Dark Matter class, 25-01-2024

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We stopped with this summary of the Oort limit and this very crude derivation of how it works in the plane parallel slab just basically imagining the stellar disk to be an infinite surface of constant surface density, which it is not, but locally you can imagine it is, and you can put a little Gaussian toolbox around it and derive all this.

And there are many levels of approximation so I want you to at least see a somewhat higher derivation, because this looks complicated. It's not as bad as it looks. A lot of that appearance is just from the choice of cylindrical coordinates, which is appropriate to the problem. And so that's where you get to pick up these funny looking terms.

Mostly, in the ORC problem, we're just concerned with the vertical contribution, not entirely, as we'll see so we're going to now drop the assumption that it's a uniform cylinder so the radial gradient does matter but we're still going to assume that it's axially symmetric. Now that's not true of real galaxies, they have spiral arms and bars, so that if you go around, At different angles, different azimuths, at a fixed radius, then the density does vary.

This term is non zero, but it's a small variation, so we're going to pretend we can ignore that. And The nice thing about Newtonian dynamics is expressed in the Poisson equation. That is, everything is linear, so you can separate it into as many components as you want. The stars, the gas, the dark matter.

And those densities all have a corresponding gravitational potential. And they can all be summed up linearly. Now remember, b squared goes as gm over r. Velocities don't add linearly. The potential goes as v squared, and it's the potential that sums linearly, so that's all and good. I was

questioning how much to really do on the board, but what you end up with, eh let's skip that, it's so dull to go through it. I'll post my notes on the webpage, but if you actually solve the Poisson equation dropping that angular term and assuming that you can separate the variables, that z is independent of r, another approximation that is okay, but not true in general then you wind up with this.

You have to integrate the Poisson equation once, which is Reduced to one derivative. And so that k sub z is the force in the vertical direction. So that's in k sub r, the radial force, that's the centripetal force. That's v squared over r, circular motion. k sub z is the restoring force per unit mass.

Always. End up talking forces, but really it's the force pre unit mass, so an acceleration. Vertical to the plane of the disk. And so this is some measurements a decade old now from Bogie and Ricks. So we can do this better now with Gaia. Problem with Gaia is that it's too much information.

You start to notice all these assumptions are wrong, so for example, we're also assuming that this is symmetric up and down, as well as azimuthally, and it's not. One side's a little noisier than the other. All these approximations are starting to, it's getting to the point where you have to do things numerically and it's challenging.

So I don't yet know of a good version of this from Gaia. Although this is still reasonably good, and estimates of The vertical force taken at different radii. And, of course, originally the ORC problem was just in the solar neighborhood. So you can do that locally still, but now you can do it at different radii.

Big surveys have been done, and by convention it's often referenced to the force at a height of 1. 1. So a parsec's above the plane. That's a bit of an historical fluke from the just the way how far out people could do it. So there's some central bulge bar component. There's a stellar disk with some modest thickness, about 8 to 1 ratio length to height.

And we're at some point here looking at stars moving up and down. And so it's that. Jittery up and down motion locally that Oort was trying to confine. And so most of that restoring force to the plane is due to the stellar disk. Locally, it's the stars that dominate the mass density. But, if we have a dark matter halo, a spherical thing, then eventually you add up some column of dark matter too, and so that's one reason why this 1.

1. The APC number is interesting. Probes high enough where you start seeing some effect from the dark matter halo. This is one of the things that gives you a constraint on how oblate the halo is, because you know how much you need how much dark matter you need to get the rotation curve. But the same distribution that gives you, that could be spherical or it could be squashed, and so it'll con contribute differently to this.

It's really hard to tell though 'cause most of the restoring force is to the, is due to the stars and the disc that's the dominant mass component locally and being very diffused up, being spherical. The dark matter is very diffused, even if it adds up to a lot, the density is low and so you have to go to pretty.

High Z before you start to really notice a contribution from that column of dark matter. That's the picture of what we're trying to do here, is measure that vertical force. It's just a choice where you reference it. Really, you would like to map out the whole force field. Now that I say that, I just now remember a very recent paper that does a Decent job of doing that locally, so I'll try to get an updated plot from that when I remember.

But anyway, you see there's this variation, and this variation is what you expect from an exponential disk, which is that fitted line there. That blue line is some complicated model of mine, which you don't need to worry about right now. But basically that is what you expect from an exponential disk, and as you go towards the center of the disk, the stellar surface density is higher, and the restoring force is greater.

And that's exactly what's seen in the data. The trick is figuring out what else that tells you on top of this. Now you'll see a lot of these different numbers for this. Throughout the notes, because there have been lots of different estimates over time, this is what Flody and Ritz got. That's astronomy, right?

Different people attempt to measure the same thing to get slightly different answers. 38, 40. It's in that ballpark. Another thing that you have to start worrying about at this point is that the solar neighborhood the traditional work problem is very local, whereas the galaxy, a lot of the math basically depends on The integral, so the galaxy has some spiral arms, something like that.

The way our galaxy looks. Nothing like that, but whatever. We're talking about some radius, and we at the moment happen to be in some space between the spiral arms. And so We can measure a local surface density in stars, 38 whatever, but what we really want to know is the average surface density averaged around that ring, right?

When you go through an arm, you pick up an extra term above what's here, so it may be 38 locally, but the whole thing might be 41, right? So some of the differences that we see And the literature may, in fact, be real, that you're sampling different parts of the disk in different ways. And there is some variation, and that's why there is that complicated blue line.

