ANITA Lecture - Radio Astronomy and Interferometry Fundamentals – David Wilner

So welcome everybody to today's Anita um lecture. It's with great pleasure that I welcome David Wilner, who is from the Harvard Smithsonian Center for Astrophysics. David is the associate director for the Radio and Geoastronomy Division, and he's visiting Swinburne University at the moment. And it was, uh, wonderful to be able to get him to give our first Anita lecture of 2014, uh, 2015, sorry. The year has progressed.

Um, just want to let you know that we have a great year coming up, and we'll have a lot of great speakers, and we'll advertise, uh, those in the meantime. Um, but until we do that, I'm going to hand over to David now, and he's going to tell us, uh, give us the first of two, two-lecture series. The next one will be next week. First one in radio astronomy and interferometry. Interferometry fundamental. So I'll hand it over to you, David.

Great. Thank you, Darren. Uh, pleased to give, uh, these two lectures here as part of the Anita program. The, these, this two-lecture series will be about radio astronomy and interferometry. The first lecture concentrates on, uh, radio astronomy and interferometry fundamentals. Um, so this will be a general overview of radio astronomy and interferometry. And the second lecture, we'll focus on a more detailed, uh, discussion of imaging and deconvolution in radio interferometry.

Um, the, uh, outline of this first talk, um, will be a brief introduction to radio astronomy and radiation mechanisms at radio wavelengths, a discussion of single antenna telescopes and synthesis array telescopes, and then a more in-depth discussion of interferometry, visibilities, and, uh, Fourier transforms. So this is that line for today's lecture.

Um, my intent in, in these lectures is not, uh, to be entirely rigorous, but to give you, uh, fundamental grounding in the most important concepts in these areas, and also to allow you to develop some intuition about some of the more subtle aspects, uh, particularly of interferometry. If you're looking for rigor, I'm going to give you a list of references where you can get as much rigor as you desire. So let's go into that.

Just before we begin, I'd like to highly recommend, uh, Essential Radio Astronomy. This is a complete one-semester course, uh, developed by Jim Condon and Scott Ransom at the National Radio Astronomy Observatories in the US. All of the course notes are online. It's very detailed, very descriptive, and, and just a fantastic resource.

Um, for interferometry, there is a, uh, reference, Thompson, Moran, and Swinson, called Interferometry and Synthesis in Radio Astronomy. This is a, a difficult, uh, book to wade through, but if there's any topic that you're interested in and you need some fundamental derivation, it is surely in there. And so that is where you should go, um, for that rigor.

There are also wonderful summer school lectures, um, available on the web. Um, in particular, the NRAO Synthesis Imaging Workshop proceedings, um, and the, uh, VLA interferometry school proceedings. Some of my slides, in fact, are from my own lecture at the NRAO Synthesis Workshop, and you can go there and see, uh, see more about that. And you can find many other, uh, useful pedagogical presentations about these topics online. The ATNF here in Australia, um, the ALMA primer, and, uh, and many others.

All right. So radio astronomy, what's special about radio wavelengths? Well, I'm sure we're all familiar with the electromagnetic spectrum, light described by its wavelength, going from very short wavelengths, gamma rays, X-rays, ultraviolet, through the visible part of the spectrum, to the infrared, the radio, um, and, and the very longest radio waves. So radio wavelengths, kind of loosely defined, are, are wavelengths from a few hundred microns, um, and beyond, out to many meters, or frequencies from a terahertz, um, or, or longer.

And what's special about radio wavelengths, as you can see in this chart, is that like visible light, radio waves are observable from the surface of the Earth. So they're very important for radio astronomy because we can build large instruments on the surface of the Earth to detect this electromagnetic radiation from cosmic sources. Um, moreover, it's a very wide spectral window. Um, it covers, you know, several decades in wavelength. It's much wider than the optical, in fact. And so there are a number of very important emission processes that come and go in various parts of the spectrum that I'll describe in a moment.

A brief history. Um, like most aspects of astronomy, radio astronomy was a serendipitous discovery. Karl Jansky, um, was employed by the Bell Telephone Company, and he was investigating in the early 1930s, uh, noise in shortwave communications across the Atlantic Ocean. And with his antenna, pictured here in New Jersey, um, he discovered, uh, some noise. Um, and it was a periodic signal that he was able to locate with this rotating antenna, um, to a specific, um, part of the sky. He thought it might have something to do with the sun, um, but, uh, after discussing his discovery with an astronomer, he realized that the periodicity of his signal was not 24 hours, but 23 hours and 56 minutes, the sidereal rate. It was something that was actually fixed in the distant sky and not the sun. And it turned out to be the area of the galactic center.

