RADIO INTERFEROMETRY AT X BAND February 28, 2017

Contents

1 GOALS 2 2 SCHEDULE 2 3 USEFUL IDL PROCEDURES FOR PRECESSION, FINDING SUN AND MOON,

AND RUNNING THE INTERFEROMETER 3 4 OUR INTERFEROMETER 4 5 CONTINUUM ‘POINT’ SOURCES 4 6 THE SUN AND MOON 6 7 SOME ASTROPHYSICS 6 8 MEASURING B cos δ 7

8.1 Getting the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8.2 The Fringe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 8.3 The Local Fringe Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 8.4 Least-Squares Fitting the Fringe to the baselines and/or the Declination . . . . . . . 10

8.4.1 The Brute-Force Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8.4.2 The Proper Nonlinear Least-Squares Technique . . . . . . . . . . . . . . . . . 11

9 MEASURING 1-D BRIGHTNESS DISTRIBUTIONS: THE FIRST STEP OF

MAKING MAPS 11 10 MEASURING THE DIAMETER OF A CIRCULAR SOURCE 16

10.1 Theory and Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

10.2 Important Mathematical and ¡¡¡PRACTICAL!!! Point! . . . . . . . . . . . . . . . . . 16

11 REFERENCE READING on INTERFEROMETRY AND APERTURE SYN-THESIS 17

The purpose of this lab is, broadly speaking, to learn radio interferometry—the basis for most of modern radio astronomy and, we predict, forefront future IR and optical astronomy. We’ll cover the basic principles, the interferometric fringe, the response to a point source, and the response to an extended source (which is the basis of high-angular resolution mapping with interferometry). We will employ least-squares fitting and Fourier transforms to measure accurate positions for a few sources and also accurate angular diameters for the Sun and Moon. This lab runs for four weeks and clearly there’s a lot of new material!

1. GOALS

• Learn how diffraction theory applies to a real radio interferometer. Fringes, their amplitudes and phases. The fringe as a ‘giant sine wave in the sky; the fringe as a ‘giant DSB mixer in the sky’; and the fringe as the autocorrelation function of the interferometer pair.

• Fourier transforming the fringe; Fourier filtering the fringe.

• Obtain horizon-to-horizon data on our various astronomical sources—point sources, and the Sun and Moon.

• VLBI nonlinear least-squares fringe fitting to determine the most accurate source declination and/or baseline projected lengths.

• Use nonlinear least-squares fitting to obtain accurate diameters for the Sun and Moon. This illustrates the general Fourier transform relationship between interferometer response and sky brightness.

2. SCHEDULE

1. Week 1 (1 Mar-7 Mar): Each group uses rotation matrices to calculate when objects of interest are up. It observes the Sun for a short time to confirm that it sees fringes. Each group does a horizon-to-horizon observation of a point source, which requires writing an observing script to run automatically. Each person calculates Fourier power spectra of the Sun data and also the point-source data. She/He calculates the range of expected local fringe frequencies (equation 14) and compares the observed spectra with these expectations. Be ready for show and tell!

2. Week 2 (8 Mar-14 Mar): Each group observes horizon-to-horizon the Sun and, if possible, the Moon with the interferometer. Each person derives accurate declinations for the point-source data using least-squares fitting as outlined in §8.

3. Week 3 (15 Mar-28 Mar): Finish the Moon observations. Spring break. Each person derives accurate angular diameters for the Sun and Moon using least-squares fitting as outlined in §9.

4. Week 4 (28 Mar-4 Apr): Each person finishes all calculations and writes (and hands in!) the lab report.

3. USEFUL IDL PROCEDURES FOR PRECESSION, FINDING SUN AND MOON, AND RUNNING THE INTERFEROMETER

STOW points the telescopes to the zenith. This minimizes wind forces to
help keep the telescope from being destroyed by strong winds. When
no-one is using the telescope, it should be stowed.  Please stow the
telescope when you are not using it!
SERVICE points the telescope to 'service position' to facilitate working
on the feed. alt=20, az=180.
GETALTAZ prints the telescopes' current alt and az.
PRECESS, ra, dec, equinox1, equinox2. Both RA and DEC must be in
DEGREES; see the documentation. This procedurre is from GSFC.
ISUN (see the documentation). An easier versun of sunpos.
SUNPOS, jd, ra, dec, longmed, oblt, RADIAN = radian
IMOON (see the documentation. An easier version of moonpos, and ***most
importantly*** imoon corrects for parallax!
MOONPOS, jd, ra, dec, dis, geolong, geolat, RADIAN = radian
FOLLOW, ra, dec, moon=moon, sun=sun, filename=filename causes the
interferometer dishes to track a given RA, DEC. If SUN or MOON is set,
it tracks the sun or moon; for these, no RA or DEC needs to be
specified. Writes out a save file (filename) with relevant
information. This is our main telescope control recording program for

