2 The Radio Moon: Brightness and Temperature

1. The Moon at Radio Wavelengths:  Mapping, Photometry, Calibration, and Temperature

Okay, you’ve imaged and studied the moon at optical wavelengths, and learned about its surface.  Next, let’s try radio – the other wavelength range that you can observe from the ground.  (Almost) everything else must be done from space.

“Small” radio telescopes are still huge.  Skynet is currently integrating a bunch of these, though right now we only have one – the 20-meter diameter telescope at Green Bank Observatory (GBO) in West Virginia.  It’s 6 stories tall, 150 tons, and has a collecting surface that’s ≈4× that of the world’s largest optical telescope.  And that’s considered small!  (The world’s largest single-dish radio telescope has a collecting surface that’s ≈625× bigger yet!)

Skynet 20m-diameter telescope at GBO.

Radio telescopes are both similar and different from optical telescopes.  Similar in that it’s a parabolic reflector, bouncing radio waves to a prime focus detector.  But different in three primary ways:

1. You can do radio astronomy during the day – the sky doesn’t scatter around long-wavelength light (e.g., radio waves) like it does short-wavelength light (e.g., blue, optical light).  You can even do it through clouds.  The only things that can shut you down are (1) lightning, (2) high winds, (3) snowfall, and (4) maintenance – which occurs frequently with such large, complex machinery.

1. The equation for calculating a circular telescope’s angular resolution / blurring scale / diffraction limit is:  𝜃= 1.22𝜆/D radians.  For 𝜆 = 21.5 cm = 0.215 m and D = 20.0 m, that’s 𝜃= 0.0131 radians =0.751 degrees.  So at this wavelength, the GBO 20m blurs everything over a scale that’s larger than any of our optical telescope’s FOV!

1. The space at prime focus is not much bigger than the wavelength of the incoming light – which means there’s only room for a single-pixel detector!  What can you do with such an instrument?

The answer is lots:

1. You can point it at something that’s varying in time (e.g., a pulsar) and measure these variations in real time, without having to extract the data from images after the fact, as in optical.  Plus you can do it fast – the GBO 20m can read out every millisecond (most optical cameras take seconds to read out).

1. Furthermore, it reads out not just the brightness of that point in the sky, but:

1. It does it in two, oppositely polarized channels – something we’ll make use of in Module 4.

1. It reads out a full spectrum of that point in the sky.

Optical telescopes can do both of these, but not without very specialized equipment.  Radio telescopes do it automatically!

1. You can move your single pixel around, e.g., over a grid pattern, called a raster mapping, collecting data “on the fly” (i.e., as you go).  You can then turn this into an image, of a large piece of the sky, after the fact.  This is tricky for all sorts of reasons, but Skynet’s produced an algorithm that might be the best ever for doing this:  https://arxiv.org/abs/1808.06128.  Read Sections 1.2 and 2.1 for an introduction.

We’ll do all of these – a, b.i, b.ii, and c – this semester.  Let’s start with c.

1. Skynet Observations

Note:  We’re going to make a total of four radio maps.  Since radio maps are costly, submit only one observation of each per group – split these observations up between the members of your group.  (Don’t worry, everyone will have access to, and use, all of the maps, once they come back.)

Target Page:  Go to My Observatory, Radio Observing, Add New Observation, and look up the moon.  The moon’s RA and Dec will fill in automatically (and will update to the current coordinates before the exposures are dispatched).  For example:

For the remaining boxes in this section, set Min Sun Separation to 10 and Min Target Elevation to 20.  (These are good settings for any radio target, except for the sun).

Proceed to the Receiver page.

Receiver Page:  This page is analogous to the Filters page for optical observing, except that we select the frequencies that we’re going to receive, and how we’re going to receive them, electronically (instead of by placing a specially made piece of glass in front of the detector, as we do with optical telescopes).

We will be using the GBO 20m’s L-band receiver, which detects radio waves with frequencies between 1300 MHz and 1800 MHz.  We want to be in low spectral resolution mode, with the “HI” bandpass filter.  This filter narrows the detector’s response to only 1355 – 1435 MHz (i.e., to 80 MHz centered on 1395 MHz).  This removes frequencies that are known to have a lot of human-made interference.  Channels should be set to 1024 (the only option in low spectral resolution mode) and Pulsar Mode should be off.

Proceed to the Path page.

Path Page:  This page is analogous to the Exposures page for optical observing, except instead of having a sensor with a fixed FOV, we have to (get to!) design the FOV of the observation, and how this “one-pixel” telescope will move around to efficiently sample this FOV.