I was trying to ask what happens if there are spiral arms that give you an enhancement here, and there's a lower depression there, and so on. And At least to first order. That model is also consistent with the data. It's not right here and that it is not right there is interesting because The places where the data are sampling the disk is very different, in this case, from where that is sampling the disk.

And so again, it might be part of the signal rather than a limit in the the galactic astronomy is really receiving a huge invigoration at this point from Gaia, and one of the first things we learn is that this is Already not good enough approximation. That's one step above the approximation we had made before where, whoop, where it's just that, right?

So there's this extra term

That basically is what comes from the rotation curve of the gradient. So it's dv squared of r, because v squared is what's representing the potential, and that's just how the math works out. Sometimes you'll see this written in terms of A and B constants, but that's what it boils down to. And so there is an increasing term with z, as you look to higher and higher z, that depends on the gradient of the rotation curve.

So the biggest impact that the dark matter halo has here is, you would, we were hoping to see it in the column, that contributes directly to sigma there, the dynamic mass density, but there's this sort of Fictitious force from just what the slope of the rotation curve is. And if the rotation curve were really Keplerian, if we didn't have dark matter or something, then the rotation curve would be declining, and that term would be negative.

So it would reduce the effect of dynamical surface density. Since rotation curves are approximately flat, this term is Roughly zero, and so this ends up being a little more than the actual surface density of stars and gas and stuff. And so that is the indirect signature of the dark matter halo, its effect on the overall rotation curve, as opposed to the direct detection in the column above z.

And as the data get better, hopefully we can Do that as well. Okay. Questions about that before I move on? Okay. That was tidying up the the ORT discrepancy, which, like I say, along with Wiggy, was one of the very girly but very different indications that there was a need for dark matter, every time I say dark matter, really there's a discrepancy of where, at an acceleration scale, where Newtonian dynamics, and at some level, GR breaks down. So you actually might change the laws at that scale, or you need some extra mass that you cannot see. Okay, so a quick reminder I want to go on and talk about disk stability now.

If you've been through 323 last semester, this is all very familiar to you. The Hubble tuning fork sequence. People tend to invest a lot in the physical meaning of this. I take a pretty broad view of that. It's just a useful shorthand for what galaxies look like. And the important things is that early type galaxies, the ellipticals, are the bulges of bright spirals.

Are pressure supported the ratio of circular velocity to velocity dispersion is small. That's the pressure support. And then and those are 3D blobs of stars with orbits oriented every which way to fill out some bloboid in three dimensions. Whereas spiral galaxies are mostly confined to two dimensions all the stars are orbiting approximately circular, not exactly circular orbits, but quasi circular orbits in pretty much the same plane, some oscillation in and out of the plane, as we just discussed.

So these things are rotationally supported and if You look at the kinetic energy budget, a lot more of it is invested in the rotational motion than in pressure. V over sigma is large for those things. That's the key difference between things on this side of the forum and things on that side of the forum.

Then there's also, whoo, whether there's a bar or not, which we will talk about in a moment. Now, a lot of times this is Depicted as a change in bulge to disc ratio. Think of ellipticals as all bulge, and late type spirals as all disc. And that's true, but it's an oversimplification. Because strictly the Hubble sequence is what Hubble's eye said, Oh, that's an S C galaxy, and that's an S D galaxy.

And other people have learned to reproduce that, but it's a visual classification, and it doesn't always have to do just with the Bulge fraction. But that's what we care about, because we care about the dynamics here. So bulge is shorthand for a dynamically hot component. It doesn't have to be perfectly spherical, it can be squashed, it can have some Partial rotational support, so you can have a mix, but that's the crude distinction.

Pressure support for early types versus rotational support for late types. And so here are a couple of examples of spiral galaxies.

And they both have spiral arms, right? Nice pretty spiral arms. But this one is more or less has a musli symmetric, aside from the spiral arms. Whereas this has a whopping huge bar at central component. Why is that different? What's going on? And that's actually the third big thing after Oort and Bojewiki that motivated the dark matter problem.

So I want to explain that because it's a forgotten thing, it seems, in a lot of discussions, but it was really important at the beginning of the modern era. And in fact, we live in a barred spiral. This is actual data. It's what it looks like. And you notice the central region is sort of peanut shaped, there's this depression in the middle.

People have mapped that out very carefully and then asked, okay, if you have a barred galaxy, right? So there's our artist's rendition of what the Milky Way might look like phase on. And say we're looking here towards the center. There's an accumulation of stars at either end of the bulge, and that basically gives you this peanut like thing.

And in fact, people have inferred that we're somewhere. That the bar axis, major axis is about 20 to 30 degrees from our line of sight. So we're not here, we're not here, we're somewhere in there. And that, I think that the Milky Way is a bar spiral is quite well established at this point, but people always like to argue about these things, so I'm sure you can find somebody to do that.

Okay, but that there are bars ever, or not always, is a bit of a problem. And this was brought to life by Oestreicher and Peebles in a very influential paper in 1973. Oestreicher had been working on analytic models of fluids, and wanted to be able to treat a stellar body as a fluid, i. e. a galaxy.

And there are tens to hundreds of billions of stars in a galaxy like the Milky Way, so it's easy to imagine treating those as the molecules in a liquid. And it does in fact turn out that the fluid dynamic approximation is very good. That is, you ask what the relaxation time is from two body encounters.