And this made headlines around the world. I have a little clip here from The New York Times, "New Radio Waves Traced to the Center of the Milky Way." "Direction is unchanging." And my favorite comment, "Its intensity is low." That's something that radio astronomers have been dealing with ever since.

Jansky is immortalized with a unit of flux density, 10 to the minus 26 watts per square meter per hertz. Um, and this discovery opened an entirely new window on, on cosmic sources and phenomena that are either difficult or completely impossible to detect at any other wavelengths. And so a real new window on the universe. Um, just as a postscript, this is all Jansky did in radio astronomy. He was then assigned to other topics at Bell Labs and never returned to it.

Okay. So I'm going to go through a series of the most important emission mechanisms that we see at radio wavelengths. Um, one of the most important is synchrotron radiation. So this is a, a continuum process from charged particles that spiral along magnetic field lines. So say, when an electron is spiraling along a magnetic field line, it accelerates. That creates electromagnetic radiation from Maxwell's equations. And this, uh, radiation tells you something about the plasma. The spectral index, that is, the variation of the strength of the emission with frequency, can tell you something about the distribution of the electron energies. And, um, the polarization properties of the radiation can tell you something about the magnetic field direction, because there is a preferred direction of the electrons spiraling. Unfortunately, it looks like we lost the beautiful picture of Cygnus A on this slide, which is too bad.

Okay. Um, let's go on to the next radiation mechanism, Bremsstrahlung, or braking radiation. This is another continuum process from electrons accelerated in a plasma. Only in this case, the electrons are, are encountering protons, and they're, they're being braked. And so that acceleration creates some electromagnetic radiation. This beautiful picture here is a star-forming region at 3.6 cm wavelength with the Very Large Array. All of these little peaks here are sources of plasma, ionized gas, which are being ionized by an embedded hot star, which has enough ultraviolet radiation to knock the electrons off the, the protons. And from the intensity of this, uh, of this emission, you can determine the mass of the ionized gas, the density of the electrons, and indeed the rate of the ionizing photons. So something about the embedded stars, which are completely invisible, uh, at really any other wavelength.

Another very important process, particularly at the shorter radio wavelengths, is dust emission. Another continuum process where fluctuations in the charges on grains produce, uh, essentially a form of, of thermal emission, a modified black body. This is most important in the radio for cold dust, T to 100 Kelvin. And in this case, the emission is proportional to the dust mass times the temperature of the dust. So one can learn something about the properties of the material, and the spectrum of the dusty emission, the change as a function of wavelength, can tell you something about the grain properties, in particular, the distribution of the grain sizes. Here on the right is a, a recent, um, image of dust emission from an edge-on disk around a nearby star. This is sort of a Kuiper Belt around the star AU Microscopii, imaged in millimeter dust emission. And on the left is a NASA artist's conception of what the system looks like.

All right. Equally, if not more important than these continuum emission processes that I've discussed, are emission from spectral lines. So these are discrete, low-energy transitions from atoms and molecules. So, um, two of the most prominent spectral lines in the radio band that, uh, are observed, I have illustrated on the right. One is the atomic hydrogen spin-flip transition at 21 cm. Very famous. You can see in this, uh, side-by-side comparison of the M81 group that in the optical, one can see the two galaxies, you know, very nicely. But in the radio, in this 21 cm line, which is showing you the neutral hydrogen, the, the galaxies show these incredible tidal tails, and you can trace back how they've interacted in the future, which is very difficult, if not impossible, to do from optical starlight alone.

A second molecular line that's also, a second spectral line that's very important is the molecular line of carbon monoxide, CO. CO is found in regions with molecular hydrogen. Molecular hydrogen, when it's cold, has no spectral lines that emit, so it's very difficult to see. But collisions with the molecular hydrogen and CO excite the CO line, and so it's a way to trace regions of interstellar medium that are molecular. And so with probes like this, one can measure things like the gas chemical composition from different transitions of, of atoms and molecules, get at temperatures and densities and other physical properties. Extremely important. Um, one can measure the Doppler shift of these discrete transitions, which gives information about line-of-sight velocities and kinematics. So you actually have a three-dimensional view of the M81 group in the sense that there's another dimension of velocity here. And for some particular transitions, one can measure the Zeeman effect, which gives information on the magnetic field strength, another physical property that's very difficult to get at any other way.