 the interferometer.
STARTCHART1 displays the incoming interferometer output and writes data
 into a save file. This is our main data-taking and data-recording
 program. The output Filename and plot parameters are prompted for. When
 observing the Sun or Moon, specifying the appropreate keyword will
 write the object's position into the data structure.

4. OUR INTERFEROMETER

With our interferometer, which operates at about 12 GHz, our attention focuses on small sources. Our interferometer has a relatively short baseline and the fringe spacing is larger than the size of all sources (except the Sun and Moon). So for our interferometer all of these sources look like “point sources”, for which all the radiation appears to come from a single point—just like a star in optical astronomy. The output of the interferometer is a sinusoidal-like signal called the “fringe”. All of the information resides in the frequency, amplitude and phase of the fringe.

If you know the baseline, the fringe properties are direct indicators of the point-source declination. With our ∼ 20-m baseline B we can get a fringe spacing \frac{λ}{B} ∼ 5′, and with a horizon-to-horizon measurement we can measure the declination 100 times more accurately (depending...). In addition, we can achieve partial angular resolution on the Sun and Moon and measure their diameters to a fraction of a percent—this turns out to be more interesting than it sounds!

We have a multiplying interferometer: the signals from the two telescopes (that’s E_{1} and E_{2}) are multiplied together, using one of the mixers you used in the previous lab, producing the product E_{1}E_{2}.^{1} We repeatedly measure and record the time average ⟨E_{1}E_{2}⟩ over a suitable interval (a few seconds). Using the cross product is important because its average value is zero unless there’s a source, so any detected signal has to come from the sky instead of the instrument. So when we look at a source, the fringe amplitude has no zero offset and the amplitude is directly proportional to the source flux (if it’s a point source).

5. CONTINUUM ‘POINT’ SOURCES

For the first part of the lab we will concentrate on ’point’ sources that lie outside the Solar system. Our telescopes are small, so there are only a few sources that are powerful enough for us; they are listed below. You’ll need to precess the coordinates to the current equinox. If you don’t, the incorrect r.a.’s will affect your fits. To do the precession, use the GSFC IDL procedure

1This multiplication is equivalent to adding the signals, squaring [(E_{1} + E_{2})^{2} = E_{1}^{2} + E_{2}^{2} + 2E_{1}E_{2}], and then subtracting the two self-products E_{1}^{2} and E_{2}^{2}.

precess.

name  r.a.(2000)  dec(2000)  S_{Jy}
3C144 (Crab Nebula) 05^{h}34^{m}31.95s +22^{∘}00′52.1′′ ∼ 496
Orion Nebula  05^{h}35^{m}17.3s  −05^{∘}23′28′′ ∼ 340
M17  18^{h}20^{m}26s  −16^{∘}10.6′  ∼ 500
3C405 (Cygnus A) 19^{h}59^{m}28.357s +40^{∘}44′02.10′′ ∼ 120
3C461 (Cas A)  23^{h}23^{m}24s  +58^{∘}48.9′  ∼ 320
SUN  varies  varies
MOON  varies  varies

As astronomers, our goal is to measure the absolute declinations as accurately as possible. As geologists, our goal is to measure the interferometer baseline as accurately as possible (to see, for example, the displacement of tectonic plates by earthquake activity). If you look at equations 11 and 12, you see that we don’t measure these quantities separabely, but rather the products B_{ew} cos(δ) and B_{ns} cos(δ), where B_{ew} and B_{ns} are the east-west and north-south components of the interferometer baseline. These products can be measured on an absolute basis with interferometry by least-squares fitting the fringe phase to the hour angle; see §8 and equation 12 below. We already know the source positions accurately, from previous intererometrric meaasurements; that’s where the positions in the above table come from. So, for this lab, we’ll emphasize the geologic aspect and measure baselines. We should be able to measure baseline components to better than 1%.