And it's very long. Many Hubble times. So stars remember the orbits that they were born on, and they do get scattered over time by close encounters with other stars and time colloquial clouds and things like that. But that's a slow enough process that it has not had time to scramble the original orbital information.

So that relaxation doesn't Doesn't set in quickly, it's much longer than the age of the universe. And what that means is that the fluid approximation is pretty good. We don't have to worry about the short length, scale length interactions between individual stars. We can look at the overall body as a fluid and model it as continuous fluid, instead of a whole bunch of discrete particles.

Of course, we know it is a bunch of discrete particles, so they said, Okay we know how to solve the inverse square problem so we'll do that. And this was one of the very early in body codes. There's a whopping 300 points there. We do this with millions of stars nowadays. You could probably do that on your phone nowadays, but in 1973 this was the best you could do.

And if you go even back Pull out the picture but people would actually try to do this with an analytic computer by putting light bulbs on the floor, using a light meter to measure the strength of the inverse square effect, one light on everything else. Calculating the force by hand, coming back and adjusting the positions and redoing it all and iterating that, I think.

It's very clever, but sounds like a big pain. Anyway Ostreicher had been finding in his analytic work that it was really hard to have a stable fluid that looked like a spiral galaxy. Spiral galaxies are thin they're all orbiting in nearly, all the stars are orbiting in nearly circular orbits in the same plane.

And that situation is wildly unstable. And the numerical simulation showed the same thing, that on a fairly short amount timescale, even if you started out with some reasonable distribution of stars and notice they're more concentrated in the center and further out. That's what real galaxies are distributed like.

Very quickly they fall apart, and you don't have anything that looks like. A spiral galaxy anymore after just a few orbital periods. So

Nope. It's okay.

Alright, so here's a simulation of this that Professor Meehaus made a long time ago, just a very short one to give you an example. Boop. So that's the bar instability. Notice that there are only a few orbital time periods. You start out as a disk, you quickly form a bar, and then you just end up with something that doesn't look like a disk a spiral disk anymore.

So this is basically what Hausbrecher and Beebles found. And think about the time scales. The orbital period of the Sun, is about 200 million years. The universe is about 14 billion years old. We know there are stars in the galaxy that are at least 10 billion years old, probably more like 12 and 13.

So the Milky Way's been around for lots and lots of those orbital time periods. It's stable. It's not gonna disappear in just a few orbital time periods. So how can the universe be full of spiral galaxies when they should self destruct on this really short time scale? And the answer was to embed The galaxies in a dark matter halo.

That's the quick version. Here is the YouTube version. Advertising gravity at me thanks, yeah. Let me do that. Okay, so this is a more detailed simulation that's been done. And what you're seeing on the left is the stars, and on the right, the dark matter. And this is phase on, this is edge on.

It's a fairly squashed distribution, but whatever. And you see this is pretty stable. It's not doing anything. This is rotating around, and for many orbital time periods, it does not fall apart. It does form a bar, and that actually looks quasi realistic in that you get that shape for the orbital structure of bars.

You get a pile up of stars at the end. You tend to get spiral arms coming out at the edge of that. Oh, and look at that. You see the peanut shape going round and round from the side. Examples of edge on galaxies that look like that, including our Milky Way, right? Right there. Right there. That's cool. But that only works

If the stellar disk is embedded in a gravitational potential of something that is spherical, or quasi spherical. It's a stupid YouTube thing, isn't it? I have no idea what their algorithm is, but we don't need it. When YouTube started, a lot of scientists posted useful things on that, and there were no ads, so you could go there and get it.

Now they've taken our content and monetized it, and so if you want to use your own content, you have to put up with that crap.

It's not my video. That was the way we shared videos. I could, but up until now, it's been fine to use YouTube. I usually play it first, so it doesn't do the pre ads, right? But I didn't realize it was going to keep going. Yes, Joy? Whenever right there, where it says the bar instability, you're not saying that the bar is instability.

Stable in that dark by ourselves that it's that If left alone disc spiral galaxies will form a bar and know the disc is unstable. Yes, exactly Okay. I have another question. Yeah, so it seems like When things fall in like the gravitational collapse we expect like the stars to start like Orbiting in circles, like a spiral galaxy, right?

So why is it that dark matter halo, whenever that collapses into the blob of dark matter, why is it just sitting there, and why doesn't it do something similar to stars? Because gas dissipates. Okay. Dark matter by construction doesn't interact that way, and so if you have non interacting point particles, after they virialize, they're just on those random orbits.

And so you can think of the same thing for elliptical galaxies, even the stars do the same thing as dark matter particles. If you make the stars really fast, then you end up with a 3D blob. In order to get a thin spiral disk, you have to form that gas disk first, and then form the stars and the disk.

Really important points, I'm glad you asked. Because people tend to forget that. But, once you make a star, it's just a point particle, and it's just going to keep on whatever orb it is. And so the only way to get a thin disk is to first form it with gas, and the gas will dissipate and settle into the preferred plane defined by whatever net angular momentum the system has.

And if there's dark matter, it's the, orientation of the dark matter spin vector. So this is something we may get up to later on, but The idea early on was that, the whole universe is expanding. Within that, you can just find some volume that's going to expand and re collapse into a galaxy.

And of course, that's going on all over the place. And Peebles, way back in 69, addressed this question and said, Okay these things can torque each other. And it turns out most of the torque from one dark matter halo onto another, which is Spherical regions of one on another, it's maximized when they're near maximum expansion.

