Radio Astronomy

Sp.-V/AQuan/1999/10/07:17:14 Page 121

Chapter 6

Radio Astronomy

Robert M. Hjellming

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

6.1 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.

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Table 6.1. List of radio source catalogs.

Type of Source Contents ◦ ◦ 4C Radio Survey [1, 2] 4844 56 cm sources > 2Jy,−7 ≤ δ ≤ 80 ◦ ◦ Bologna B2 Sky Survey [3Ð5] 408 MHz, 21.7 ≤ δ ≤ 40 ◦ ◦ Bologna B3 Sky Survey [6, 7] 408 MHz, 37 ≤ δ ≤ 47 ◦ 20 cm N. Sky Catalog [8] 365, 1400, 4850 MHz, 0 ≤ δ ≤ 75 ◦ ◦ ◦ ◦ 6 cm Gal. Plane Survey [9] Parkes 64 m, |b|≤2 , l = 190 − 360 − 40 ◦ ◦ 6 cm South & Tropical Surveys [10, 11] Parkes 64 m, −78 ≤ δ ≤−10 11 cm All Sky Catalog [12] Bright sources ◦ ◦ ◦ 11 cm N. Sky Catalog [13] 6483 sources, |b|≤5 , 240 ≤ l ≤ 357 Bright Galaxies [14] All radio data pre-1975, normal galaxies 20 cm Gal. Plane Survey [15] VLA B Conf. Survey ◦ ◦ ◦ 21 cm Gal. Plane Survey [16] Eff. 100m, |b|≤4 ,95 ≤ l ≤ 357 Galaxy HI [17, 18] Galaxies, vradial ≤ 3000 km/s Atlas of Galactic HI [19] 21 cm H emission line profiles Molecular lines [20, 21] Frequencies, other molecular data Pulsars [22] Catalog of known pulsars Opt. Pos. Radio Stars [23] 221 radio stars ◦ 20 cm Radio Sources δ>−5 [24, 25] 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. Furst,¬ 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., & Furst,¬ 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. Requi« eme,` 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 Wm−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

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108

Active Sun 107

Sky Background 106 Jupiter (radiation belts) 105 Quiet Sun

104 Moon Crab Cas A Pulsar Cyg A (unpulsed) 103 Crab M31 Virgo A Nebula 3C295 102 Flux Density [Jy] 3C273

101 Orion Nebula Jupiter Mars 100 BL Lac SS433

10-1 Crab Pulsar (pulsed) Cyg X-3 (non-flaring)

-2 10 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, α Sν ∝ ν , 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 ATMOSPHERIC WINDOW AND SKY BRIGHTNESS

6.2.1 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 low- frequency 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

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1 1

1 mm PWV

Atmospheric Transparency 8 mm PWV

0 0 0.01 0.1 1 10 100 1000 0 200 400 600 800 1000 Wavelength [cm] 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 ν/ of a black body of temperature T is Bν = (2hν3/c2)(1/[eh kT − 1]), where h and k are the Planck and Boltzmann constants. For small values of hν/kT, the RayleighÐJeans 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

λ2 T ≡ Bν. (6.1) b 2k Equation (6.1) is often used even when black body or long wavelength approximations are not hν/kT applicable. Because of this, the concept of radiation temperature (Tr ≡ [hν/k]/[e − 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

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] 100000 -1 Earth's Galactic 10000 cm Atmosphere Background

-1 100 1000 str -2 100 Earth's Atmosphere 10 COBE [W cm [W 10 15 1 Cosmic 0.1 Background 0.01 Galactic Background COBE 0.001 1

Cosmic Brightness Temperature [K]

Brightness x 10 0.0001 Background 100 10 1 0.1 100 10 1 0.1 Wavelength [cm] 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 RADIO WAVE PROPAGATION 6.3.1 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]

e2πiν|R−r|/c Pν(r) = , (6.2) |R − r| 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  e2πiν|R−r|/c Eν(r) = E(R) dS, (6.3) |R − r| where dS 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(Eθ Hφ − Eφ Hθ )/2 = Eθ Hφ. The power dP at distance r Sp.-V/AQuan/1999/10/07:17:14 Page 126

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passing through solid angle d = dA/r 2 is the Poynting flux through dA at distance r. From the field distribution over the surface of an antenna we compute dP/d which allows us to calculate the important properties of antennas. The normalized antenna pattern is defined by dP/d Pn(θ, φ) = (6.4) (dP/d)max 2 so an infinitesimal current element has Pn(θ, φ) = sin θ. The beam solid angle of an antenna is  = (θ, φ)  = π/ a 4π Pn d , and the directivity is D 4 a [2]. For real antennas we integrate over the collecting surface and from that compute dP/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

Tsys Tnoise = √ , (6.6) ν t 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

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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 /v of a pulse of radiation is tpulse 0 ds group 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  dt e2 L pulse =∼ . 3 Ne ds (6.7) dν 4π0mcν 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 2 delays induce Faraday rotation of the field vectors by an angle φ = λ 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  5 Rm = 8.1 × 10 Ne B|| ds (6.8)

