13 Measuring the Masses of Galaxies: Doppler Spectroscopy

MODULE 6:  Radio Doppler Imaging and Spectroscopy:  Weighing Galaxies

Color can be used in so many ways!

In Module 1, we used color at optical wavelengths to study the surface and cloudtop compositions of planets and moons in the solar system (including our moon).

In Module 2, we learned how to use color to “see” the temperatures of stars, correcting for the reddening effects of dust to get it right.

In Modules 3 and 5, we also used color to highlight heated gas and dust, and then used them as probes of temperature and density across the nebula.

In Module 4, we used color at radio wavelengths to help identify sources of continuum emission, be it thermal, bremsstrahlung, or synchrotron, and hence whether temperature, density, and/or magnetic fields were responsible for the emission.

In this module, we’re going to again use color in a new way, and again at radio wavelengths.  Where Module 4 focused on continuum emission, in Module 6, we’re going to focus on line emission.  Specifically on the 21-cm, “HI” emission line.

We’ve seen this in our integrated spectra – and have been cutting it out! – since Module 1B.  But what is it?  And how can we use it?

Like the H⍺ emission line at optical wavelengths, the HI emission line at radio wavelengths is produced by hydrogen.  But under very different conditions, and in a very different way.  The H⍺ emission line is produced by hot, excited hydrogen, when hydrogen’s sole electron drops from the second-excited to the first-excited state, emitting a 656-nm photon.

But the HI emission line is produced by cold hydrogen, with hydrogen’s electron in the ground state the entire time.  How can energy be emitted if the electron doesn’t change energy levels?  It turns out there are two, subtly different configurations of the ground state, with a (very!) small energy difference between them.

Both the proton and the electron are spinning charges.  And when a charge spins, it produces a magnetic field.  So both particles are simultaneously little bar magnets.  And you’ve played with bar magnets before.  It takes more energy to hold them together when their fields are aligned, and less energy when they’re oppositely aligned.  The same is true for these particles – so there are actually two distinct energy levels in the ground state itself.

How does cold hydrogen get into the higher of these two, ground-state configurations?  It usually happens collisionally – a gentle jostle from bumping into another atom’s all it takes.  Then what happens?  Well, nothing for about 11 million years!  But given enough time, it randomly decays to the lower-energy, oppositely aligned configuration, releasing the small energy difference between them as a 𝜆 = 21.106 cm, or 𝜈 = 1420.4 MHz, radio wave.

So this doesn’t happen often.  But hydrogen is – by far! – the most abundant element in the universe, and most of it is cold / in the ground state.  This is more than enough to make up for the slow de-excitation rate – we haven’t taken a single L-band data set where we didn’t see at least some HI emission.

But what can we do with it?

As with optical emission lines, we can use it to measure motion.  If an object’s moving toward us, its waves get compressed, and we measure a shorter wavelength.  If an object’s moving away from us, its waves get stretched out, and we measure a longer wavelength.  (And if an object’s moving perpendicular to the line of sight between us, we detect no wavelength shift).

This is called the Doppler effect, and can be used to measure the component of an object’s velocity along the line of sight, given by:

v = c × (𝜆observed – 𝜆emitted) / 𝜆emitted

or by:

v = c × (𝜈emitted – 𝜈observed) / 𝜈emitted

where negative values mean motion toward us, and positive values mean motion away from us, and where 𝜆emitted = 21.106 cm, and 𝜈emitted = 1420.4 MHz.

As we’ll learn in Module 7, spiral galaxies are full of cold, hydrogen gas.  In this module, we’re going to measure the HI emission line in two galaxies, and use it to determine:  (1) how fast the galaxy is moving toward or away from us; (2) how fast the galaxy is rotating on its axis; and (3) the mass of the galaxy.

That’s right, we’re going to weigh whole galaxies – let’s get started!

1. First Skynet Observation

Our first target is Andromeda – our sister galaxy!  Andromeda is big enough, and close enough, that we can resolve it with our radio telescope, despite the telescope’s 0.75° beam width / blurring scale.  So let’s map it!

