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References and Resources for this lecture

Full reference information can always be found in the syllabus, under “References Materials.”

• Essential Radio Astronomy Ch 1: emission mechanisms, relevant astrophysical objects
• Tools of Radio Astronomy: radio windows, units, flux densities
• Interferometry and Synthesis in Radio Astronomy: single-dish observations, units, flux densities

Course Overview

Welcome to AS 5003: Contemporary Astrophysics, on radio astronomy and interferometric imaging!

• Syllabus overview regarding format and tutorial schedule

Astrophysics at radio wavelengths

Emission mechanisms

• Radio synchrotron (continuum), non-thermal emission from relativistic electrons in magnetic fields
• Bremsstrahlung (a.k.a. free-free) emission (continuum), thermal emission from ionized gas (H II regions)
• Thermal emission from cold (< 100 K) media, like dust (continuum)
• Atomic hyperfine splitting (“21-cm” line corresponding to neutral hydrogen)
• Molecular emission lines, primarily from rotational transitions (e.g., CO \(J=2-1\))

The earth’s atmosphere is very transparent in the radio region of the electromagnetic spectrum, especially compared to optical windows. Only towards the microwave region (wavelengths around 1 mm), does the atmospheric transmission start to decline.

Astrophysical objects

In truth, almost everything these days!

• quasars
• gamma ray burst (GRB) afterglows
• fast radio bursts (FRBs)
• pulsars
• supernovae remnants
• cosmic microwave background (CMB)
• galaxies, including molecules at high redshift sources
• dust/interstellar medium
• the Sun
• planets (e.g., Jupiter, Uranus)
• protoplanetary disks
• molecular clouds (molecular emission)
• black hole accretion disks (EHT)

Single-dish radio telescopes

In this section, we’ll cover existing single-dish radio telescopes and refresh our memory of basic telescope performance.

The same $$ \theta \approx \frac{\lambda}{D} $$ applies for radio antennas. Take a \(\lambda = 1\;\mathrm{cm}\) observation, for example. Compared to an optical \(\lambda = 500\;\mathrm{nm}\) telescope the same size, the resolution will be a factor of $$ \frac{1\;\mathrm{cm}}{500\;\mathrm{nm}} = 20,000 $$ worse. Yikes!

Radio astronomers are constantly trying to find ways to increase angular (spatial) resolution.

• One way is to build bigger telescopes, such as the Green Bank Telescope (100m diameter)

• Another way is to work at higher frequencies (shorter wavelengths), e.g. sub-mm radio astronomy (IRAM 30m telescope)

• A final way is to use interferometry, sometimes also at sub-mm wavelengths (ALMA), which will be the main focus of this course

It’s easier to build larger telescopes at longer wavelengths because the tolerances required for the reflecting surface are less strict than at optical wavelengths. Though typically one must keep surface tolerances to within $$ \sigma = \frac{\lambda_\mathrm{min}}{16}, $$ otherwise the efficiency of the antenna starts to decline substantially. For the 100 m diameter GBT operating at it’s highest frequency (100 GHz) or 3 mm, this translates to \(200\;\mu\mathrm{m}\), which is the thickness of two sheets of paper! That’s quite an engineering challenge, and is the reason why large, steerable dishes are difficult to build.

Keeping telescopes fixed is one way to build a little bit bigger, such as FAST, which is a five hundred meter diameter fixed telescope in China. See also Arecibo, which unfortunately collapsed in December 2020. Eventually, though, the materials/engineering cost to building large single dish telescopes becomes prohibitive.

Single dish observations

The “beam” of a receiving antenna, or power pattern as a function of direction, can be calculated using the reciprocity theorems for transmitting and receiving antennas. These state that the far field electric field pattern \(f(l)\) is the Fourier transform (much more in lectures 2 and 3!) of the electric field illuminating the aperture of the telescope \(g(u)\).

For large apertures, the nulls at \( l = \pm 1, 2, \ldots\) appear at the angles \(\theta \approx \lambda/D, 2 \lambda/D, \ldots\). In two dimensions, for a circular aperture, this is an Airy pattern.

This is an idealized representation, but is still helpful. The beam can pick up power through sidelobes at a range of angles. Directional antennas help concentrate power in the main beam, but antennas with secondary stages (and thus supports, like most telescopes) create opportunities for ground radiation to reflect into the receiver.