2 −3 in units of radians/m , where B|| is in Gauss, Ne is in cm , 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 ∼ −3 ∼ using mean values for the Galactic ISM such as Ne = 0.003 cm , 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, −3/5 11/5 ◦ ◦ −3/5 11/5 ◦ 0.5(| sin b|) λ √milliarcsec for 0. 6 < |b| < 4 ,13(| sin b|) λ 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 ∼ −6 −16 14 −3 estimated from the above formulas by Ne = (1.55R + 2.99R ) × 10 m 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)

−τ0,ν X (z) I (z,ν)= I0,νe , (6.9) Sp.-V/AQuan/1999/10/07:17:14 Page 128

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1012 F2

] Day -3 F1 F 1011 E

1010 Night E 109 D Electron Concentration [m

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.ForX < 5 the formula X(z) =−0.0045+1.00672 sec z−0.002234 sec2 z−0.000 624 7 sec3 z −4 ∼ has an error less than 6 × 10 . 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 RADIO TELESCOPES AND ARRAYS 6.4.1 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    π 2  D θ  2J1 λ sin  (θ) = , Pn  π  (6.10)  D θ  λ sin

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

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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 Definition Uniform Circular Ap.   =  Source solid angle s source d   = (θ, φ)  Effective source solid angle s source Pn d  Bν (θ, φ)P d = main lobe n Surface brightness Bν  main lobe Pn d ◦ Half power beam width Pn(HPBW) = 1/2 HPBW = 1.02(λ/D) = 58 (λ/D) ◦ Beam width at first nulls Pn(BWFN) = 0 BWFN = 2.44(λ/D) = 140 (λ/D)   = (θ, φ)  Beam solid angle A 4π Pn d   = (θ, φ)  Main beam solid angle M main lobe Pn d Directivity 4π/A 2 Effective area Ae = λ /A 2 Aperture efficiency A = Ae/(π D /4) Beam efficiency M = M/A

6.4.2 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 Culgoora, Australia 6 km EW Arr. 1 + (6) 22 m; 0.3Ð24 cm Parkes, Australia 64 m; 0.7Ð75 cm Basovizza-Solar Radio Stn. Trieste, Italy 10 m; 38Ð130 cm solar Bleien Radio Ast. Obs. Zurich, Switzerland 5 m; 30Ð300 cm solar 7m;10Ð300 cm solar Caltech Submillimeter Obs. Mauna Kea, Hawaii 10.4 m; 1.3 mmÐ350 µm Crawford Hill Obs. Holmdel, New Jersey 7 m Horn Tunable; 21 cm 7 m Tunable; 1.3Ð3mm Decameter Wave Radio Obs. Gauribidanur, India 1.5 km T Arr.; 200Ð900 cm Deep Space Network Sta. Goldstone, California 34 m; 3.6Ð13 cm 70 m; 3.6Ð13 cm Deep Space Network Sta. Robledo, Spain 34 m; 3.6Ð13 cm 70 m; 3.6Ð13 cm Deep Space Network Sta. Tidbinbilla, Australia 34 m; 3.6Ð13 cm 70 m; 3.6Ð13 cm Sp.-V/AQuan/1999/10/07:17:14 Page 130

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Table 6.3. (Continued.)

