Radio Astronomy Lecture Notes

Preface

This set of lecture notes is adapted from Prof. Dale Gary's Phys 728: Radio Astronomy course, which he taught at NJIT as a graduate-level course until Spring 2019.

Lecture 1: Introduction to Radio Astronomy

Overview of Radio Emission from Astronomical Objects

The Radio Sky

When we look at the sky at night with our unaided eyes, we see about 2000 stars of various levels of brightness, and if we are far from city lights we may see the faint band of the Milky Way, which is the light from billions of stars making up our galaxy. But if our eyes were able to see radiowaves, the sky might look like the image below.

(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

It may appear similar to the starry sky, but in fact most of the point-like objects are not stars, but luminous radio galaxies billions of light years away. The larger sources are ionized clouds of hydrogen, or supernova remnants.

Looking toward the center of our galaxy, our radio eyes would see a large variety of strange features, most of which are not visible in other wavelengths.

Credit: Gizmodo Australia

Owens Valley Long Wavelength Array Movie

The Electromagnetic Spectrum

You should all be aware of the different types of radiation that make up the electromagnetic spectrum, but it is worthwhile to list the names of the different types of emission, in energy, or frequency) order:

gamma rays (> ~1 MeV) hard X-rays (10-1000 keV) soft X-rays (1-10 Å) EUV (~100 Å) UV (~1000 Å) visible (4000-7000 Å -- 400-700 nm) near IR (~1 micron) IR (10 microns) THz (~100 microns—3000 GHz) submillimeter (300 GHz - 700 GHz) millimeter (30 GHz - 300 GHz) microwave (3 GHz - 30 GHz) decimeter (300 MHz - 3 GHz) ("cable" TV/UHF band) meterwave (30 MHz - 300 MHz) (TV/FM/HF band) dekameter (3 MHz - 30 MHz) (Shortwave AM band (0.5 MHz - 1.7 MHz) etc.

File:Spectrum opacity.GIF

Note that the units change as we go from top to bottom--use energy units near the top, then switch to wavelength units, then switch to frequency units. This is purely a matter of convenience and convention. We could stick with energy, or wavelength, or frequency throughout, but the range of 6 or 7 decades makes it inconvenient to stick with one measure. The relationships among energy, frequency, and wavelength are, of course:

$E=h\nu =hc/\lambda$.

For the purposes of this course, we will be concentrating on techniques of interferometry and synthesis imaging that work for the range from submillimeter to dekameter, although there are practical difficulties at both extremes, and it is currently most common to use interferometry in the millimeter to meterwave range. There are on-going efforts to extend interferometry to both higher frequencies (submillimeter—ALMA) and lower frequencies (space arrays).

Why Observe At Radio Wavelengths?

There are many reasons why it is advantageous to observe at radio wavelengths.

Advantages of Radio

• Radio waves reach the ground
• Can observe objects or phenomena that are difficult or impossible to detect in other wavelength ranges
• Can use radio emission for quantitative physical diagnostics of object parameters

The first reason is simply that it is possible to observe radio waves from the ground. As shown in the figure below, spacecraft are needed to observe astronomical objects in gamma rays, X-rays, UV, and IR, while ground observations are possible in the visible, some parts of the near IR, and the radio. NJIT has solar observatories exploiting all of these ground windows.

File:Transgraph colorized.jpg

Credit: NASA/IPAC

Note that the window closes at the long-wavelength end of the spectrum--not because of the atmosphere, which remains transparent to long-wavelength radio waves--but rather due to the ionosphere, which reflects the radiation.

A second reason is that some objects and phenomena are invisible or hard to detect in other wavelengths, and can only be seen, or can be seen with greater sensitivity, in the radio. Here are a few of many many examples from which we could choose:

File:M81mosaic.gif

Neutral hydrogen traces interactions among galaxies in the M81 group. (c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

File:Cena overlay.jpg

Centaurus A -- peculiar galaxy with radio lobes. From HST web site.

