Dark Matter class, 30-01-2024
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Good morning. The first homework is due a week from Thursday. I know it's February 8th, so it's still January, so that seems like a long time away, but it will sneak up on you.

Any questions before I launch in today? Okay I want to keep talking about how we characterize the baryonic components of galaxies. We can subtract that out before we can talk about what's dark. And so we had ended with this relative sizes of galaxies from big through Milky Way, which is not terribly small.

But it is compact, too tiny, and so galaxies exist. Oops, that was quick. Over this huge dynamic range of both mass and thighs. So this is the stellar mass. Basically the luminosity. How much starlight there is, plus some prescription for the stellar mass to light ratio. Over this range, it doesn't matter too much what that is it just goes down to a couple hundred thousand, that's a modest sized globular cluster, but if it's out in the middle of intergalactic space, all by itself, you call it a galaxy, so this is, these are tiny galaxies, all the way up to giants. Several, not quite to a trillion solar masses in stars, but close. Showing both early and late types, so the round things are late types, spirals, irregular things that rotate. And triangles are the early types. elliptical galaxies. Those tend to be the most massive things, so they populate the very most massive edge in the way spirals don't.

Those massive spirals like that big one on the previous slide, you should see 2885 or maybe two or three. Times 11th. Our nearest neighbor, Andromeda, is about 1. 4 times 10 to the 11th, somewhere in that ballpark, so it's right up in there. The Milky Way is 5 or 6 times 10 to the 10th, so it's right up in here.

And so that big blob of points are the classical bright galaxies that define the Hubble sequence. Those are the things that are easy to see. But like I say, they exist over six, almost seven orders of magnitude. Huge range in mass, stellar mass. And you would naively expect there to be an associated range in the dark matter halo mass, the things that reside there.

That's the natural assumption. We'll see as we go along, that's not quite true. That was the obvious thing to assume and we worked all through the 90's to try to avoid having to break that assumption and the odds we struggled with. Maybe we had to, and by the teens it was like, oh yeah, that's what happens.

I say that just so you know what the mental progression is, because it matters. In this problem, the history of things matters, especially when it's not a solved thing. It's one thing to teach the history of something like, Oh, Michelson and Morley didn't see the shift in the speed of light, so they knew there was no ether.

That grossly oversimplifies the actual history of what went on. Maybe it's not too bad to oversimplify it that way, since it's now pretty well established. That's the end game. But that's not at all. A good description of how it actually happened. And we are still in that process of figuring it out. Ah, okay Some of the smallest galaxies we know about are also early to types, these dwarf spheroidals, in the sense that they're pressure supported blobs.

Ah, but there are some late types that are known there. And the reason there are a lot more dwarf spheroidals known is that those are the satellites of Milky Way. And Andromeda, they're really close to us and those tend to be clustered around bright galaxies. Whereas the late tides, those are the least clustered population you can find.

You have to find themselves, by themselves, off in the middle of nowhere, so they're a bit harder to survey for. But there are examples of that. Very tiny things. So galaxies exist all over this range. There's no reason to think that less massive things couldn't exist, right? It's just hard to see them if you don't see them.

If you had brighter things, bigger galaxies, you would see them. So this is a real upper limit to the mass. Same thing for size. The half light radius, just that radius that includes half the light. It ranges from, a couple hundred parsecs up to tens of kiloparsecs, but most galaxies are a few kiloparsecs in typical size.

There's not quite as huge a dynamic range in that.

And these lines show lines of surface brightness, so that's high surface brightness, 100 solar masses per square parsec. Down to one solar mass per parsec. That's pretty low, so you don't see many things. And then a lot of the ones you do are these so called ultra diffuse galaxies, which are basically just really hard to see, hence the name.

Okay, so The half light radius is a very crude way of approximating the light profile because you can have different shapes. So there's a galaxy with a very nearly pure exponential disk.

Low surface brightness is a function of radius,

something like that, but you could also have a galaxy with a bright central ball, something like that, and then dips down and then has a really extended, very low surface brightness disk, and the half light radii of these two different light profiles might be pretty close to the same, depending on how a lot of the light is in that central component.

So it's very compact. So that, as a size, can be a little misleading. You'd want to have some other information. Concentration, like the radius, ratio of the 80th to 20th percentile, something like that. And if you do that, in fact, the ellipticals, the early types and late types don't sit on top of each other so well.

Half light babies just does that. Okay, but so here's one galaxy, it's in fact this exact galaxy that's doing this and one can make a fit to that. And this is the so called exponential disk, comes up a lot in the discourse because it's a handy shorthand for describing the light distribution of a galaxy like this.

You fit ellipses, each of those ellipses corresponds to one of these points because it has some major axis radius. And the surface brightness is an isophote, a constant brightness that goes down as you go further out into that plane. Now it's not perfectly exponential there are little bumps and wiggles here, and those matter to dynamics, but it's a first order equation.

The exponential is a good approximation, and it's a simple function. It just has a central surface brightness, i. e. that intercept, and a scale length. Basically, the inverse of that scale length is the slope of that line. That gives you a different way of Quantifying the sizes of galaxies, and like I say, late type galaxies, spirals, late type spirals, irregulars, are pretty well described usually by such an exponential disk.

Reasonable approximation. Not perfect. And bright spirals often have central bulges, so you see an upward part of that, and you can fit both components, et cetera, et cetera. Now I've written this here in terms of linear units, solar luminosities per square parsec. Those are both physical units. How much luminosity, how much physical size.

And that is related to the observer's magnitude per square arcsecond by this funny formula, which you can find on the useful numbers page. The zero point depends on the definition of the bandpass, so that's the absolute magnitude of the Sun, in whatever bandpass you're working in, we'll change that.