All right. I'll make a quick remark on, on units in radio astronomy because I, I have found in my experience that people find this very confusing. Um, intensity is an often used unit, um, uh, quantity, I mean, and its units are watts per square meter, you know, energy per steradian per hertz. So this intensity is effectively a brightness, and it's independent of source distance. So if you imagine looking at, say, the surface of the sun, if you resolve a little piece of that surface, it has some kind of intensity, and that's true no matter how far away it is. It may be 5,800 degrees, no matter, you know, how far away it is.

Um, radio astronomers use a kind of peculiar unit called Rayleigh-Jeans brightness temperature, which converts the intensity into a temperature using the Rayleigh-Jeans approximation. And so for thermal emission processes, this brightness temperature is just the temperature of the emitting body. So we were looking at dust emission before, um, it might have a temperature of 10K or something like that. Um, for other cases, particularly all of those non-thermal processes I mentioned, like synchrotron emission, um, radio astronomers still talk about brightness temperature, which you can determine with this formula. That's the temperature a black body would need to have to be that bright, but it doesn't correspond to any physical temperature. So that can be confusing.

And then finally, the most important, uh, unit is probably flux density, and this is just the integral of intensity over some solid angle. So density decreases with source distance squared, just as solid angle does. Okay. So if you're worried about units, um, and you get confused, you might refer back to this slide.

All right. What about a radio telescope? So this is a very abstract view of a radio telescope on this slide, and I'll come to a more concrete example in a moment. But if you just consider a distant source, uh, of emission, which has some intensity I in some direction, then the power from this emission passes through some kind of sensor, some radio sensor, and that will increment, um, power in, in the sensor. It's the intensity times some differential frequency and area and solid angle. So power in watts. Um, and so the power that you collect from the intensity of this distant source is a suitable integral, right, over these, uh, these, uh, quantities of frequency, area, and angle. And if you want to understand what that is, then for your telescope, of course, you have to understand what the response of the telescope is in all these areas, how much collecting area it has, what the shape of the frequency, um, sensitivity is, and, and, and whatnot.

So here's a real radio telescope. A very typical one. This one kind of looks like, like an optical telescope in a way. It has a parabolic primary mirror, and that is a focusing paraboloid with high gain in some direction that, um, collects all the radio waves and sends them to a focus. It's steerable, so it can move in two directions and track sources across the sky. It has a secondary mirror, which in radio astronomy is often called a subreflector, which is in front of the focus, and it sends the radio waves through the hole in the middle of the dish to a receiver that's located behind the primary, typically in a, a spacious or relatively spacious cabin where one can put a lot of equipment.

Um, this particular telescope, you can see, is on-axis in the sense that the antenna axis is the same as the optical axis. It's symmetric. It need not be that way. And the receiver in the back of this telescope is a little waveguide horn, which is just a little piece of metal which effectively collects the electromagnetic radiation and sends it to an amplifier, which in this case is cryogenically cooled, so that it reduces any background noise. Um, and this is kind of the general scheme of most radio telescopes, but many, many variations are possible.

Um, so many variations that I don't even want to start to get into it, but I'll just describe one, um, because it's kind of a fascinating one. You may have seen it before. It's, uh, currently the largest radio telescope in the world, the Arecibo radio telescope in Puerto Rico. This is a 300-meter diameter spherical primary, so it's not a paraboloid, which means it focuses to a line and not to a point. So that's an interesting thing to deal with at the receiver level. Um, it's made of an incredible number of perforated aluminum panels, more than 38,000 of them, and, and it can reflect radiation as short as a few centimeters. Now, it reflects the radiation up, um, to a receiver platform, which is suspended 150 meters above the dish by cables, and that can move around to, uh, to track the source over a 40-degree cone of the sky around the zenith, straight up. So you can see that, you know, this is, uh, got all the same elements of that abstract radio telescope that I mentioned, um, but it's a very different sort of implementation. And there are many, many other variations which don't even use dishes but use dipoles, um, on the ground, and, and whatnot, that, uh, are, are even completely different.

Okay. Now, that's a very big telescope, but there's a big issue in radio astronomy, and that is angular resolution. So I'm sure you're all familiar with Rayleigh's criteria that, um, the diffraction limit of a telescope is given by the number of wavelengths divided by the diameter of the telescope, um, in units of radians. So if you think of your eye as a telescope, you're observing in the optical at, say, half a micron, and the diameter of your pupil is a couple of millimeters, then the resolution of your eye is something like, it's about 50 arcseconds, about one arcminute. And if you think about, say, looking at the moon with your eye, you can, you know, start to resolve some small features, but they're, they're limited.