File:Jupiter lo.jpg	File:SUN dulk gary.jpg
Jupiter's Radiation Belt	The Sun

(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

The third important reason to explore astronomical objects in radio wavelengths is that the emission properties provide quantitative physical information about conditions in the source. We will see that radio emission is produced in a large number of ways. The low-energy radio photons are relatively easy to produce, which makes radio emission sensitive to a great many parameters. However, the number of mechanisms is itself a problem. Before one can use the emission to give information, one must first determine which radio emission mechanism is responsible for the emission. In practice, the most accurate way to determine the emission mechanism is to have spectral information, since different emission mechanisms have different characteristic spectral properties. In addition to helping to determine the emission mechanism, quantifying spectral properties such as peak brightness, peak frequency, spectral slopes, etc., also provides quantitative diagnostic parameters.

For all of these reasons and more, the radio range of wavelengths is as essential as gamma ray, X-ray, UV, optical, and IR for providing a complete picture of the physical nature of astronomical sources.

Overview of Radio Instrumentation

What Is Different About Radio Instrumentation?

Astronomical telescopes that work in the radio range look and operate very differently from the more familiar optical instrumentation. In fact, the "radio" range is so broad (6 or 7 orders of magnitude) that instruments at the low end of the frequency range look very different from those at the high end. We will take a brief look at the differences, using some existing or planned telescopes as examples.

Single Element Instruments

The term "single element" means either single parabolic dishes, or in some cases single dipole elements. Here are a few pictures:

File:ARECIBO.jpg

Arecibo: The largest single dish in the world, 306 m (c) Cornell University / National Science Foundation

File:GBT.jpg

Green Bank Telescope (GBT): The largest fully steerable single dish in the world, 100 x 110 m (c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

File:RATAN.jpg	File:Metsahovi.jpg
RATAN 600: Diameter 600m, part of a "dish" reflecting surface	Metsahovi: Large mm dish

File:Bruny.JPG

Bruny Island Radio Spectrograph

Single radio elements have limited spatial resolution (the diffraction limit of the telescope). This diffraction limited resolution is proportional to wavelength (as it is also for optical telescopes, but it seems more extreme for radio telescopes due to the huge range of wavelengths over which they are typically used). The diffraction limit for a circular aperture of diameter D is $\theta \approx 1.22\lambda /D$, where $\theta$ is the angular diameter of the Airy Disk at the half-power point (the full-width-half-maximum, or FWHM) in radians. At a frequency of 5 GHz, even the Arecibo dish has an angular resolution of only about 50 arcseconds. The fully-steerable GBT has a resolution at this frequency of only 150 arcseconds.

Because of the limited spatial resolution of single element telescopes, sophisticated techniques have been developed to combine single elements into multiple-element arrays, which work together to form a single telescope. In such arrays, the spatial resolution is determined not by the size of the individual elements, but rather by the maximum separation between elements, which is referred to as the baseline length, B. With an interferometer, the diffraction limit is $\theta \approx \lambda /B$, where B can extend to many (even thousands of) km.

Interferometers

We now show some examples of interferometer arrays:

File:VLA lo.jpg

Close-up of VLA (Very Large Array) (c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

File:VLAWye1 lo.jpg

Aerial view of VLA in its most compact configuration. (c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

File:2014-05-20 1.jpg

Ten antennas of NJIT's 13-antenna Expanded Owens Valley Solar Array (EOVSA)

File:Lwa1.jpg

The Long Wavelength Array (LWA) station at the VLA in New Mexico (See LWA-TV) (Univ. of New Mexico)

File:LWA OVRO.PNG

The Long Wavelength Array (LWA) station at Owens Valley in California (Gregg Hallinan, Caltech)

File:Rh ns2.jpg

North arm of the Nancay Radioheliograph (Meudon, Observatoire de Paris)

File:Norh.gif

Nobeyama Radioheliograph (National Astronomical Observatory of Japan)

File:OwensValley lo.jpg

One of 10 antennas of the Very Long Baseline Array (VLBA) -- this one at OVRO. (c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

File:VLBA1 lo.jpg

Site locations for the entire 10-station VLBA (c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

• Low Frequency Array (LOFAR)

File:Lofar-14aug2009-19.jpg

• Atacama Large Millimeter Array (ALMA)

File:ALMA potw eso 130415.jpg

Atacama Large Millimeter Array (ALMA) (c) ALMA (ESO/NAOJ/NRAO)

Future arrays, now under consideration/construction:

• Frequency Agile Solar Radiotelescope (FASR)

File:FASR Closeup.jpg

• Square Kilometer Array (MeerKAT (South Africa))

File:2017 meerkat 01-1030x578.jpg

Brightness Temperature and Flux Density

Planck Function

You should all be familiar with the basics of black-body radiation. By simply observing how hot objects behave, 19th century scientists came up with the following empirical laws.