You can think of single-dish telescopes (unless they have a sophisticated, multi-pixel receiver) essentially as single-pixel devices. So to make a map of the sky, you would need to raster scan the telescope across the region of interest, reading out antenna temperature as a function of RA, Dec. To make a good (i.e., scientifically accurate) map, you should focus on Nyquist sampling the sky to a uniform sensitivity, usually through a hexagonal pattern of dithering. More advanced instruments may have an array of “feeds” in a focal plane (mirroring a set of “pixels”), but this is still a small number of pixels compared to a typical CCD (e.g., 25 or 36 compared to \(2046^2\)).

Image Units

Radiative transfer recap

What are the units of the sky, and the images we make as representations of it?

One of the most useful quantities from radiative transfer is \(I_\nu\).

\(I_\nu \) is the specific intensity of radiation, you can think of it as the energy carried along by an infinitesimal “bundle” of rays.

It has dimensions of: $$ \mathrm{energy}\; (\mathrm{time})^{-1} \;(\mathrm{area})^{-1} \;(\mathrm{solid\,angle})^{-1} \; (\mathrm{frequency})^{-1} $$ in CGS units, we would write $$ \mathrm{ergs}\;\mathrm{s}^{-1}\;\mathrm{cm}^{-2}\;\mathrm{ster}^{-1}\;\mathrm{Hz}^{-1} $$ In astronomical settings, I’ve always seen \(I_\nu\) referred to as the “specific intensity.” In non-astronomy settings, I’ve seen “spectral intensity.” If \(I_\nu\) is integrated over all frequencies, it’s called the radiant intensity \(I\).

\(I_\nu \) can be a little mind-bending to think about… it can be a function of

• 3D space \(\vec{x}\)
• direction \(\vec{\Omega}\)
• frequency

Intensity itself is not a vector quantity; rather it is a scalar field that is a function of position and direction \(I_\nu(\vec{x}, \vec{\Omega})\). Rybicki and Lightman write the angular direction vector as \(\vec{\Omega}\) and the solid angle surrounding that vector as \(\mathrm{d}\Omega\).

If we have a defined reference frame, we would probably write \(\vec{\Omega}\) as a vector in spherical coordinates and define the components along \(\hat{\phi}\) and \(\hat{\theta}\) and let \(\mathrm{d}\Omega = d\phi \sin \theta d\theta\).

When we are making astrophysical observations from the earth, though, we are making or acquiring images of regions of the celestial sky. So we generally talk of \(I_\nu(\alpha, \delta)\), where \(\alpha\) and \(\delta\) are R.A. and declination offsets from some direction, respectively. We’re always looking from the same place (at least compared to the size of the universe), so we don’t worry about specifying position within 3D space. But if we went to the Andromeda Galaxy and started mapping the celestial sky, we would need to, then. In the end though, images are have the same units because they represent specific intensity. It’s very common to refer to \(I_\nu(\alpha, \delta)\) as the surface brightness.

Flux

Once you’ve defined \(I_\nu\), then it’s relatively easy to calculate quantities like energy density, flux, momentum, etc, as integrals of the specific intensity field.

Flux is where you integrate out the angular dependence:

$$ F_\nu = \int I_\nu \cos \theta d \Omega $$ (intensity passing through some differential area \(dA\), lowered by the effective angle).

\(F_\nu\) has units of $$ \mathrm{ergs}\;\mathrm{s}^{-1}\;\mathrm{cm}^{-2}\;\mathrm{Hz}^{-1} $$ (i.e., angular dependence has been integrated out).

Most astrophysical sources produce significantly less energy in radio waves compared to higher frequency bands, and so the raw CGS unit can be quite cumbersome. To make this easier, astronomers use a unit called the “Jansky,” which is defined as

$$ 1\,\mathrm{Jy} = 10^{-23}~\mathrm{ergs}\;\mathrm{s}^{-1}\;\mathrm{cm}^{-2}\;\mathrm{Hz}^{-1} $$

The Jansky is a unit of flux.

How do we report specific intensities/surface brightnesses for radio sources, then? We can, reintroduce the “per solid angle” to the Jansky, for example $$ \mathrm{Jy}\;\mathrm{arcsec}^{-2}. $$

Later on the course, we’ll talk about \(\mathrm{Jy}\;\mathrm{beam}^{-1}\), which is another unit of surface brightness/specific intensity.