To make a square, or rectangular, map, select the “Map” path type.  Since we’re only mapping a single source, the map doesn’t have to be too big, which is good – mapping is time consuming / expensive!  For maps around sources, we measure their width and height in “beam widths”.  1 beam width is the scale over which the telescope blurs 68.3% of a point source’s signal (in 1D; it’s 0.6832 = 46.6% in both dimensions).  2 beam widths capture 95.4%, and 3 beam widths capture 99.73% – almost all:

But then you need a few more beam widths to establish the background level.  So, select beam widths and enter 6 × 6.  The interface will then convert this into degrees in RA and degrees in Dec.  (Note, the width in RA will be greater than the height in Dec, especially as you approach Dec = ±90, because lines of constant RA converge at the celestial poles, just as lines of constant longitude converge at the Earth’s poles.)  For example:

Now that you’ve set the size of your map (your FOV), you have to design how the telescope will “raster” or “sweep” around to efficiently sample this FOV – back and forth in RA, or up and down in Dec?  It doesn’t really matter, though I usually sweep in RA, because it’s less that the weight of the telescope has to fight against gravity.

A horizontal raster map, where the telescope “sweeps” back and forth in RA, with each sweep at an incrementally greater Dec.

Once you’ve set this, you have to select how much of a gap you want between sweeps, and how often it will record data along the sweeps.  Technically, you need data every 0.4 beam widths, in both directions, to reconstruct what’s there without losing information.  But in reality, rasters are seldom perfect – wind and rapid turnarounds can distort the raster pattern, creating larger than expected gaps.  If you’re going to make a precision measurement – and we are! – I recommend gaps of 1/5 beam widths in both directions.  For example:

Lastly, you have to decide how long to integrate (or expose) at each grid point.  However, since the telescope never stops moving, you are instead determining how quickly it moves from one grid point to the next, also known as its slew speed.  I normally select an integration time of 0.3 seconds, which for this grid spacing corresponds to a slew speed of ≈0.5°/sec.  This is a good speed – the raster pattern begins to fall apart if executing turnarounds at speeds above 1°/sec.

Only 4.5 minutes to complete – not bad.  Go ahead and submit!

Note:  If your target is up now, you can watch the raster execute, and the data come in live, here:  https://skynet.unc.edu/telescopes/view?id=34.  This is one of Skynet’s cooler features!

Calibration Observation:  When your image comes back, you’re going to measure the moon’s brightness (technically called flux density).  However, it will be in arbitrary units, and these arbitrary units change slowly with time.  To convert these to standard units, you also have to map a “calibration” source, around the same time.  This is a source of known flux density.  The three brightest ones are Cyg A (3C 405), Tau A (3C 144), and Vir A (3C 274), and they are distributed fairly equally around the sky.

Using the “SkyViewer” on Skynet’s (radio) Target page, find whichever of these is closest to the moon right now, and submit an identically designed observation of it.  It’s important that your target observation (in this case, of the moon) and your calibration observation take place within the same day, and ideally within a few hours of each other.

Rinse and Repeat:  Lastly, we’re going to repeat both of these observations a few days later, once the moon’s phase has changed.  (Note – use the same calibration source both times.)  Consult a moon phase calendar online, and pick an upcoming date with a different phase.  You can either wait and put these observations in then, or enable “Delay” on the Target page and select the date.  For example:

Enter your observation’s information into your group’s “Obs Info” sheet, so your groupmates can easily access it (they will put theirs there too).

Check Your Images:  Check your observation’s page each day, and preview the preliminary data products provided by GBO.  Here’s what they are:

1. Upper Left:  The brightness detected in each of the telescope’s two, oppositely-polarized channels vs. time.  To make this plot, they summed the brightness over all observed frequencies.

Note:  These brightnesses are not well calibrated – you could observe an always-the-same-brightness source at different times and get different brightness values, depending on the performance of the electronics.  Consequently, we automatically turn on, and then off, an always-the-same-brightness radio “noise” maker in the telescope’s receiver, at the beginning and end of each observation.  If the electronics are under- or over-performing, its brightness values will be low or high, respectively, in the same proportion as the rest of your brightness values.  Consequently, in the image-processing section below, we divide all of our data by the measured brightness of the noise source, canceling out these instrumental effects, and putting everything into consistent units.  We call these noise-source units.

Note:  In the example below, both channels are working (this isn’t always the case!), but only the “pre” noise-source measurement was carried out.  In the image-processing section below, we will use whichever noise-source measurement is available, and interpolate between them if both are available.

1. Upper Right:  The brightness detected in each of the telescope’s two, oppositely polarized channels vs. frequency.  To make this plot, they summed the brightness over all observed times.

