virgo A; BL Lac is a blazar, while 3C273 and 3C295 are strong quasars. Defining the convention that the spectral index α is the power law index for the flux density, Sν ∝ ν α , then positive spectral indices indicate optically thick emission while negative spectral indices indicate optically thin emission. Spectral indices near zero indicate optically thin emission for thermal sources, while similarly flat spectra for synchrotron radiation sources indicate optically thick emission from a wide range of optical depths. 6.2 6.2.1 ATMOSPHERIC WINDOW AND SKY BRIGHTNESS Atmospheric Window The low-frequency end of the radio window is set by ionospheric absorption. Since the ionosphere has diurnal variations, solar cycle variations, variations depending upon the effect of particle storms impacting the atmosphere, and variations with Earth latitude and longitude, the lowest frequency where the atmosphere is transparent varies considerably over the range 10 to 30 MHz. From that lowfrequency cut-off, up to about 22 GHz, the Earth’s atmosphere does not absorb radio waves, although variation in the index of refraction affects the phase of incoming radio waves at the higher frequencies. Sp.-V/AQuan/1999/10/07:17:14 Page 124 Atmospheric Transparency 124 / 6 R ADIO A STRONOMY 1 1 1 mm PWV 8 mm PWV 0 0.01 0 0.1 1 10 100 1000 0 Wavelength [cm] 200 400 600 800 1000 Frequency [GHz] Figure 6.2. The radio window for the Earth’s atmosphere shown in terms of plots of atmospheric transparency as a function of both wavelength and frequency. The solid line is for excellent conditions with 1 mm PWV and the dashed line is for normal conditions with 8 mm PWV. At the lowest frequencies plasma effects in the ionosphere induce variable Faraday rotation in incoming radio waves. In Figure 6.2 we plot models for the transparency of the Earth’s atmosphere for two values of precipitable water vapor (PWV), 1 and 8 mm, representing extremely low and normal levels, respectively. 6.2.2 Surface Brightness and Brightness Temperature All astronomy is based upon measurements of surface brightness on the celestial sphere: Iν (α, δ, t), where t is time, (α,δ) is position specified by right ascension and declination, ν is frequency, and I is the specific intensity which can be any of the four Stokes parameters. The surface brightness of a black body of temperature T is Bν = (2hν 3 /c2 )(1/[ehν/kT − 1]), where h and k are the Planck and Boltzmann constants. For small values of hν/kT , the Rayleigh–Jeans approximation is valid, and Bν (T ) ∼ = 2kT /λ2 ; this is valid at longer radio wavelengths. For reasons related to the antenna measurement equation, radio astronomy often uses brightness temperature rather than surface brightness to describe measurements, where the brightness temperature is Tb ≡ λ2 Bν . 2k (6.1) Equation (6.1) is often used even when black body or long wavelength approximations are not applicable. Because of this, the concept of radiation temperature (Tr ≡ [hν/k]/[ehν/kT − 1]) is commonly used; where T is a “true” physical temperature. 6.2.3 Sources of Background Radiation At each wavelength there are sources of background radiation detectable by radio telescopes, principally the Earth’s atmosphere, the cosmic radiation background, and diffuse emission from our Sp.-V/AQuan/1999/10/07:17:14 Page 125 100000 Galactic Background Earth's Atmosphere 10000 100 1000 Earth's Atmosphere 100 10 COBE 1 10 Cosmic Background Galactic Background 0.1 0.01 0.001 COBE Cosmic Background 0.0001 100 10 1 0.1 100 Wavelength [cm] 10 1 1 Brightness Temperature [K] 15 -2 -1 -1 Brightness x 10 [W cm str cm ] 6.3 R ADIO WAVE P ROPAGATION / 125 0.1 Wavelength [cm] Figure 6.3. Apparent brightness (left) and brightness temperature (right) plotted as a function of wavelength for: the Earth’s atmosphere (dashed line, model; open circle, measurements); cosmic background radiation from COBE (solid line) and other instruments (filled circles); and galactic background radiation (dotted line). Galaxy. In Figure 6.3 we plot data and models for these sources of radiation. At wavelengths shorter than ∼ 1 cm the Earth’s atmosphere dominates the background, while above 13 cm the galactic background dominates. Between 1 and ∼ 13 cm the cosmic background dominates. However, in many cases man-made interference, or solar radio emission, can be more important “background emission,” particularly at wavelengths longer than 10 cm. 6.3 6.3.1 RADIO WAVE PROPAGATION Radiators-Absorbers, Fields, and Coupling Equations Radiation measurement at radio wavelengths allows direct measurement of the amplitude and phase of electromagnetic fields; this is the reason radiation field analysis in radio astronomy is usually done in terms of fields. If we identify a distant point in a radio source with the vector R, and the location of the oscillating current element with the vector, the fields between the two points are proportional to the propagator [1] Pν (r) = e2πiν|R−r|/c , |R − r| (6.2) so by Huygens’ Principle, and the fact that all radiation appears as if it originates from the celestial sphere, the field on the celestial sphere at r is the superposition E ν (r) = E(R) e2πiν|R−r|/c d S, |R − r| (6.3) where d S is a surface element on the celestial sphere. The radiation flux for propagating fields is determined by the real part of the Poynting flux averaged over time, so Sr = E × H = Re(E θ H φ − E φ Hθ )/2 = E θ Hφ . The power d P at distance r Sp.-V/AQuan/1999/10/07:17:14 126 / 6 Page 126 R ADIO A STRONOMY passing through solid angle d = d A/r 2 is the Poynting flux through d A at distance r . From the field distribution over the surface of an antenna we compute d P/d which allows us to calculate the important properties of antennas. The normalized antenna pattern is defined by Pn (θ, φ) = d P/d (d P/d)max (6.4) so an infinitesimal current element has Pn (θ, φ) = sin2 θ . The beam solid angle of an antenna is a = 4π Pn (θ, φ) d, and the directivity is D = 4π/ a [2]. For real antennas we integrate over the collecting surface and from that compute d P/d from which antenna parameters may be computed. 