If you think of a really sharp optical telescope, like the Hubble Space Telescope, it's observing at the same wavelength as your eye, half a micron, but instead of being two millimeters in diameter, it's two meters in diameter. And so it's a hundred times better in resolving power, down to 50 milliarcseconds. So that's really pretty good. And that's great for astronomy because a lot of things are small.

Now, how about that Arecibo radio telescope? So suppose you're observing at six centimeters, and now the effective area that you can use that actually reflects up to the, to the receiver is something like 200 meters. The resolution of that is 60 arcseconds. It's just about the same as your eye. So, um, that's about as big a radio telescope as you can imagine making, and its resolution is no better than your eye. So if you want arcsecond or better resolution at radio wavelengths, you need kilometer-size telescopes. And, uh, that was realized early on, but it's a big challenge because it's hard to make a single telescope that's a kilometer or many kilometer size. And that's why synthesis telescopes came to be.

Um, I should mention that the radio jargon for the, the resolution of a single element, um, like, like the Arecibo telescope, is the primary beam. So you'd say the primary beam size is 60 arcseconds for Arecibo.

So synthesis telescopes. Well, interferometry is a technique, um, to use distributed small apertures, use lots of little telescopes to synthesize a larger telescope, a larger aperture. So I have two examples here on this slide. On the left is the Very Large Array. This is a very famous radio telescope located in New Mexico in the United States. It operates at wavelengths from a few meters down to a little bit less than a centimeter, and it, uh, can extend to resolve to, to, to be 35 kilometers in diameter. So if you imagine you're observing at 21 centimeters, that was the wavelength of the, the hydrogen spin-flip line, then you could make images with a resolution of, of 1.4 arcseconds. So now you're getting down to an interesting amount of resolution, but as you can see, you need a telescope that's 35 kilometers across. That's pretty big.

And the newest, uh, of these synthesis telescopes, these large synthesis telescopes, is ALMA, which operates in the millimeter and submillimeter part of the spectrum. It has a maximum size of 15 kilometers, and at the shortest wavelength, that's down to five milliarcseconds, actually an order of magnitude already better than the Hubble Space Telescope, if you're interested in, in certain kinds of small sources.

Now, the way synthesis telescopes work is not exactly the same as the way single-element telescopes work, um, because obviously you're not capturing all of the same information when you only have a small fragment of the, of, of the telescope, um, aperture. So indeed, what happens here is that you're inferring properties of the source in the sky from certain characteristics of the received electric field that are not just the intensity. Um, so this is quite a bit less intuitive than direct imaging, but I want to give you a feel for that.

Oh, I have one more example. Example first. The Very Long Baseline Array. This is, uh, this is a fascinating telescope. So it has a maximum size of the size of the Earth, 8,000 kilometers. And so if you're observing at a wavelength of 1 centimeter, now the resolution is already 0.2 milliarcseconds, which is quite incredible. Now, for this to work, you can imagine you can't actually physically connect all of these telescopes, but you can record the information at each telescope and then combine it later to achieve this resolution, um, and infer certain properties of the sources.

All right. So now we're going to transition and talk in more detail about interferometry. Here's our schematic two-element interferometer. So you can imagine each one of these telescopes, one and two, are abstract radio telescopes that do some sort of, uh, collection of the radio waves and detect them. So these telescopes are pointing at a source, um, at a, at a direction which is labeled by this vector S0, which is at an angle theta from the zenith. And the way that the two-element interferometer works is that the telescope on the left, you can see the signal gets there a little bit later than the telescope at the right. There is a time delay, which is labeled on the figure as B sin theta over C. And so what one does is one delays the signal, um, from telescope 2, in this case, that's the tau that's listed there, and you multiply the signals and integrate them. And what you measure, and I'm sorry that my font was converted here, is something called coherence, um, of the signals. So what the interferometer measures is not intensity directly, but coherence.

We haven't really had too many equations in this talk. Here's the one slide that has an important equation. All right. So what is this coherence? Um, I want to give you some feel for that. Um, there's something that radio astronomers call the visibility. It's the complex visibility function. And what this is effectively is the two-dimensional Fourier transform of what the emission is on the sky. And if you want a derivation of where this comes from for the two-element interferometer, one can look at that, uh, that reference I mentioned, Thompson, Moran, and Swinson, in chapter 14. There are a few different ways one can derive this. I think this is one of the more transparent ones, but it still takes several pages of equations, and so I'm not going to go into that for this talk. Um, there are limitations as to when this applies and when it doesn't.