Other units of surface brightness that you might encounter at other wavelengths include \(\mathrm{mag}\;\mathrm{arcsec}^{-2}\) (optical) and \(\mathrm{MJy}\;\mathrm{sr}^{-1}\) (infrared).

Questions for review:

• What is the name of \(I_\nu\), and what are its units?
• What is the name of \(F_\nu\), and what are its units?
• If we made an astronomical observation of a “point source,” would we report \(I_\nu\) or \(F_\nu\)?
• What about for a spatially resolved source?
• Is a Jansky a unit for \(I_\nu\) or \(F_\nu\)?

The many temperatures of radio astronomy

From Cosmic Sources only

• brightness temperature: \(T_B\), the temperature such that a blackbody would emit with the observed specific intensity. N.B.: One needs to be very careful of the context when using brightness temperature. Classic radio astronomers will define brightness temperature using the Rayleigh-Jeans definition (Essential Radio Astronomy by Condon and Ransom, Tools of Radio Astronomy by Wilson et al.) $$ T_B \equiv \frac{c^2}{2 k} \frac{1}{\nu^2} I_\nu $$ even when the Rayleigh-Jeans version of the blackbody formula is not valid for the given temperature and observing frequency. This may not be physically accurate, but has the advantage that intensity and brightness temperature are always linearly related (i.e., a 20% increase in \(T_B\) corresponds to a 20% increase in \(I_\nu\)).

Other, more physics-minded resources will define brightness temperature using the full form of the Planck formula (Physics of the Interstellar and Intergalactic Medium, Draine). $$ I_\nu = \frac{2 h \nu^3}{c^2} \frac{1}{\exp{(h \nu / k T_B)} - 1} $$ such that $$ T_B \equiv \frac{h \nu / k}{\ln[1 + 2 h \nu^3 / c^2 I_\nu].} $$ If the source is in thermal equilibrium, this form has the advantage that \(T_B\) does correspond to a physical temperature. Draine uses the term antenna temperature to refer to brightness temperature arrived at using the Rayleigh-Jeans definition.

Unfortunately, this ambiguity makes communicating using brightness temperature (a great concept!) confusing and error-prone. If there is any ambiguity about whether the Rayleigh-Jeans approximation is valid \(h \nu \ll k T_B \) for all observed regions in your field of view, then you should state which form of the blackbody formula you are using in your figure/article.

• antenna temperature: if you are in a radio astronomy context, you will also come across this terminology, which is functionally the same as the classic radio astronomer’s definition of brightness temperature. Key is that specific intensity is linearly related to antenna temperature and makes it easy to substitute one for the other using the Rayleigh-Jeans approximation $$ T_A(\nu) = \frac{c^2}{2 k \nu^2} I_\nu $$ Typically, antenna temperature is used to describe a measured temperature from the instrument in the broader context of other noise sources.

In the field of radio astronomy, be aware that one frequently combines temperatures in other interesting ways. One can express random noise power in terms of an effective temperature $$ P = k T \Delta \nu $$ where \(\Delta \nu\) is the bandwidth of the observation. Here the power is equal to the noise power delivered to a matched load by a resistor at physical temperature \(T\). By matched load, we mean we connect a resistor to the input terminals of a linear amplifier. The fact that this resistor has some temperature (i.e., we haven’t cooled it to absolute zero…) means that the thermal motion of the electrons will produce a random, variable current \(i(t)\) input to the amplifier. The mean value of this current is zero, but the root mean squared value is non-zero, and this represents a non-zero power. I.e., you can draw (some) power from a resistor at room temperature, purely from thermal motions. The situation is not dissimilar to the random walk of a particle in Brownian motion. For more details, see Tools of Radio Astronomy, Chapter 1.8.

Including noise sources

• Antenna temperature \(T_A\) the component of the power received by the antenna from cosmic sources. It has the same interpretation as before (though we’ll talk about beam dilution in a second).
• Receiver temperature \(T_R\) the component of the power from internal noise of the receiver components themselves, ground radiation, atmospheric emission, etc…
• System temperature \(T_S = T_A + T_R\) is the sum of receiver temperature and antenna temperature. It’s the one power number coming out of the backend of your telescope. It’s up to you to calibrate \(T_R\) accurately enough to measure \(T_A\).