6.3.2 Radio Noise and Detection Limits The measured quantities for radio telescopes are total power or correlated total power at some point in the signal path of the electronics. This is one of the sources of the predominance of using temperature as a measurement variable, because of Nyquist’s law which relates the total power Pa to an antenna temperature Ta by Pa = 4kTa ν, (6.5) where k ν is the frequency range (or bandwidth) over which the total power measurement is made. Pa is the measure of the sum of the power due to radiation sources external to the antenna and the power generated in the antenna and its electronics system. In modern systems the latter is dominated by the receiver temperature, but is generally called system temperature (Tsys ) which constitutes an irreducible minimum level of noise in the measured signal. The power being measured is that of a fluctuating voltage, originally induced by the external electric fields in the form of oscillating currents in the antenna’s collecting surface, which radiate with a focus on either a feed at the prime focus of the antenna or on one or more subreflectors directed at a feed. The feed absorbs radiation and produces an output signal which is transferred to an electronic receiver followed (usually) by a complicated series of amplifiers, frequency converters, and other electronics elements. Then the fluctuating total power is then measured over a specific frequency range with a bandwidth (ν), and over a specific time interval t. For a system temperature Tsys the noise component of the measured signal is Tnoise = √ Tsys ν t , (6.6) which indicates how noise changes with bandwidth and integration time. The same principle applies in the case of interferometry where signals are either added or (almost all the time) correlated (or multiplied), so it is the correlated power from two antennas that is being measured. The noise component of the system temperature is usually the limiting factor in the measurements being made. The fundamental limit to electronic noise temperatures is the quantum limit: hν/k = 0.048νGHz K, where νGHz is the frequency in units of GHz. Real system temperatures must be above this limit. In the 1990s the state of the art of modern electronics is roughly capable of producing noise temperatures of 4hν/k. It is expected that at the beginning of the twenty-first century noise temperatures will be ∼ 2hν/k, but it is unlikely that they will ever get very close to the quantum limit. 6.3.3 Effects of Ionosphere, ISM, and Source Environment Because radio signals are best dealt with by an analysis based upon electric fields, one should think of radio sources as three-dimensional regions radiating electric fields. Once radiation that will eventually Sp.-V/AQuan/1999/10/07:17:14 Page 127 6.3 R ADIO WAVE P ROPAGATION / 127 reach a radio telescope on, or near, the Earth is produced, one then must think of propagation phenomena which affect these fields. Absorption and re-radiation by free electrons, ions, and atomic or molecular species can change the original radiation field into one modified by many effects. One can decompose this process into propagation effects: inside radio sources; in the intergalactic and interstellar medium (IGM and ISM); in the interplanetary medium of the solar system; and in the Earth’s atmosphere and ionosphere. The ionized component of the intervening medium between a radio source and an observing telescope affects propagation by introducing time delays and Faraday rotation. The time of arrival L of a pulse of radiation is tpulse = 0 ds/vgroup where L is the propagation path length, ds is a segment of the line of sight, and vgroup is the group velocity. The delay per unit frequency introduced by the electron concentration Ne is approximated by L dtpulse ∼ e2 Ne ds. (6.7) = dν 4π 0 mcν 3 0 The pulsed emission of radio pulsars allows the measurement of the changing time of arrival of a pulse L with frequency, so the dispersion measure, Dm = 0 Ne ds, is routinely measured for the lines of sight to pulsars in our galaxy. If coexistent with magnetic fields, the same electrons that cause propagation delays induce Faraday rotation of the field vectors by an angle φ = λ2 Rm where λ is the wavelength. Rm is the rotation measure, which depends on the product of the local electron density, Ne , and the component of the magnetic field parallel to the line of sight, B|| , and is given by Rm = 8.1 × 105 Ne B|| ds (6.8) in units of radians/m2 , where B|| is in Gauss, Ne is in cm−3 , ds is in parsecs, and the integral is over the path length along the line of sight. An estimate for the magnitude of these effects in can be made using mean values for the Galactic ISM such as Ne ∼ = 0.003 cm−3 , B|| ∼ = 2 µG, and typically ◦ 2 Rm ∼ −18| cot b| cos(l − 94 ) radians/m , where l and b are the galactic longitude and latitude. The scattering of radiation from electrons in the ISM, which produces interstellar scintillation, also has the effect of increasing the apparent size of point sources of radio emission. This effect produces an angular size due to scintillation which is roughly 7.5λ11/5 milliarcsec for |b| ≤ 0.◦ 6, 0.5(| sin b|)−3/5 λ11/5√milliarcsec for 0.