So basically, you have this proto galaxy, it hasn't yet decoupled, when it's about to turn around and collapse, separate from the Hubble flow, that's when they torque each other. And so you get dark matter haloes with a distribution of spins. There's not much, right? They twerk each other a little bit so that the overall system, the dark matter halo, has a modest spin, and he, people's defined a spin parameter, which I don't have in the current notes, but they'll come up, it'll come up at some point.

A dimensionless spin parameter that It was of order 5 7 percent is the typical number, but with a distribution, there were smaller ones, there were bigger ones. And so an important concept in galaxy formation is that, okay, you start out with that mostly spherical, mostly pressure supported dark matter halo, but it's got a little bit of angular momentum.

Then the gas within that subsequently cools, and in fact, you actually have to have infall because the dark matter starts forming before the baryons, but that's a higher order concern. Basically, you can think of the gas within that dark matter halo as dissipating and it shrinks in radius. It has to do that to make a galaxy that's very small compared to the dark matter halo.

And as the gas dissipates, it'll also settle into the dark matter. Preferred plane. So one idea for why you have this difference between early and late types is a late type basically managed to do that mostly in one big column sedate. It has to be a gradual process. And it can't get disturbed.

Or in early time had this going on in lots of little chunks and they all merged together and you got crazy stuff. But so yes, you need to format cold disk. And keeping disk cold is what lost record peoples were worried about. And what they found is that happens if you have some dominant, externally imposed dark matter potential.

Just let me write down the math, because the, if nothing else, the notation's a little confusing. Where did it go? Oh, there it is. Alright, that's jolly.

The kinetic energy they rotational and pressure terms. Pressure, Is basically what's represented by the velocity dispersion, basically random motions in whatever direction, all directions. And rotation, it's a quotation of the kinetic energy in invested in that circular motion in the plane of the disk.

You have some total kinetic energy, and then the virial theorem tells you that k has to be about half of the potential. Half of the total. And so they so you can define this quantity, this little t up there, as the ratio of the portion of the kinetic energy that is invested in rotation to the potential, and the theoretical maximum for that is a half, just from the baryon theory. It's all kinetic energy, and this number will be a half. In the Milky Way, the circular velocity in the solar neighborhood is of order of 230 kilometers per second.

The velocity dispersion is of order 20 kilometers per second. That depends on what Population of stars that you'd look at. Some of them are colder, 10. Some are hotter, 40. But, the point is that the average sigma is much smaller than the velocity. So this is the property of being dynamically cold.

And, of course, the kinetic energy goes as the square of these quantities. So the square of 230 is a lot bigger than the square of titty. And so this number t is pretty close to that limit of a half. It's about 0. 45 locally. That's actually being generous. And that's typical of spiral galaxies in general.

And yet, if you run a simulation at that level, it falls apart. And so what they did was vary the amount of W by imposing a static potential. So when you're doing a simulation, You can calculate the forces between particles, but you can also just add an analytic potential, like a Hermite's potential or something, and say, okay, there's also this force that you respond to.

And so that's what they did. Probably a plumber potential, excuse me, at the time. And they just kept cranking that up until they got a thing that was stable, and they found Just by playing around with the numerics, that it was about at that point. So you needed t to be less than about 0. 14, when the stuff you could see was more like 0.

45, hence you needed some spherical potential imposed by an invisible mass component that gave you that potential. It's the need for dark matter. And for this to work, the potential has to be Quasi spherical, right? The dark matter halo cannot be squashed like a disk or else you're just putting all the, you have the same problem, basically.

You have to put everything in a disk. Now, in detail there are much better simulations now. Selwood wrote a really magnus opus about all this, and in detail all this is wrong. But it gives the right idea, the gist of it is true. And it's really only the details are wrong. For example, you can get away with a stable disk if it's Rotation curve rises deeply enough and that's basically saying that it's the shape of the gradient of the potential that matters, but it amounts to the same thing.

And there are plenty of galaxies that do not have that kind of rotation curve, so you still end up needing something like that. One of the things Elwood did was just systematically ask, okay you need a halo to prevent the bar from just becoming huge and taking over. What happens if you vary the mass of the halo relative to the mass of the disk?

And quantify how we measure that. So what's That's what's done is the way people do this in general is they break they do a Fourier analysis and break things into modes, and so you basically have m there as the number of the mode, and a bar is a m equals 2 mode, right? And basically the ratio of the amplitude of that mode to the base mode, which is what the average surface density is is what's shown here.

And without a dark matter halo this explodes and it goes to one and it doesn't look like it is anymore. But if you add in a halo, then you do get a finite bar strength, and it settles out after a while. So one of the things that Southwood often emphasizes in A for a Cold Star is no matter how careful you are Setting up your disk in your halo and having orbits that ought to be in equilibrium and stable with each other.

They're not. You gotta let the simulation run for a while until it finds that. And so you see that time variation. But it does stabilize. So what you see here, when the halo is a mere factor of two more than the disk, is that the bar grows pretty rapidly at first, but it does stabilize at something less than one.

It's pretty strong, this is 0. 3 about, that's a strong bar. But it doesn't explode. Yay. That's a necessary criterion for spiral galaxies existing over Hubble time. And this line of thought was really influential in the seventies and is one of the things that kick started the, for 40 years before that or to work and or Z wiki's work had been screaming in the wind.