But if you look at the figure on the right, this, this describes the basic geometry. And I can explain the equation that comes on the left mathematically. So our two-element interferometer, there is on the ground, and there are two coordinates, U and V. U points East, and V points North. And you measure the separation of these telescopes on the ground in terms of number of wavelengths in U and V. So they're the East-West and North-South spatial frequencies, they're called, measured in wavelengths on the ground. Then if you go to the sky plane, there's an intensity distribution, which is called T_lm on this figure. Now, L and M are angles on the sky, and so they're the East-West and North-South angles in the plane that's tangent to the curved sky.

Now, what is this visibility? So this visibility as a function of U and V on the ground is this double integral of T_lm on the sky times this exponential factor, e to the 2 pi i (UL + VM), then integrated over these angles. So if you remember this e to the 2 pi i times these other factors is just a series of sines and cosines. e to the iX is cos x + i sin x. I'm sure you remember that. So this visibility is the sky brightness multiplied by a bunch of signs and cosines and then integrated. And what that actually is in detail is a two-dimensional Fourier transform. And at the bottom of the slide, I have a shorthand notation for this equation that V_uv is this Fourier transform, that little arrow with the F of T_lm. And one of the properties of the Fourier transform that I'll describe in a moment is that it also works the other way. The sky brightness distribution is the Fourier transform of the visibilities. So this interferometer is measuring coherence, it's measuring the visibility function, and so for a particular separation of antennas at a point U and V, one gets some measure of the Fourier transform of the sky brightness T_lm.

All right. So you can see Fourier transforms are important. So I want to spend a few minutes discussing in more detail the Fourier transform. So I'm sure some of you know everything about Fourier transforms, some of you may know nothing. So hopefully for those that know something, this will be a good review.

Basically, Fourier theory states that any well-behaved signal, that includes images, can be expressed as the sum of sinusoids. And here's a picture of, of our hero, Jean-Baptiste Joseph Fourier, Frenchman from the 18th century. If you imagine a signal like a square wave, you can represent a square wave, kind of, by four sinusoids. If you sum them up appropriately, you get something that kind of looks like a square wave. And in fact, you may know from your Fourier analysis classes that you can approximate the square wave by an infinite number of, uh, sine waves. It's just that to get those sharp edges, you need a lot of little tiny waves, so you need a lot of terms in your function. But the Fourier transform, at its heart, is just a mathematical tool that decomposes a signal, any signal, into a sinusoidal component. And what's important to remember is that the Fourier transform of a function of a signal contains all of the information of the original signal. So if you, if you know the Fourier transform, then you actually know the original.

Okay. So if you want to do radio astronomy, and in particular interferometry, you really want to acquire some comfort with the Fourier domain. If you look at old textbooks, functions and their Fourier transforms are said to occupy upper and lower domains, as if, and this quote is from Bracewell, it's one of my favorites, "Functions circulated at ground level and their transforms in the underworld." And you can decide whether it's the visibility or the sky brightness that's in the underworld at the end of the lecture.

So some properties of the Fourier transform are that, um, it's linear. So I've written here, the small g is the Fourier transform of the large G. So if you have two functions, say G and H, if you add those two functions together and get a new function, the Fourier transform is just the addition of the Fourier transform of the individual functions. So they simply add. So that's useful.

There's also a scaling property. If you multiply the coordinate by some, some value alpha here, you scale it, then the Fourier transform is also scaled, but by the inverse of that alpha. There's a shift theorem. If you shift the coordinate, then you just add a phase term to the Fourier transform. You just change the phase of the sinusoids. And there's a convolution multiplication theorem that is, if a function is the convolution of two functions, then its Fourier transform is the product of those, of the Fourier transforms of those functions. And finally, I'll just mention, uh, one version of the sampling theorem. If you have a function that's restricted to a domain, say theta, it's completely determined if the Fourier transform is sampled at intervals of one over theta. So you don't need to sample every part of it. You just need to, to satisfy the sampling theorem to completely determine the function. So those are all mathematical properties of the Fourier transform that are useful to keep in mind. And in some examples that are coming up, you'll see how these, uh, how these come into play.

Okay. Another word about visibilities. Um, as I mentioned, each visibility, so at some UV point, contains information on the sky brightness everywhere, right? Because, because of the sinusoidal nature of the, uh, of the phase term, each point in the UV plane, as it's called in the visibility domain, has information on the sky brightness everywhere in that domain, right? Not just at one place in, in the sky brightness distribution or within a particular region, but everywhere. The other thing is that you remember, each visibility is a complex quantity. So it's often expressed as a real and imaginary part, um, as a real part and an imaginary part, or as an amplitude and a phase. Either one works.