◦ 6 < |b| < 4◦ , 13(| sin b|)−3/5 λ11/5 milliarcsec for 15◦ > |b| > 4◦ , and 15λ2 / | sin b| milliarcsec for |b| > 15◦ , where b is the galactic latitude and λ is the wavelength in meters. At long wavelengths, and for source line of sights close to the Sun, the solar corona and solar wind contribute very strongly to propagation delay, Faraday rotation, and scintillation. This can be estimated from the above formulas by Ne ∼ = (1.55R −6 + 2.99R −16 ) × 1014 m−3 for R < 4, and 11 −2 −3 ∼ Ne = 5 × 10 R m for 4 < R < 20, where R is the radial distance from the center of the Sun. The scattering size for an unresolved radio source due to the interplanetary medium is approximately 50(λ/R)2 arcmin where λ is in meters. The ionosphere of the Earth is a major, and highly variable, contributor of electrons along the line of sight that affects the propagation of radio waves at long wavelengths. Figure 6.4 shows typical, but idealized, distributions of the ionospheric electron density for night and day, during sunspot maximum, and at temperate latitudes. The troposphere of the Earth’s atmosphere also has the effect of absorbing the radiation from distant sources. If I0,ν is the surface brightness of a source seen from outside the Earth’s atmosphere, and τ0,ν is the optical depth at the zenith, then at a zenith angle (z) I (z, ν) = I0,ν e−τ0,ν X (z) , (6.9) Sp.-V/AQuan/1999/10/07:17:14 Page 128 128 / 6 R ADIO A STRONOMY F2 Day -3 Electron Concentration [m ] 1012 F F1 1011 E 1010 Night 109 E D 108 100 1000 Height [km] Figure 6.4. The electron concentration profile of the Earth’s ionosphere for day and night plotted against height above the Earth’s surface, at solar maximum, and at mid-latitudes. where X (z) is the relative air mass in units of the air mass at the zenith. To first order, X (z) ∼ = sec(z) = 1/ cos z. For X < 5 the formula X (z) = −0.0045+1.00672 sec z−0.002234 sec2 z−0.000 624 7 sec3 z has an error less than 6 × 10−4 . Also, τ0,ν ∼ = 0.12, 0.05, and 0.04 at 20, 6, and 2 cm, but can be very large and variable at mm wavelengths. Even more important than air mass in affecting radio observations are the delay and scattering effects in the Earth’s troposphere that produce the radio equivalent of seeing disks. However, since the 1970s radio astronomers have devised algorithms that allow so-called self-calibration of phase variations produced by the troposphere for radio interferometry measurements with four or more antennas in an array. Phase self-calibration by interferometric arrays are methods by which one can, for any observed field with a strong source, solve for the differences in atmospheric phase variations over each antenna, and then remove these phase variations from interferometric data. 6.4 6.4.1 RADIO TELESCOPES AND ARRAYS Properties of Antennas The normalized antenna pattern Pn (θ, φ) (Equation (6.4)) is used to define many of the important antenna properties. Although real antenna patterns cannot be exactly described by a mathematical function, many are close to that for a uniformly illuminated circular aperture of diameter D for which 2 πD 2J1 sin θ λ Pn (θ ) = , (6.10) πD sin θ λ where J1 is a first-order Bessel function. In Table 6.2 we list various definitions of antenna and source properties that depend upon the antenna pattern, and give the approximate values for uniform circular Sp.-V/AQuan/1999/10/07:17:14 Page 129 6.4 R ADIO T ELESCOPES AND A RRAYS / 129 apertures. Since some of the directly measured quantities for radio sources are dependent upon the antenna pattern, some of these are also listed in Table 6.2 [2]. Table 6.2. Antenna properties. Quantity Source solid angle Effective source solid angle Surface brightness Half power beam width Beam width at first nulls Beam solid angle Uniform Circular Ap. s = source Pn (θ, φ) d Bν (θ, φ)Pn d Bν = main lobe main lobe Pn d HPBW = 1.02(λ/D) = 58◦ (λ/D) Pn (HPBW ) = 1/2 Pn (BWFN ) = 0 A = 4π Pn (θ, φ) d BWFN = 2.44(λ/D) = 140◦ (λ/D) Directivity M = main lobe Pn (θ, φ) d 4π/ A Effective area A e = λ2 / A Aperture efficiency Beam efficiency A = Ae /(π D 2 /4) M = M / A Main beam solid angle 6.4.2 Definition s = source d Major Radio Telescopes and Arrays There are many tens of radio telescope systems in the world that are, have been, or are about to be, important in radio astronomy. Many are listed in Table 6.3, where the size of the telescope or array, its main operational wavelengths, and its type or specialized role may be listed in abbreviated form. For a complete list of most of these telescopes and arrays, see [3]. Table 6.3. Radio observatories. Name of Observatory/Inst. Location Description Australia Tel. Nat. Facility Basovizza-Solar Radio Stn. Bleien Radio Ast. Obs. Culgoora, Australia Parkes, Australia Trieste, Italy Zurich, Switzerland Caltech Submillimeter Obs. Crawford Hill Obs. Mauna Kea, Hawaii Holmdel, New Jersey Decameter Wave Radio Obs. Deep Space Network Sta. Gauribidanur, India Goldstone, California Deep Space Network Sta. Robledo, Spain Deep Space Network Sta. Tidbinbilla, Australia 6 km EW Arr. 1 + (6) 22 m; 0.3–24 cm 64 m; 0.7–75 cm 10 m; 38–130 cm solar 5 m; 30–300 cm solar 7 m; 10–300 cm solar 10.4 m; 1.3 mm–350 µm 7 m Horn Tunable; 21 cm 7 m Tunable; 1.3–3 mm 1.5 km T Arr.; 200–900 cm 34 m; 3.6–13 cm 70 m; 3.6–13 cm 34 m; 3.6–13 cm 70 m; 3.6–13 cm 34 m; 3.6–13 cm 70 m; 3.6–13 cm Sp.-V/AQuan/1999/10/07:17:14 130 / 6 Page 130 R ADIO A STRONOMY Table 6.3. (Continued.) Name of Observatory/Inst. Location Description Dominion Radio Astro. Obs. Penticton, BC, Canada Dwingeloo Radio Obs. Eur. Incoh. Scatt. Fac. Dwingeloo, Netherlands Kiruna, Sweden Fleurs Radio Tel. Five College Radio Ast. Obs. Giant Metrewave Radio Tel. Hat Creek Radio Ast. Obs. Haystack Obs. Hiraiso Solar-Terr. Res. Ctr. Humain Radio Ast. Stn. Fleurs, NSW, Aus. Quabbin Res., Mass. Pune District, India Cassel, California Westford, Massachusetts Nakaminato, Ibaraki, Japan Humain, Belgium High Time Resolving Tel. Interplanetary Scint. Obs. Instituto Argentino de Rad. Inst. de Radio Astron. Mill. IPS Fleurs Solar Obs. Itapetinga Radio Obs. Nanjing, China Toyakawa, Japan Parque Pereya Iraola, Arg. Plateau de Bure, France Pico Veleta, Spain Fleurs, NSW, Aus. Atibaia, Brazil James Clerk Maxwell Tel. Kashima Space Res. Ctr. Kisaruzu College Obs. Maryland