Nobody paid much attention. It's actually really fun to go read the preface he wrote to the CGCG's Catalog of Galaxies and Clusters of Galaxies because it just is a big Middle finger to everybody in the Astronomical Society. You fools at the Academy! And he names, and he's it's hilarious to read.

Don't do that, try to be a little bigger than that. But, he had pointed out a number of important problems that the luminaries of astronomy at the time had pretty much blown off. He was not happy. So it was in the 70s that problems like this really brought the problem to the fore.

Of course, that's when Vera Rubin was doing her rotation curve work as well, and so everything started to snowball then. But so as you increase the halo mass a factor of 2, 3, 4, 5 over the mass of the disk, then the rate of growth slows, still get pretty big in this case. But if the halo mass ratio gets too big, it saturates at a pretty weak level.

That's only 0. 02. Another thing to be wary of this is that you can't have too much halo, at least within the radius of the disk. Or else you never see bars or spiral arms. Those are m equals two modes with a pitch angle. And so that's a tension that we still haven't really settled on.

Because at the time they did this, all disks were high surface brightness, and you just needed to explain the upper limit. You didn't worry about that regime. I I studied low surface brightness galaxies and A lot of that reasoning doesn't hold, and so one of the first times, I think it is the first time that Professor Meehuis and I collaborated was on a simulation of this pair of high and low surface brightness galaxies, and we picked them because they had more or less the same rotation curve, We normalize in scale rate, so this is the circular velocity as a function of radius, normalized to the exponential scale length of the disk.

So the low surface brightness galaxy has a much larger scale length and a much lower surface brightness, but they integrate up to the same stellar mass. More or less in the same flat velocity, same spot on tully fisher. But the stability properties should be different because of that. And so here are a couple of other more sophisticated than the simple T of Ostericher and Peebles.

One due to Goldreich and Tremain is x parameter that is a measure of global stability against forming bars, and so it's not. And there's also the famous Tumorate Q parameter, which is a local stability that appears in a lot of theories of star formation. When does a galaxy form stars?

When the local disk goes unstable according to the Q parameter. Now we had trouble doing this one, and they looked identical, but that's by assumption because we didn't know what the velocity dispersions were. So we just assumed they scaled. So the fact that these were the same was more or less put in by assumption.

And it gets worse if you assume something else, so that was a conservative difference. And the things that go in there is the radial velocity dispersion, so the speed in and out of the disk. Kappa is the epicyclic frequency you can approximate the orbit of a star. I should draw this picture.

With the epicycle approximation, which sounds terrible, but it's not in this case. When I say a star in the galaxy has a quasi circular orbit, it really does trace out as a sort of a splatter graph pattern,

going between a maximum and a minimum radius. But it's not closed, right? So that path doesn't come back and meet. So it just fills in that region in there until something does it. And so you can approximate that by having the radius of the diving sitter, The main radius is the center of an ellipse, that is what the star is going at.

So the combination of motions along this little list, and of course the big thing, is a good way of approximating that motion, what's going on. And so the epicyclic frequency says how many times do you go around in that epicycle, right? What is the frequency of that oscillation between inner and outer radii, basically.

And that is not even fraction of the overall period. If it were, it would be closed, right? So a closed orbit requires you to come back to the same spot, so you'd have to have This to be exactly half of the total period, or some even number ratio, and it's not, in general. Okay that's what the epicyclic frequency is, and it also appears in the Woldreich Intramain criterion.

R is just the radius, g is Newton's constant, sigma is the surface density of the disk, and m is the mode number, so you can write down each one of these modes, and here x That's the mode of a bar. And right away, we were like, geez, the low surface brightness galaxy should be really stable against bar formation.

Which is counterintuitive because you look at them and they're fragile, and you'd think it'd be easy to push them into instability. But, the reason this, the opposite happens, is exactly that. In order for these things to have the same rotation curve, even normalized by scale length, you need a much larger enclosed halo to disc ratio for the low surface brightness galaxy than you do for the high surface brightness galaxy.

Right away we anticipated that would be a more stable thing, which was a counterintuitive thing, so it's so Chris was like let's do a simulation. And this is the simulation he did, right? This is old timey, we don't have a movie, just have steps. And the time step is the number and the thing, so the time steps going along that way.

The high surface brightness galaxy up there, the low surface brightness galaxy up there. Now, the bar instability means that the the disk of stars is predisposed to make a bar all by itself. It doesn't need any Outer influence to do that. But you can also do it by whacking it with a hammer. And so that's what we did.

We actually had an interaction between these things where they passed by each other pretty close on probrate orbits. So you're basically having an external force that's saying, please make a bar. This is how it was set up. And so you can do that. Early on, nothing has happened, and then whoosh, you make this huge bar.

But it does settle down into something reasonable. There's a bar, there's spiral arms coming off the end of the bar. Real galaxies look like that. So okay, great, and that's a pretty strong bar. The amplitude's 4 or something. The LSB was like ooo. It gets really Non circular.

These are face on views, so it doesn't look like a circle, even though that's what it started out with. And so there is an A equals 2 mode, but it's not a classic bar the way this thing is. And that's entirely because we whacked it real hard. If we just let it sit alone, it wouldn't do anything at all.

It would just sat there, orbiting around and making little bitty spiral. Light features, but very little that was bar like. So we were like, okay, that's what's going on. There really is more dark matter in the LSBs, and so you get this difference. And so here's the amplitude as a function of that time. And there's the LSB thing, so the model goes by paracenter passage, they whack each other, basically.