And of course, you see the minus 2. 5 log, that's just the magnitude definition. You can derive that yourself it's not Too hard, but it's useful to see where the heck that comes from. A weird zero point, that's just how it works out, given the funny units we work in. And I mention all that because sometimes you'll see magnitudes per square arcsecond, sometimes you'll see solar luminosities per square parsec.

Those are equivalent. In the local universe, surface brightness is distance independent because as an object gets further away the size shrinks at the same rate, that is, the surface area shrinks at the same rate that the observed flux does, so surface brightness is a conserved quantity in a Euclidean geometry.

Of course, we don't live in a universe of the Euclidean geometry, and as you get to largest redshifts, then you have to worry about 1 plus z corrections, i. e., the fact that Euclidean geometry is only A valid approximation pretty nearby to us. So this ceases to be distance independent, but it's good to keep in your mind that in the Euclidean limit of nearby galaxies, that's distance independent.

Okay. This is just an example of those ellipse fits. There's a picture of a, in this case, an elliptical galaxy, and one fits ellipses. They go all the way in, all the way out, and you get a lot more than just the surface brightness out of it. You get the position angle axis ratios, et cetera, which tells you something about the intrinsic orientation and shape of the galaxy.

Now this is projected on the sky, of course, so you don't know the full 3D shape, you just know that from our viewing point it has this shape. In the z axis it could be very long, along our line of sight, or it could be very compressed. We don't see that right away, though, if you try to build models that include Stars that have, that are on orbits that would support such a structure, you can find that there are subtle effects like a twisting of the isophotes might indicate that there's some triaxial structure there, at least proglade or coglade beyond what that's a subtle business. The only point here is that Each ellipse is a line of constant surface brightness, a so called isophobe. And so when we make these plots of surface brightness against radius each of these points corresponds to one of these ellipses, the surface brightness around that ellipse.

And you can do it in different bandpasses, so this is the same galaxy, in the J band, in the V band. And they look Similar, but the profiles are not the same. That means there's a bit of a color gradient there. So the properties of the stars are changing somewhat as you go out. In this case probably the metallicity changes as a function of radius.

It could also be age. Okay, so that's just how we go about measuring the light distribution. There are other things.

Galaxies are made of stars as well as gas. So this is a spiral galaxy NGC 6946 in terms of in different bands. The optical image, it's getting hard to see, isn't it? Oh, seems to me there'd be too little light in here. There you can see that better now. So that's the optical image. This is the near infrared image.

That's a pretty good map of where the stellar mass is. Alright, so there's no bandpasses perfect for relating the starlight to the stellar mass. But that's pretty much as good as it gets. So you can see there's a strong concentration in the center. It's only about 5 percent of the total light is in that little central bulge there.

But, as you'll see, it actually has a noticeable effect. Impact on the kinematics, that mass and its distribution matters. You can see that there is mass in the spiral arms. It's not just a completely axis symmetric thing. And here you can see dust flames that are much less prominent there. Stellar mass.

And then there's the atomic gas, so this is the 21 centimeter flux on the same scale, right? So it does tend to go rather farther out than just the stars. That's a one to one scale there. And the arms tend to continue, even if you don't see them anymore in the optical. So now this is a plot. of the cold gas mass, which is usually dominated by that atomic gas.

Against the stellar mass again, so again you see this range of a factor of a million in gas. Not quite as big a range here, I've made it the same, so it's one to one. And that line is the one to one line. Classical spirals like the Milky Way are these red points. These are the so called early type spirals.

And you see they're mostly stars, right? They're below the one to one line of gas distiller mass. The blue things are late types, SCDs, Irregulars, basically. And they're all down here. Notice they're mostly gas. On average, they're two thirds gas. So a few of them are more stars than gas, but most of them are above the line.

And that's just typical once you get below this mass of a few times ten to the nine. And then there are also the early types. The ellipticals tend to be devoid of cold gas, or have very little. So the squares are actual detections, and then the triangles are just upper limits. So these things are all just raining down.

What's missing here?

There's a giant gaping hole here, right? And that's just a selection effect. The survey from which this comes just doesn't see anything below 10 to the 10 solar masses. There are surely elliptical galaxies down here. We just haven't gone out and measured them in any systematic way. Same thing, these irregular galaxies, while they seem to dominate this plot, they're faint, and so they're rare in surveys.

If you just do A straight up survey, magnitude limited survey. You see the bright things, and that's those things. So there's been a lot of work done on those things. To quantify that,

here's the same thing. Yearly type giant spirals like the Milky Way. The so called dwarf irregulars, big range of mass. Dwarf is a terrible bit of nomenclature, so much bad nomenclature in astronomy. But people, when they say a dwarf galaxy, what they mean by that could differ by a couple of orders of magnitude.

So Usually, they mean low mass or low luminosity, but how low depends on the context. Again be aware for local variations in that dialect. Okay, but galaxies exist over this entire range of Stellar mass and gas mass. And you know the logarithmic nature of this hides a bit the fact that you couldn't live too far away.

You already have galaxies here that are more than 90 percent gas, more than 90 percent stars, so that's a huge range in the gas fraction. Basically get all the above. Okay, so if you go out and do a big survey like the Sloan Digital Sky Survey, This is the raw numbers of galaxies that you find as a function of stellar mass.

They're all up here, right? These things. And that is a selection effect. Basically a survey will always find the brightest objects because you can see them the farthest away. That's all there is to it. So

We all get to Earth and we absorb some part of the sky. There's some really bright thing out here, some really faint thing here. And you can calculate the flux that you receive from that, right?