Point Obs. Staz. Rad. di Medicina Mauna Kea, Hawaii Kashima, Japan Kisaruza chiba, Japan Riverside, Maryland Medicina, Italy Metsahovi Obs. Radio Sta. Molonglo Obs. Synth. Tel. Mount Pleasant Radio Obs. Metsähovi, Finland Hoskinstown, NSW, Aus. Cambridge, Tasmania, Aus. Mullard Radio Ast. Obs. National Astro. and Ion. Obs. National Obs. Ast. Center National Radio Ast. Obs. Cambridge, England Arecibo, Puerto Rico Yebes, Spain Green Bank, West Va. NRAO mm Telescope Very Large Array Very Long Baseline Array Netherlands Found. Res. Ast. Nobeyama Solar Radio Obs. Kitt Peak, Arizona Socorro, New Mexico Socorro, New Mexico Dwingeloo, Netherlands Nobeyama, Japan Nobeyama Radio Obs. Nobeyama, Japan Nobeyama, Japan Noto, Sicily England Jodrell Bank, Cheshire Jodrell Bank, Cheshire Wardle, Cheshire Defford, Worchestershire Darnhall, Cheshire Knockin, Shropshire Pickmere, Cheshire Jodrell Bank, Cheshire Floirac, France Plateau de Bure, France 600 m EW Arr. 4 9 m; 21,90 cm 26 m; 18–21 cm 25 m; 6–29 cm 32 m; 32 cm 4 40 × 120 m; 130 cm EW Arr. 32 5.8 m + 6 14 m; 21 cm 13.7 m; 1–7 mm 25 km Irr. Arr. 36 45 m; 21–790 cm 3 + (3) T Arr. 6 m; 1–7 mm 36 m; 0.7–18 cm 2 10 m + 6 m + 1 m; 300–950 cm 7.5 m; 50 cm solar T Arr. 48 4 m; 74 cm solar 2 m; 3.2 cm solar 100 × 20 m; 92 cm 2 30 m; 17–21 cm 0.4 km T Arr. (3) 15 m; 1.3–4 mm 30 m; 0.8–4 mm 2.5 m + 12-Yagi; 6,11,20, 120 cm 13.7 m; 0.3–28 cm 1.5 m; 4.3 cm solar 15 m; 1.3 mm-300 µm 26 m; 3–150 cm 1.5 m; CO 2.6 mm Line 26 m; 0.7–90 cm 32 m; 1.3–20 cm VLBI T Parab. Cyl.; 74 cm 14 m; 0.3–1.3 cm 1570 × 12 m EW Cyl. Par.; 36 cm 14 m; 30–50 cm pulars 26 m; 2.5–50 cm 5 km EW Arr. (4) 13 m; cm λ 305 m; λ > 6 cm 14 m; mm λ 43 m; 1.3–600 cm 100 m; 0.7–90 cm 12 m; 0.9–7 mm 1–35 km Y Arr. (27) 25 m; 0.7–90 cm 5000 km Arr. 10 25 m; 0.7–90 cm VLBI 22 m Ant. 0.5 km T Arr. 16 1.2 m, 1.8 cm solar T Arr 17 6 m, 190 cm 45 m, 0.26–3 cm 0.7 km T Arr. (5) 10 m, 0.25–1.3 cm 32 m; 0.3–1.3 cm VLBI 134 km Irr. MERLIN Arr. 7 Ant.: 76 m; 5–200 cm 38 × 26 m Par.; 1.3–200 cm 38 × 26 m; 17–370 cm 25 m; 6–200 cm 25 m; 1.3–200 cm 25 m; 1.3–200 cm 25 m; 1.3–200 cm 13 m; 20–50 cm pulsars 2.5 m; 2.5–4 mm 2.5 m; 1.3 mm Noto Radio Ast. Sta. Nuffield Radio Ast. Labs. Obs. de Bordeaux Obs. de Grenoble Sp.-V/AQuan/1999/10/07:17:14 Page 131 6.5 R ADIO E MISSION AND A BSORPTION P ROCESSES / 131 Table 6.3. (Continued.) Name of Observatory/Inst. Location Description Obs. für Solare Astr. Obs. Radioastron. de Maipú Ohio State Radio Obs. Ondrejov Ast. Obs. Tremsdorf, Germany Maipú, Chile Delaware, Ohio Ondrejov, Czech. Onsala Space Obs. Onsala, Sweden Ooty Radiotelescope Ooty Synthesis Radiotelescope Owens Valley Radio Obs. Ootacamund, India Big Pine, California Purple Mountain Obs. Purple Mt. Obs. Solar Fac. Delingha, China Nanjing, China Puschino Radio Ast. Sta. Puschino, Russia Radioobs. Effelsberg RATAN-600 Solar Radiospec. Obs. Sta. de Rad. de Nancay Effelsberg, Germany Zelenchukskaya, SU Ravensburg, Germany Nancay, France Swedish-ESO Submm. Tel. Tonantzintla Solar Radio Int. Toyokaya Obs. La Silla, Chile Puebla, Mexico Toyokawa, Japan U. of Mich. Radio Obs. URAN-1 Interferometer UTR-2 Array Westerbork Synth. Rad. Tel. Yunnan Obs. Dexter, Michigan Kharkov, SU Kharkov, SU Westerbork, Netherlands Kunming, China 1.5 + 4 + 10.5 m + Yagi; 3.2–750 cm 160 × 73 m; 670 cm 100 × 30 m; 1.9 cm 3 m; 1.5–3 cm solar 7.5 m; 37–120 cm solar 7.5 m; 25–300 cm solar 20 m; 3.7–270 cm 25 m; 30–260 cm 530 × 30 m Par. Cyl.; 90 cm (ORT) + 8 23 × 9 m; 90 cm 0.4 km T Arr. (4) 10 m; 1.3–3 mm 40 m; 0.7–90 cm Arr. of 2 27 m; 4–30 cm 13.7 m; 0.3,1.3 cm 1.5 m; 3.2–11 cm solar 2 m; 6 cm solar 22 m Tel. Cross-type 1 km decimetric array 18 acre decimetric phased array 100 m; 0.6–49 cm 0.6 cm circle of 895 elem.; 0.8–30 cm 7 m + 8 dipoles; 30–1000 cm 16 1 m Parab.; 3.2 cm 24 Parab.; 67–200 cm solar 16 + 2 Parab.; 67–200 cm solar 299 × 40 m; 9–21 cm 144 Con. Log. Per.; 380–2000 cm 15 m; 0.8–3 mm 2 1.1 m Parab.; 4 cm solar 0.85 m; 3.2 cm solar 1.5 m; 8 cm solar 2 m; 14 cm solar 3 m; 2.8 cm solar 3 arrays; 3.2,7.8 cm solar 25 m; cm λ Linear Arr, Xed dipoles; 1200–3000 cm T Arr 1800 × 54 m + 900 × 54 m; 1200–3000 cm 4 km EW Arr. 10 + (4) 25 m; 6–90 cm 2.5, 3, 3.2, & 10 m; 8.1–130 cm 6.5 6.5.1 RADIO EMISSION AND ABSORPTION PROCESSES Source Models and Prediction of Observables The relationship between the emission and absorption coefficients, which are used to describe the microphysics of radiation processes, and the theoretical quantities corresponding to the direct observables, is important in discussing the principle physical processes in radio astronomy. There are three principal observables: surface brightness Bν (and the related brightness temperature Tb,ν ), the integrated flux density Sν for a source with a closed boundary of solid angle source , and Vν the Sp.-V/AQuan/1999/10/07:17:14 132 / 6 Page 132 R ADIO A STRONOMY coherence (or visibility) function measured by radio interferometers. All are computed as integrals over the true sky brightness, or specific intensity, Iν , and are given by Iν dsource Bν = source , (6.11) source dsource 2k Sν = Iν dsource ∼ Tb dsource , (6.12) = 2 λ source source (where the approximation is valid for the Rayleigh–Jeans limit and most radio wavelengths) and Vν (u, v) = Iν (α, δ)Pn (α − α0 , δ − δ0 )e−2πi/λ[L j −Lk ]·[s(α,δ)−s0 (α0 ,δ0 )] d, (6.13) where L j and Lk are vector locations of antennas j and k, and s is a unit vector pointing to locations on the celestial sphere such as (α, δ) or the reference position (α0 , δ0 ). For a spherically symmetric brightness distribution the latter equation simplifies to a Hankel transform: Vν (u, v) = Iν (θ )Pn (θ )J0 (2πqθ)2π θ dθ, (6.14) where (L j − Lk ) · (s − s0 )/λ = ux + vy + wz, q = (u 2 + v 2 + w 2 )1/2 ∼ = (u 2 + v 2 )1/2 , and 2 2 2 1/2 2 2 1/2 ∼ θ = (x + y + z ) = (x + y ) . In Table 6.4 a number of models and the associated observables are