Now let's talk about the kinematics of the Milky Way, and its rotation, and how that provides compelling evidence for the existence of dark matter. This is of course just artist's conception. The dark halo is suddenly blue. The important thing about the rotation of the galactic disc is it's a differentially rotating disc. Meaning the angular velocity is a function of radius. It's not like a solid body disk that will have constant angular velocity everywhere. Inner parts are spinning faster. Now, that has a number of interesting consequences. One of which is that wherever you are, there'll be things passing you by or lagging behind you, on either of the radii. And if those things are say, giant molecular clouds, with a lot of mass, that provides nice tidal shocks with which to dissipate freshly made clusters. But it also changes other things, like enables spiral density waves. So, as I mentioned we use normally, mostly, H 121 cm line. Although we can also use nonmolecular lines to map the kinematics. The problem is we're just sitting in a plane, and we have just radial look out. So, when you look at this, right? They'll be some spinning discs, they'll be some spiral arms, but you're looking along the given line of sight. So, you'll be seeing gas on all range of different velocities. There'll be piled up gas when you, say, cross the spiral arm, or some giant cloud or something like that. But without knowing how far that is, there isn't much you can do. So we have to measure distances, and the way that's done is we can't tell the distance to H1 cloud. But if we can say, well there is a big conversation of hydrogen over there. And I also see star clusters in the same direction, chances are pretty good that they are close to each other. Because stars are born by the same clouds, and we know how to measure distances to star clusters. Like from HR diagram fitting of the main sequence. So you have to make these assumptions, that you see something that's associated with a given blob of gas. And sometimes it's vastly wrong, and you find out sooner or later, but most of the time it's okay. And so what radio telescope gives you is a plot like this. You go around in galactic longitude. Now each longitude you have a spectrum of neutral hydrogen, doppler shift corresponds to velocity and different intensity of different velocity. So they look like this. They're called LVD diagrams. And you can actually see there some systematic thing that corresponds to spiral arms. So after doing this in great and gory detail, then associating distances to particular spots in it. You have rate, you have radii, you have velocities, you know there is differentially rotating disk you can solve it and find out what the rotation as a function of radius is. And here it is. This is what the observed rotation curve of the Milky Way looks like, rotation curve means radial velocity I'm sorry, linear velocity. In this case tangential velocity, as a function of radius. And there is more in the middle, there is a lot more mass, then kind of goes down, then flattens out, with some bumps and wiggles. To a good approximation, outside the very middle, it's roughly flat. Now If you had purely Keplerian motions, you'll be dropping dramatically with the radius because your rotational potential will be declining. And therefore kinetic energy has to go down. And so, right away there is going to be an issue here. So if all the mass was squished in the middle, it will be just like solar system. Every star would be zipping around a gigantic black hole, but that's not the case. So when you actually decompose this, you add up all the stars and gas that we see. And ask, okay, there is this much mass I'm now going to compute gravitational potentials of function of radius. You do that and you ask what will be the velocities needed to balance the gravity of the stuff that we see? You come up with the prediction that is shown here as a blue line which is vastly wrong. So the only way to bring velocities up to what's observed, is to add some mass that you haven't seen. And so this is the origin of the dark matter detections. Now this is commonly seen in essentially every spiral galaxy there is. And also in elliptical galaxies except it's not rotation, it's random motion. So, couple things to note here. Our first is that the importance of the dark halo made out of some mysterious substance increases with radius. Near the middle, regular stuff, stars, gas, dominant galactic irridational field. The further out you go the stars thin out exponentially, but the stark material probably declines but gets to be dormant. And one thing we can tell is that it couldn't possibly be all in the disk. Because if you were to smush it all in the disk, say make it out to black holes flying around or something. Then that will cause dynamical heating of stars the disks will puff up because there will be all this mass in the disk to be compensated with random motions. This is not observed so this is why pretty sure it's some sort of spherical or almost spherical distribution. Well the interpretation of this is pretty straight forward, right? It's the usual centrifugal force equals centripetal force. And so if you measure velocity, and remember it's circular motion so it's just one variable, radius. If you measure velocity