Coverage for soouthern sources may be interrupted by the Campanile—check (by going up to the roof periodically and looking)! And coverage for nortern sources, especially Cas A, is interrupted by Evans Hall.

We’ll want each group to pick a source and obtain a horizon-to-horizon observation of the fringes. Each group should pick a different source. Observing a source takes at least half a day. With four groups and less than a week to do the measuremets, you need to cooperate in scheduling the interferometer. Organize yourselves!

We’ll want each person to measure the source’s declination by least-squares fitting the horizonto-horizon track of fringe phase and amplitude. Compare your results with other members of your group. Help each other out, but each person should write her/his own software.

6. THE SUN AND MOON

The Sun and Moon are special cases, for two reasons. First, their positions change from day to day—and for the Moon, the change in just an hour is significant^{2}. Second, they are both extended sources, not point sources. This causes their fringe amplitudes and phases to change with time, in a manner that depends on their brightness distribution^{3}; this makes the determination of accurate positions a bit tricky, but we’ll ignore that detail for now.

You can get the Sun and Moon positions from our IDL procedures isun and imoon. For the Moon, you should be aware that it is nearby so that you have to correct for your location on the Earth’s surface. The parallax effect is far greater for the Moon than the Sun—so large that unless you correct for it, the Moon will probably lie outside the telescope main beam! imoon takes care of this automatically unless you tell it not to^{4}. A previous undergrad lab TA, Erik Shirokoff^{5}, worked very hard to get the moon position, including parallax, really right; he wrote imoon. When observing the Sun or Moon using follow, the positions will be calculated and recorded automatically.

7. SOME ASTROPHYSICS

Here’s a bit of physical information. M17 (“M” for Messier) and Orion are HII regions—places where hot stars have produced warm (T ∼ 10^{4} K) ionized gas, where the electrons flying past the protons get deflected and produce free-free (bremsstrahlung) radiation. The Sun also emits free-free radiation, just like the HII regions.

The other sources, and sources paired with some of the HII regions, radiate in synchrotron radiation—relativistic electrons gyrating in a magnetic field. For these sources, the source designation 3CXYZ designate source number XYZ from the third Cambridge (England) catalog; in the early 1960’s, Cambridge radioastronomers produced the first reliable comprehensive catalog of strong radio sources in the Northern hemisphere. The Crab Nebula (also called Taurus A) is powered by the Crab pulsar, and is a ∼ 1000 yr-old supernova remnant in the Galaxy about 1 kpc distant. Cas A is another supernova remnant, not powered by a pulsar; rather, the relativistic electrons are produced behind the fast shock wave produced by the explosion. Cas A is ∼ 300 yr old and about 2.5 kpc distant. Both of these supernova remnants are expanding rapidly, as befits their young ages, and Cas A (in which a pulsar does not constantly replenish the electrons) is gradually getting dimmer.

2Use what you already know to make an order-of-magnitude estimate of how far the Sun and Moon move in one day!

3These changes in fringe properties are exactly what’s necessary to map the sources!

4If you specify the geocentric option, imoon gives the lunar position as seen from the center of the Earth. You don’t want to use this option!

5Erik became a Berkeley grad student in physics, later a Caltech postdoc in astrophysics, now runs his own lab at University of Chicago (http://kicp.uchicago.edu/ shiro/)

In external galaxies, the ultimate source of the electrons involves acceleration of electrons near the black hole at the center; the electrons are then spewed out to extragalactic space in narrow collimated jets, and produce large “emission lobes” at the end of the jets. Cygnus A is a giant elliptical galaxy 220 Mpc distant. Cyg A is a powerful radio source—it’s 10^{5} times further than Cas A and just as bright! The study of the mechanism by which this enormous power is generated, which implies enormous energies, has led to the current awareness of and interest in high-energy astrophysics. For information on both types of source and some beautiful pictures, see chapters 10 and 13 in Galactic and Extragalactic Radio Astronomy, second edition (1988, ed. G.L. Verschuur and K.I. Kellermann).

The Moon is a completely different story. Contrary to what you might expect, at radio wavelengths it doesn’t shine by reflected sunlight. Rather, its emission is blackbody radiation from its solid surface. Its surface is heated by sunlight, and at short wavelengths (but not at long ones) there’s a big difference between the temperature of the sunlit and dark parts of the Lunar surface. You can tell a lot about its surface properties from the polarization of the radiation and also from its time variability as the surface heats up from sunlight and cools off from darkness—just like the Sahara.