listed. Table 6.4. Observables for simple source models. Model Tb (θ )/Tb,max 2 Gaussian e−4 ln 2(θ/θH ) Uniform disk 1, θ ≤ θH /2 0, θ > θH /2 Limb-brightened 2 − θ 2 ), 2θH /(θH (shot glass) “Thin” ring θ ≤ θH /2 0, θ > θH /2 δ(θ − θH /2) Sν (J y) Vν (q)/Sν 2 (arcsec) Tb,max (K )θH 1360λ2 (cm) 2 2 e−(π 4 ln 2)(qθH ) 2 (arcsec) Tb,max (K )θH 1961λ2 (cm) 2 (arcsec) Tb,max (K )θH 981λ2 (cm) 2 (arcsec) 420Tb,max (K )θH λ2 (cm) 2J1 (πqθH )/(πqθH )a sin(πqθH )/(πqθH ) J0 (πqθH ) Note a J is a Bessel function of order n, and assuming that source is at the center of the field, then n θ = r/d (d is the distance to the object) and θH is the half-intensity point in these spherically symmetric models. The resolution of an instrument of size D is set by diffraction theory to be λ/D, and for radio astronomy the best resolution of an antenna is determined by its diameter (Dm in units of meters) and the shortest wavelength it can observe (∼ [surface rms]/16). For arrays the size is the maximum separation of antennas (Dkm in units of km). Thus resolution = λcm λcm λ = 2 . = 34 D Dm Dkm (6.15) Sp.-V/AQuan/1999/10/07:17:14 Page 133 6.5 R ADIO E MISSION AND A BSORPTION P ROCESSES / 133 The resolution of the instrument and the characteristics of the radio source, which we will discuss in terms of the models in Table 6.4, together with the other parameters determining the sensitivity of the antenna or array, determine the minimum surface brightness that can be detected. For modern paraboloid-shape antennas the 5σ detection level for flux density during a time interval tsec is 20Tsys 1 σdetection = Jy, (6.16) √ 2 FK/Jy Dm νMHz tsec where FK/Jy is a fixed, empirically determined constant for each antenna–receiver combination, while for an array of N antennas of this size, 2.5Tsys 1 σdetection = Jy, (6.17) √ 2 ηc a D m νMHz tsec N (N − 1)/2 where ηc is the correlator efficiency (0.82 for 3-level correlations), a is the aperture efficiency (∼ 0.5 for simple paraboloids, but 0.6–0.7 for specially designed antennas), νMHz is the bandwidth in MHz, and tsec is the integration time in seconds. 6.5.2 Thermal Free–Free and Free–Bound Transitions For thermal bremsstrahlung radiation, the emission and absorption coefficients jν ρ and κν ρ are proportional to ρ 2 , where ρ is the mass density. The source function jν /κν , at radio wavelengths, is the Planck function for an electron temperature Te . Thermal bremsstrahlung or free–free radiation processes are caused by interactions between free electrons and positive ions in a partially or fully ionized plasma. The emission coefficient for free–free emission is given by jff,ν ρ = 5.4 × 10−39 Ne2 Te 1/2 gff ehν/kTe ∼ = 7.45 × 10−39 Ne2 Te0.34 ν −0.11 , (6.18) where we have used 3/2 31/2 Te −0.11 ∼ gff (ν, Te ) = 17.7 + ln = 1.38 × Te−0.34 νGHz π ν (6.19) for the free–free Gaunt factor, with an approximation valid at radio wavelengths [4, 5]. In (6.18)– (6.19), Ne is the electron concentration in units of cm−3 , and Te is the electron temperature. The Planck function relates the emission and absorption coefficients for black body radiation, i.e., ν 2 ν 2 1 jν 2hν 3 ∼ Sν = = 2 hν/kT , (6.20) 2kTe ≡ 2kTr = e −1 κν c c c e ∼ Te is valid for longer radio wavelengths, where the approximation that the radiation temperature Tr = but becomes invalid at mm wavelengths. For bound–free transitions the emission coefficient and Gaunt factor are given by −3/2 jfb,ν ρ = 1.72 × 10−33 Ne2 Te ∞ n −3 gfb (ν, Te , n)e157890/n 2T e , (6.21) n=m where n is the principal quantum number and m is the integer portion of (3.789 89 × 1015 Hz/ν) and gbf (ν, Te , n) = 1 + 0.1728(u n − 1) 0.0496(u 2n + 43 u n + 1) − , n(u n + 1)2/3 n(u n + 1)4/3 where u n = n 2 (ν/3.289 89 × 1015 Hz) [6]. (6.22) Sp.-V/AQuan/1999/10/07:17:14 Page 134 134 / 6 6.5.3 R ADIO A STRONOMY Spectral Lines—Thermal Bound–Bound Transitions For bound–bound transitions between any two levels with energies E n = E 1 , ..., E ionization and statistical weights gn , radiation absorption and emission involves photons of energy hνnm = E m − E n . The properties of a line transition are determined by the Einstein coefficients Amn , Bmn , and Bnm , which are related by Bmn = c2 Amn 2hν 3 and Bnm = gm Bmn . gn (6.23) The emission and absorption coefficients for line transitions are jbb,νnm = hνnm Nm Amn φmn (νnm ) 4π (6.24) κbb,νnm = hνnm Nn gn φmn (νmn ) Nm Bnm φmn (νnm ) 1 − , 4π Nm gm φmn (νnm ) (6.25) and where φnm (νmn ) and φmn (νmn ) are absorption and emission “line” profile functions, which are usually equal to each other. Note that in general the source function can be expressed as Sνnm = 1 jbb,νnm 2hν 3 . = 2 N g φ (ν ) κbb,νnm c n n mn mn −1 Nm gm φmn (νnm ) (6.26) If the line is in Local Thermodynamic Equilibrium (LTE) with temperature T , then Nn /Nm = (gn /gm ) exp(hνnm /kT ) and S = Bν so κbb,LTE,νnm = hνnm Nm Bnm φmn (νnm ) 1 − e−hνnm /kT . 4π (6.27) Most of the known spectral lines from molecules in the interstellar medium have been detected at radio wavelengths. Many are in Table 6.5. For current lists and information about the parameters of various molecular species see the Internet URL http://spec.jpl.nasa.gov. Table 6.5. Known interstellar molecules. Name of molecule Chemical symbol Wavelength region Date and telescope of discovery methyladyne cyanogen radical methyladyne ion hydroxyl radical ammonia water formaldehyde carbon monoxide hydrogen cyanide cyanoacetylene hydrogen methanol formic acid formyl radical ion CH CN CH+ OH NH3 H2 O H2 CO CO HCN HC3 N H2 CH3 OH HCOOH HCO+ 4 300 Å 3 875 Å 4 232 Å 18 cm 1.3 cm 1.4 cm 6.2 cm 2.6 mm 3.4 mm 3.3 cm 1013–1108 Å 36 cm 18 cm 3.4 mm 1937 – Mt. Wilson 2.5 m 1940 – Mt. Wilson 2.5 m 1941 – Mt. Wilson 2.5 m 1963 – Lincoln Lab. 26 m 1968 – Hat Creek 6 m 1968 – Hat Creek 6 m 1969 – NRAO 43 m 1970 – NRAO 11 m 1970 – NRAO 11 m 1970 – NRAO 43 m 1970 – NRL rocket 1970 – NRAO 43 m 1970 – NRAO 43 m 1970 – NRAO 11 m Sp.