So this has a really big luminosity, this has a really tiny luminosity, so they produce the same flux, and your survey is flux limited typically at a very different distance. And so basically the volume You could have seen this thing out to as small compared to the volume out to which you can see that thing.

So as a practical matter of flux limited things, lots of these things just because they're sensitive to them over a much larger volume. That's all there is to it. So this upper limit is a real physical cutoff. There just aren't any galaxies brighter than 10 to the 12th. And in fact, the number density is falling exponentially.

But this low tail to low luminosities is entirely That selection effect, that you're seeing fewer and fewer of these low mass things, just because you can't see them very far away. And so what you'd really like to know is the intrinsic number of distances per unit volume. So you have to correct for that effect, which you can calculate.

Getting it right is technically tricky, but you can do it.

And note that we know about galaxies down to below 10 to the 7th solar masses in this plot, and really we know them down below 10 to the 6th, but even in Sloan, doing this vast area of the sky, there are almost none of these galaxies present here. Most of these things are known from other smaller but deeper surveys.

And so that's a huge correction factor that has to be made. And so here is an example of calculating the luminosity function. So that thing up there is the total luminosity function from the gamma survey. Moffat et al. And I picked that one to illustrate it because they use near infrared magnitudes that are good tracers of the stellar mass.

So this is a reasonable estimate of the stellar mass function of the stellar masses. And so I've dropped on top of that the raw numbers, and you can see this is tiny, Whereas the intrinsic numbers are large, so this is, once you make a correction for that volume, which is, as I say, you can calculate, but it's a really touchy calculation, because it's a huge volume correction.

Once you do that, then you have the number density of galaxies per unit volume and you typically find that those numbers go up, but only slowly to lower luminosities. There are a lot of these things. There are more of these things, but not a lot more. And people have worried a lot about exactly what that faint in slope of the luminosity function is, because the corrections get large here, so it's easy to go wrong.

But, people pretty consistently find that. This is steep, but not so steep as to put a lot of mass there. That is when you integrate over this

mass, is the convolution of the mass per galaxy times the number density per galaxy. And the mass wins. So basically, most of the mass in galaxies resides at the bright end. It would have to be steeper than this for the small things to be competitive. That's important. Because we can also calculate what we expect for the mass function of dark matter halos.

And that's very steep. And approximately, roughly, equals of mass in different decade bins. So where this has this characteristic shape, an exponential cutoff at high mass, a relatively low fainting slope. The dark matter mass function is approximately a power law and a steeper one. And that right away tells you that the assumption of a one to one mass halo to disk is not satisfying.

You can't have that. If every dark matter halo formed an equal number of its baryons into stars, then this would just be a shifted version of dark matter. Matter mass function. And it's not. Understanding that is one of the reasons we need to understand how baryons cool and condense, how they form stars, how feedback can impact The continued formation of stars, and that's generally the reason we think that there's not a whole lot the reason we think feedback is important to studying this mass function.

That, we'll get on to that in more detail, but there's a huge difference here. And there's also a difference between early and late tides. So we saw in that earlier plot that there were more early tides at the high mass edge, and you can see that in the difference in the mass function.

Once you get to the most massive things, it's all ellipticals. And then as you get to low mass things, it's all those or irregulars that were gas rich enough as they should. We even be talking about the stellar mass at that point because there's more gas than stars. So you have to bear in mind all those things that, that people talk about are going on.

Okay

as well as the stars, there's the interstellar medium. I'm going to talk a bit about what goes on there. You have the atomic gas, the molecular gas, and 

Ionized gas as well as dust in the Milky Way. We measured all these things pretty well. The dust to gas ratio has some approximately fixed value, which is of order 1%. The dust doesn't weigh a lot. But if you measure a column of gas, you can be pretty sure that there's an associated amount of dust with it.

Most of the mass is in atomic H1. The molecular gas and that's more extended than the stars. Molecular gas is more or less distributed like the stars, it tends to be in the spiral arms, and it tends to fall off exponentially, as quickly as the stars do, unlike the atomic gas. And the ionized gas is actually most of the volume, it's little of the mass, but it's most of the volume.

These things tend to be in denser clouds and there's basically a hierarchy of structures in the interstellar medium. It's a big, messy place. Okay. These are useful links if they're still there just about the details of Line formation for both atomic gas and molecular gas but just so you have a clue about the physics of what we're detecting here the atomic gas is traced by the 21 centimeter line, which comes from the spin flip transition.

The upper level is when the electron and proton have their spins aligned. There's a slightly lower energy level when they're anti aligned. And so that transition emits a photon in the radio band. It's a very fine transition, right? A 21 centimeter photon is not very much energy. So there's not a lot of motivation for this transition to happen.

And so the rate is low. Oh, I'm getting ahead of myself. The frequency you can Work out to high precision from what we know about the Irish Atom. So it's 1420 megahertz on your radio dial. And that's literally on your radio dial. That's why people have to, The government established radio quiet zones around Green Bank in West Virginia and out in New Mexico where the Very Large Array is, that's where the radio telescopes work.

It's become very hard to enforce those in the days of cell phones. Anyway here's another example of how the atomic gas is typically more extended than the stars. So this is your exponential. Disc with some scale length, like that, and that does not, that same scale length does not explain the gas disc.

So you have to measure those components separately, and the 21 centimeter line is a great way of doing that. So here's where I was getting on to the transition rate, how often this happens. Again, you can Bit of quantum mechanics, haha, just a bit. It's a silly number. And for any one atom, the half life of this transition is 11 million years.

Doesn't happen very often. But in the conditions that the interstellar medium is at Where you see this stuff it's basically an equilibrium just from collisions, right? There's a little bit of thermal energy. Atoms bump into each other once in a while, and every once in a while some of that kinetic energy gets imparted to flipping the spins.