So here's an example of a Fourier transform of an image. We have here an astronomer, you know, represented as sky brightness T of LM. If you Fourier transform that image, you get information, visibility at a whole bunch of UV points. You have U and V, and you have phase. So you see that's, uh, one way to do it. Or it could be real and imaginary. In this case, we've chosen to use amplitude and phase.

So I'm going to go through a few examples of, of Fourier transforms to give you a little more intuition about them. Um, a delta function is kind of the narrowest, uh, function you can have. So if you imagine sky brightness that's really peaky, a delta function, if you Fourier transform it, it's a, a constant. Right? It has the same visibility amplitude everywhere. A Gaussian, um, transforms into a Gaussian. And a narrow Gaussian transforms into a wide Gaussian. So narrow features transform into wide features, and vice versa. So if you see something that's narrow in one domain, it's going to be broad in the other domain. So here's an elliptical Gaussian. You can see it's, uh, narrow in one dimension and broad in the other. And if you look at its Fourier transform, it's broad in the opposite dimension and narrow in the other.

Okay. Here's a uniform disk. When you Fourier transform that, you get a Bessel function. So hopefully you can see the series of rings that come around from the Bessel function. And, you know, what does this mean? The, uh, the disk has a sharp edge, right? And so if you want to represent that sharp edge, it means you need a lot of high spatial frequencies. You need values, um, in the Fourier transform out at large U and V, in order to represent that sharp edge. So this is analogous to that square wave that I showed you earlier. Um, if you really want to get the sharp edge, then you need, excuse me, you need all of, of, of the, uh, visibility plane.

Here's another example. You might think about what's the Fourier transform of this look like. [Music] Think about that for a minute. Here's the answer. It looks a lot like the Bessel function above. So why is that? That's because the spatial frequencies you need to make three disks are pretty much the same as the spatial frequencies you need to make one. And it's the phase information that I'm not showing here that's telling you where that stuff is located. So this is, um, I think perhaps the biggest trap in Fourier transforms and radio interferometry is understanding the, the nature of amplitude and phase. And so try to give you a little bit of an example here. Amplitude tells you how much of a certain spatial frequency you need. So here's our, our little Gaussian on the left. It Fourier transforms into a big Gaussian. And the phase is constant. Why is the phase constant? It's telling you where these spatial frequencies are located. Our little Gaussian is in the center of this image, and so the phase information is kind of not, not very interesting.

If I shift the source over a little bit away from the center, then the amplitudes of the visibility are exactly the same, right? You need the same sine waves to make the small Gaussian, whether it's shifted or not. But the phase is different. Oh, great. And unfortunately, the slide isn't showing that, but you can imagine that the phase is different. And if I were to move the source in a different direction and do the Fourier transform again, the amplitudes are the same, and the phase that you can't see is different. Okay, have to fix that.

All right. So let's review a little bit this concept of visibilities. This is the one equation. Remember the visibility as a function of U and V, so those are spatial frequencies on the ground, measured in wavelengths, is this double integral over sine waves of the sky brightness distribution, um, where L and M are angles on the sky. So there are a few things to notice. One is, if you put zero in for U and V, right, so you're at the origin of the UV plane, then what you get is just the integral of the sky brightness distribution. And in radio astronomy, that's just the total flux density. So the visibility, right, at the center of the UV plane is just the total flux density.

Another thing to notice is that, um, because sky brightness is real, right, it's not imaginary, um, if you put in minus U and minus V, then you get the conjugate version of V_uv. So V, the visibility, is said to be mathematically Hermitian. And that means you get two visibilities for one measurement. That is, if you make a measurement at U and V, you also get an answer at minus U, minus V. So that's, uh, convenient for radio astronomy.

All right. So now this little illustration is to kind of show you what this equation means. So I've said it in words a number of times, I'll say it again. The visibility is the sky brightness multiplied by a bunch of sound wave, sine waves, and then integrated. So this little smiley face here is going to be our sky brightness distribution, our T of LM. Now, if we have an interferometer, then it effectively puts a sine wave on the sky, a fringe pattern, as it's called, and it multiplies the sky brightness distribution by this fringe pattern. So you can see the pattern of light and dark is plus and minus of the sine wave. And so you do this multiplication and you get a number, and that's the visibility. So that's all there is to it.

