Finally, let us turn to the examination of correlations between Galaxian properties
or scaling relations which, in my opinion, is probably the most important and most
interesting part of this whole thing. This is what he's really telling us of
something about formation of galaxies and why they are the way they are.
The Column Scaling Laws, because they're power-laws that galaxies can be scaled as
the power of something like luminosities proportional to some power velocity
dispersion for example. And importance of these correlations is
that they are telling us something about how galaxies form, what is the physics
behind it. In some cases, either correlations between
distance-dependent and distance-independent quantities, they are
used as distance indicator relations, like Tully-Fisher or fundamental plane.
And this is a quantitative way of distinguishing physically distinct
families of galaxies as opposed to something like the superficial appearance
and images taken of the particular wavelength.
These are physical properties and they're [unknown] you saw this in the last module,
like for dwarf ellipticals, which are not elliptical and real elliptic.
So classic example of this type of relation is the Tully-Fisher relation
between circular speed of galactic disks, and the total luminosity of a galaxy.
And it goes roughly as luminosity goes as the fourth power of the circular speed,
but power differs a little bit and depending on the wavelength and so on.
But, that's roughly what it is, and we can measure circular speed either optically
through, say spectra along the major axis of the galaxy or through neutral hydrogen.
Or even if you just take the whole galaxy into one spectrum, the broadening of H121
centimeter line tells you what is the amplitude of the circular speed.
Note that in order to get the intercept of this relation correct, you have to know
the distances. But to get slope, you don't need to do
this. You need to just know the relative
distances. And interesting thing about Tully-Fisher
relation is that it's a very good one. It, its intrinsic scatter is maybe 10%,
maybe even less in some cases. This is what it looks like in five
different filters, blue, visual red, and very near infrared, and near infrared.
And, you can notice an interesting trend. The further red you go, the better
correlation you see. The scatter is smaller.
We can understand this because blue light is susceptible both to extras from regions
of young star formation, which are unusually bright and decreases due to the
galactic extension. Blue light is more sensitive to the
galactic dust, whereas in infrared, you bypass much of these two problems.
So you can think of the infrared Tully-Fisher relation as being the more
indicative of intrinsic one. The slope also changes and that's slightly
different story. And the reason why this is so and the
reason why this is so interesting is that, circular speed is a basically a property
of the dark halo. And, luminosity is a product of the
integrated stellar evolution in the galaxy or Hubble time.
So somehow, dark halo mass seems to regulate star formation history of a
galaxy. That in itself is interesting, but even
more interesting is the small scatter, because we can think of any number of
reasons why the scatter can increase. By changing ratios of dark matter to
luminous one by tweaking up star formation differently in different environments by
different kinds of merging and so on. Yet somehow, in the end, we end up with
what's almost a perfect correlation. This has been probably many different
ways. You may recall that there is this whole
family of low surface bright discs which do have normal amounts of dark matter and
gas, but just not very many stars. And so, even they follow Tully-Fisher
relation, which is saying, this is not so much relation between starlight and
property of dark halo, but between total baryonic mass and the dark halo.
Now, that begins to make a little more sense.
An equivalent relation for elliptical galaxies is called Faber-Jackson relation.
Because, the rotational speeds are not important in ellipticals, by and large,
velocity dispersion is, that's where kinetic energy is.
Similar thing applies, that luminosity is proportional to roughly the fourth power
of the velocity dispersion. Here, playing the role of the circular
speed for the discs. Its a pretty good correlation, but it has
large scatter which cannot be explained by the measurement pairs alone.
There was something that was causing the scattered circle's second parameter and
now we know what that is. The other ingredient for elliptical
galaxies is so called Kormendy Relation, which relates the effective radii of
galaxies with their mean surface brightness.
And it goes in the sense of mean surface brightness being smaller for the more
large, for the larger, therefore, also more luminous ellipticals.
This is reflected also in the fact that, the, the more luminous the larger ones
tend to have more diffused surface brightness profiles.