Basically, this is in equilibrium, just thermally. And it's not too sensitive to the temperature, which is nice. What that means is that for typical conditions of temperature and density, which in principle the flux could depend on a lot, It doesn't depend on too much, and so you can just write down a formula that the mass of atomic gas follows from the flux that you measure, and the distance, and some constant that depends on all those atomic constants that we just flashed up there, and nobody wants to think about.

This is really handy, right? For the Stellar luminosity, we measure that, and then we have to worry about some stellar population model to assign a mass to light ratio. Here, the mass to light ratio is basically given by the atomic physics. So you count 21 centimeter photons, you're basically counting How many hydrogen atoms there are out there, and you know how much each one weighs, so that gives you the mass in atomic gas.

So this, the units here just gives you solar masses for distances and make it parsecs. And FLOX, which is radio astronomers Right? Every branch of astronomy has to do its own silly thing. The units that are typically used here are Jansky kilometers per second. Jansky was the original radio astronomer, so that's a luminosity.

But over, integrated over Now, there's a definition of a Jansky of 10 to the minus 26 watts per square meter per hertz, and because galaxies are rotating, one side's coming at you, the other side's going away, and so that kilometers per second is there because you have to integrate over the red shifts.

And so here's an example of how this is done. Just with a single dish radio telescope. A large single dish radio telescope like the former Arecibo telescope or the current Green Bank telescope. Is sensitive to this and can detect it in, a few seconds often, certainly a few minutes.

But it has a big beam. It just is a blob on the sky. So what you do is you scan your receiver through frequency until you hit the redshift that the galaxy is at, and then you see this characteristic thing. So the Jansky's at watts per square meter per hertz. Is how bright the thing is, the brightness, temperature, technically, as the radio astronomers talked about.

But it's spread out in frequency, which we convert to kilometers a second. And so as you scan through, you basically integrate over that profile, and that's what that flux integral means in Jamski kilometers per second, summing up all the this double horn profile is pretty classical shape for spiral galaxies.

Why do you reckon that is?

Think about what's going on. You see some spiral galaxy rotating on the sky, and you're basically seeing the side that's coming at you. And the side that's going away from you. And if you further think about that distribution of atomic gas, it goes way far out. But it's not very bright in the center.

Unlike the stars, it has its peak brightness in the center. That's not true here. There's actually often a dip in the center where there's not a whole lot of Atomic gas, or if it tends to be in molecular gas, in the center parts of those galaxies. And so you're basically seeing one side and the other, and so you get that double side thing.

And the width of this beat is a crude measure of the rotation speed of the galaxy, basically how fast it's going around. Yeah? I don't really

So you're looking at the galaxy from the side, and you're seeing it rotating like one side coming towards you and one going away from you? Yeah, so this is we always write this in terms of kilometers per second because we're used to thinking of the Hubble expansion. But this is really frequency.

And so you're tuning the radio telescope in frequency. There's nothing there. Oh, you hit the redshifted 21 centimeter line. And so Let's see if there's a nice We

can come back to this. But this is now what a resolved atomic gas disk looks like, observed with An array of radio telescopes acting together as an interferometer. So that plot we were looking at before, that's just single dish, doesn't resolve any of this, it's all a blob to it. So all it sees, as you scan in frequency, and that's what the colors are here, it's redshift and blueshift.

So this side is coming at you. As I was going away, and as you scan along, there's nothing at this frequency, nothing at this frequency. Oh! There's a big signal there. So that's that first peak. And then, at a higher redshift longer wavelength, shorter frequency, whoo, there's the other peak. So that's what we're seeing as we scan along.

And those are both radio astronomy at the same

frequency, but they're rather different kinds of observations. This can be done rapidly with a single ditch telescope, the other requires an interferometer and longer integrations, typically hours to days. And so what that means is this is cheap, and so we have this kind of data for lots and lots of galaxies, whereas the interferometric data we don't have so much for, a couple hundred at this point.

Yeah? Is this much higher that just , not quite symmetric or yep. The galaxy is asymmetric and there are statistics as to this and sometimes they are nicely symmetric, but this is pretty common and about half of galaxies are asymmetric at sort of 10% level in flux. If you split up here side and.

About 20 percent are grossly asymmetric, and that ratio is more than 10%. Asymmetries like that happen all the time.

Which is really annoying, as you might imagine, for building a mass model. We, of course, pretend it's symmetric, and go on. One way around that is to just take the galaxies that are more symmetric and say I know something's goofy here, I'm not going to worry about it. Or, you can engage with that and say, okay, I'm going to approximate it as axisymmetric, and then what you find is that the scatter in your various relations goes up, in the sense that you would expect, having included some have tolerated some amount of asymmetry in the sample.

Okay, so that's just the atomic gas. The molecular gas is traced in a variety of ways. The traditional way to do that is with carbon monoxide emission. That just happens to be a molecule that is relatively easy. to detect in regions that are conducive to the formation of molecules in interstellar space.

So those regions tend to be 30 Kelvin or less, so very cold. And in fact, it's it's something funny in the history of cosmology. We celebrate the discovery of the microwave background, which is really the millimeter background. It just happened to be noticed first in microwaves. And that was in the mid 60s.

If you go back a decade before that, in the 50s, people were doing this. And I think it was McMullen, noticed that, geez, I did a range of temperatures in the molecular gas, but it never seems to go below a few Kelvin. There's a reason for that. The whole universe is flooded with 3 degree radiation.

And he was seeing that gas was in equilibrium with that. In retrospect, that was a good detection of the microwave background, but just nobody knew what it meant at the time. Okay, so more typically, they're 10 to 30 Kelvin. This is the densest, coldest part of the interstellar medium, so this is where stars form.