Name of Observatory/Inst. Location Description

Dominion Radio Astro. Obs. Penticton, BC, Canada 600 m EW Arr. 4 9 m; 21,90 cm 26 m; 18Ð21 cm Dwingeloo Radio Obs. Dwingeloo, Netherlands 25 m; 6Ð29 cm Eur. Incoh. Scatt. Fac. Kiruna, Sweden 32 m; 32 cm 440× 120 m; 130 cm Fleurs Radio Tel. Fleurs, NSW, Aus. EW Arr. 32 5.8 m + 614m;21cm Five College Radio Ast. Obs. Quabbin Res., Mass. 13.7 m; 1Ð7mm Giant Metrewave Radio Tel. Pune District, India 25 km Irr. Arr. 36 45 m; 21Ð790 cm Hat Creek Radio Ast. Obs. Cassel, California 3 + (3) T Arr. 6 m; 1Ð7mm Haystack Obs. Westford, Massachusetts 36 m; 0.7Ð18 cm Hiraiso Solar-Terr. Res. Ctr. Nakaminato, Ibaraki, Japan 2 10 m + 6m+ 1m;300Ð950 cm Humain Radio Ast. Stn. Humain, Belgium 7.5 m; 50 cm solar T Arr. 48 4 m; 74 cm solar High Time Resolving Tel. Nanjing, China 2 m; 3.2 cm solar Interplanetary Scint. Obs. Toyakawa, Japan 100 × 20 m; 92 cm Instituto Argentino de Rad. Parque Pereya Iraola, Arg. 2 30 m; 17Ð21 cm Inst. de Radio Astron. Mill. Plateau de Bure, France 0.4 km T Arr. (3) 15 m; 1.3Ð4mm Pico Veleta, Spain 30 m; 0.8Ð4mm IPS Fleurs Solar Obs. Fleurs, NSW, Aus. 2.5 m + 12-Yagi; 6,11,20, 120 cm Itapetinga Radio Obs. Atibaia, Brazil 13.7 m; 0.3Ð28 cm 1.5 m; 4.3 cm solar James Clerk Maxwell Tel. Mauna Kea, Hawaii 15 m; 1.3 mm-300 µm Kashima Space Res. Ctr. Kashima, Japan 26 m; 3Ð150 cm Kisaruzu College Obs. Kisaruza chiba, Japan 1.5 m; CO 2.6 mm Line Maryland Point Obs. Riverside, Maryland 26 m; 0.7Ð90 cm Staz. Rad. di Medicina Medicina, Italy 32 m; 1.3Ð20 cm VLBI T Parab. Cyl.; 74 cm Metsahovi Obs. Radio Sta. Metsahovi,¬ Finland 14 m; 0.3Ð1.3 cm Molonglo Obs. Synth. Tel. Hoskinstown, NSW, Aus. 1570 × 12 m EW Cyl. Par.; 36 cm Mount Pleasant Radio Obs. Cambridge, Tasmania, Aus. 14 m; 30Ð50 cm pulars 26 m; 2.5Ð50 cm Mullard Radio Ast. Obs. Cambridge, England 5 km EW Arr. (4) 13 m; cm λ National Astro. and Ion. Obs. Arecibo, Puerto Rico 305 m; λ>6cm National Obs. Ast. Center Yebes, Spain 14 m; mm λ National Radio Ast. Obs. Green Bank, West Va. 43 m; 1.3Ð600 cm 100 m; 0.7Ð90 cm NRAO mm Telescope Kitt Peak, Arizona 12 m; 0.9Ð7mm Very Large Array Socorro, New Mexico 1Ð35 km Y Arr. (27) 25 m; 0.7Ð90 cm Very Long Baseline Array Socorro, New Mexico 5000 km Arr. 10 25 m; 0.7Ð90 cm VLBI Netherlands Found. Res. Ast. Dwingeloo, Netherlands 22 m Ant. Nobeyama Solar Radio Obs. Nobeyama, Japan 0.5 km T Arr. 16 1.2 m, 1.8 cm solar T Arr 17 6 m, 190 cm Nobeyama Radio Obs. Nobeyama, Japan 45 m, 0.26Ð3cm Nobeyama, Japan 0.7 km T Arr. (5) 10 m, 0.25Ð1.3 cm Noto Radio Ast. Sta. Noto, Sicily 32 m; 0.3Ð1.3 cm VLBI Nuffield Radio Ast. Labs. England 134 km Irr. MERLIN Arr. 7 Ant.: Jodrell Bank, Cheshire 76 m; 5Ð200 cm Jodrell Bank, Cheshire 38 × 26 m Par.; 1.3Ð200 cm Wardle, Cheshire 38 × 26 m; 17Ð370 cm Defford, Worchestershire 25 m; 6Ð200 cm Darnhall, Cheshire 25 m; 1.3Ð200 cm Knockin, Shropshire 25 m; 1.3Ð200 cm Pickmere, Cheshire 25 m; 1.3Ð200 cm Jodrell Bank, Cheshire 13 m; 20Ð50 cm pulsars Obs. de Bordeaux Floirac, France 2.5 m; 2.5Ð4mm Obs. de Grenoble Plateau de Bure, France 2.5 m; 1.3 mm Sp.-V/AQuan/1999/10/07:17:14 Page 131

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Table 6.3. (Continued.)

Name of Observatory/Inst. Location Description

Obs. fur¬ Solare Astr. Tremsdorf, Germany 1.5 + 4 + 10.5m+ Yagi; 3.2Ð750 cm Obs. Radioastron. de Maipu« Maipu,« Chile 160 × 73 m; 670 cm Ohio State Radio Obs. Delaware, Ohio 100 × 30 m; 1.9 cm Ondrejov Ast. Obs. Ondrejov, Czech. 3 m; 1.5Ð3 cm solar 7.5 m; 37Ð120 cm solar 7.5 m; 25Ð300 cm solar Onsala Space Obs. Onsala, Sweden 20 m; 3.7Ð270 cm 25 m; 30Ð260 cm Ooty Radiotelescope Ootacamund, India 530 × 30 m Par. Cyl.; 90 cm Ooty Synthesis Radiotelescope (ORT) + 823× 9m;90cm Owens Valley Radio Obs. Big Pine, California 0.4 km T Arr. (4) 10 m; 1.3Ð3mm 40 m; 0.7Ð90 cm Arr.of227m;4Ð30 cm Purple Mountain Obs. Delingha, China 13.7 m; 0.3,1.3 cm Purple Mt. Obs. Solar Fac. Nanjing, China 1.5 m; 3.2Ð11 cm solar 2 m; 6 cm solar Puschino Radio Ast. Sta. Puschino, Russia 22 m Tel. Cross-type 1 km decimetric array 18 acre decimetric phased array Radioobs. Effelsberg Effelsberg, Germany 100 m; 0.6Ð49 cm RATAN-600 Zelenchukskaya, SU 0.6 cm circle of 895 elem.; 0.8Ð30 cm Solar Radiospec. Obs. Ravensburg, Germany 7 m + 8 dipoles; 30Ð1000 cm Sta. de Rad. de Nancay Nancay, France 16 1 m Parab.; 3.2 cm 24 Parab.; 67Ð200 cm solar 16 + 2 Parab.; 67Ð200 cm solar 299 × 40 m; 9Ð21 cm 144 Con. Log. Per.; 380Ð2000 cm Swedish-ESO Submm. Tel. La Silla, Chile 15 m; 0.8Ð3mm Tonantzintla Solar Radio Int. Puebla, Mexico 2 1.1 m Parab.; 4 cm solar Toyokaya Obs. Toyokawa, Japan 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 U. of Mich. Radio Obs. Dexter, Michigan 25 m; cm λ URAN-1 Interferometer Kharkov, SU Linear Arr, Xed dipoles; 1200Ð3000 cm UTR-2 Array Kharkov, SU T Arr 1800 × 54 m + 900 × 54 m; 1200Ð3000 cm Westerbork Synth. Rad. Tel. Westerbork, Netherlands 4 km EW Arr. 10 + (4) 25 m; 6Ð90 cm Yunnan Obs. Kunming, China 2.5, 3, 3.2, & 10 m; 8.1Ð130 cm