The only differences from before are:

1. We want more spectral detail on the HI line.  So instead of low-resolution spectral mode, we’re going to use high-resolution spectral mode.  This narrows the bandwidth from 80 MHz to 15.625 MHz, and actually collects not one, but two spectra.  You get to pick the central frequency of each spectrum, but for this observation the defaults (1406.25 MHz and 1421.875 MHz) are good.

1. Andromeda’s big, so 6 beam widths × 6 beam widths isn’t going to cut it.  Instead, let’s map 10 degrees × 10 degrees.  This should be a 23-minute observation, so let’s submit only one per team.

Submit your observation!  (And add it to your team’s “Obs Info” sheet.)

1. Image Processing

Your observation is complete!  But before turning it into an image, let’s check out GBO’s preliminary data products, and see how they differ in high-resolution spectral mode.

First difference is that there is now not one, but two spectral plots, each showing a time-integrated spectrum for each of the detector’s two channels.  The higher-frequency spectral plot is on top, and in it we should see the HI emission line.

This is actually a combination of our galaxy’s HI line and Andromeda’s.  We’ll see how to separate them out in a moment.

2 × 2 = 4 spectra, so 4 images.  They should look something like this:

You can really see Andromeda’s extent in the top two, higher-frequency images.  This is because these include the HI line, and consequently, we’re mapping out Andromeda’s nearly edge-on disk (see optical image above), where the cold, hydrogen gas is located.

The bottom two, lower-frequency images do not include the HI line, and consequently aren’t mapping out Andromeda’s cold, hydrogen gas, but rather bremsstrahlung and synchrotron emission from all of the SFRs and SNRs near its center (Module 4A) – see how Andromeda’s more compact in the lower-frequency images?

We’re going to focus on the line emission, so the higher-frequency data.  Since Andromeda’s pretty close to edge-on, one side of that ellipse is spinning toward us, relative to the motion of Andromeda’s center, and the other side is spinning away from us, again, relative to the motion of Andromeda’s center.  So – one side is blueshifted relative to the center, and one side is redshifted relative to the center.

In Module 4, we learned how to make natural color images at radio wavelengths.  Now, we’re going to learn how to make Doppler color images at radio wavelengths.  Where the former transforms an object’s continuum emission into color, the latter will transform an object’s motion into color.  We just have to pick good frequency ranges over which to average the spectrum.

To that end…here is what’s called an “on/off” spectrum, taken with the telescope pointed at Andromeda’s center.  We’ll learn all about (on-off)/off spectra below, but in short, it’s a way to bring out / better see the HI line emission.

The complicated spike between 1420.4 and 1420.7 MHz is due to cold, hydrogen gas in the Milky Way – which we have to look through to see Andromeda.  The broader distribution between 1420.5 and 1422.9 MHz, and centered on 1421.7 MHz, is due to cold, hydrogen gas in Andromeda.  We’re going to take these frequencies, and break them into thirds:  (1) we’ll average over 1420.5 – 1421.3 MHz to create our red layer; (2) over 1421.3 – 1422.1 to create our green layer; and (3) over 1422.1 – 1422.9 MHz to create our blue layer.  Note – this avoids almost all of the HI emission from the Milky Way, cutting it out of / making it invisible in our images!

Radio Cartographer:  First, we’re going to make an image covering the full range of Andromeda’s HI line:  1420.5 MHz – 1422.9 MHz.  We will use this image as a luminance layer for the other three, reducing noise in our combined image.  But more importantly, we’re going to use it to calculate the best-possible time-delay correction, and then use this value to correct the other three images – it’s important that all of these images use the same time-delay correction.  So on your observation’s page on Skynet, your Image Processing Basics settings will probably look like this:

There is unlikely to be any RFI in this, very narrow frequency range, so next you just have to enter the minimum and maximum frequencies, like this:

Then, under Contaminant Cleaning and Surface Modeling Details, (1) increase the 1D Background Subtraction Scale to 8 beamwidths, to accommodate for Andromeda’s greater width, (2) set the 2D RFI Subtraction scale to the Faint Target preset, but then (3) change the Surface Model Weighting Scale to “2/3 (display-quality image)”.  The result will look like this:

Let’s Gooo!  Give the algorithm a moment to run, and then refresh the page.