as a function of radius, you can tell how much mass is inside of that radius in order to create a centripetal force needed to balance it. And, the mass for constant. Enclosed mass will be like this. But, now, if you were to assume that, say, density of the total material, goes as a power law. This turns out to be a good assumption. So it goes as radius to some power minus alpha. And then plug this in, and we ask in what circumstances, for what value of alpha is V(r) equal constant? And the answer is, if density goes as a minus second power of radius. Now, if it were pure power law, from zero coordinate to infinity then that will imply infinite density in the middle, which is clearly not the case. So this idealized model of pure dense, pure power law of dense distribution is called singular isothermal sphere. Singular because its got singularity point of infinity in the middle. Isothermal because velocity is constant, meaning kinetic energy is constant. And it's a reasonable approximation expect right in the middle, and a real large radius. So, now if you put this in a formula, you find out that oh, because mass goes as 4 pi r squared, density of radius times the r. And the density goes radius minus second power, you find out that mass is linearly proportional to radius as you integrate it. So, the further out you go the more mass you get. And the question is where does it stop? And the answer is, nobody actually knows, but we're pretty sure it stopped by about 100 kiloparsecs radius. Because at that point, you start running into halos of other galaxies and so on. So, what is this mysterious stuff? How do we know this is actually true and it's not just to say we got something measured, something wrong, or maybe Newtonian gravity doesn't work. I mean, there are proposals to do that as well. And it turns out, there's several independent lines of evidence that give us the same result. You can do the same thing as you've done in this galaxy for elliptical galaxies, except use random motions, not rotational. But then if you look at clusters of galaxies, turns out they're filled with hot gas, x-ray gas, temperatures at millions or tens of millions of kelvin. If you asked, what are the kinetic energies of those protons and electrons, and how much mass there has to be in order to keep the cluster together. You get, again, huge amount of dark matter. This was actually how Fritz Zwicky first discovered dark matter, but he used galaxies as test particles, and not x-ray gas. A completely different method uses gravitational lensing. Remember mass bends light rays. And so if you look behind the massive cluster of galaxies you'll see the galaxies in the background distorted into little arches and stuff. And so you can invert this, and that tells you how much mass of any kind is there inside that radius that requires you to make those gravitational optics. Completely different physics, completely different measurements. And amazingly enough, produces exact same results as measurements from x-ray gas or galaxy motions. A modern thing is looking at Cosmic Microwave Background fluctuations, which is arena of precision cosmology to which we'll come later in the class. And that too implies that there is a large amount of some non-barionic mass. That's mass that's not the regular protons, and electrons, and stuff. And, again, in exactly the right amount. Finally, there's one more argument, and this is the kinematics and evolution of large scale structure in the universe. Are also implying presence of dark matter in order to account for depth dynamics, and how the large scale structure forms. Because if you have more dark matter it's going to accelerate the collapse of structures, and so on. That too matches everything. So we have several different approaches based on different physics, completely different measurements, and they all agree. And this is why we're really convinced that there really is such a thing as dark matter. That's okay. That's not complete fabrication. You know already there are things that very much exist, like neutrinos, but do not interact with electromagnetic field, right? So this is some other kind of particles. Well, we don't know what it is. Turns out that standard model of particle physics does not predict existence of suitable particles to account for dark matter. So this is physics beyond the standard model which is why physicists are all excited about it. And there is some hope that this will be seen in accelerators. But there also super precise measurements in lab deep underground that try to capture dark matter particles scattering off of the atoms. And after many years, and hundreds of different possibilities for what it could be. Things have now narrowed down to essentially two types of hypothetical particles. One is called weakly interacting massive particles or WIMPS, and the other is called axions. And it's most likely WIMPS. And WIMPs probably have masses of 100, 200 giga-electron-volts, something like that. And there is every expectation that they, too, will be seen in Large Hadron Collider, just like Higgs particle was. So, my guess is that before too long, we will actually know what dark matter is. And you'd know the one kind of dark matter. Could be 92 different kinds of dark matter, but at least this is a start.