6.5 RADIO EMISSION AND ABSORPTION PROCESSES

6.5.1 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 Page 132

132 / 6 RADIO ASTRONOMY

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 d = source source , Bν  (6.11) source d source   ∼ 2k ν = ν  =  , S I d source 2 Tb d source (6.12) source λ source (where the approximation is valid for the RayleighÐJeans limit and most radio wavelengths) and 

−2πi/λ[L j −Lk ]·[s(α,δ)−s0(α0,δ0)] Vν(u,v)= Iν(α, δ)Pn(α − α0,δ− δ0)e 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)2πθ dθ, (6.14)

2 2 2 1/2 ∼ 2 2 1/2 where (L j − Lk) · (s − s0)/λ = ux + vy + wz, q = (u + v + w ) = (u + v ) , and θ = (x2 + y2 + z2)1/2 =∼ (x2 + y2)1/2. 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 Sν (Jy) Vν (q)/Sν ( )θ2 ( ) − (θ/θ )2 Tb,max K arcsec −(π2 )( θ )2 Gaussian e 4ln2 H H e 4ln2 q H 1360λ2(cm) ,θ≤ θ / ( )θ2 ( ) 1 H 2 Tb,max K H arcsec a Uniform disk 2J1(πqθH)/(πqθH) 0,θ>θH/2 1961λ2(cm) ( )θ2 ( ) 2 2 Tb,max K H arcsec Limb-brightened 2θH/(θ − θ ), sin(πqθH)/(πqθH) H 981λ2(cm) θ ≤ θ /2 (shot glass) H 0,θ>θH/2 ( )θ2 ( ) 420Tb,max K H arcsec “Thin” ring δ(θ − θH/2) J0(πqθH) λ2(cm)

Note a Jn is a Bessel function of order n, and assuming that source is at the center of the field, then 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

λ  λcm  λcm resolution = = 34 = 2 . (6.15) D Dm Dkm Sp.-V/AQuan/1999/10/07:17:14 Page 133

6.5 RADIO EMISSION AND ABSORPTION PROCESSES / 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 σ = √ Jy, (6.16) detection 2 ν  FK/Jy Dm MHz tsec

where FK/Jy is a fixed, empirically determined constant for each antennaÐreceiver combination, while for an array of N antennas of this size,  2.5Tsys 1 σ = √ Jy, (6.17) detection η  2 ν  ( − )/ c a Dm MHz 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 0.6Ð0.7 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 jνρ and κνρ are proportional to ρ2, 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 freeÐfree radiation processes are caused by interactions between free electrons and positive ions in a partially or fully ionized plasma. The emission coefficient for freeÐfree emission is given by − / ν/ − . − . ρ = . × 39 2 1 2 h kTe =∼ . × 39 2 0 34ν 0 11, jff,ν 5 4 10 Ne Te gffe 7 45 10 Ne Te (6.18) where we have used   1/2 3/2 3 T − . − . g (ν, T ) = 17.7 + ln e =∼ 1.38 × T 0 34ν 0 11 (6.19) ff e π ν e GHz

for the freeÐfree Gaunt factor, with an approximation valid at radio wavelengths [4, 5]. In (6.18)Ð −3 (6.19), Ne is the electron concentration in units of cm , and Te is the electron temperature. The Planck function relates the emission and absorption coefficients for black body radiation, i.e.,     ν3 ν 2 ν 2 S = jν = 2h 1 ≡ ∼ , ν ν/ 2kTr = 2kTe (6.20) κν c2 eh kTe − 1 c c ∼ where the approximation that the radiation temperature Tr = Te is valid for longer radio wavelengths, but becomes invalid at mm wavelengths. For boundÐfree transitions the emission coefficient and Gaunt factor are given by ∞ − − / − / 2 ρ = . × 33 2 3 2 3 (ν, , ) 157890 n Te , jfb,ν 1 72 10 Ne Te n gfb Te n 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 . ( − ) 0.0496(u2 + 4 u + 1) (ν, , ) = + 0 1728 un 1 − n 3 n , gbf Te n 1 2/3 4/3 (6.22) n(un + 1) n(un + 1) 2 15 where un = n (ν/3.289 89 × 10 Hz) [6]. Sp.-V/AQuan/1999/10/07:17:14 Page 134