8. MEASURING B cos δ

We measure the product B cos δ from the fringe frequency, which depends on the baseline orientation, baseline length, declination, and hour angle (all these go into the projected baseline). If we observe a source horizon-to-horizon, the projected baseline changes a lot, and so does the fringe frequency. We first do a ‘sanity check’, taking the Fourier transform of the data to see if we see the fringe and measure its range of frequencies. If it all looks good, we take those data, together with the known declination, and do a least-squares fit to derive the most accurate baseline components from our data.

8.1. Getting the Data

FIRST WEEK: Before doing the weak sources in the Table, do the Sun for a much shorter time, say an hour. This will give you confidence that the system works (or so we hope). There should be an easily-recognizable signal that you can look at visually, think about, and derive the approximate declination with pencil and paper. Then later you can write software to do the same, and make sure you get the right answer. Also, during this first week, do the horizon-to-horizon track of one of the sources from the Table.

8.2. The Fringe

The two interferometer telescopes have different distances from the source. The difference can range from zero (if the source is overhead) to nearly the the full baseline (if the source is near the horizon). This distance difference is tiny compared to the distance to the source, but it’s important!

It’s convenient to think of the different distances in terms of relative path delay in time units for the two telescopes; we call this the geometrical path delay τ_{g}. But don’t forget! The signals travel through a lot of electronics before they get multiplied and the two paths aren’t of equal length, so there is an additional relative delay from the difference in cable length τ_{c}. The total relative delay is the sum of the two,

τ_{tot} = τ_{g}(h_{s}) + τ_{c} .  (1)

We explicitly include the fact that τ_{g} is a function of time—that is, the hour angle of the source h_{s}. In contrast, τ_{c} is independent of time (unless somebody changes the cable setup…).

We don’t know τ_{c} (but the least-squares process can tell us what it is). However, we do know τ_{g} because it’s just geometry—the geometry discussed in the reading. For the east-west baseline component with length B_{ew}, we have

τ_{g,ew}(h_{s}) = [\frac{B_{ew}}{c} cos δ] sin h_{s} .  (2)

and for the north-south component

τ_{g,ns}(h_{s}) = [\frac{B_{ns}}{c} sin L cos δ] cos h_{s} − [\frac{B_{ns}}{c} cos L sin δ] ,  (3)

where L is the terrestrial latitude. This has two terms. The first is the delay perpendicular to the Earth’s axis, which changes as cos h. The second is the delay parallel to the Earth’s axis, which has no h dependence; since it is independent of h, we’ll lump it into the cable delay τ_{c}. Accordingly, we define

τ′_{g}(h_{s}) = [\frac{B_{ew}}{c} cos δ] sin h_{s} + [\frac{B_{ns}}{c} sin L cos δ] cos h_{s}  (4)

and the B_{ns}-modified cable delay as

τ′_{c} = τ_{c} − [\frac{B_{ns}}{c} cos L sin δ] .  (5)

The output of the interferometer is the product of what the two telescopes see. If they are looking at a monochromatic source then the voltages for the two telescopes are

E_{1}(t) = cos(2πνt)  (6)

E_{2}(t) = cos(2πν[t + τ_{tot}]) .  (7)

and the product is the interferometer fringe output

F (t) = cos(2πνt) cos(2πν[t + τ_{tot}]) .  (8)

There’s a trigonometric identity that allows us to write this in terms of the sum and difference of the two arguments^{6}. The sum term varies rapidly with time and averages to zero; it’s the difference term we want, so if we exclude the sum term (and forget about the factor \frac{1}{2}) we get

F (h_{s}) = cos(2πν[τ′_{g}(h_{s}) + τ′_{c}]) .  (9)

We want to use a least-squares fit to find the values of quantities that comprise the argument of the cosine. If you have done least-squares fitting before (and if you remember anything about it!) you’ll realize that the straightforward least-squares fitting technique won’t work on this type of problem, because the unknown quantities appear inside nonlinear functions. We can simplify things for the fitting process by using another trig identity^{7} and writing this as the sum of two trig functions

F (h_{s}) = cos(2πντ_{c}) cos (2πντ′_{g}) − sin(2πντ_{c}) sin (2πντ′_{g}) .  (10)

This may not look simpler! But it is, because for the purposes of least-squares it involves only the single variable τ′_{g} in the trig function arguments (we are assuming that we know the right ascension well enough to get a good value for h_{s}, which appears in τ′_{g}).