• the peak wavelength shifts proportional to temperature $\lambda _{\text{max}}T=2.898\times 10^{-3}\,{\text{m K}}$. (Wien Displacement Law)
• the intensity increases as the square of the frequency at low frequencies (Rayleigh-Jeans Law)
• the intensity decreases exponentially at high frequencies (Wien Law)
• the flux of radiation emitted by a blackbody increases as the fourth-power of the temperature (Stefan-Boltzmann Law)

A plot of this behavior for several temperatures as a function of wavelength is shown in the figure below.

Blackbody Spectrum

The functional form that corresponds to these curves is called the Planck function, which was derived by Max Planck by introducing the Planck constant $h=6.63\times 10^{-34}\,{\text{J s}}$, and postulating that photon energies were quantized in units of $h\nu$. This was the beginning of the quantum theory, which was later extended to matter as well as radiation. This function has two forms, written below, which are related by $B_{\lambda }(T)d\lambda =B_{\nu }(T)d\nu$.

$B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{hc/\lambda kT}-1}}$         (wavelength form) (1)

$B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}}$         (frequency form) (2)

• To derive the Wien displacement law, find the maximum of the function by setting $dB_{\lambda }(T)/d\lambda =0$, to get $\lambda _{\text{max}}T=hc/5k=2.898\times 10^{-3}\,{\text{m}}$
• To derive the Rayleigh-Jeans law, expand $e^{h\nu /kT}$ in $B_{\nu }(T)$ for $h\nu \ll kT$ to get $B_{\nu }(T)=2kT\nu ^{2}/c^{2}$
• To derive the Wien Law, expand $e^{h\nu /kT}$ in $B_{\nu }(T)$ for $h\nu \gg kT$ to get $B_{\nu }(T)=(2h\nu ^{3}/c^{2})e^{-h\nu /kT}$
• To derive the Stefan-Boltzmann Law, integrate $B_{\lambda }(T)$ over all wavelengths--hint: use the relation

$\int _{0}^{\infty }{\frac {u^{3}du}{e^{u}-1}}={\frac {\pi ^{4}}{15}}$

--to get the flux $F=\sigma T^{4}$, where $\sigma =2\pi ^{5}k^{4}/(15c^{2}h^{3})=5.669\times 10^{-8}\,{\text{W/m}}^{2}{\text{/K}}^{4}$ is the Stefan-Boltzmann constant.

Rayleigh-Jeans Limit

The Rayleigh-Jeans limit is the Planck function in the limit of low-energy photons, $h\nu \ll kT$, which we argue is the relevant limit for any source that produces radio emission. To see this, take a relatively high radio frequency, say 100 GHz, and ask how cool the source must be in order that the above condition be violated (i.e. for $h\nu \approx kT$). We have

$T=h\nu /k=(6.63\times 10^{-34}\,{\text{J s}})(1\times 10^{11}\,{\text{s}}^{-1})/(1.38\times 10^{-23}\,{\text{J/K}})=4.8\,{\text{K}}!$

So even very cold sources at high frequencies still meet the Rayleigh-Jeans criterion. This turns out to be especially useful for radio astronomy, which we will discuss in a moment. But first, let's look at another plot of the Planck function, with axes suitable for a visual appreciation of the Rayleigh-Jeans limit.

File:Planck.jpg

By plotting $B_{\nu }(T)$ on a log-log plot, the part of the curve that obeys the Rayleigh-Jeans Law,

$B_{\nu }(T)={\frac {2kT\nu ^{2}}{c^{2}}}$               (3)

is very obvious--it is the straight line portion with a slope of 2. Here you can see that changing the temperature over many orders of magnitude just increases the intensity linearly, and that it is valid over the entire range of radio frequencies all the way to THz ($10^{12}$ Hz).

Surface Brightness (Intensity) and Flux Density

The monochromatic intensity $I(\nu )$ has units of J m^-2 s^-1 Hz^-1 sr^-1, where sr = sterradians is the unit of solid angle, $d\Omega$. By comparing these units with the Planck function, we see that they are the same. The Planck function gives the monochromatic intensity of the blackbody that it represents. The intensity, or surface brightness, is then integrated over all frequencies:

$I=\int I(\nu )d\nu$ (units: J m^-2 s^-1 sr^-1).