It's just like ringing a gong in a dynamical sense. And then, for the laser's brightest galaxy, it saturates out at point two. That's its most that we can induce on it. Whereas the HSB it goes up to a half or something. That's a really strong bar. I say all this because it's a clue that helps you constrain the mass ratio of the halo to disk.

It's not just whatever the That's what your mass model to a rotation curve does. You can't arbitrarily make the disk too small, just as you can't arbitrarily make it too big. Make it too big, it's unstable and falls apart, you make the disk mass too small, you never form a spiral arm or a bar or anything interesting that we see.

And so this was pointed out Early on by ulu and Bosma and their collaborators way back in 1987. And so they did this mogul analysis. So again, this is how you do the four a d composition of the moats. Now you can read it and how it's not tipped on its side, right? That's the mode number two for a bar or for.

A two arm spiral. A two arm spiral differs from a bar for having a finite P. That's the pitch angle at which the arms are. And so the arms can be tightly wound close to circular, or they can open up pretty wide. So that's a big pitch angle, that's a small pitch angle. There are, ways of doing this, but people usually resort to simulations.

Because it's a complicated mess. But there are some analytic approaches, and they did that Athanasoulou did that at all, and they found that, in fact, you could not have too small a disk by a fairly sensitive factor. There was only a factor of two between the disk explaining all the rotation. And half of it, and you had to have, be in that band in order to not suppress the kind of spiral structure that we saw.

It's a lot more complicated than this. So Selwood talks a lot about this in his 2016 review. The early simulations just imposed, including the one Chris and I did, we just imposed a halo potential from outside, an analytic potential to get the rotation curve but the growth rate depends on whether or not the halo is live, because when the halo is live, i.

e. individual particles that are, you're properly calculating the effects of the inverse square law on, just like you are for the stars, they interact with each other, they interact with the disk, and that affects the the growth modes. And whether it enhances or suppresses the growth mode depends on the properties of the dark matter halo.

Like I say, it gets complicated. It depends on the spherical mass distribution. It depends on the squashing. It just depends on everything. Still, you can do this kind of analysis. And so for example, Fuchs did this over a range of surface brightness because our concern was that, maybe you had too much stability from the high mass halo of the LSB.

And it turns out that's true, and Fuchs went through this for a whole bunch of galaxies. He found that the stellar disks of low surface brightness galaxies needed to be systematically heavier than you would have expected for The stellar populations. It's basically saying you need some disk gravity to drive spiral arms.

Spiral arms and bars are the signature of disk self gravity. That they don't just rip the galaxy apart is the suppression from the dark matter halo. And you gotta get the balance right to get a galaxy like what there are all sorts of things like what we see there, and I have to explain that whole distribution.

Okay. Yes? A quick technical question. Where does this form the thing? I'm having a hard time understanding the formulas. So theta and r are like axisymmetric coordinates? Is that correct? Yes. So this is the radius and theta is the azimuth and so it seems to be the function that is transformed, missing, or.

Yeah, it hasn't been written there, and in fact, I forget it offhand, so I Okay, so there should be like a density of theta r or something. Yeah, you have to know the density distribution, and you generally assume it's exponential. And it doesn't have to be logarithmic here. That is an empirical choice that seems to match the distributions of spirals in a crude way.

Okay, so that should be like a density in the Fourier transform with respect to the angular and the radial sum. Yeah, so you're trying to map out, okay, if I go around an azimuth, how does the density vary? And b equals zero just gives you just, okay, it's always the same angle, so it's a bar. And in fact, it's worth mentioning that bars are supported by a known orbit family.

So there's a whole theory of orbit structure which I'm not going to delve into for the most part. But in bars, you basically have an orbit family, x1, and a lesser field one, x2 orbits. Really clever names, I know. But it's that x1 orbit that stars are on, and those are closed orbits. In a rotating frame, right?

So this is looking at the orbits in a rotating frame, and they more or less go around that same oval track, and that is what the bar is, and the whole bar has some pattern speed that it tumbles around with. That's another constraint that we can learn, because the amplitude, basically the stronger the amplitude, the faster the bar turns.

And that patter speed is another measure of the another constraint on how much the dark matter halo is impeding it, because if you have a very cuspy mass distribution, that is the mass of dark matter continues, the density of dark matter continues to increase strongly towards the center, where the bar is, then it's basically a source of dynamic friction.

The bar you can think of is this rod rotating around in this soup of dark matter. And there's dynamic friction between the two, and so a cuspy halo will cause the bar's pattern speed to slow down fairly quickly. And that's As you can observe, most bars seem to be fast, which is an argument for a more accord distribution of dark matter that there's less density of dark matter in such a distribution, and so you don't get as much dynamical friction, and so a bar, once formed, can stay As they call it, fast, that is, its pattern speed can be consistent with what one sees.

Clear as mud, right? I wanted to talk you through all that, because those were the big things that Kicked off the whole dark matter problem, aside from, we'll talk about rotation curves plenty, we'll talk about gravitational lensing, talk about cosmology, but it was really these things that got the ball rolling.

And the clusters of galaxies, like I said Zwicky pointed that out, and it was a huge discrepancy, and people ignored it for 40 years. The Oort discrepancy was a very subtle thing, and he worried about it some off and on for 30 years, and it wasn't until the 60s. It really started to come and then it played into this stability argument.