To proceed with least-squares, replace cos(2πντ_{c}) and − sin(2πντ_{c}) by two “unknown constants” A and B, respectively; assume that they are unrelated and solve for them using the standard least-squares process. Also, it’s convenient and intuitive to make the substitution

ντ′_{g}(B_{ew}, B_{ns}, δ, h_{s}) = [\frac{B_{ew}}{λ} cos δ] sin h_{s} + [\frac{B_{ns}}{λ} sin L cos δ] cos h_{s}  (11)

which expresses the delay in units of wavelength, and thus the “number of turns” or phase. These substitutions give the Fringe Amplitude for a point source

F (h_{s}) = A cos (2πντ′_{g}) + B sin (2πντ′_{g}) .  (12)

6cos(A) cos(B) = \frac{1}{2} [cos(A − B) + cos(A + B)]

7cos(A + B) = cos(A) cos(B) − sin(A) sin(B)

8.3. The Local Fringe Frequency

We want to develop the concept of a local fringe frequency f_{f}. In equation 12, the argument of the trig functions depends on h_{s}, which is the hour angle and increases monotonically with time, so we can regard it as time^{8}. Now it’s not h_{s} itself in the equation, but rather sin(h_{s}) and cos(h_{s}), which are nonlinear functions of time. They multiply the baseline in units of wavelength, which is large, so the product rapidly oscillates back and forth, but at a frequency that depends on time as h_{s} changes. This is the local fringe frequency.

To see this, expand the hour angle terms sin(h_{s}) and cos(h_{s}) each into its own Taylor series, centered on the current hour angle of the source h_{s,0}. For example, for the sin(h_{s}) term we have

sin(h_{s}) = sin(h_{s,0}) + Δh_{s} \frac{d sin(h_{s})}{dh}|_{h_{s,0}} = sin(h_{s,0}) + Δh_{s} cos(h_{s,0})  (13)

The local fringe frequency is contained in the second term because, for a small region around h_{s,0} in equation 12, F (h_{s}) varies as f_{f} Δh_{s}, where the local fringe frequency f_{f} is

f_{f} = [\frac{B_{ew}}{λ} cos δ] cos h_{s,0} − [\frac{B_{ns}}{λ} sin L cos δ] sin h_{s,0} (14)

This is the local fringe frequency in cycles per radian on the sky. If you want to turn it into cycles per hour coming out of the interferometer’s multiplier, multiply by \frac{dh_{s}}{dt} = \frac{2π}{24} ; or cycles per minute, multiply by \frac{2π}{60×24}; etc. Our interferometer is almost (but not quite) east-west with B_{ew} ≈ 20 m, B_{ns} ∼ 0, and λ ≈ 2.5 cm. At the meridian (h_{s,0} = 0), the fringe frequency is [f_{f} ≈ (\frac{B_{ew}}{λ} cos δ) ≈ 0.015 cos δ] cycles per second and the period is \frac{1}{f_{f}} ≈ \frac{18}{cos δ} seconds. ; check these numbers to make sure you understand this calculation!)

Use the above to calculate the range of local fringe frequencies that you should see in your data. Does it work? Look at the Fourier transform of your data to check! And try Fourier filtering!

8.4. Least-Squares Fitting the Fringe to the baselines and/or the Declination

Now let’s return to the least-squares solution of equation 12 for the unknown quantities Q_{ew} = [\frac{B_{ew}}{λ} cos δ] and Q_{ns} = [\frac{B_{ns}}{λ} sin L cos δ]. This is a nonlinear least-squares problem because these quantities sit inside nonlinear functions. There are two ways to attack this: the conventional way and the ‘brute-force’ way. We’ll do the latter first.

8Except that its units are radians. This is no problem: 24 hours is 2π radians.