This too is a pretty good correlation, roughly goes a slope of minus 0.8, but
it's not perfect one. It too has a scatter as larger than what
measurement error suggest. Again, implying there is a second
parameter involved. But let's just play with Kormendy relation
for a moment to illustrate how you can use these correlations to glean something
about galaxy formation. A simple Virial Theorem argument is that
mass with proportional to the radius and velocity of dispersion squared.
Now, if we assume that mass-to-light ratios are roughly constant, which is not
such a bad assumption after all then, you can use luminosity as a proxy for the
mass. And therefore, you can use projected mass
density instead of surface brightness. So if you form ellipticals through
dissipationless merging, just pure gravity.
No cooling, nothing like that. Then, kinetic energy per unit mass remains
constant. Kinetic energy per unit mass is directly
proportional to the velocity dissipation squared.
And now, going back to the relation between velocity dispersion, mass, and the
radius, we infer that radius should be proportional to the surface brightness,
projected mass distribution if you will, to the minus 1 power, which is too steep
relative to observations. Now, assume instead that elliptical form
entirely through dissipative collapse, there is no extra stuff being added in
galaxy, galaxy just shrinks. And because the mass is conserved in this
time, the radius goes down surface brightness to go up by square, and so,
radius will be proportional to, inversely proportional to square root of surface
brightness, which is now too shallow for Kormendy relation.
So, it's in between. And that suggests that both dissipative
collapse and dissipationless merging play a role in formation of elliptical galaxies
and that's almost certainly true. Now, the resolution of the second
parameter problem was discovery of the so-called fundamental plane, a family of
correlations for elliptical galaxies that unify any number of their properties into
two-dimensional scaling relations. Often expressed between radius, velocity
dispersion, and surface brightness, but you can use luminosity in place of radius
as well. And so, here, the four different
projections, the Kormendy relation, the upper left.
And you, and you see the little arrow bar symbol in the corner and that tells you
what the errors are, so quickly there is extra scattered involved, the
Faber-Jackson relation on the right, top right.
The plot of velocity dispersion, which is like kinetic temperature of stars versus
surface brightness, like projecting density in the lower left, which is the
coordinates as the cooling diagram for galaxies.
And there, we see absolutely no correlation whatsoever.
Naively, you might expect that galaxies with higher kinetic temperatures would
also need to have higher densities, but that's simply not the case.
This set of plots immediately tells you that there are at least two independent
coordinates here, which is why there is no correlation in the bottom left.
But, since there is some correlation chances are that this is a flattened
distribution in multidimensional space. And indeed, if you combine quantities in
such a way to look at this new two-dimensional correlation edge on, it
looks essentially perfect within measurement errors, that is the
fundamental plane. It is plane, because there are two
independent variables and it's fundamental, because it connects
fundamental properties of ellipticals. Well, can we understand where these
correlations come from? Yeah, up, up to some point.
So we can always start with the Virial Theorem that always has to apply, and we
can then substitute mass for luminosity using mass-to-light ratios.
Now, here is the crucial part. If all galaxies were homologous, had just
same kind of structure, only scaled versions of each other, then we can relate
the important three-dimensional values of, say mean radius and mean kinetic energy,
appearing mass to the observed lines to some constant of proportionality K.
And we can account for density structure again with some proportionality constant,
because we just assume they all have same shape.
Then we can simply, from the Virial Theorem, come up with the expression that
radius should scale as measure of velocity to the second power.
That could be velocity dispersion, surface brightness to the minus 1 power, and mass
light ratio to the minus 1 power. Likewise, we can deduce that luminosity
will scales of fourth power of velocity or velocity dispersion, just like in
Tully-Fisher or, or Faber-Jackson, minus 1 power of surface brightness and minus
second power of mass-to-light ratio. So you can see that, in Tully-Fisher
relation, that suggest that surface brightness and mass-to-light ratio somehow
together play the role of the second parameter.
For Faber-Jackon relation, the same. For Kormendy relation as its velocity
dispersion and mass-to-light ratio play the role of second parameter.
So this is what Virial Theorem plus homology imply.
Any deviations of observed correlations from these, are telling you something
about their assumptions. One of them or another is wrong.
We will talk more about this in the next module.