And it's interesting to go back and read a book like Bob Sanders book on dark matter, because he was a grad student at the time that Oestreicher started advocating for this at Princeton, and he relates the story of, they were sitting around having a chat about things, and Oestreicher came and said, Ah, what do you think about there being dark matter halos ten times as massive as a galaxy?

And everybody's Get out! That's crazy talk. Because, you don't want to invoke invisible entities, like angels dancing on the head of a man. So it took a while, but this is where it started getting taken seriously in the modern time. And then all these things went together.

So I've tried to just list out those, the basics of those things so that it's all in one place in the notes. Okay. So with the time left, I wanted to give a little update on the baryonic contents of things. We've already talked about this a little. But like I say, you have to account for what you can see before you worry about what you can't.

And there are Some basics

that you should know about how we go about that. Again, we have the dichotomy between early and late type galaxies, that's the distinction between pressure support, where the stars in this galaxy are on very eccentric radial orbits, and they're distributed all over the place, and that's what fills out a quasi spherical shape.

So they're pressure supported, most of the kinetic energy is invested in that random motion. Light type galaxies, which include spiral galaxies, irregular galaxies, basically anything that rotates, most of that kinetic energy is Invest in circular motion, so as I drew on the board, the individual orbits are it's only circular, not perfectly.

But for the most part, stars are just going around in circles. And they're doing so in the same plane, so when you see these things edge on, then they are, in fact, thin. And you can quantitatively measure the distribution of light. Perpendicular to this, parallel to this, and the typical ratio is like 8 to 1.

Sometimes you see ultra thin galaxies that are like 20 to 1. Sometimes you see fat galaxies that are 4 or 5 to 1. But typically it's 8 or 10 to 1, something like that. And of course in detail, like we see in our own galaxy, depends on the population of stars that you're looking at. So for our purposes, we want to look at a population of stars that traces the stellar mass.

It's often like the red giants or the long lived low mass stars. Which don't produce any light, so they're not very useful that way. So here's the Hubble sequence again expanded a little. Hubble sequence ended at SC's, and we had to have tacked on other types as we go along. SD's, SM, irregular galaxies.

Physically, these are all the same beast. Physically, rotating, rotationally supported, and you get a rich variety of morphology for all the reasons we were just talking about with disk stability. So that's all I really care about. The Hubble sequence for, is it shorthand for, is the galaxy pressure supported or is it rotationally supported?

Now what's in a galaxy's, it's the baryonic mass of the stuff you can see is the sum of all those different components. The stars and the gas, and then that equals this formula. Oh, okay. Where the heck does that come from? Okay, so the first part is the stellar mass. You get that from observing the luminosity.

really what you do is measure flux. So you go to the telescope. Measure how many photons you get from a galaxy. You look at a standard star to calibrate that arrival rate. You get a magnitude, or equivalently a flux. If you know the distance to a thing, then you can turn that flux into a luminosity. That is the power emitted by a galaxy, how many hertz per second, photon per second, is coming from all the stars.

And so that luminosity is a proxy for the stellar mass. The more stars, the more luminosity, right? So you want to be able to take that observation Tells you how much light's coming from a galaxy, and turn that into a stellar mass, because this is always a problem we face. We get what we see, we count photons, but we want to know about the mass for the physics.

And that's, an observational problem that can be done. Then there's this additional step of turning the luminosity into a mass. We do that by assigning some mass to light ratio, typically by building a stellar population model. I won't belabor that, but we know a lot about stars, and so we can do that.

As I mentioned last time, the first order, this is a number of order one. But by looking at the colors of the galaxy, or the spectrum, or some other indicator, you can refine that somewhat, and it depends on what band pass I you're looking in. So you can look at the B band, that's the blue part of the spectrum, where young stars come and go rapidly, so it's actually a, not a good tracer of the stellar mass.

But if you look at longer wavelengths the I band, or the k band, 3. 6 microns, and that's a pretty stable relation between stellar mass and luminosity because most of that light is produced by red giants and those come into a quasi equilibrium with the overall stellar mass once a star formation has cooked along for a long enough time.

Long enough being a few giga years. But galaxies are typically over 10 giga years old, so it's long enough. That more or less works. Okay, so that's the stellar mass. Observe the light, do some modeling work to guess the mass to light ratio, and that number is always uncertain at some level, and people argue a lot about that.

We can do it to a factor of 2. I think at this point we can do it to 20%, but that's a controversial statement. A lot of people who would not grant more than a factor of 2. So that's the ballpark. Other stuff. There's also gas in the interstellar medium, and so that's that second term. And we typically talk about the hydrogen, because that is most of the mass, but it is not all.

There's also helium. The primordial helium fraction is about a quarter. So right away, just the hydrogen is only three quarters of stuff. And on top of that, there are some metals, right? Everything else in the periodic table is usually only one or two percent, but that affects this hydrogen fraction. So x is the Fraction of the gas mass that is hydrogen.

So by observing the 21 centimeter line, and we're measuring the amount of atomic hydrogen that's out there, and then we apply this correction factor on the assumption that hydrogen is also representative of the appropriate amount of helium and everything else in the periodic table. So there's some, Is it really 0.

73? It's 0. 75, 0. 7. So it's uncertain at that level. At least it's not a factor of two. And there's a really easy formula for the atomic gas. That the mass in neutral hydrogen is just this number times the observed flux times the square of the distance, right? So any mass dependent thing is always going to depend on the square of the distance.

That's just getting to luminosity, basically, in the 21 centimeter line. And then this conversion factor comes out of the atomic physics of the spin flip transition, plus some generalized assumptions about the temperature of the interstellar medium. The line is excited by collisions, and so there's some quasi thermal equilibrium this comes into.