8.4.1. The Brute-Force Technique

We’ll first assume that B_{ns} = 0, which is not unreasonable because our interferometer baseline is close to east-west, and solve for Q_{ew}. Make a guess at the proper value of Q_{ew} and, if you don’t know the right ascension α accurately, adopt a reasonably accurate value for it (required so that you can compute h_{s} from the local sidereal time LST). With this, you know the arguments of the trig functions and this means that you can solve for A and B using the standard least-squares process; be sure and save the sum of the squares of the residuals, which we will call S. Then change the guessed-at value of Q_{ew} and do it again. Do this a number of times and plot the sum-of-squares S versus the guessed-at value of Q_{ew}. The best value of Q_{ew} is where the sum-of-squares is a minimum. What is this? It’s brute-force least-squares!

Now do the same for Q_{ns}. The best value of Q_{ew} depends on the value of Q_{ns}, so you’ll need to develop a 2-d array for S versus the assumed values of Q_{ew} and Q_{ns} so that you can find the true global minimum.

How about the uncertainties? You can get these from the curvature matrix [α], which is easily derived from the behavior of S around the minimum. The covariance matrix α^{−1} is the inverse of the curvature matrix. See §2 and §3 of the ‘Lite’ version of the Aficionado’s handout.

8.4.2. The Proper Nonlinear Least-Squares Technique

The usual technique for nonlinear least-square fitting is this: you guess values for unknowns Q_{ew} and Q_{ns}, expand the equation for F (h_{s}) in a one-term Taylor series about these guesses, and solve for the corrections. The procedure is discussed in detail in §7 of the ‘Lite’ version of the Aficionado’s handout.

Our particular problem is not well-suited to the ‘proper’ technique because the initial guessed values need to be quite accurate. If they are not accurate enough, then the proper technique converges to a local, non-global minimum.

The advantage of using the proper technique is that it provides the uncertainties, and in particular the covariance matrix. If you have time, try both methods–the ‘brute-force’ and ‘proper’ methods, using the ‘brute-force’ result as input to the ‘proper’ method, and compare the results, in particular the uncertainties.

9. MEASURING 1-D BRIGHTNESS DISTRIBUTIONS: THE FIRST STEP OF MAKING MAPS

Disclaimer This section assumes an east-west interferometer. Generalizing to the case of B_{ns} ≠ 0 can be regarded either as an ‘exercise for the student’ or as an unnecessary complication.

When we think of a time-variable signal, we think of frequency as being cycles per second—and its inverse, the period, is in seconds, the number of seconds that separates adjacent peaks of the sine wave.

The interferometer projects a giant sine wave on the sky. Its frequency, which changes with position, is measured in cycles per radian—and its inverse, the period, is the angular separation of adjacent peaks, measured in the angular units of radians. You can, of course, also think of frequency in terms of cycles per degree or cycles per arcminute, with the corresponding periods (“fringe separation”) in units of degrees or arcminutes.

When we observe with a range of baseline lengths and orientations, the giant sine waves in the sky have corresponding ranges of frequencies and orientations. We sample brightness of the sky in Fourier space. The fringes at each baseline length and orientation have amplitudes and phases. To recover the brightness of the sky in real, angular space, we measure as many Fourier components as we can and take their Fourier transform. If we had complete sampling in Fourier space, we would recover the true brightness distribution. In real life, we have incomplete sampling, so we recover a distorted representation of the true distribution. There is a whole literature of techniques for minimizing this distortion, the most prominent being “cleaning” and “maximum entropy”. Full-fledged research arrays, such as the Very Large Array (VLA) in New Mexico, rely on these techniques to map the sky.

In our case we have just two dishes along an east-west line. The effective baseline length changes as the source rises higher in the sky, and if the source is away from declination δ = 0^{∘} the orientation of the baseline also changes, at least to some degree. We will map the Sun and Moon, which never get very far from δ = 0^{∘}, so effectively we have only a 1-d sampling of the source with a range of baseline lengths.

Figure 1 illustrates this 1-d concept. It presents a generic circular source of uniform brightness in the sky, which we call the MUN—a bastardization of the MOON and the SUN^{9}. The top panel shows the MUN in the sky as it really is: the fringes cover the 2-d object. At the bottom, we integrate along vertical strips to get the 1-d brightness distribution—the vertically-integrated 1-d equivalent, in which both the brightness distribution and the fringes depend on only one coordinate.

This one coordinate is the horizontal direction in the bottom panel of the Figure. In real life this is hour angle because we have an east-west interferometer and our source is at low declination— meaning that the baseline projected on the sky is mainly east-west, the direction of hour angle. Thus we denote this direction by the letter h.