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6.5.3 Spectral Lines—Thermal Bound–Bound Transitions

For boundÐbound transitions between any two levels with energies En = E1, ..., Eionization and statistical weights gn, radiation absorption and emission involves photons of energy hνnm = Em − En. The properties of a line transition are determined by the Einstein coefficients Amn, Bmn, and Bnm, which are related by

2 = c = gm . Bmn 3 Amn and Bnm Bmn (6.23) 2hν gn The emission and absorption coefficients for line transitions are

hνnm j ,ν = N A φ (ν ) (6.24) bb nm 4π m mn mn nm and   ν φ (ν ) κ = h nm φ (ν ) − Nn gn mn mn , bb,νnm Nm Bnm mn nm 1 (6.25) 4π Nm gmφmn(νnm)

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  

j ,ν 2hν3  1  S = bb nm =   . νnm   (6.26) κ ,ν c2 Nn gnφmn(νmn) bb nm − 1 Nm gmφmn(νnm)

If the line is in Local Thermodynamic Equilibrium (LTE) with temperature T , then Nn/Nm = (gn/gm) exp(hνnm/kT) and S = Bν so ν h nm −hν /kT κ , ,ν = N B φ (ν ) 1 − e nm . (6.27) bb LTE nm 4π m nm mn nm 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 Chemical Wavelength Date and telescope molecule symbol region of discovery

methyladyne CH 4 300 Aû 1937 Ð Mt. Wilson 2.5 m cyanogen radical CN 3 875 Aû 1940 Ð Mt. Wilson 2.5 m + methyladyne ion CH 4 232 Aû 1941 Ð Mt. Wilson 2.5 m hydroxyl radical OH 18 cm 1963 Ð Lincoln Lab. 26 m ammonia NH3 1.3 cm 1968 Ð Hat Creek 6 m water H2O 1.4 cm 1968 Ð Hat Creek 6 m formaldehyde H2CO 6.2 cm 1969 Ð NRAO43m carbon monoxide CO 2.6 mm 1970 Ð NRAO11m hydrogen cyanide HCN 3.4 mm 1970 Ð NRAO11m cyanoacetylene HC3N 3.3 cm 1970 Ð NRAO43m hydrogen H2 1013Ð1108 Aû 1970 Ð NRL rocket methanol CH3OH 36 cm 1970 Ð NRAO43m formic acid HCOOH 18 cm 1970 Ð NRAO43m + formyl radical ion HCO 3.4 mm 1970 Ð NRAO11m Sp.-V/AQuan/1999/10/07:17:14 Page 135

6.5 RADIO EMISSION AND ABSORPTION PROCESSES / 135

Table 6.5. (Continued.)