By dense, we're talking greater than a hundred. Maybe a thousand, maybe ten thousand molecules per cc. That's still a very hard vacuum to you and me, right? It's denser than the atomic gas, where it's maybe one, ten atoms per cubic cc. But still, as the interstellar medium goes, this is dense, and so this is places where gravity can start to win out over gas pressure, and you start to form stars.

And not surprisingly, the gas in this phase tends to trace out the spiral arms, which is where you see stars forming, mostly in. And spiral galaxies. So those things are all tied together. Fortunately, as a mass component, it doesn't weigh too much. It's not zero. But it also traces out where the stars are, so at some level it just gets absorbed into the the uncertainty in the stellar mass to light ratio.

Now, CO is not the greatest tracer because it depends on the metallicity. And it forms in conditions that are not exactly the same as molecular hydrogen, which is presumably most of the mass. So we're using carbon monoxide as a proxy because it's a line we can detect. But, converting that to what you want to know, the total gas mass there, the mass of molecular hydrogen is a bit dodgy.

So this is one formula that's offered, again, the same structure as what for the what you saw for the atomic gas. Depends on the square of the distance, of course, depends on the flux that you absorb for CO, but this constant is not as well determined. As it is for atomic gas. The atomic gas is just fixed by the the atomic physics of the hydrogen atom, relatively straightforward, whereas here you have to do some conversion how much CO flux corresponds to how much molecular hydrogen.

And that's surprisingly hard to do. Diatomic molecules like molecular hydrogen tend to have very boring radio spectra. It takes a lot more energy than that to excite them. So they're essentially invisible to radio telescopes. And in fact, you have to go well in the UV to CH2 directly, and then you only see it in absorption.

It's just a big pain in the watcher's hoochie to see one of the most common molecules out there. This number, this X factor, XCO, has been calibrated by looking at giant molecular clouds in the Milky Way and nearby, like the Large Magellanic Cloud trying to estimate their dynamical mass, so let's hope there's no dark matter there and so that dynamical mass is dominated by the just the gas and mostly the hydrogen.

And so then you can say, okay, there's this dynamical mass of hydrogen, there's this much flux in Carbon monoxide, and you can get some number out like this. This number, people argue about all the time, right? It's uncertain, for the reasons that should be obvious from what I just said. And at the factor of two level, and the thing that's amazing to me is that it's It doesn't vary more than that because you would expect it to vary a lot with metallicity.

Also, it depends on the UV flux. These molecular clouds can only be sustained where there's enough dust to shield. The molecules from the interstellar starlight, the ultraviolet, just the ambient ultraviolet, will break them up, so you need a fairly high opacity. The dust also, those grains, play an important role in forming the molecules because individual atoms will stick to the grain, meet up with something else, and say, oh, I'm now a carbon monoxide molecule, and go back off into the gas phase.

So a lot of really rich molecular physics. As it goes on in the interstellar medium. Which we won't care about because we just want to have some idea of how much mass there is. Okay, but you should be aware that there are all those issues. And there are other proxies. Carbon monoxide is just one proxy.

Another proxy is the dust luminosity at 100 microns. Basically, you have so much dust in it. It's associated with molecular gas, so that's another proxy. Calibrating it has all the same issues, but in principle you can do it that way. Another way to do it is star formation. So this is a star formation rate measured from H alpha.

There, the astrophysics is that when stars form, some of them form with high enough mass that they're surfaced. Surface temperature produces enough UV photons to ionize the surrounding hydrogen, and those show up in various ways, including Balmer emission, and again, if you go through it, you need to ionize a hydrogen atom in order to have a Balmer H alpha photon come out.

And so you can calculate the rate. It's by counting H alpha photons, you're basically counting ultraviolet photons. And then you can relate that to how many stars do I need to make that happen. That's also very uncertain because it depends on the initial mass function. What is the spectrum of masses of stars?

How UV radiation? Only a small fraction of the population does that, so then there's, even if you get that part right, there's a big extrapolation to all the stars that are associated with those massive stars. There are entire reviews written about this, but that star formation is going on in association with molecular mass, so you need The molecular gas there in order to form stars.

It's basically a sequence of condensation. You can have atomic gas, and if that gets dense enough and dusty enough, it can become molecular gas. And if that cools enough, it can start to form. Not tools that form stars. So here the proxy is the star formation rate. And that correlates with stellar mass.

These black points are things where CO is measured. These blue things are things where the CO is not measured. Because the star formation rate is so low, if you ask if the ratio is the same as up here, What would I see? And the answer is you wouldn't be able to see it. So this is the proxy here.

So this is now the molecular hydrogen mass, function of stellar mass. Those galaxies are all measured. But if we extrapolate along, you get the same thing. And the interesting thing is that what this works out to be is that on average, there is some real scatter there. But on average, the molecular gas mass is about 7 percent of the stellar mass.

Like I say, the stars are already a proxy for the molecular gas from which they were born. Okay and But as I say, the uncertainty in the stellar mass to light ratio is 10 to 20 percent at best, so this is in the noise at that level. All right so far we've just talked about the hydrogen associated with the hydrogen, be it either in the atomic or the molecular phase.

There's also helium, most of which is primordial, and there's also everything else in the periodic table. So the stellar folks started the Convention of referring to the fraction, the mass fraction of each thing. So x is the mass fraction in hydrogen, y is the helium fraction, and z is everything else.

So if somebody tells you that the solar metallicity is 2% that's what they mean. Everything above hydrogen and helium in the periodic table is 2 percent of the mass of the Sun. And the primordial helium fraction is very close to a quarter. Primordial hydrogen fraction is very close to three quarters.