As the Earth rotates, the source moves through the fringe pattern to give the fringe response R(h_{s}). For a point source, R(h_{s}) = F (h_{s}) (see equation 12); for an extended source, we have to integrate over the extent of the source. Let I(h − h_{s}) be the 1-d intensity distribution in the sky.

9In truth, it represents neither, because neither the Sun nor the Moon has uniform surface brightness.

Fig. 1.— The 2-d and 1-d MUN. At the top, we see the situation in the sky as it really is: the fringes cover the 2-d object. At the bottom, we integrate along vertical strips to get the 1-d brightness distribution and, also, the fringe amplitude (which goes from –1 to +1).

The source center is the intensity-weighted mean of the position, i.e.

h_{s} = \frac{∫ I(h)hdh}{∫ I(h)dh} .  (15)

That is, on the bottom panel of Figure 1, I(h − h_{s}) is the intensity of the source (vertical direction) and Δh = h − h_{s} the horizontal coordinate—the hour angle h relative to the hour angle of the source center h_{s}.

We can express the interferometer response R(h_{s}) using equation 12, which is for a point source; for an extended source, we imagine I(Δh) as being composed of little slices in hour angle, with each little slice of the source characterized by its position offset Δh and its intensity I(Δh), so we just integrate:

R(h_{s}) = A ∫ I(Δh) cos [2π (\frac{B}{λ} cos δ) sin h] dΔh+B ∫ I(Δh) sin [2π (\frac{B}{λ} cos δ) sin h] dΔh (16)

Now express cos [2π (\frac{B}{λ} cos δ) sin h] in terms of the local fringe frequency (equations 13 and 14), which replaces cos [2π (\frac{B}{λ} cos δ) sin h] by a sum of two terms, which we temporarily denote α and β^{10}. Now, as usual, we use trig identities [cos(α + β) = cos(α) cos(β) − sin(α) sin(β) and sin(α + β) = sin(α) cos(β) + cos(α) sin(β)]. The first (cosine) term of equation 16 becomes

R^{cos}(h_{s}) = A cos(α) ∫ I(Δh) cos(2πf_{f}Δh) dΔh − A sin(α) ∫ I(Δh) sin(2πf_{f}Δh) dΔh (17)

with an equivalent, similar expression for R^{sin}(h_{s}).

For our source (the MUN), we assume that I(Δh) is symmetric (This also retains our algebraic sanity.). This means that in the above equation the second term, which is antisymmetric, integrates to zero. Similarly, the antisymmetric term in the equivalent equation R^{sin}(h_{s}) also integrates to zero, so we end up with R(h_{s}) = R^{cos}(h_{s}) + R^{sin}(h_{s}), or

R(h_{s}) = \underbrace{F (h_{s})}_{P oint−source F ringe} × \underbrace{∫ I(Δh) cos(2πf_{f}Δh) dΔh}_{F ringe Modulator}  (18)

Note the structure of equation 18. It consists of two factors. The first “Point-source Fringe” term is identical to equation 12—it’s the response to a point source located at Δh = 0. The other modulates (multiplies) this function.

Generally, the modulating function is the Fourier transform of the source intensity distribution on the sky. Here, we assumed a one-dimensional symmetric source, which means that the sine portion of the Fourier transform is zero; this is why equation 18 is only a cosine Fourier transform instead of a full one. More generally, the Fringe Modulator depends on the two-dimensional map of intensity on the sky, so it’s a double integral instead of a single one.

10where α = 2π ( \frac{B}{λ} cos δ) sin h_{s} and β = 2πf_{f} Δh .

Fig. 2.— Examples of 1-d brightness distributions and their Fourier transforms. Top panel: the brightness distributions. Bottom: the Fourier transforms (Fringe Amplitude vs. \frac{Source Diameter}{F ringeP eriod}). In both, panels, the solid line is for a flat brightness distribution and the dashed one for a cosine distribution.

Figure 2 (top panel) shows two examples of 1-d brightness distributions, a flat and a cosine distribution. The bottom panel shows the Fourier transforms. Both of the modulating functions are trig functions. In particular, for the flat distribution the modulating function is \frac{sin(2πf_{f} R)}{2πf_{f} R}. It can (and does!) go through zero. The locations of these zero points provide crucial information about the source structure. The zeros occur for f_{f} = \frac{n}{2R}. It’s more intuitive to express the zeros in terms of fringe period (equal to \frac{1}{f_{f}}): the zeros occur at P eriod = \frac{2R}{n}. There’s a zero whenever there’s an integral number of fringe periods over the source width. This makes perfect sense, because then the source contributes equally to the negative and positive portions of the fringe and the net integral is zero.