-V/AQuan/1999/10/07:17:14 Page 135 6.5 R ADIO E MISSION AND A BSORPTION P ROCESSES / 135 Table 6.5. (Continued.) Name of molecule Chemical symbol Wavelength region Telescope of discovery formamide carbon monosulfide silicon monoxide carbonyl sulfide methyl cyanide isocyanic acid methyl acetylene acetaldehyde thioformaldehyde hydrogen isocyanide hydrogen sulfide methanimine sulfur monoxide diazenylium ethynyl radical methylamine NH2 CHO CS SiO OCS CH3 CN HNCO CH3 CCH CH3 CHO H2 CS HNC H2 S CH2 NH SO N2 H+ C2 H CH3 NH2 dimethyl ether ethanol sulfur dioxide silicon sulfide vinyl cyanide methyl formate nitrogen sulfide cyanamide cyanodiacetylene formyl radical acetylene cyanoethynyl radical ketene cyanotriacetylene nitrosyl radical confirmed ethyl cyanide cyano-octatetra-yne methane nitric oxide butadiynyl radical methyl mercaptan isothiocyanic acid thioformyl radical ion protonated carbon dioxide ethylene cyanotetraacetylene silicon dicarbide propynylidyne methyl diacetylene methyl cyanoacetylene tricarbon monoxide silane protonated HCN (CH3 )2 O CH3 CH2 OH SO2 SiS CH2 CHCN CH3 OCHO NS NH2 CN HC5 N HCO C2 H2 C3 N H2 CCO HC7 N HNO 6.5 cm 2.0 mm 2.3 mm 2.7 mm 2.7 mm 3.4 mm 3.5 mm 28 cm 9.5 cm 3.3 mm 1.8 mm 5.7 cm 3.0 mm 3.2 mm 3.4 mm 3.5 mm 4.1 mm 9.6 mm 2.9–3.5 mm 3.6 mm 2.8, 3.3 mm 22 cm 18 cm 2.6 mm 3.7 mm 3.0 cm 3.5 mm infrared 3.4 mm 2.9 mm 2.9 cm 3.7 mm 1.9 mm 2.7–4.0 mm 2.9 cm infrared 2.0 mm 2.6–3.5 mm 3 mm 3 mm 3 mm 3 mm infrared many (cm) many (mm) many (3 mm) several (cm) several (cm) 2 cm infrared 3, 2, 1 mm cyclopropynylidene — hydrogen chloride protonated water confirmed C3 H2 H2 D+ ? HCl ? H3 O+ 1971 – NRAO 43 m 1971 – NRAO 11 m 1971 – NRAO 11 m 1971 – NRAO 11 m 1971 – NRAO 11 m 1971 – NRAO 11 m 1971 – NRAO 11 m 1971 – NRAO 43 m 1971 – Parkes 64 m 1971 – NRAO 11 m 1972 – NRAO 11 m 1972 – Parkes 64 m 1973 – NRAO 11 m 1974 – NRAO 11 m 1974 – NRAO 11 m 1974 – NRAO 11 m 1974 – Tokyo 6 m 1974 – NRAO 11 m 1974 – NRAO 11 m 1975 – NRAO 11 m 1975 – NRAO 11 m 1975 – Parkes 64 m 1975 – Parkes 64 m 1975 – Texas 5 m 1975 – NRAO 11 m 1976 – Algonquin 46 m 1976 – NRAO 11 m 1976 – KPNO 4 m 1976 – NRAO 11 m 1976 – NRAO 11 m 1977 – Algonguin 46 m 1977 – NRAO 11 m 1990 – FCRAO 14 m 1977 – NRAO 11 m 1977 – Algonguin 46 m 1977 – KPNO 4 m 1978 – NRAO 11 m 1978 – NRAO 11 m 1979 – BTL 7 m 1979 – BTL 7 m 1980 – BTL 7 m 1980 – BTL 7 m 1980 – KPNO 4 m 1981 – Algonquin 46 m 1984 – BTL 7 m 1984 – BTL 7 m 1984 – Haystack 37 m 1984 – Haystack 37 m 1984 – Haystack 37 m 1984 – KPNO 4 m 1984 – NRAO 12 m 1984 – Texas 5 m 1985 – many 1985 – KAO 1985 – KAO 1986 – NRAO 12 m 1990 – CSO 10 m CH3 CH2 CN HC9 N CH4 NO C4 H CH3 SH HNCS HCS+ HOCO+ C2 H4 HC11 N SiC2 C3 H CH3 C4 H CH3 C3 N C3 O SiH4 NCNH+ 3, 2 mm 0.62 mm 0.48 mm 1.0 mm 0.8 mm Sp.-V/AQuan/1999/10/07:17:14 136 / 6 Page 136 R ADIO A STRONOMY Table 6.5. (Continued.) 6.5.4 Name of molecule Chemical symbol Wavelength region Telescope of discovery pentynylidyne radical hexatriynyl radical phosphorus nitride C5 H C6 H PN radio radio 3, 2, 1 mm — — acetone sodium chloride aluminum chloride potassium chloride aluminum fluoride methyl isocyanide cyanomethyl radical C2 S C3 S (CH3 )2 CO ? NaCl AlCl KCl AlF CH3 NC CH2 CN silicon carbide propynal SiC HCCCHO — phosphorus carbide propadienylidene SiC4 CP H2 CCC butatrienylidene silicon nitride silylene (pend. conf.) — carbon suboxide H2 CCCC SiN SiH2 ? HCCN C2 O isocyanoacetylene sulfur oxide ion ethinylisocyanide magnesium isocyanide protonated HC3 NH+ — carbon monoxide ion sodium cyanide — nitrous oxide magnesium cyanide prot. formaldehyde octatetraynyl radical octatetraynyl radical protonated hydrogen hexapentaenylidene ethylene oxide hydrogen flouride HCCNC SO+ HNCCC MgNC NC3 H+ NH2 CO+ NaCN CH2 D+ N2 O MgCN H2 COH+ C8 H C7 H H+ 3 H2 C6 c − C2 H4 O HF 3, 7 mm 3, 7 mm 3.4 mm 2, 3 mm 2, 3 mm 2, 3 mm 2, 3 mm 3 mm 6.5 mm 1.3 cm 2 mm 8 mm 1.6 cm 3.5, 7.5 mm 1.2, 3.0 mm 3 mm 1.4 cm many (3 mm) 1.1, 1.4 mm 0.8, 1.0 mm 3 mm 7 mm 1.3 cm 7 mm 1.3, 3 mm 1.1, 0.8, 0.64 cm radio radio 0.645 mm 1.3, 0.8 mm 3, 2.4 mm 0.8–4.0 mm 2.0–4.0 mm radio radio radio radio IR-3.67 µm K-band K, Q, 3 mm,1 mm 121.7 µm 1986 – IRAM 30 m 1986 – IRAM 30 m 1986 – NRAO 12 m 1986 – FCRAO 14 m 1986 – Nobeyama 45 m 1986 – Nobeyama 45 m 1987 – IRAM 30 m 1987 – IRAM 30 m 1987 – IRAM 30 m 1987 – IRAM 30 m 1987 – IRAM 30 m 1987 – IRAM 30 m 1988 – Nobeyama 45 m 1988 – NRAO 43 m 1989 – IRAM 30 m 1989 – Nobeyama 45 m 1989 – NRAO 43 m 1989 – Nobeyama 45 m 1989 – IRAM 30 m 1990 – IRAM 30 m 1990 – NRAO 43 m 1990 – IRAM 30 m 1990 – NRAO 12 m 1990 – CSO 10 m 1991 – IRAM 30 m 1991 – Nobeyama 45 m 1991 – NRAO 43 m 1992 – Nobeyama 45 m 1992 – NRAO 12 m 1992 – Nobeyama 45 m 1992 – Nobeyama 45 m 1993 – Nobeyama 45 m 1993 – CSO 1993 – NRAO 12 m, CSO 1993 – NRAO 12 m 1993 – NRAO 12 m, CSO 1994 – NRAO 12 m 1995 – NRAO 12 m 1995 – Nobeyama 45 m 1996 – IRAM 30 m 1996 – IRAM 30 m 1996 – UKIRT 1997 – NASA 34 m 1997 – NRO, SEST, Haystack 1997 – ISO Magneto-Bremsstrahlung Emission and Absorption The other radiation process of dominant importance in radio astronomy is magneto-bremsstrahlung emission and absorption which results from the interaction between fast-moving electrons and the ambient magnetic field. There are three major varieties of magneto-bremsstrahlung emission. When the moving electrons are very relativistic, the emission process is the very efficient, highly beamed synchrotron emission which dominates the emission from many galactic and most extragalactic ra- Sp.