But as time goes on that gas gets processed through stars into metals that get returned to the interstellar medium. At nauseam, it's more stars are born. And so This hydrogen fraction slowly goes down from its primordial value to something slightly smaller. And it's a very subtle effect, I think it's 0.

73 in the sun, something like that. So anyway one should correct the hydrogen masses that you're estimating for everything else, and this formula is an approximation that does that, because the oxygen abundance Also depends on the stellar mass. So again, we're using the stellar mass as a proxy to estimate these other things, the molecular gas mass and also the correction factor to get up to the total mass associated with the hydrogen gas mass.

And so here's some estimates of the oxygen abundance as a function of galaxy stellar mass. And there's a nice result from Sloan. Boop, M1991, that was me. I'm old. That was like the first big paper I wrote, but in fact, I'm old enough that the definition of solar oxygen abundance was higher, which is most of the difference in that.

Anyway anyway It's actually, it's, I want to mention that because at a time 12 plus log O to H, we add this 12 here so that the numbers aren't negative, so that a bigger number is more oxygen. I refolded against that in this paper, but since then I've been beaten down. So at the time, solar was 8.

9. And Milky Way is somewhere in this range. And my number made sense at the time, because the Milky Way is about a solar metallicity galaxy, like a star. The Sun is pretty typical. The Sun got, we have struggled to get the oxygen abundance right in the Sun. And that's a big deal, because one, it's what we can measure in other galaxies.

Two, it's the third most abundant thing after hydrogen and helium, the periodic table. Hydrogen, helium, oxygen, carbon, neon, nitrogen. And then iron. Just, whoop, skip on to the iron. Anyway Since 1991, the definition of solar change from here to 8.6, that's a drop of 0.3 decks. That's a factor of two.

So we didn't know in modern times the oxygen abundance of the sun to the factor of two. Since then, it's migrated back up to 8.7, 8.8, somewhere in that range. So this does a pretty good job of giving a solar metallic for. Milky Way like galaxy. You can also see a lot of these things are crazy high. Milky Way is abnormally metal poor by those things, so it's hard to do this, and it's hard to be sure you've got it right.

It's the factor of two level. For this purpose, we don't care. The variation on this is modest, and then the change That it causes the next is really modest, because you're just going from no metals to 2 percent metals, maybe a little more, a little less, right? And that's what's all in here. And so if you look at this, the power law is small, about to the 1 5th power, and its mass scale is huge compared to galaxies, to the 24, it's the 10 to the 11.

And so it's basically just a tweak on this thing. And I mention all that because, we've worked out these scaling relations, so if you really want to try to estimate these things right, you can do that. And the data are getting precise enough that you can start to pretend to worry about that.

And then dust. We've talked about the different gas phases. The ionized gas within the disk is negligible, takes up a lot of volume. It has really low density, so it doesn't add up to much mass. And then the dust itself also has a negligible mass, but it has a whopping powerful effect on the luminosity that we observe.

So these are spectral energy distributions for galaxy models over a huge range in wavelength. So this is basically the spectrum that you would expect for a galaxy. From the UV, just a thousand angstroms there all the way out into the pretty far infrared, 100 microns, is where Dust glows. It is dust absorbed starlight, as well as scattering it.

It warms up and responds, it then re radiates in the infrared. And so there's this huge amount of luminosity that comes out that depends on how actively the galaxy is forming stars. So you see all these little blue lines? That's the emergent spectra, and there's an example of data that these models are trying to describe.

There it is in the infrared. In order to explain this infrared luminosity, you need the original spectrum to be up here. So this galaxy These galaxies are, these are all model galaxies, there's a lot of data there, but in order to explain this infrared luminosity, you've got to have a whole lot of ultraviolet luminosity.

And you don't see that directly. You see some UV gets out, but most of it has gotten absorbed by the dust, which heats up and re radiates in the infrared. This is one reason why you really don't want to use the blue part of the spectrum to estimate stellar mass. A lot of the UV light has been redistributed.

Absorbed, re emitted in the infrared. The energy has to come out somewhere, right? You don't just gobble it up and it never comes out. But it, it goes from 2, 000 angstroms to coming out at 100 microns. It's a bit of a switch. So if you measure this luminosity you're missing by almost an order of magnitude What you would infer for the stellar mass in this one example.

But the other thing you notice is that all these models run together. And so the optimal place to measure the mass to light ratio is about 2 microns, 3 microns. And for star forming galaxies, that's basically happens because you After a few giga years of steady star formation, you establish an approximate equilibrium between the red giant branch and the rest of the population.

The red giant stars are producing all this light and they themselves do not represent much of the stellar mass, but they're in a good equilibrium ratio. It's not terrible in the optical part of the spectrum, but you see the lines start to diverge here and so you expect more scatter if you estimate mass to light ratio there than you do down here.

And you can see that in the, Kinematic data. So that's what we understand from stellar populations. Okay, so how many of you remember stellar populations from Professor Mijos course? See, this is the thing. It's always good to tell you the same thing twice. It doesn't hurt. But I don't want to belabor it too much, so I do want to go over it.

But I'm not gonna gonna try not to say too much because stellar populations and modeling thereof is a very rich and active subject. We mostly understand things, but there are still important things that we don't. Like how much light comes from the asymptotic giant branch as opposed to the

Red giant branch. Okay, basics. The way to, that modeling usually proceeds is to build a galaxy, a composite stellar spectrum, out of a lot of individual simple single populations. And so Star clusters are examples of that. Places where a population of stars all form more or less at the same time from the same cloud of gas.