Name of Chemical Wavelength Telescope of molecule symbol region discovery

formamide NH2CHO 6.5 cm 1971 Ð NRAO43m carbon monosulfide CS 2.0 mm 1971 Ð NRAO11m silicon monoxide SiO 2.3 mm 1971 Ð NRAO11m carbonyl sulfide OCS 2.7 mm 1971 Ð NRAO11m methyl cyanide CH3CN 2.7 mm 1971 Ð NRAO11m isocyanic acid HNCO 3.4 mm 1971 Ð NRAO11m methyl acetylene CH3CCH 3.5 mm 1971 Ð NRAO11m acetaldehyde CH3CHO 28 cm 1971 Ð NRAO43m thioformaldehyde H2CS 9.5 cm 1971 Ð Parkes 64 m hydrogen isocyanide HNC 3.3 mm 1971 Ð NRAO11m hydrogen sulfide H2S 1.8 mm 1972 Ð NRAO11m methanimine CH2NH 5.7 cm 1972 Ð Parkes 64 m sulfur monoxide SO 3.0 mm 1973 Ð NRAO11m + diazenylium N2H 3.2 mm 1974 Ð NRAO11m ethynyl radical C2H 3.4 mm 1974 Ð NRAO11m methylamine CH3NH2 3.5 mm 1974 Ð NRAO11m 4.1 mm 1974 Ð Tokyo6m dimethyl ether (CH3)2O 9.6 mm 1974 Ð NRAO11m ethanol CH3CH2OH 2.9Ð3.5 mm 1974 Ð NRAO11m sulfur dioxide SO2 3.6 mm 1975 Ð NRAO11m silicon sulfide SiS 2.8, 3.3 mm 1975 Ð NRAO11m vinyl cyanide CH2CHCN 22 cm 1975 Ð Parkes 64 m methyl formate CH3OCHO 18 cm 1975 Ð Parkes 64 m nitrogen sulfide NS 2.6 mm 1975 Ð Texas5m cyanamide NH2CN 3.7 mm 1975 Ð NRAO11m cyanodiacetylene HC5N 3.0 cm 1976 Ð Algonquin 46 m formyl radical HCO 3.5 mm 1976 Ð NRAO11m acetylene C2H2 infrared 1976 Ð KPNO 4 m cyanoethynyl radical C3N 3.4 mm 1976 Ð NRAO11m ketene H2CCO 2.9 mm 1976 Ð NRAO11m cyanotriacetylene HC7N 2.9 cm 1977 Ð Algonguin 46 m nitrosyl radical HNO 3.7 mm 1977 Ð NRAO11m confirmed 1.9 mm 1990 Ð FCRAO 14 m ethyl cyanide CH3CH2CN 2.7Ð4.0 mm 1977 Ð NRAO11m cyano-octatetra-yne HC9N 2.9 cm 1977 Ð Algonguin 46 m methane CH4 infrared 1977 Ð KPNO 4 m nitric oxide NO 2.0 mm 1978 Ð NRAO11m butadiynyl radical C4H 2.6Ð3.5 mm 1978 Ð NRAO11m methyl mercaptan CH3SH 3 mm 1979 Ð BTL7m isothiocyanic acid HNCS 3 mm 1979 Ð BTL7m + thioformyl radical ion HCS 3 mm 1980 Ð BTL7m + protonated carbon dioxide HOCO 3 mm 1980 Ð BTL7m ethylene C2H4 infrared 1980 Ð KPNO 4 m cyanotetraacetylene HC11N many (cm) 1981 Ð Algonquin 46 m silicon dicarbide SiC2 many (mm) 1984 Ð BTL7m propynylidyne C3H many (3 mm) 1984 Ð BTL7m methyl diacetylene CH3C4H several (cm) 1984 Ð Haystack 37 m methyl cyanoacetylene CH3C3N several (cm) 1984 Ð Haystack 37 m tricarbon monoxide C3O 2 cm 1984 Ð Haystack 37 m silane SiH4 infrared 1984 Ð KPNO 4 m + protonated HCN NCNH 3, 2, 1 mm 1984 Ð NRAO12m 1984 Ð Texas5m cyclopropynylidene C3H2 3, 2 mm 1985 Ð many + — H2D ? 0.62 mm 1985 Ð KAO hydrogen chloride HCl ? 0.48 mm 1985 Ð KAO + protonated water H3O 1.0 mm 1986 Ð NRAO12m confirmed 0.8 mm 1990 Ð CSO 10 m Sp.-V/AQuan/1999/10/07:17:14 Page 136

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Table 6.5. (Continued.)

Name of Chemical Wavelength Telescope of molecule symbol region discovery

pentynylidyne radical C5H radio 1986 Ð IRAM 30 m hexatriynyl radical C6H radio 1986 Ð IRAM 30 m phosphorus nitride PN 3, 2, 1 mm 1986 Ð NRAO12m 1986 Ð FCRAO 14 m — C2S 3, 7 mm 1986 Ð Nobeyama 45 m — C3S 3, 7 mm 1986 Ð Nobeyama 45 m acetone (CH3)2CO ? 3.4 mm 1987 Ð IRAM 30 m sodium chloride NaCl 2, 3 mm 1987 Ð IRAM 30 m aluminum chloride AlCl 2, 3 mm 1987 Ð IRAM 30 m potassium chloride KCl 2, 3 mm 1987 Ð IRAM 30 m aluminum fluoride AlF 2, 3 mm 1987 Ð IRAM 30 m methyl isocyanide CH3NC 3 mm 1987 Ð IRAM 30 m cyanomethyl radical CH2CN 6.5 mm 1988 Ð Nobeyama 45 m 1.3 cm 1988 Ð NRAO43m silicon carbide SiC 2 mm 1989 Ð IRAM 30 m propynal HCCCHO 8 mm 1989 Ð Nobeyama 45 m 1.6 cm 1989 Ð NRAO43m — SiC4 3.5, 7.5 mm 1989 Ð Nobeyama 45 m phosphorus carbide CP 1.2, 3.0 mm 1989 Ð IRAM 30 m propadienylidene H2CCC 3 mm 1990 Ð IRAM 30 m 1.4 cm 1990 Ð NRAO43m butatrienylidene H2CCCC many (3 mm) 1990 Ð IRAM 30 m silicon nitride SiN 1.1, 1.4 mm 1990 Ð NRAO12m silylene (pend. conf.) SiH2 ? 0.8, 1.0 mm 1990 Ð CSO 10 m — HCCN 3 mm 1991 Ð IRAM 30 m carbon suboxide C2O 7 mm 1991 Ð Nobeyama 45 m 1.3 cm 1991 Ð NRAO43m isocyanoacetylene HCCNC 7 mm 1992 Ð Nobeyama 45 m + sulfur oxide ion SO 1.3, 3 mm 1992 Ð NRAO12m ethinylisocyanide HNCCC 1.1, 0.8, 0.64 cm 1992 Ð Nobeyama 45 m magnesium isocyanide MgNC radio 1992 Ð Nobeyama 45 m + + protonated HC3NH NC3H radio 1993 Ð Nobeyama 45 m — NH2 0.645 mm 1993 Ð CSO + carbon monoxide ion CO 1.3, 0.8 mm 1993 Ð NRAO 12 m, CSO sodium cyanide NaCN 3, 2.4 mm 1993 Ð NRAO12m + — CH2D 0.8Ð4.0 mm 1993 Ð NRAO 12 m, CSO nitrous oxide N2O 2.0Ð4.0 mm 1994 Ð NRAO12m magnesium cyanide MgCN radio 1995 Ð NRAO12m + prot. formaldehyde H2COH radio 1995 Ð Nobeyama 45 m octatetraynyl radical C8H radio 1996 Ð IRAM 30 m octatetraynyl radical C7H radio 1996 Ð IRAM 30 m + µ protonated hydrogen H3 IR-3.67 m 1996 Ð UKIRT hexapentaenylidene H2C6 K-band 1997 Ð NASA 34 m ethylene oxide c − C2H4O K, Q, 3 mm,1 mm 1997 Ð NRO, SEST, Haystack hydrogen flouride HF 121.7 µm 1997 Ð ISO