-V/AQuan/1999/10/07:17:14 Page 137 6.5 R ADIO E MISSION AND A BSORPTION P ROCESSES / 137 dio sources. However, in some stars and certain solar system environments, two other magnetobremsstrahlung processes occur when the moving electrons are nonrelativistic or only mildly relativistic. When mildly relativistic (γLorentz ≡ (1 − (v/c)2 )−1/2 ∼ = 2–3) electrons undergo interactions with magnetic fields, the emission process is called gyro synchrotron, which is much less beamed and much less efficient at producing radio emission than synchrotron processes; however, it tends to be highly circularly polarized, a characteristic commonly found in stellar radio emission. Even less efficient is the magneto-bremsstrahlung resulting from less relativistic particles, with γLorentz ≤ 1, which is called cyclotron or gyro resonance emission. The details of nonrelativistic (cyclotron) and mildly relativistic (gyro synchrotron) emission and absorption processes are more complicated than the highly relativistic case. They are summarized by [7], and extensively discussed by [8] and [9]. Much of the analysis of radio source data uses very simple models for the behavior of synchrotron radiating sources. This section summarizes formulas used in such analysis. We assume that the density of relativistic electrons can be described by a power law distribution in energy, N (E) = K E −γ in the energy range E to E + d E, where γ is a constant. If these electrons are mixed in with a uniform distribution of magnetic fields of strength H , which can be described as having uniformly random directions on a large scale, then the emission and absorption coefficients are given by jν ρ = 1.35 × 10 and −22 a(γ )K H (γ +1)/2 6.36 × 1018 ν (γ −1)/2 (6.28) κν ρ = 0.019g(γ )(3.5 × 109 )γ K H (γ +2)/2 ν −(γ +4)/2 , (6.29) where H is in gauss and all other variables are in cgs units, so that the source function is Sν = a(γ ) γ /2 −1/2 5/2 jν = 2.84 × 10−30 ν erg s−1 cm−3 2 H κν g(γ ) (6.30) [10]. Table 6.6 gives some of the values of the functions a(γ ), g(γ ), the brightness temperature source function Ts , and the angular size (0 ) of a uniform source with flux density S0 and magnetic field HmG in units of milligauss. Table 6.6. Incoherent synchrotron emission constants. γ a(γ ) g(γ ) α 1.0 15 2.0 2.5 3.0 4.0 5.0 0.283 0.147 0.103 0.0852 0.0742 0.0725 0.0922 0.96 0.79 0.70 0.66 0.65 0.69 0.83 0.0 0.25 0.50 0.75 1.0 1.5 2.0 −1/2 1/2 Ts νGHz HmG 1.9 × 1012 1.0 × 1012 6.6 × 1011 4.9 × 1011 3.6 × 1011 2.3 × 1011 1.7 × 1011 −1/2 0 S0 K K K K K K K −1/4 5/4 HmG νGHz 0.015 mas 0.021 mas 0.026 mas 0.030 mas 0.035 mas 0.044 mas 0.051 mas The specific intensity is given by Iν = 0 L Sν (1 − exp(−τν )) dτν , where dτν = κν ρ ds in the above integral, which is along the line of sight of length L. (6.31) Sp.-V/AQuan/1999/10/07:17:14 138 / 6 Page 138 R ADIO A STRONOMY We see from (6.30) that the important case where jν /κν is constant occurs if the magnetic field is uniform in strength and geometry. Equations (6.28)–(6.30) also tell us that for the optically thick and thin limits, Iν is proportional to H −1/2 ν 5/2 and K H (γ +1)/2 ν −(γ −1)/2 L, respectively, where L is a line-of-sight path length. Defining the spectral index α by Sν ∝ ν α , then α = 2.5 and −(γ − 1)/2 in the optically thick and thin limits, respectively. The values of α in the latter case for various values of γ are listed in Table 6.6. The evolution of the relativistic electron energy distribution N (E, t) is the primary factor in the evolution or synchrotron radiation sources. In its most general form this evolution is described by ∂ N (E, t) ∂ dE + N (e, t) = (E, T ) − (E, t), (6.32) ∂t ∂E dt where (E, T ) and (E, t) are functions describing particle “injection” and “escape,” with escape time scales T , respectively. Usually and are zero, as they will be for all models discussed below, or applies to only small portions of the radio sources. The equation for evolution of N (E, t) is supplemented by the energy loss equation: dE = φ(E) = −ζ − ηE − ξ E 2 , dt (6.33) where the first two terms are due to losses by ionization in the ambient, medium, and free–free interaction with this medium. The last term, which is usually the most important, includes both energy loss due to synchrotron radiation and energy loss due to inverse Compton scattering off photons in the local radiation field. The coefficients of the first two terms in (6.33) are: E E −20 ζ ≈ 3.33 × 10−20 n H 6.27 + ln + 1.22 × 10 73.4 ln − ln n (6.34) n e e mc2 mc2 and E η ≈ 8 × 10−16 n H n e 0.36 + ln , (6.35) mc2 where n H is the concentration of neutral hydrogen atoms and n e is the concentration of thermal electrons. For synchrotron radiation losses ξsynch ∼ = 2.37 × 10−3 H⊥2 , where H⊥ is the magnetic field perpendicular to the velocity vector of the radiating relativistic electron. The time scale for synchrotron losses is tsynch ≡ E/(−d E/dt) = 1/(ζsynch E) ∼ (5.1 × 108 /H⊥2 )(mc2 /E) s. Synchrotron radiation losses dominate in regions with large magnetic fields and low radiation fields. For inverse Compton losses ξIC ≈ 3.97 × 10−4 u rad , where u rad is the radiation energy density. Inverse Compton losses dominate when brightness temperature approaches Tb ≤ 1012 K. Above 1012 K all relativistic electrons loss their energy rapidly due to this process. For steady-state synchrotron radiation sources where one assumes (E, t) = (E), and N (E, t) = N (E), and ∂ N /∂t = 0, the relativistic particle energy equation has the solution N (E) = φ −1 (E) (E) d E. So if one can assume a power law for the particle injection, (E) = 0 E −γ , then N (E) = 0 (γ − 1)−1 E −γ (ζ E −1 + η + ζ E)−1 . (6.36) The relativistic particle spectra and observable radio spectrum for such steady-state radio sources can be described by three segments of power laws as shown in Figure 6.5. The principle effect of more complex models is replacement of the sharp spectral breaks with smooth transitions. Sp.