If we separate our antennas a little farther on the ground, then we get a different set of sine waves. We get a higher spatial frequency, right, because in this case, we've made U bigger, so we have a faster sine wave. And you can see if you multiply this sine wave by the little smiley face, you get a number. And in fact, that number is essentially the same as the number from the previous set of antennas, right? And then if we do it again, this time we turn the antennas, um, a little bit, so we have an angle. Now, the fringe pattern is rotated. And if we do the same multiplication of the sky brightness by the sine wave, we get a number again. It's almost the same number, if not exactly the same number. So what is that telling you? That means any separation of the antennas is giving you the same visibility. That's telling you that the source is small, right? That it's a point source. Remember the Fourier transform of a delta function is a constant. And so this is telling you that there's not much information besides that the source is smaller than the resolution of your antenna baselines.

So by contrast, here's a bigger source, another happy source, a bigger one. If we start with the antennas close together, we get the broad sine wave. If we do this multiplication, you can see we get a number. But you can see part of the source is on the negative part of the wave, and part of it's on the positive part, and part of it's on the negative part. So if we do this multiplication and integrate, then we get a number which is smaller than the total flux. That's to say, the source is resolved.

If we move the antennas farther apart, and we do the same exercise, now we have a bunch of sine waves across the source. So we do the multiplication and we get a number that's now, you know, probably small, except for the little variations on the source like the eyes and the smile. And if we turn, uh, to a different baseline, and we do this exercise again, we're going to get a different number. And a lot of the difference in this case is going to be due to the fact that there's this small-scale structure on the source, the eyes and the mouth. And so that's the name of the game here is that if you can make enough different measurements of the visibility, then you can learn about the structure of the sky brightness. If you made enough different antenna pair measurements, then you could reconstruct the information that you have a smiley face on the sky.

So that's aperture synthesis, right? The basic idea is to sample the visibility function at enough UV points using these distributed small aperture antennas to synthesize the resolution of a large aperture, which goes out to the maximum U and V that you can measure. And as I said, one pair of antennas, one baseline, gives you two samples because the visibility is Hermitian. At a time, so two samples is not very many if you want to build up all that information. If you had to do it one pair of antennas at a time, it would be pretty painful.

Um, so if you have N antennas, well, then you have N times N minus one pairs, so you get N times N minus one on samples at a time. And if you're really clever, you'll note that the Earth is turning whether you want it to or not, and so your U and V on the ground are changing whether you want them or not for fixed antennas. And so Earth rotation actually tends to fill in the UV plane, um, no matter what you do. And in fact, the, uh, the realization that one could take advantage of this and make aperture synthesis measurements, um, using a number of antennas and Earth rotation, was a big part of the, the 1974 Nobel Prize in Physics. Sir Martin Ryle, British, had to balance the, uh, the French from earlier, 1974.

So if you only have N antennas, then you probably have to move them around to get more samples. That is, you can reconfigure their physical layout. And another thing you can do, if your source emission is the same at a bunch of wavelengths, which you know, maybe it is, maybe it isn't, you'd have to know, um, you can observe at multiple wavelengths at once, um, because remember U and V are measured in wavelengths. So if you were measuring, if you were observing at multiple wavelengths, you can cover different parts of the UV plane at the same time. And so this is called multifrequency synthesis. But it, it only works if you understand the source spectrum and you can characterize it.

Now, your source is varying while you're making different measurements of U and V, of visibility, then you're in trouble in terms of aperture synthesis. So one of the assumptions is that while you're collecting all these samples of the Fourier transform, the source isn't changing.

Here are a few examples of aperture synthesis telescopes, in particular ones that that work at millimeter wavelengths. I already showed you the VLA and and ALMA. Here on the left, one of my favorite telescopes is the Submillimeter Array that our institution helped build up on Mauna Kea. The Australia Telescope Compact Array is a synthesis telescope, the IRAM Plateau de Bure in Grenoble, and the CARMA array in California. You can see they're all different configurations of, of multiple antennas, all of which can be moved around to collect visibilities and ultimately to make images of celestial objects.

All right. So let me give you an example of UV sampling. This is from the Submillimeter Array. So this picture here on the left are the 24 stations that one can put, each of the eight antennas of the SMA. Um, in this case, we're only using six of them at a frequency of 345 gigahertz at a declination of 22 degrees. And we put those six antennas in those places, and if we let the Earth turn, then this is the sampling of, of the UV plane that one gets out of this particular configuration for this source at this frequency. So you can see the span is about a thousand kilo-lambda, thousand, thousand wavelengths in both U and V. And, uh, what else to say about this? There are lots of little gaps in these elliptical tracks. That's when you're not looking at this target source of interest, but you're looking at a calibrator. Um, and there are some longer gaps, maybe when something went wrong. You could then move the antennas around.