6.5.4 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 RADIO EMISSION AND ABSORPTION PROCESSES / 137

dio sources. However, in some stars and certain solar system environments, two other magneto- bremsstrahlung processes occur when the moving electrons are nonrelativistic or only mildly rela- 2 −1/2 ∼ tivistic. When mildly relativistic (γLorentz ≡ (1 − (v/c) ) = 2Ð3) 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 effi- cient 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) = KE−γ in the energy range E to E + dE, 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   (γ −1)/2 . × 18 ρ = . × −22 (γ ) (γ +1)/2 6 36 10 jν 1 35 10 a KH ν (6.28) and 9 γ (γ +2)/2 −(γ +4)/2 κνρ = 0.019g(γ )(3.5 × 10 ) KH ν , (6.29)

where H is in gauss and all other variables are in cgs units, so that the source function is (γ ) jν −30 a γ/2 −1/2 5/2 −1 −3 Sν = = 2.84 × 10 2 H ν erg s cm (6.30) κν g(γ ) [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 (γ ) (γ ) α ν−1/2 1/2  −1/2 −1/4ν5/4 a g Ts GHz HmG 0 S0 HmG GHz 1.0 0.283 0.96 0.0 1.9 × 1012 K 0.015 mas 1 5 0.147 0.79 0.25 1.0 × 1012 K 0.021 mas 2.0 0.103 0.70 0.50 6.6 × 1011 K 0.026 mas 2.5 0.0852 0.66 0.75 4.9 × 1011 K 0.030 mas 3.0 0.0742 0.65 1.0 3.6 × 1011 K 0.035 mas 4.0 0.0725 0.69 1.5 2.3 × 1011 K 0.044 mas 5.0 0.0922 0.83 2.0 1.7 × 1011 K 0.051 mas

The specific intensity is given by  L Iν = Sν(1 − exp(−τν)) dτν, (6.31) 0

where dτν = κνρ ds in the above integral, which is along the line of sight of length L. Sp.-V/AQuan/1999/10/07:17:14 Page 138

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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)Ð(6.30) also tell us that for the optically thick − / / (γ + )/ −(γ − )/ and thin limits, Iν is proportional to H 1 2ν5 2 and KH 1 2ν 1 2 L, respectively, where L is a α line-of-sight path length. Defining the spectral index α by Sν ∝ ν , then α = 2.5 and −(γ − 1)/2in 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 E2, (6.33) dt where the first two terms are due to losses by ionization in the ambient, medium, and freeÐfree 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:     −20 E −20 E ζ ≈ 3.33 × 10 nH 6.27 + ln + 1.22 × 10 ne 73.4ln − ln ne (6.34) mc2 mc2 and   −16 E η ≈ 8 × 10 nHne 0.36 + ln , (6.35) mc2

where nH is the concentration of neutral hydrogen atoms and ne is the concentration of thermal electrons. ∼ −3 2 For synchrotron radiation losses ξsynch = 2.37 × 10 H⊥, where H⊥ is the magnetic field perpendicular to the velocity vector of the radiating relativistic electron. The time scale for synchrotron 8 2 2 losses is tsynch ≡ E/(−dE/dt) = 1/(ζsynch E) ∼ (5.1 × 10 /H⊥)(mc /E) s. Synchrotron radiation losses dominate in regions with large magnetic fields and low radiation fields. −4 For inverse Compton losses ξIC ≈ 3.97 × 10 urad, where urad is the radiation energy density. 12 12 Inverse Compton losses dominate when brightness temperature approaches Tb ≤ 10 K. Above 10 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) = −1 −γ φ (E) (E) dE. So if one can assume a power law for the particle injection, (E) = 0 E , then