In any observation, you will have your cosmic signal of interest and several contributions of noise (see Essential Radio Astronomy, Ch. 3.6.1) $$ T_S=T_\mathrm{cmb}+T_\mathrm{rsb}+T_A + [1−\exp(−\tau_A)] T_\mathrm{atm}+T_\mathrm{spill}+T_r+ \ldots $$ such as the CMB, other galactic background sources, the atmosphere, spillover radiation from the ground, the temperature of the radiometer itself (hopefully cryogenically cooled), etc.

In the limit that \(T_A \ll T_S\) (most astronomy situations, unfortunately!), we have $$ S/N \approx C \frac{T_A}{T_S} \sqrt{\Delta \nu \Delta t} $$ where \(C\) is a constant of proportionality greater than or equal to 1, and \(\Delta t\) is the integration time. If we let \(\Delta \nu \approx 1\;\mathrm{GHz}\) and \(\Delta t \approx 1\;\mathrm{h}\), then we can get \(\sqrt{\Delta \nu \Delta t} \approx 10^6\), allowing us to detect a signal which is less than \(10^{-6}\) the system noise. A great illustration of this capability is the COBE satellite that studied CMB anisotropies with brightness temperatures \(< 10^{-7}\) that of the system temperature. To achieve these contrasts, however, it’s important to keep systematics under control, otherwise the S/N scaling won’t hold!

Beam dilution

In previous lectures, we’ve been talking about specific intensity \(I_\nu(\Omega)\) as a “known” quantity of direction (e.g., R.A., Dec.) and used antenna temperature \(T_A\) as a linear proxy for its specification. For the following discussion, we’re going to move into the realm of observations, and discuss the ways \(T_A\) can be an unfaithful proxy for the “true” specific intensity distribution or brightness temperature. we’ll use the symbol \(T_b(\Omega)\) to denote the “true” brightness/antenna temperature, assuming we’re in the Rayleigh-Jeans limit, and redefine \(T_A\) to mean the response of the telescope to the cosmic radiation.

When we’re doing observations, we don’t always have access to the highest resolution version of \(T_b(\Omega)\), but rather we have access to a quantity which is the true \(T_b(\Omega)\) convolved with the beam of the telescope, which is the implication of \(T_A\) for this discussion.

large (fully resolved) source

In the limit that we are observing a source that subtends a solid angle much larger than the beam of the antenna, $$ \Omega_S > \Omega_A $$ the convolution of the beam doesn’t matter, we’re still sensing approximately the same \(T_b(\Omega)\) such that $$ T_A(\Omega) \approx T_b(\Omega). $$ If the source is in LTE, then we also have that \(T_A \approx T\).

small (unresolved) source

If the source is more compact than the antenna beam, $$ \Omega_S < \Omega_A, $$ then the measured antenna temperature is basically the “true” intensity averaged over the area of the main beam. This lowers the measured antenna temperature by a factor $$ \frac{T_A}{T_b} = \frac{\Omega_S}{\Omega_A} $$ where the ratio \(\frac{\Omega_S}{\Omega_A}\) is called the beam filling factor.

For example, you could have a compact source with \(T_b = 10^4\) K, but if it only fills 1% of the beam solid angle then you would measure an antenna temperature of 100 K. If you took your observations at face-value (and assumed LTE), then you would incorrectly conclude that the source is 100x cooler than it actually is.

Beam dilution also applies to observations of sources that have structure on spatial scales below the observable limit, which, to be honest, is going to be most astrophysical sources of interest. For example, consider a gas filament in a star-forming region.

Radio-bright, spatially concentrated regions will be “smeared out” by the beam. If you wanted to use antenna temperature (and assume LTE) to estimate the physical conditions of the gas filament, you do so at the peril of measuring incorrect temperatures. The unfortunate reality here is that, without higher resolution images to guide you (which sometimes exist at optical or infrared wavelengths), it’s quite difficult to estimate how badly your measurements are affected by beam dilution.

For more useful single-dish guidance, see these notes by James Jackson, or Ch 3.1.6 of Essential Radio Astronomy.

Introduction to interferometric arrays, ALMA, VLA, SMA, VLBA

Now that we’ve covered some of the fundamentals around radio telescopes and single-dish antennas, we’ll move on to discussing how we combine the signals from multiple antennas to do interferometry. Here are some of he interferometers operating today: