Finally, let's talk a little bit about so-called scaling relations, and this turns out to be a very interesting subject. What's shown here are two different projections in a parameter space of radius, density and kenetic temperature of elliptical galaxies less [INAUDIBLE] and nebula clusters sure to reflect two pictures I've shown you before. But here it's in 3D space. And in some cases they're excellent correlations and it's an interesting thing to understand. Now scaling laws are called that way because they're than their power laws. Take log of two quantities they scale in a linear fashion. And they're built there by galaxy physics and galaxy formation. So trying to understand where they come from is teaching us something important. They're also different, for ellipticals, and spirals and vortices and so on, so that tells us that they really are kind of different families of galaxies they are isn't really continuum. And interestingly enough, if you can correlate something that depends on distance, like luminosity or absolute radius, with something that does not correlate with distance, say, rotational speed. Then, you can measure the distance in dependent quantities and figure out how far something is. This is just like using HR diagram to measure distances to clusters, where luminosity is distant dependant, but color and temperature is not. And so by measuring color of a star, you know from the calibrated background what it's true luminosity is, you compare it with the apparent luminosity, the distance. So this is how we can do that for galaxies. Well there are two important correlations, one is for spirals, one for ellipticals. For spirals, there is a correlation between luminosity and maximum circular speed. It's called Tully-Fisher Relation and rules of luminosity is roughly is the fourth power of circular velocity because of mostly flat rotation curve. It gets a little better as you get from bluer filters into the near infrared. And that's probably because you just don't get fluctuations from young stars or dust. And it's an amazing correlation. It's good to about 10% of intervening scatter and maybe even zero. Now, why is this interesting? Well because the circular speed is the property of the dark halo. And luminosity is a product of the stellar evolution of the disk. And the fact that they are essentially perfectly correlated tells you that somehow history of star formation in the disk Is governed in some way by the dark halo, even though dark matter by itself does not interact with regular stuff in any other way than gravity. And understanding this is going to be a very interesting thing. So we don't know why it's so good, but it is, because anything we could think of would just spoil that correlation. For elliptical galaxies, there's something similar, called Faber-Jackson relation. But the really interesting one is bivariate correlation, which means, as much of properties like radii, luminosity, densities, velocities, dispersions, what have you and any two of them can predict the third one. So just like Talley Fisher relation is like a correlation on xy plane of luminosity and velocity. This one is a plane embedded in three dimensional space of say measure of size like radius or mass. Measure of temperature like velocity dispersion and measure of density. And that's why it's call Fundamental Plane and it's usually expressed in this form. Scaling of radius with velocity dispersion and surface brightness. The two pictures here show what happens if you rotate this parameter space. The one on the left, you're looking on nearly face-on. There is no correlation whatsoever. And then you tilt your point of view and look at it edge-on, you see there is a beautiful correlation. And so this is actually pretty amazing that any number of important properties of ellipticals are united into just two numbers. And why just that? We don't know again. Why is it so good? Now there is no mystery why there are these correlations, and you can go from just Virial Theorem to a potential in kinetic energy. Say what you observe is not same thing as mean mass radius and so on. Lump all this together and you say that for a galaxy bound by newtonian gravity, radius truth scales velocity squared, surface brightness minus one and master light ratio to minus one. And luminosity scales of velocity to the fourth power, hallelujah. We recovered Tully-Fisher. Surface brightness of minus one power and master ratio minus two power. Now if any of those deviate, from perfection then you go and create some scatter. Now we know that in reality mass to light ratios, surface brightness, they all differ systematically or randomly. And yet somehow things composite to make perfect relations. So for Tully-Fisher, we got the right slope for power of velocity, for the fundamental plane we did not. And that's due to the systematic deviations of structure of ellipticals from being perfectly homologous. Meaning ellipticals are not just scaled version of each other, bigger and smaller but same structure. Now there is a systematic change in the way they're arranged. Internally as a function of mass, and that tilts the plane from Virial Theorem into what's observed. So if you have galaxies bound by gravity, that means Virial Theorem applies, and they're almost but not quite homologous version of each other, then you expect to recover something like this. But any process that we can think of just spoils up these correlations. It would puff them up into pure randomness. And somehow, that doesn't happen. So instead of that, the opposite happens. And just like H-R diagram was this two-dimensional space of luminosity and temperature in which stars were on one-dimensional sequences, the main sequence, the red giant branch, and what have you. For galaxies, there is one extra dimension. The three dimensions, one of which measures size, radius, mass Another one measures density and the third one that measures kinetic energy. Rotational speed or velocity dispersion. And in that space, they're not in lines, they're on sheets. And different galaxies in different position. And so no matter where they start from, they end on a lot of these correlations. And nowhere else, so there's preferred canonical states. And so the hope is that we can use this in the same way that H-R diagram was used to develop theories of stellar structure in evolution and test the models. Our numerical simulation of galaxy formation can reproduce this correlations, but again nobody knows why that and not something else.