This is a spectrum, but note – the undulations you see are not physical, but a product of the telescope’s electronics.  However, this plot can be useful for identifying “radio frequency interference”, or RFI.  Although this telescope is in a protected, National Radio Quiet Zone, sometimes (and more frequently everyday!) human-made transmissions come through, either from elsewhere on the ground, or from satellites in orbit.  They are usually at specific frequencies, which we can then cut out, in the image-processing section below.

However, the “blip” near 1420 MHz is not RFI.  This is an astronomical emission line, caused by cold hydrogen.  Since cold hydrogen is the most abundant thing in the universe (well, except for dark matter and dark energy…), we’ll see this a lot, and make use of it in Modules 5 – 7.

Here’s an example with RFI – you can see it at ≈1380 MHz, and from the plot on the left, you can also see when it occurred, near the end of the observation (also note that RFI is often polarized, appearing differently between the two channels):

• Lower Left:  This plot shows the path that the telescope actually took.  Just make sure it didn’t do anything crazy!

The green circle shows you the size of the telescope’s beam (it’s one beam width across).

• Lower Center and Right:  These are preliminary images.  They include all sorts of defects, as described in more detail in Section 1.2 of https://arxiv.org/abs/1808.06128.  Most obvious in these particular images is a time-delay problem:  There appears to be a delay between when signal was measured and when the telescope’s position was measured.  This causes signal to appear in the wrong position, and since the telescope keeps changing directions, this causes everything to get “zig-zagged”.  This – and all of the defects described in Section 1.2 – are problems that Skynet’s single-dish image-processing algorithm (called “Radio Cartographer”) is designed to correct, below.

As long as you have at least one good channel, with at least one good noise-source measurement, and any RFI is localized to a single frequency, you are good to go.  If not, you will need to resubmit the observation (and its calibration observation, since they must be carried out close together in time).  But you might want to check with your instructor first, since these are relatively expensive observations.  (Your instructor should also report any problems to Prof. Reichart at UNC, who will then communicate them to GBO – these telescopes really are complicated, cutting-edge pieces of machinery and electronics; aspects of them fail and must be repaired frequently!)

1. Image Processing

Enough of these preliminary images – let’s make the real deal!  This begins on the observation’s page in Skynet.

Image Processing Basics:  If both channels look good for both your target observation (in this case, the moon) and your calibration observation (Cyg A, Tau A, or Vir A), set Channel to “sum”, which will sum both channel’s values.  However, if these two observations have only one good channel in common, select it (“left” or “right”) instead, for both observations.

Gain Calibration sets whether you will use the “pre” noise-source calibration, the “post” noise-source calibration, or interpolate between both of them.  You should almost always be able to leave this set to “interpolated” – even if one looks messy, or is missing!  Radio Cartographer is good at detecting such things, and cleaning these measurements, so you get the best noise-source calibration possible.

Image Coordinates should be set to “equatorial”, which means the image will be made in the RA/Dec coordinate system.  Time-Delay Correction should (almost always) be selected.  As should be Include Raw Image – this is Radio Cartographer’s version of the preliminary images you looked at above.  Selecting this approximately doubles the processing time, but it’s often good to compare before vs. after, once we get this into Afterglow.

Frequency Range:  For this investigation, we’re going to sum the measured brightnesses over the full frequency range.  However, if you noticed RFI in GBO’s spectral plot, this is your chance to cut it out.  You can use the “Live Spectrum” tool (link below GBO’s spectral plot), to measure minimum and maximum skip frequencies, which you can then enter here.

Note:  No need to cut out the cold-hydrogen emission line near 1420 MHz.  Its emission is probably pretty uniformly spread across this (small) map, and will be background-subtracted out.

Contaminant Cleaning and Surface Modeling Details:  This part’s a bit technical, but it’s important to know what Radio Cartographer is doing, behind the scenes, to make your final image.

• 1D Background Subtraction Scale:  First, Radio Cartographer runs along each sweep of the map, separating sources from background, and removing the background – the background is contaminated by emission from the Earth leaking into the receiver, as well as by changes in the noise characteristics of the receiver as the telescope is slewing it back and forth.  As long as the telescope is well-focused, 6 beam widths is a good scale for this – smaller structures (like astronomical sources) survive; larger structures (like background variations) are wiped out.

• 2D RFI Subtraction Scale:  You might have gotten rid of all of the RFI when you cut out obviously contaminated frequencies above.  But some might have been too low-level to see in the summed spectrum.  Or it might have been broadband, contaminating all of the frequencies – lightning is a good example of this.  But don’t worry, Radio Cartographer has another way of detecting RFI – from its shape in the map.

Everything from outer space is smeared out by the telescope by ~1 beam width – in all directions.  But RFI (usually) isn’t from outer space.  It contaminates your map for as long as it continues – along a sweep.  Sometimes along two.  But seldom more.  And since our sweeps are only 1/5 of a beam width apart, RFI is seldom “wider” than this in your map, making it easy to separate from legit signal, which is ~1 beam width wide.

“Raw” image, in which strong RFI contaminates up to 2 sweeps of a large map.  But it’s easy to distinguish, and disentangle, from the astronomical sources, because it’s too narrow (at least in one direction) to be from space.  (There are also two, brief bursts of RFI below the main source, Vir A.)

When you’re imaging something bright (like the moon, or our calibration sources), select the “Bright Target with Airy Rings” preset, and it will select a scale that wipes out RFI, but not the Airy (diffraction) rings that well-focused telescopes (optical or radio) produce around bright sources.

Airy rings around a bright source.

Or, when you’re imaging something too faint to detect its Airy rings, select the “Faint Target” preset, and it will select an even more aggressive RFI removal scale – but not one big enough to remove point sources (or anything wider, like the moon).

• 2D Surface-Model Weighting Scale:  Lastly, we have to fill in the gaps between our measurements.  Other algorithms do this by averaging the data within ≈2 beam widths of each pixel in the final image.  However, this blurs an already blurry image even more!  Instead, we fit a “surface” model to the data within 2 beam widths of each pixel in the final image, weighting the data closer to the center more.  We give you three weighting options – weighting the central 1/3, 1/2, or 2/3 beam widths most strongly.  The 1/3 option is most accurate, so choose this when you’re going to make brightness measurements, like we are going to do here.  However, the 2/3 option yields a (slightly!) smoother-looking image, and will be useful once we’re ready to do radio astrophotography.

Examples of our surface model, fitted about a pixel near the center of a source (left) and near the base of a source (right).  Only the central (red) value is retained.  This is then repeated for every pixel in the final image.

1. 2D Surface Model:  For our surface model, we use a 3rd-order polynomial in RA multiplied by a 3rd-order polynomial in Dec.  If you selected the Bright Target with Airy Rings preset, we (automatically) add a “noise prior” to this, which helps us to better model the tricky region between the main source and its first Airy ring.  If you selected the Faint Target preset, you don’t need this addition, and it’s turned off automatically.

Lets Gooo:  This is what my programmer Reed Fu says when he is excited.  His two other emotional states are surprise (“Oooo”) and disappointment (“Booo”).  He coded this page, so he got to name the button.

Press the Lets Gooo button and this will initiate the Radio Cartographer algorithm.  For a map this small, it should only take a few seconds to complete.  Refresh the page, and if complete, download it onto your computer.

Note:  If you want to remake the image with other parameters, press Create New Cartographer Job.

In Afterglow, press Open Files and navigate to your Workspace.  Upload the radio file you just downloaded onto your computer, and open it.

It will have many layers.  The primary layer is labeled “main”.  Another interesting layer is “path”, which shows you the location of each measurement (1) after the time-delay correction, and (2) after some measurements are removed by the RFI cleaning process.  Also, “raw” shows you what the image looks like without any corrections or cleaning.  (You can ignore the other layers here.)

IMPORTANT:  The colors in these images are completely arbitrary and artificial!  This is Afterglow’s “rainbow” color palette, in which purple means faint and red means bright.  We’ll learn how to make radio images with “natural” colors in Module 1D.

Hey – you just imaged the invisible universe!

Rinse and Repeat:  Repeat these steps with your other three observations, and open them in Afterglow.

1. Brightness Measurements

You have four images – let’s use them to measure the brightness of the moon, at two different phases.

Aperture Photometry:  The process of measuring the brightness of an astronomical source is called “photometry”.  The simplest type of photometry is called “aperture” photometry, and consists of (1) drawing a big-enough circle (the aperture) around the source, and summing all of the pixel values inside of it, and (2) drawing two more circles around the source, even farther out, and measuring the background level between them.  This region is called the “annulus”.  The background level is subtracted off of each pixel value in the aperture before summing their values.  For example:

Aperture and annulus.  We position the aperture in the gap between the main source and its first Airy ring.  We position the annulus beyond the first Airy ring, so it doesn’t contaminate the background measurement.  Note – the background measurement is fairly sophisticated, and is not thrown off by (some) stray emission in the annulus.

Note:  If you want to see the Airy rings, go to Display Settings (top tab on the right), and change Stretch Mode from “linear” to “hyperbolic arcsine” or “square root”.  Also, I like using the Bright Target preset for these single-source images.

To set the sizes of the aperture and annulus, go to Settings on the top menu, and then Aperture Photometry on the left.  The default settings are for optical sources, which are much smaller.  Aperture Mode should be set to “Constant Aperture”.  Change Centroiding Radius and Aperture to 28 pixels, Annulus Inner to 50 pixels, and Annulus Outer to 70 pixels.  Everything else can be left the same.

Return to the Afterglow Workbench, and open the Photometry tab (fourth from bottom on the right).  Make sure Source Marker and Source Aperture are enabled, and click near the center of the source.  It will draw the aperture and annulus, and measure the source’s flux density – in noise-source units, which we’ll turn into physical units using your calibration source, below.

Note:  The flux-density error bars are underestimated here, because the pixels in radio images are not independent of each other.  We have code to produce better error bars in this case, but haven’t implemented it yet.

Rinse and Repeat:  First, go to the Source Catalog tab (fifth from bottom on the right), and disable “Include sources from other files” – else you’ll see each image’s aperture and annulus in each of your other images.

Then, repeat these steps for your other three images, recording the flux density for each.  Here’s a table to help:

Flux-Density Calibration:  Final step – we need to convert these flux densities into physical units.  At radio wavelengths, the unit of choice is the Jansky (Jy):  1 Jy = 10-26 W/m2/Hz.

First, if you divide your moon flux density by its corresponding calibration source’s flux density, you’ll know how many times brighter the moon was than your calibration source (or equivalently, you’ll know the moons flux density at that time in calibration-source units, instead of in noise-source units).

Then you just need to multiply by the known flux density of the calibration source.  You can get that here, under “Radio”.  Select your calibration source, and enter the start and stop frequencies that you used to make the image (1355 MHz and 1435 MHz, in this case).  Cyg A and Vir A are truly constant, so you don’t need to enter the observation’s date.  Tau A, however, is fading, so you do need to enter the date – but it’s fading so slowly that you can round off to the nearest year.  For example:

Then press Compute.

Calculate the moon’s flux density in Janskys for both of your moon observations.  Here’s a table to help:

Portfolio Entry

When we look at the moon at optical wavelengths, we see reflected sunlight.  And the amount that we see changes a lot as the moon’s phase changes.

What about at radio wavelengths?  If it’s also reflected sunlight, the moon’s radio brightness – or flux density – should also change a lot with phase.  Is this what you found?

If this flux density is not changing – significantly – with phase, then you’re not looking at reflected sunlight, but at the moon’s thermal emission, from deep enough into the lunar soil that the temperature is always the same, despite changes in illumination (just like underground caverns on Earth hold their temperature all day, and all year, long).

Use Stellarium to look up the moon’s distance away on the date and time of your first moon observation (remember to switch the time zone to UT).  Its temperature is then given by:

where S is the flux density of the moon (which you measured), 1 Jy = 10-26 W/m2/Hz, 1 W = 1 J/s, k = 1.38 × 10-23 J/K is Boltzmann’s constant, c =  3.00 × 108 m/s is the speed of light, 𝜈 = 1395 MHz = 1.395 × 109 Hz is your central frequency, d is the distance to the moon during your observation, and R = 1737 km is the radius of the moon.  (Check out this equation’s derivation here.)

Look up the moon’s distance away on the date and time of your second moon observation, and repeat this calculation.  (These should be even more constant than your flux-density measurements, because they compensate for the moon’s changing distance away.)  Here’s a table to help:

Compare your measurements to a simple model, and to professional measurements, here:  Lunar Radio Temperatures.  Set the frequency to 1 GHz and turn on each measurement.  Also check out 2 GHz (your frequency, 1395 MHz = 1.395 GHz, is between these two).  How’d you do?

In your portfolio/blog entry, present your favorite moon and calibration-source images.  Describe your observations, including telescope, frequency, and mapping pattern.  Describe how you processed and calibrated the resulting images.

Explain whether you’re looking at reflected or absorbed and thermally re-emitted sunlight, and why.  Present your calculations of the moon’s (subsurface) temperature, and how you fared against the pros!

Remember, your blog audience is other astro-interested students!

Reading

Astronomy Today

Section 5.5:  Radio Astronomy

Section 5.2:  Telescope Size

Resolving Power

Section 1.5:  The Motion of the Moon

Lunar Phases

Skynet Algorithm for Single-Dish Radio Mapping I: Contaminant-Cleaning, Mapping, and Photometering Small-Scale Structures

Section 2.1:  Green Bank Observatory 20-m

Section 1.2:  Single-Dish Mapping