So here's seven antennas of, of the array at different stations, and the UV coverage looks like this. So now the antennas are closer together, so U and V are smaller. And then here's another configuration, seven antennas moved into an even more compact configuration, and that's the coverage you get out of that. And then you can combine them all. Right. And these three observations, each of which runs for some eight hours, gives you this coverage of the UV plane. So all of these different samples of the Fourier transform of the source that we were interested in.

So what are the implications of this kind of coverage? Right? The samples of the visibility function are limited entirely by the number of antennas you're using and by the geometry of the Earth and the target position in the sky. So one typically has a UV coverage like this, in the sense that the properties are, there's an outer boundary. So there's no information on size scales that are smaller than what's sampled by the outer boundary. So this corresponds to the resolution limit. So this is, you know, the size of our telescope. It's, um, 500,000 wavelengths in, in every direction.

Interferometry also usually ends up with an inner hole, right, in the, the center there of the UV coverage. That you can only put the antennas so close together in an interferometer. So that means there's no information on larger scales than that, and extended structures that are too big are completely invisible. That's not something that we tend to have a lot of intuition about, um, but it's a very important feature. And then in between this inner hole and the outer boundary, we have irregular coverage based on where we could put the antennas and, and where, you know, how the Earth was rotating, where the source was in the sky. And so typically the sampling theorem is violated, and information is missing.

So all this sounds terrible, um, but it's not so bad. And the Imaging and Deconvolution lecture will talk about, well, how you can recover from some of these problems. Just to give you a little more intuition about these two issues, here's our, our Fourier transform of an astronomer again. Remember the amplitude and phase. If we were to, um, get rid of all the high spatial frequencies, right, just apply this mask so we only measured the ones in the middle there, then we just get a blurry version of the astronomer, right? Our resolution is limited. So just like observing with a small telescope instead of a big one, or with your eyeball instead of the Hubble Space Telescope, things are blurrier.

But the opposite is also true. If we were to lose the low spatial frequencies, the ones in the middle, the Fourier transform of that is a highly spatially filtered version of the astronomer, right? We only see the edges, right? The, the total flux is gone because that was in the middle, right? Total flux density gone, but you pick up all the small-scale features. So maybe like the, uh, the smiley face in my example, but in this case, an astronomer's smiley face.

So Fourier transforms are great. They're fundamental to interferometry. And if you're not familiar with XKCD, I hope you will be. You can take a look at this cartoon at your leisure. But if you accidentally Fourier transform your cat, you're in trouble.

And to summarize, uh, what have I talked about today? I've talked a little bit about radio astronomy. The things to remember are that radio wavelengths from a few hundred microns to tens of meters reach the ground. Um, there are many different kinds of celestial sources and radiation mechanisms in this wavelength range, uh, including, I described synchrotron, Bremsstrahlung, dust emission, atomic and molecular spectral lines. And all these give windows on the universe that are not available, say, in the optical, which is the only other wavelength regime that reaches the ground. And if you want high angular resolution at radio wavelengths, well, then you need interferometry. And what interferometry does is it samples visibilities that are related to the sky brightness by the Fourier transform. And I encourage you to acquire some comfort with the Fourier domain so that you can learn a little bit more about interferometry. And that'll be the subject of lecture two, um, coming up. Thank you.

Okay. So Darren here again. I just want to say once more, thanks to David for giving this lecture. Um, we're following up next week with the second one on Imaging and Deconvolution, uh, in radio interferometry. That will be slightly more advanced, but still contain a lot of great material for students and people working in this field to use as a reference, or if you just want to bring yourself back up to speed. So watch out for that. For people who are, are watching this, uh, on YouTube, um, in the future, just click through to that next lecture. It'll be available. Um, and just to emphasize that all the Anita lectures go online and they're available to you to watch, um, at your leisure, and to direct your students or colleagues towards. So go to the Anita web page, you can find the links to that, and I encourage you to do so. One of the reasons why we're hosting, or one of the main reasons we're hosting these lectures, is so that we can build up a resource, um, that can be used by students and professional astronomers and interested amateur astronomers. So with that, thanks again, David, and we'll call this, I'll call this lecture to a close. Thank you very much for listening in.