Integrate this again over angular area to get the flux F:

$F=\int Id\Omega$ (units: J m^-2 s^-1, or W m^-2).

which is just the power per unit area. In radio astronomy, we often discuss a related quantity called the flux density, which is the monochromatic intensity (or the Planck function) integrated over solid angle:

$S=\int I(\nu )\,d\Omega$   (units: W m^-2 Hz^-1)         (4)

In fact, the flux density is a fundamental quantity measured by radio telescopes, and is the basis for two different units:

• 1 Jansky (Jy) = 10^-26 W m^-2 Hz^-1
• 1 Solar Flux Unit (sfu) = 10^-22 W m^-2 Hz^-1 = 10,000 Jy.

The quantity that a radio telescope measures is the flux density over some band $\Delta \nu$, so the strength of radio sources in the sky are often specified in Jy. Likewise, the strength of solar sources, especially solar radio bursts, are often specified in sfu.

Brightness Temperature

We are now ready to show a great conceptual simplification that the Rayleigh-Jeans limit gives to the discipline of radio astronomy. We have so far been talking about blackbodies, which are by definition optically thick and in thermal equilibrium. What if a source is not optically thick? In that case, its emission will appear weaker (lower intensity) than if it were optically thick. Whether or not a source is optically thick is a function of frequency. As it turns out, many radio-emitting plasmas are optically thick at low frequencies, but optically thin at high frequencies. In this case, the brightness follows the Planck function up to some frequency, then begins to fall away as it becomes more and more optically thin with frequency. Schematically, it looks something like this:

File:Planck thin.jpg

Radio spectrum for a 10⁶ K plasma that is optically thick below about 10 GHz, and optically thin at higher frequencies. The brightness below 10 GHz corresponds to a million degree blackbody.

We will discuss optical depth in more detail in two weeks, when we discuss radiative transfer. For now, we just want to develop the idea of brightness temperature.

In the Rayleigh-Jeans limit, a blackbody has a temperature given by the Rayleigh-Jeans Law, eq (3), i.e.

$T={\frac {B_{\nu }(T)c^{2}}{2k\nu ^{2}}}$

so as long as the plasma in the above figure is optically thick, we can use the brightness of the emission to determine the plasma temperature. But when it is optically thin, the brightness, or intensity, is less than the Planck function. Nevertheless, we can still talk about a brightness temperature, or the equivalent temperature that a blackbody would have in order to be as bright. The brightness temperature is the same as the true temperature only for an optically thick blackbody. We designate the brightness temperature as $T_{b}$. Using this notation, the flux density measured by a radio telescope becomes:

$S=\int {\frac {2kT_{b}\nu ^{2}}{c^{2}}}d\Omega ={\frac {2k\nu ^{2}}{c^{2}}}\int T_{b}d\Omega$                 (5)

where we have substituted $B_{\nu }$ for $I(\nu )$ in (4), and used (3). So the flux density measured by a radio telescope is just the brightness temperature integrated over the source, times some fundamental constants and frequency-squared.

So far, eq (5) pertains only to thermal emission, but we can extend it to all radio emission simply by considering non-thermal sources as having an effective temperature $T_{eff}$. For a single electron of energy E, its effective temperature is just its kinetic temperature $T_{eff}=E/k$.

To summarize, then, the brightness temperature is the equivalent temperature a black body would have in order to be as bright as the observed brightness. It is important to realize that this is a useful concept only for radiation that obeys the Rayleigh-Jeans Law.

One last point to make is the limit of the integral in eq (5). We earlier mentioned the resolution of a single dish antenna of diameter D, as $\theta \approx 1.22\lambda /D$. This is also the width of the field of view of the antenna--only a source in an area of the sky within this angular distance can be seen. The field of view is also called the beam. Let's look at some consequences of this.

• Unresolved (point) Source: If the antenna is pointing at a source with a very small size, the flux density that it measures is the brightness integrated over the entire source, i.e. the total flux density of the source. In this case, the integral (5) is to be taken over the source.
• Extended Source: If the antenna is pointing at an extended source, part of which falls outside the field of view, it will measure only the flux from the source within the beam. In this case, the integral (5) is to be taken over the beam size.