And it's pretty robust, but it You know, you can't imagine violating that once in a while, but for the most part this seems to work pretty well. Those are number one and two for most galaxies, mostly stars and atomic gas, and number three is usually the molecular gas you don't see the molecular hydrogen directly one common way of Doing that is the using the carbon monoxide line, which is more readily observable.

As a tracer saying, hey, there are molecules here. So again, we're presuming where there's emission from a molecule, like carbon monoxide, there is the appropriate amount of hydrogen associated with that. And people have worked hard to calibrate this, and it's really controversial. There's the so called x factor, and people, again, it's not quite a factor of two variation, but there's some debate about what that right number is.

For us, it doesn't matter too much, because, like I say, this is number three on the hit parade here, so an error here does not have a big impact on most things that we would like to do. In general, however, it's a lot more complicated because we've only been able to calibrate this where we can measure it in nearby giant molecular clouds, mostly in the Milky Way, some in the Magellanic Cloud, and lo and behold, they're different, because the metallicities of the gas are different, and right or wrong, right?

CO is going to depend on metallicity, you have to have both carbons. And oxygen, right? It's not going to be a basal tracer if the hydrogen is going to depend on the metallicity. It also depends on the opacity, so it depends on the dust content, because this stuff is broken up by UV light, which tends to be harder from the stars and lower metallicity galaxies.

So you really expect this thing to vary a lot with metallicity. It's not obvious it does, but it's not obvious that we've measured it, and then in the places that were no or very low metallicity, you can't see it at all. There's no CNO. It's, there's no really great way to measure the molecular gas.

There are other proxies that people have come up with. I got annoyed with this, so I sat down with it myself, and I came to the conclusion that on average, the molecular gas mass is about 7 percent of the stellar mass. Factor or two scatter in that, maybe more. But it, that worked surprisingly well, and the distribution is also similar to that.

Stars are made out of molecular gas, and on average, they seem to have the same exponential radial distribution. Atomic gas tends to be more extended. Then beyond that, there's also ionized gas in the disk. That's the the pretty H2 regions. Whoops! Do we have any H2 regions there? Yeah, there we go.

Some nice H2 regions. So like I say, those are the neon lights of the universe and you can see them very well, but there's not really any mass in them. If you go further out, there is this circumgalactic medium, but you have to go way, way far out in order for that to add up to a significant amount of mass, the amount of ionized gas within the luminous radius of most galaxies is basically negligible. We're talking millions instead of billions of solar masses. It might add up to something, if you go out, 100, 200 kiloparsecs, but that's much larger than the, that's comparable to the radius of the dark matter halo.

It's much bigger than the galaxy itself. Okay Just late type galaxies alone live over a huge dynamic range. So people say, oh, it's just galaxies. That's a big statement. That's a huge range of physical objects. So luminosities, just for rotating disks, never mind the local group dwarfs, those range from 10 million to 500 billion.

In this particular band, 3. 6 micron band of Spitzer. You got five decades right there. From teeny tiny galaxies to really big ones. And so I've put a scale bar on all this. This is one of the most massive galaxies in the local universe, and that's also a large galaxy, 100 kiloparsecs.

That's 10 kiloparsecs on this sort of Milky Way sized galaxy. Milky Way is comparable to that in size and mass. And okay, 10 kiloparsecs, that's down there. They look Similar, but physically they're very different, and keep going down to lower and lower mass. As well as a big range in stellar mass and luminosity, there's a big range in gas mass.

The upper limit on the atomic gas mass is smaller than that on stars. It's interesting about how galaxies form the most, or evolve. Most massive galaxies seem to have formed. They've gone through most of their gas already. But there are low mass things that have more gas than stars. The big range of surface brightness.

Surface brightness quantifies how spread out or concentrated the starlight on. So here's a low surface brightness galaxy. The stars are pretty far out. And the high surface brightness things are very concentrated towards the center. And it almost looks like one star there. It gets so bright, but it's not.

It's just all the stars are that, jammed that close together. So we need to understand all this differences. The gas fraction is basically none to all. And you get this huge range of circular velocities from the smallest thing that I'm really convinced by is 15 kilometers a second, but it's hard to find such small things, so probably smaller things exist.

There is an upper limit around 300 kilometers per second, of which this is right up. And in fact, Berit Rubin once offered a prize, just, she would endow herself to anybody who found a disc that rotated faster than 300 kilometers a second. And she paid out because she didn't spell out that she meant the flat part of the rotation curve of a spiral galaxy.

And somebody found a disc orbiting a central black hole and that went higher than that. It's yeah, okay, I guess you win, but that's because I screwed up in how I phrased it. This still seems to be pretty, maybe there are things edging a little beyond that, but the number density of galaxies with velocities higher than that really takes a nosedive.

Very strong exponential cutoff among late type galaxies. Now I've shown this with these scale bars, that does not really give you the visceral impression of how the size differs, so I've taken those sine images and scaled them appropriately. There's the big galaxy. And if you didn't see the little dwarf galaxy there, you wouldn't notice it.

Alright, and in fact, it looks like just a patch in the outer parts of the big galaxy. And this one here, that's sort of Milky Way size, and Milky Way itself is not a small galaxy. But you get things that are much, much bigger, you get things that are much, much smaller. And so you can quantify that size, and we'll pick it up there next time.