−1 −γ −1 −1 N(E) = 0(γ − 1) E (ζ E + η + ζ E) . (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 RADIO EMISSION AND ABSORPTION PROCESSES / 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 Sν α absorption + 1/2 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, freeÐfree, 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 −1 equation d(E )/dt = ζsynch which leads to a spectrum described by E = E0/[1 + ζsynch(t − t0)E0]. −γ −γ γ −1 Then if at t = t0, N(E) = N0 E , then at later times N(E) = N0 E (1 − ζsynch Et) for E < ET(t), and N(E, T ) = 0 for E > ET(t), where ET(t) = 1/(ζsyncht). The frequency above ν ≈ −3 −2 which synchrotron losses dominate is b BGausstyear GHz. 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 near- analytic equations. For a spherical region which “suddenly” has a distribution of relativistic particles −γ given by N0 = K0 E and which expands with an outer radius r2(t) the flux density of the source is described by the quantities in Table 6.7 [11Ð13]. 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 z0 the radius of the cross section of the jet is r0 (jet Mach number M0 = z0/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 z2 =∞ and z1 = z0; 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 = ( / )−1 −2/3 E E0 r2 r0 E = E0[r2(z2)/r0] = ( / )−2 −1 H H0 r2 r0 H = H0[r2(z2)/r0] = ( / )−(γ +2) −2(γ +2)/3 K K0 r2 r0 K = K0[r(z2)/r0] τ = . (γ )( . × 9)γ (γ +2)/2 0 0 038g 3 5 10 K0 H0 r0 same Sp.-V/AQuan/1999/10/07:17:14 Page 140

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Table 6.7. (Continued.)

Expanding sphere Conical jet

ξ(τ; τ>20) = 1 ξ(τ; τ>20) = 1 ξ(τ; τ<20) = 0.66584 + 0.09089τ ξ(τ; τ<20) = 0.78517 + 0.06273τ − 0.009989τ 2 + 0.0005208τ 3 − 0.007242τ 2 + 0.0003905τ 3 − 0.00001268τ 4 + 0.000000115τ 5 − 0.00000973τ 4 + 0.00000009τ 5       τ ν −(γ +4)/2 ( ) −(7γ +8)/6 −(2γ +3) −(γ +4)/2  0 r2 z2  r ν τν = τν = τ sin() ν r 0 r ν 0 0 0 0  ν 5/2 − . γ −1/2 5/2 ν = . 0 65 ν 3 Sν = S M sin() S 1 42e H0,mG GHz mas 0 0 ν 0    × − (−τ ) × ξ τ ) z2(t) 3/2   1 exp ν ( ν × ζ 1 − exp(−τν ) ξ(τν ) dζ z1(t)   t z2 r2(t; free exp.) = r0 r2(t; free exp.) = r0 t0 z0   2/5 1/2 t z2 r2(t; E cons.) = r0 r2(t; E cons.) = r0 t0 z0   1/4 1/3 t z2 r2(t; Mom. cons.) = r0 r2(t; Mom. cons.) = r0 t0 z0

z2 = v2(t − tstart) z1 = 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 [1] Properties of radio telescopes [2] Interferometry and aperture synthesis [3] Aperture synthesis [4, 5] Observation-oriented radio astronomy textbooks [6, 7] Synchrotron radiation physics [8] Radio astrophysical theory [9] General astrophysical theory [10]

References 1. Verschuur, V.L., & Kellermann, K.I. 1988, Galactic and Extragalactic Radio Astronomy (Springer-Verlag, New York) 2. Christiansen, W.N., & Hogbom,¬ 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

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REFERENCES

1. Clark, B.G. 1989, Synthesis Imaging in Radio Astron- sity of Chicago Press, Chicago), Chapter 9 omy, A.S.P. Conference Series, 6, p. 1 7. Dulk, G.A. 1985, ARA&A, 23, 169 2. Christiansen, W.N., & Hogbom,¬ J.A. 1985, Radio Tele- 8. Kundu, M.R. 1965, Solar Radio Astronomy (Inter- scopes (Cambridge University Press, Cambridge) science, New York) 3. Price, R.M. et al. 1989, Radio Astronomy Observatories 9. Zheleznyakov, V.V. 1970, Radio Emission of the Sun (National Academy Press, Washington, DC) and Planets (Pergamon Press, Oxford) 4. Karzas W.J., & Latter, R. 1961, ApJS, 6, 167 10. Ginzburg, V.L., & Syrovatskii, S.I. 1965, ARA&A, 3, 5. Hjellming, R.M., Wade, C.M., Vandenberg, N.R., & 297 Newell, R.T. 1979, AJ, 84, 1619 11. van der Laan, H. 1966, Nature, 211, 1131 6. Aller, L.H. & Liller, W. 1968, Nebulae and Interstellar 12. Kellermann, K.I. 1966, ApJ, 146, 621 Matter, edited by B. Middlehurst & L.H. Aller (Univer- 13. Hjellming, R.M., & Johnston, K.J. 1988, ApJ, 328, 600 Sp.-V/AQuan/1999/10/07:17:14 Page 142