Now, press “Create New Cartographer Job” – we’re going to run it again, but with two changes.  First, change Time-Delay Option to “custom”.  It should default to the value that was just measured.  For example:

Second, change the minimum and maximum frequencies to those for our red layer (1420.5 – 1421.3 MHz), like this:

Let’s Gooo!  Then to those of our green layer (1421.3 – 1422.1 MHz).  Let’s Gooo!  Then to those of our blue layer (1422.1 – 1422.9 MHz).  Let’s Gooo!  (You can run all three of these simultaneously.)

Once each completes, click on it at the bottom, and then download it.

Afterglow:  Upload your four images into your workspace, and open them.  Check the main image in each.

If all looks good, right-click on each main image and select “Remove Image From File”.  Then, group them into their own group.  Drag the luminance layer above the other three, and set its color to gray and its blend mode to “luminosity”.  Then, color the red layer red, the green layer green, and the blue layer blue.

Almost there – to adjust the composite image’s brightness and contrast, go to the Display Settings tab (top tab on the right).  View the display settings for the luminance layer, and under Color Composite Tools, select Link All Layers (Percentile).  I recommend using the midtone stretch mode, and adjusting the midtone level.

You’ve made a Doppler color image of Andromeda!  We’ve colored the center green – relative to it, which side is spinning toward you and which side is spinning away?  You are seeing Andromeda rotate, in color!

Note:  A fun extra is to add the Milky Way’s HI emission (e.g., 1420.3 – 1420.5 MHz) as an additional, narrowband layer, using a different color.  And/or, the SFD dust map (essentially IRAS 100 micron), from the Skyview Virtual Observatory (see Module 4B for details).

1. Andromeda Calculations!

Line-of-Sight Speed and Collision Timescale:  First, let’s figure out how fast Andromeda is coming at us.  Use the second Doppler effect equation above, and the line’s central frequency of 1421.7 MHz, to calculate Andromeda’s speed in km/s.

Okay, but how fast is it moving across the sky (i.e., perpendicular to the line of sight)?  This is called its “proper motion”.  We’ve been taking pictures of Andromeda for a long time now – long enough to see if it’s moving across the sky, with respect to background galaxies.  And it isn’t – not one bit!

But wait – doesn’t that mean we’re on a collision course?  It does!  Let’s figure out how much time we have left.  Look up Andromeda’s distance away in km, divide by the speed you just calculated in km/s, and then divide by the number of seconds in a year (≈𝜋 × 107 s/yr).  How many (billions of) years do we have left?  (We’ll explore this further in Module 7).

Rotational Speed and Enclosed Mass:  Now, use the same Doppler effect equation, and either of the line’s edge frequencies – 1420.5 or 1422.9 MHz – to calculate how fast Andromeda’s moving toward us at its edge.  How much does this differ from its motion toward us at its center?  This difference is Andromeda’s rotation speed at its edge.

Okay, but how much mass has to be there, interior to this orbit, to keep material that is moving at such a speed in a circular orbit – not flying off, and not falling to Andromeda’s center?  Newton showed this “enclosed mass” is given by:

M = v2r / G

where v is the orbital / rotational speed at Andromeda’s edge, in km/s; r is the radius of the orbit / Andromeda, in km, and G = 6.67 × 10-20 km3 kg-1 s-2 is Newton’s constant.

Cool – so we just need Andromeda’s radius in km.  As you did in Module 3, use Afterglow’s measurement tool under the Plotter Tab (third tab from the top on the right) and estimate Andromeda’s radius in arcminutes.  Divide it by 60 to get it in degrees.  Then by the small-angle formula:

r in km = angle in degrees × 𝜋/180° × distance in km

Plug this into Newton’s equation to calculate Andromeda’s mass in kg.  Convert this to solar masses using 1 solar mass = 2 × 1030 kg.

For comparison, the Milky Way’s mass is roughly 1 – 1.5 trillion (1012) solar masses.  Who’s the bigger sister, Andromeda or the Milky Way?

That’s a lot of mass coming our way…

1. Skynet Observation (Spectroscopy)

Congratulations – you’ve weighed Andromeda.  Let’s try it again, for something farther out.  Pick an edge-on spiral from the following list (one per team!):

Since these galaxies are much farther away, they’ll all be smaller than our telescope’s beam width, and consequently we can’t Doppler image them like we did Andromeda.

However, we can still collect a spectrum.  Two differences from before:

1. On the Receiver Settings page, change the center frequencies to 1403.375 and 1419 MHz.  This should keep the action in the higher-frequency spectrum, for all of these targets.

1. On the Path page, switch the path type to “On/Off”, and configure the other boxes like this:

Do this and the telescope will take two pairs of spectra – one on the target, for 300 seconds, and another 2 degrees away, for another 300 seconds.  When done, it will calculate (on – off) / off at each point in each spectrum.  This will remove (most) of the instrumental oscillations, like you see in the first spectral figure in Section B, leaving (mostly) astronomical signal from the target.

Some of the on/off spectra will look like the on/off spectrum of Andromeda in Section B – incomplete cancellation of the Milky Way’s (very!) strong HI line, and a bump for your target’s HI line.  Others will look like this:

Here we see:  (1) residual – but much smaller! – instrumental oscillations; (2) incompletely canceled – but much smaller! – Milky Way HI, around 1420.4 MHz; (3) a blip in the middle of the spectrum – it’s not RFI, but an artifact of the signal processing (learn to ignore it); and (3) a “horn-shaped” HI emission line from the target galaxy.  The left horn is being emitted from the edge of the galaxy moving away from us, and the right horn is being emitted from the edge of the galaxy moving toward us.  Note – these horns need not be symmetric.

This is a 10-minute observation – again, let’s submit only one per team.  (And add it to your team’s “Obs Info” sheet.)

Once it comes back, use the Live Spectrum tool to measure your galaxy’s HI line’s minimum and maximum frequencies.  Average them to get the frequency corresponding to emission from your galaxy’s center.

1. Weigh Another Galaxy!

Line-of-Sight Speed and Hubble’s Constant:  As before, let’s start by measuring your galaxy’s motion toward or away from us.  Betcha it’s moving away from us…  Why?  Because once you’re farther away from Andromeda, nearly every galaxy is moving away from us!  Why?  Because the universe is expanding, carrying everything apart from everything else.

Want to calculate how fast the universe is expanding?  Look up your galaxy’s distance away in Mpc, and divide this number into the speed you just calculated in km/s:  H0 = v/d.  What you get is something called Hubble’s constant, which measures the rate at which space is currently expanding.  You should get something around 70 (km/s)/Mpc.

Rotational Speed and Enclosed Mass:  Now, as you did before, calculate (1) your galaxy’s rotation speed at its edge, and (2) your galaxy’s mass.

For the latter, you need your galaxy’s radius, which you could get by taking a picture of it at optical wavelengths, measuring its angular radius in Afterglow, and, knowing its distance, calculating its physical radius using the small-angle formula.  But we’ve already done this for you, and put it in the table above!

Again, convert your measurement to solar masses, and compare to the Milky Way.

Portfolio Entry

Another module, another – totally different – way to use color to do physics.  Time to enlighten the masses!

In your portfolio/blog entry:

1. Describe your observations – you did two very different observations this time.  One for something you could resolve (even at radio wavelengths!), and one for something you couldn’t.  Describe their unique spectral requirements and settings.

1. Describe how you created your Doppler LRGB image.  If you included extra layers, describe them too.  Show us your final image!

1. Describe your Andromeda calculations, including collision timescale with the Milky Way and mass.  How does the mass of Andromeda compare to the mass of the Milky Way?

1. For your more-distant, edge-on spiral galaxy, show us your (on-off)/off spectrum, and describe what’s there such that someone not used to looking at these can understand what’s going on.

1. Describe your calculations for this galaxy, including Hubble’s constant and its mass.  How does the former compare to expectations?  How does the latter compare to the Milky Way?

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