They have the same age, the same metallicity. And of course, you do that a gazillion times to make a whole galaxy. So you can model the stellar population of a whole galaxy by adding up a lot of examples of that. So we want to be able to Model those complex stellar populations, and that, of course, is what was done to generate these model spectra.

And we can anchor, oh, I have to go through the builds. And we can anchor that in real data. All right, so we know a lot about the evolution of individual stars, of simple single populations. And we can measure the spectra of those individual stars and work out how they'll change over time. But still, we have to worry about some things.

We need to know the initial mass function, that is, the spectrum of masses with which stars are formed. For every solar mass star, how many tin solar mass stars are also born, how many hundred solar mass stars, how many tenth of a solar mass stars. That's what the IMF quantifies and that's a bit of a dodgy proposition because all the light, all the metals are made by the highest mass stars and all the mass is locked into the lower mass stars.

Out of half or a third of a solar mass, just because there are lots of them. The Sun, by that standard, is a little massive. A little more massive than average. It's average, but it's a little big. But it's not so big that it's going to blow up as a supernova or anything like that. The other thing you want to know is what the birth rate is.

How often are you star, forming stars, and galaxies might form all their stars at once and be a giant single population, or they might form their stars at a more or less steady rate. Constant star formation rate is a decent approximation of a galaxy like a Milky Way. Not perfect, but good for his approximation.

There can be stochastic fluctuations in that. There can be bursts. There can be periods of inactivity. That's all the sort of thing that people try to work out over time. Okay here are all the elements that go into stellar population modeling. That initial mass function, the number of stars as a function of mass.

Saul Peter first I don't know if he was the first to think of this, but he was very influential. In the fifties he wrote this down as a differential number. Per unit mass, so you always have to look, is it the number or is it the ndm, to be sure of what you're looking at. Ah, but it's a function of mass, so there's a stellar mass there, and then there are different examples.

At the time, Saul Peter just went out and counted stars, and he mostly, you have the same selection effect, so he mostly saw bright stars, and said, okay, that looks like a power law. But if you extrapolate it to very low masses, it explodes. This is a steep power law. And so the integral to zero is diverges.

And so usually people using a solar theory math would say, Okay, the minimum mass is, when a star is a brown dwarf, about a tenth of a solar mass. When a wave's solar mass is something like that. And then there are different ideas, and something like that black line is about right. It doesn't continue as a power law all the way to zero mass.

And so there is some peak around, like I say, a third to a fifth solar mass, where most of the mass resides. You need to know stellar evolution. So there's a zero age main sequence. You form a star with that spectrum of stars. You form a stellar population with that spectrum of masses. The massive ones evolve first, and then the others peel off.

I hope you all remember that. And then At any given moment, you can look at the distribution of stars in your model HR diagram there, and say, okay I got 106 star, it has a spectrum like this, so you need a library of stellar spectra, and then you can assign spectra to each star at each stage of evolution and calculate the emergent flux from that step.

And you can do this for a whole bunch of Single stellar population. So all that just gives you one star cluster, basically. And then you need to consider the accumulation of lots and lots of star forming events by assuming some birth rate. And you can also worry about, okay, how does chemical evolution go on?

So the metallicity of the stellar population you would expect to creep up over time. What is the dust doing? Dust is so annoying, right? So here is the dust attenuation curve in the Milky Way, has this prominent bump at about 2, 200 angstroms. That bump, that feature is absent at low mid helicity in the Magellanic Clouds, so another reason why the UV is a hard place to work.

But you can do all those things. You can have some dust. But it absorbs some UV and so then it re radiates in the infrared and you can put all those things together into a predicted spectrum. There is the spectrum in blue that is emergent from the stars and then some of that gets absorbed and so what you see It's that red line if there's a lot of dust there, to open it up.

So complicated. It's messy. Fortunately, we don't care about all that. We just want the bottom line number. How much light corresponds to how much mass? And so you can use these kind of models to estimate what the mass to light ratio is in different spectral bands. So this is V band, that's smack in the middle of the optical spectrum.

It's what everybody has as a go to thing that you observe. Here in the I band, that's the reddest. The reddest band that's regularly accessible from the ground. It's really red. If you hold up an eye band filter, it's just blood red. Or at least you'd like to. It's around 7, 000 angstrom right at the edge of your eyes perception.

And then if you move in the near infrared then that's a better, Mapping, as I said, of the stellar mass to light. But it depends a lot on whose models you believe. There's a big difference there. And I think we've fixed that, but the reason That happens, that you should be aware of, is that evolution in the HR diagram.

So

remember your stellar evolution, how bright stars are as a function of their temperature. Temperature going the wrong way because this is astronomy. So you start out with stars on the main sequence. Ah, and, as Time goes on. If you keep a steady star formation rate, there'll always be some stars up here.

But they'll also be evolving away, so eventually you establish a red giant branch. So this is typical of the HR diagram. Around the Milky Way. And we can do a good job of modeling the evolution up to the top of the red giant branch. After that, it gets dodgy. Lower stellar mass stars like the Sun have a helium flash, blow off some of their mass, and then arrive on the horizontal branch.

Exactly where is hard to predict. And those things are now burning helium in their cores instead of hydrogen. This is all hydrogen burning. Then you've reached the end of the road for that, and you're burning helium. You have the same physics go on, though, that you run out of fuel, converting the helium into carbon, so you're building up ash in the core.

The core condenses and gets hotter. The outer envelope expands and responds and starts to move back to this upper right part of the diagram, what's called the asymptotic joint branch. Because we don't know how to follow it well after that, we have trouble predicting this mark. And that's why there's this huge difference in different models there.