10. MEASURING THE DIAMETER OF A CIRCULAR SOURCE

10.1. Theory and Math

Our goal is to measure and compare the diameters of the Sun and Moon. We’ll make the assumption that the sources are uniformly-bright disks of radius R, which means

I(Δh) = \frac{(R^{2} − Δh^{2})^{1/2}}{R}  (19)

To obtain the theoretical modulating function MF_{theory}, you use the integral in equation 18, which is

MF_{theory} = \frac{1}{R} ∫_{−R}^{R}(R^{2} − Δh^{2})^{1/2} cos(2πf_{f}Δh)dΔh  (20)

If you want, you can do this analytically by substituting Δ(R cos(θ) for Δh; you end up with a Bessel function.

We are running a lab class, not a math class, so let’s proceed by doing the integral numerically! To accomplish this, split I(Δh) into 2N + 1 tiny little slices (the total number is odd, which makes the slices symmetric about Δh = 0). Each slice has width δh = \frac{R}{N}, and Δh_{n} = nδh, where n runs from −N to +N. Then the integral becomes a sum:

MF_{theory} ≈ \frac{1}{R} ∑_{n=−N}^{n=+N}[R^{2} − (nδh)^{2}]^{1/2} cos(2πf_{f}nδh)δh  (21)

which we rewrite as

MF_{theory} ≈ δh ∑_{n=−N}^{n=+N} [1 − (\frac{n}{N} )^{2}]^{1/2} cos (\frac{2πf_{f}Rn}{N})  (22)

10.2. Important Mathematical and ¡¡¡PRACTICAL!!! Point!

Note the first important point that MF_{theory}…

• is a function only of the combination f_{f}R, and

• in particular, has zero crossings that occur at specific values of f_{f}R.

Note the second important point that for MF_{observed}…

• The zero crossings occur at specific measured values of f_{f}.

Thus, by comparing the zero crossing numbers for MF_{theory} and MF_{observed}, you get the radius R.

11. REFERENCE READING on INTERFEROMETRY AND APERTURE SYNTHESIS

The appropriate reference for our purposes is the article Interferometry and Aperture Synthesis, which is chapter 10 of the book Galactic and Extragalactic Radio Astronomy, First Edition. The authors are Fomalont and Wright; Melvyn Wright is a research scientist in our radio lab here at Berkeley and is a real expert. This chapter is excellent, providing the basics without excessive detail (although it has more than we need). If you want more depth than we provide here, we suggest the following sections of this chapter: (1) §10.1.3, which describes a two-element interferometer; section e of this chapter is on polarization and you can skip it; (2) §10.2.1 and §10.2.2, which describe “a working interferometer”; and (3) Appendix II, which describes the geometrical details.

There is a scaling mistake in their equation for the fringe frequency ν_{f}: their equation needs to be multiplied by the rotation rate of the Earth in radians per second. For example, for an east-west baseline of 343.8 wavelengths looking at declination zero on the meridian, the fringe frequency on the sky is 343.8 fringes per radian. This means that the fringe spacing on the sky is \frac{1}{343.8} radians or 10 arcmin; it takes the Earth 40 seconds of time to turn through 10 arcmin, so the fringe period in this case is 40 seconds and ν_{f} = .025 Hz. More generally, for this east-west interferometer the fringe frequency is ν_{f} = .025 cos δ cos h Hz, where δ is the declination and h the hour angle.

If you want to know really everything and in complete mathematical detail, read the book Interferometry and Synthesis in Radio Astronomy by Thompson, Moran, and Swenson. But such detail is more than we want and can be overwhelming. Our main interest is the geometry—how the baseline projects on the u − v plane. This is in chapter 4, and the most important sections for us are chapter 4.2 and 4.4.

With the longer baselines of research-class interferometers/arrays comes increased angular resolution, and all of our sources become finite in angular size, which means that the telescope arrays can map the sources using Fourier techniques. Some types of source are so small that mapping them requires interferometers with baseline lengths comparable to the Earth’s diameter, a technique called “Very Long Baseline Interferometry” (VLBI). And some sources, such as pulsars, are even too small for VLBI!