-V/AQuan/1999/10/07:17:14 Page 139 6.5 R ADIO E MISSION AND A BSORPTION P ROCESSES / 139 α = − (γ − 1)/2 γ−1 γ log N(E) γ+1 log E ionization losses free-free losses synchrotron + inverse compton losses α − 1/2 α log S ν α + 1/2 absorption absorption log ν Figure 6.5. Relativistic particle (top), and observable flux density (bottom), spectrum for steady-state synchrotron radiation sources. The changes in slope indicate regions where ionization, free–free, and synchrotron (+ inverse Compton) losses dominate. By each curve segment is the power law index for the energy and flux density spectra. External or self-absorption can occur at any point in the flux density spectrum; two possibilities are indicated in the figure. A synchrotron radio source with losses only due to synchrotron radiation has an energy loss equation d(E −1 )/dt = ζsynch which leads to a spectrum described by E = E 0 /[1 + ζsynch (t − t0 )E 0 ]. Then if at t = t0 , N (E) = N0 E −γ , then at later times N (E) = N0 E −γ (1 − ζsynch Et)γ −1 for E < E T (t), and N (E, T ) = 0 for E > E T (t), where E T (t) = 1/(ζsynch t). The frequency above −3 −2 GHz. which synchrotron losses dominate is νb ≈ BGauss tyear Time-dependent synchrotron radiation source models with complex geometry and complex energy loss mechanisms require extensive computation. However in the case of simple geometries and only adiabatic losses the evolution of the flux density of a radio source can be expressed in analytic or nearanalytic equations. For a spherical region which “suddenly” has a distribution of relativistic particles given by N0 = K 0 E −γ and which expands with an outer radius r2 (t) the flux density of the source is described by the quantities in Table 6.7 [11–13]. An analogous model can be expressed for a conical jet with ejection of adiabatically expanding plasma [13]. The jet ejection parameters are such that at an initial distance z 0 the radius of the cross section of the jet is r0 (jet Mach number M0 = z 0 /r0 ) and the relativistic particle density is N0 . The quantities that determine the flux density for an observer for whom the jet is oriented at an inclination angle are in Table 6.7; for a continuous jet z 2 = ∞ and z 1 = z 0 ; however, a time-dependent jet which starts ejection with velocity v2 at time tstart and ends ejection with velocity v1 at time tstop is described by the same equations with the necessary time dependence in the limits of integral for the flux density. Table 6.7. Radio source models—only adiabatic energy losses. Expanding sphere Conical jet E = E 0 (r2 /r0 )−1 H = H0 (r2 /r0 )−2 K = K 0 (r2 /r0 )−(γ +2) E = E 0 [r2 (z 2 )/r0 ]−2/3 H = H0 [r2 (z 2 )/r0 ]−1 K = K 0 [r (z 2 )/r0 ]−2(γ +2)/3 (γ +2)/2 τ0 = 0.038g(γ )(3.5 × 109 )γ K 0 H0 r0 same Sp.-V/AQuan/1999/10/07:17:14 Page 140 140 / 6 R ADIO A STRONOMY Table 6.7. (Continued.) Expanding sphere Conical jet ξ(τ ; τ > 20) = 1 ξ(τ ; τ < 20) = 0.66584 + 0.09089τ − 0.009989τ 2 + 0.0005208τ 3 − 0.00001268τ 4 + 0.000000115τ 5 ξ(τ ; τ > 20) = 1 ξ(τ ; τ < 20) = 0.78517 + 0.06273τ − 0.007242τ 2 + 0.0003905τ 3 − 0.00000973τ 4 + 0.00000009τ 5 −(γ +4)/2 r2 (z 2 ) −(7γ +8)/6 ν τ0 τν = sin() ν0 r0 τν = τ0 r −(2γ +3) ν −(γ +4)/2 r0 ν0 −1/2 5/2 Sν = 1.42e−0.65γ H0,mG νGHz 3mas × 1 − exp(−τν ) × ξ( τν ) r2 (t; free exp.) = r0 r2 (t; E cons.) = r0 r2 (t; Mom. cons.) = r0 t t0 ν 5/2 sin() ν0 z (t) × z 2(t) ζ 3/2 1 − exp(−τν ) ξ(τν ) dζ Sν = S0 M0 1 r2 (t; free exp.) = r0 t 2/5 t0 r2 (t; E cons.) = r0 t 1/4 t0 r2 (t; Mom. cons.) = r0 z2 z0 z 2 1/2 z0 z 2 1/3 z0 z 2 = v2 (t − tstart ) z 1 = v1 (t − tstop ) 6.6 RADIO ASTRONOMY REFERENCES Table 6.8 gives further references for radio astronomy. Table 6.8. List of radio astronomy references. Topic Reference Radio astronomy research field reviews Properties of radio telescopes Interferometry and aperture synthesis Aperture synthesis Observation-oriented radio astronomy textbooks Synchrotron radiation physics Radio astrophysical theory General astrophysical theory [1] [2] [3] [4, 5] [6, 7] [8] [9] [10] References 1. Verschuur, V.L., & Kellermann, K.I. 1988, Galactic and Extragalactic Radio Astronomy (Springer-Verlag, New York) 2. Christiansen, W.N., & Högbom, J.A. 1985, Radio Telescopes (Cambridge University Press, Cambridge) 3. Thompson, A.R., Moran, J.M., & Swenson, G.W. 1986, Interferometry and Synthesis in Radio Astronomy (Wiley, New York) 4. Perley, R.A., Schwab, F.R., & Bridle, A.H. 1989, Synthesis Imaging in Radio Astronomy, A.S.P. Conference Series, 6 5. Cornwell, T.J., & Perley, R.A. 1991, Radio Interferometry: Theory, Techniques, and Applications, A.S.P. Conference Series, 19 6. Rohlfs, K. 1986, Tools of Radio Astronomy (Springer-Verlag, Berlin) 7. Krause, J.D. 1986, Radio Astronomy, 2nd ed. (Cygnus-Quasar Books, Powell) 8. Ginzburg, V.L., & Syrovatskii, S.I. 1965, Ann. Rev. Astron. Astrophys., 3, 297 9. Pacholczyk, A.G. 1970, Radio Astrophysics (W.H. Freeman, San Francisco) 10. Longair, M.S. 1981, High Energy Astrophysics: an informal introduction for students of physics and astronomy (Cambridge University Press, Cambridge) Sp.-V/AQuan/1999/10/07:17:14 Page 141 6.6 R ADIO A STRONOMY R EFERENCES / 141 REFERENCES 1. Clark, B.G. 1989, Synthesis Imaging in Radio Astronomy, A.S.P. Conference Series, 6, p. 1 2. Christiansen, W.N., & Högbom, J.A. 1985, Radio Telescopes (Cambridge University Press, Cambridge) 3. Price, R.M. et al. 1989, Radio Astronomy Observatories (National Academy Press, Washington, DC) 4. Karzas W.J., & Latter, R. 1961, ApJS, 6, 167 5. Hjellming, R.M., Wade, C.M., Vandenberg, N.R., & Newell, R.T. 1979, AJ, 84, 1619 6. Aller, L.H. & Liller, W. 1968, Nebulae and Interstellar Matter, edited by B. Middlehurst & L.H. Aller (Univer- sity of Chicago Press, Chicago), Chapter 9 7. Dulk, G.A. 1985, ARA&A, 23, 169 8. Kundu, M.R. 1965, Solar Radio Astronomy (Interscience, New York) 9. Zheleznyakov, V.V. 1970, Radio Emission of the Sun and Planets (Pergamon Press, Oxford) 10. Ginzburg, V.L., & Syrovatskii, S.I. 1965, ARA&A, 3, 297 11. van der Laan, H. 1966, Nature, 211, 1131 12. Kellermann, K.I. 1966, ApJ, 146, 621 13. Hjellming, R.M., & Johnston, K.J. 1988, ApJ, 328, 600 Sp.-V/AQuan/1999/10/07:17:14 Page 142