Because people have ascribed different things to what the HEV stars do, and they produce a lot of light, and the infrared producing very little in the optical. Jim Schomburg fixed that. He just said let's look at the data, go out and measure. Look at the AGB populations in these kind of clusters and just include it numerically rather than worrying about the models.

And then you can reconcile all these different things. We do that here, it's a long story. So that gets us to

the ability to measure the baryonic mass of galaxies. It's not dark matter at all, but like I say, you gotta account for what you can see before you worry about what you cannot see. And there are two steps for stars. One is the measurement. You have to measure both the brightness and how far away it is, and you get luminosity.

In some band pass I, and then you can estimate the mass to light ratio in that band pass I with your favorite population model and that gives you a stellar mass estimate. So that's, that chunk of the baryonic mass. For the gas mass, that chunk, you need to know the hydrogen fraction, because what goes in here is the mass of atomic hydrogen, mass of molecular hydrogen.

And so this is a number of, around 1. 3, 1. 4 depending slightly on that metallicity, as I brought you with me. So that's not a huge factor. People sometimes just assume it's 1. 4 for everything. Something like that. In principle, it depends a little bit on mass because it depends a little bit on metallicity and those things are related.

And then to get the big numbers there, the atomic gas mass, molecular gas mass, we do as discussed, you measure the 21 centimeter flux and turn that into an H1 mass that way. For molecular gas, you measure the carbon monoxide flux. And do it that way. These are hard data to come by. There's a lot of atomic gas data out there.

Lots and lots of optical surveys that give you the top line. It's just hard and expensive. observationally to do the molecular gas. That's often a limiting factor, which is why I emphasize that, g's do a good, factor of the two approximation of molecular gas is just about seven percent of the stillness.

Okay, then we come on to the velocity fields, which is what we use to measure the rotation curve. And if we did that And everything that we saw was what we got, and we should be able to match them up. And that is in radio interferometer nomenclature called a moment zero map. That's basically the distribution of A 20 mission.

But the interferometers give you a data cube, and all you've done here is collapse the velocities, so you're just asking what is the map of atomic gas on the sky. But you can also project in velocity space and you get a velocity field, so one side is coming at you, right in, the other side is coming at you, from this 3D blob, which represents the emission of 21 centimeter in this 3D space, two on the sky, r, a, and x, but also the third dimension of frequency.

Velocity. Alright, so these interferometers give you these 3D data cubes of which moment zero and moment one are just projections. And traditionally, the method of doing this has been to project a velocity field and then fit that to get the one dimensional azimuthally symmetric rotation curve corresponding to that.

As computation has gotten easier, one can now model the whole thing in 3D and learn a bit more about non circular motions and things like that. But traditionally, this is how it's been done. And a big survey from a bit over a decade ago now is the Thing survey, H1 nearby galaxy survey, still some of the best atomic gas data in the literature.

So here's NGC 2403, famous nearby galaxy, with a really extended atomic gas disk. So there's the stars from Galax and Spitzer, and here's the H1 gas, and here is the velocity field. And so you see one side's going away from us, the other side's coming towards us, and so you get this 1D rotation curve corresponding to that.

And you get the approaching and receding sides separately. Those are not always symmetric, for the same reasons that the H1 horn profile is not always symmetric. And so that tells you something about non circular motions, if there's a bar in the center of the thing, and so forth.

So traditionally what's been done is to take this velocity field, and there are holes, right? You don't get emission anywhere, but you can deal with that in the data processing. Because what you do is, again, you fit ellipses. And you find now the iso velocity surface corresponding to circular motion.

This is called the tilted ring model. You're looking for the ring that best represents circular motion. And so as you go around in azimuth one of these rings, the velocity varies as a sinusoid, just because at some point you're looking at the tangent away from you, comes around, and you're looking at the tangent towards you, and you're just seeing that around without motion, so you see everything in between too.

And so the process is to fit your S sinusoid to the data. And you get out of that not just the circular speed, but if there is a distortion from a sinusoid that encodes information about non circular motion, if the orbits are a bit eccentric. It tells you, of course, where the center of each ring is. It tells you the position angle, and it also contains information about the inclination.

So you can imagine that a disc is warped. It has one inclination until you get further out, and it changes inclination, and warps and edge on galaxies like that. And so in principle, it's possible to model this. So here's one inclination, and then it warps in and out. That's a fairly common effect, but you have to be careful in applying it because if you just let all these things go free without supervision, then you find that your galaxy looks like a ruffled potato chip, and the galaxies don't look like that.

Have to be careful. So here's another velocity field. Notice this one has a very straight line. That's an indication of a solid body rotation. It's just very slowly, steadily rising a rotation curve. And this is the equation that you're fitting. So any point here depends on redshift. This is a galaxy, so there's the whole system redshift, the redshift in the center.

Depends on circular motion and the cosine around your ring. And if there are any strong non circular motions, it'll show up as a sine component. Like all those things, just how far it is, circular velocity, radial velocity. And so you can fit that tilted ring model to get your isobelocity contours. And if there is a twist in position angle, or if there's a bar in non circular motion, that'll show up as the minor axis being not necessarily orthogonal to the major axis of the velocity field.

And the tilt here, you can see, is the wiggle in the position angle. So there's a lot of information in this kind of data. And here are a whole bunch of ones from Bosno way back in 1981. And these were really important to launching the dark matter paradigm because they go way far out and they showed that the optical rotation curves that Vera Rubin had taken weren't just flat out to the end of the optical disk.

They continued way out farther beyond that where the mass of the stars was really done. Typical optical only goes out that far. The radial goes out that far. At that point, all the mass was Encompassed, and we really should have seen a falling rotation curve if we were going to use it. Okay, any questions?

All right, that's it for today. We will get on with it next time. Hopefully stop talking about those boring baryons.