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Finally, let us turn to the examination of
correlations between Galaxian properties

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or scaling relations which, in my opinion,
is probably the most important and most

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interesting part of this whole thing.
This is what he's really telling us of

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something about formation of galaxies and
why they are the way they are.

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The Column Scaling Laws, because they're
power-laws that galaxies can be scaled as

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the power of something like luminosities
proportional to some power velocity

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dispersion for example.
And importance of these correlations is

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that they are telling us something about
how galaxies form, what is the physics

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behind it.
In some cases, either correlations between

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distance-dependent and
distance-independent quantities, they are

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used as distance indicator relations, like
Tully-Fisher or fundamental plane.

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And this is a quantitative way of
distinguishing physically distinct

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families of galaxies as opposed to
something like the superficial appearance

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and images taken of the particular
wavelength.

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These are physical properties and they're
[unknown] you saw this in the last module,

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like for dwarf ellipticals, which are not
elliptical and real elliptic.

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So classic example of this type of
relation is the Tully-Fisher relation

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between circular speed of galactic disks,
and the total luminosity of a galaxy.

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And it goes roughly as luminosity goes as
the fourth power of the circular speed,

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but power differs a little bit and
depending on the wavelength and so on.

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But, that's roughly what it is, and we can
measure circular speed either optically

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through, say spectra along the major axis
of the galaxy or through neutral hydrogen.

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Or even if you just take the whole galaxy
into one spectrum, the broadening of H121

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centimeter line tells you what is the
amplitude of the circular speed.

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Note that in order to get the intercept of
this relation correct, you have to know

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the distances.
But to get slope, you don't need to do

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this.
You need to just know the relative

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distances.
And interesting thing about Tully-Fisher

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relation is that it's a very good one.
It, its intrinsic scatter is maybe 10%,

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maybe even less in some cases.
This is what it looks like in five

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different filters, blue, visual red, and
very near infrared, and near infrared.

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And, you can notice an interesting trend.
The further red you go, the better

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correlation you see.
The scatter is smaller.

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We can understand this because blue light
is susceptible both to extras from regions

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of young star formation, which are
unusually bright and decreases due to the

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galactic extension.
Blue light is more sensitive to the

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galactic dust, whereas in infrared, you
bypass much of these two problems.

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So you can think of the infrared
Tully-Fisher relation as being the more

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indicative of intrinsic one.
The slope also changes and that's slightly

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different story.
And the reason why this is so and the

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reason why this is so interesting is that,
circular speed is a basically a property

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of the dark halo.
And, luminosity is a product of the

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integrated stellar evolution in the galaxy
or Hubble time.

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So somehow, dark halo mass seems to
regulate star formation history of a

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galaxy.
That in itself is interesting, but even

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more interesting is the small scatter,
because we can think of any number of

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reasons why the scatter can increase.
By changing ratios of dark matter to

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luminous one by tweaking up star formation
differently in different environments by

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different kinds of merging and so on.
Yet somehow, in the end, we end up with

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what's almost a perfect correlation.
This has been probably many different

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ways.
You may recall that there is this whole

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family of low surface bright discs which
do have normal amounts of dark matter and

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gas, but just not very many stars.
And so, even they follow Tully-Fisher

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relation, which is saying, this is not so
much relation between starlight and

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property of dark halo, but between total
baryonic mass and the dark halo.

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Now, that begins to make a little more
sense.

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An equivalent relation for elliptical
galaxies is called Faber-Jackson relation.

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Because, the rotational speeds are not
important in ellipticals, by and large,

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velocity dispersion is, that's where
kinetic energy is.

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Similar thing applies, that luminosity is
proportional to roughly the fourth power

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of the velocity dispersion.
Here, playing the role of the circular

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speed for the discs.
Its a pretty good correlation, but it has

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large scatter which cannot be explained by
the measurement pairs alone.

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There was something that was causing the
scattered circle's second parameter and

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now we know what that is.
The other ingredient for elliptical

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galaxies is so called Kormendy Relation,
which relates the effective radii of

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galaxies with their mean surface
brightness.

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And it goes in the sense of mean surface
brightness being smaller for the more

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large, for the larger, therefore, also
more luminous ellipticals.

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This is reflected also in the fact that,
the, the more luminous the larger ones

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tend to have more diffused surface
brightness profiles.

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This too is a pretty good correlation,
roughly goes a slope of minus 0.8, but

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it's not perfect one.
It too has a scatter as larger than what

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measurement error suggest.
Again, implying there is a second

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parameter involved.
But let's just play with Kormendy relation

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for a moment to illustrate how you can use
these correlations to glean something

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about galaxy formation.
A simple Virial Theorem argument is that

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mass with proportional to the radius and
velocity of dispersion squared.

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Now, if we assume that mass-to-light
ratios are roughly constant, which is not

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such a bad assumption after all then, you
can use luminosity as a proxy for the

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mass.
And therefore, you can use projected mass

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density instead of surface brightness.
So if you form ellipticals through

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dissipationless merging, just pure
gravity.

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No cooling, nothing like that.
Then, kinetic energy per unit mass remains

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constant.
Kinetic energy per unit mass is directly

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proportional to the velocity dissipation
squared.

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And now, going back to the relation
between velocity dispersion, mass, and the

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radius, we infer that radius should be
proportional to the surface brightness,

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projected mass distribution if you will,
to the minus 1 power, which is too steep

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relative to observations.
Now, assume instead that elliptical form

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entirely through dissipative collapse,
there is no extra stuff being added in

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galaxy, galaxy just shrinks.
And because the mass is conserved in this

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time, the radius goes down surface
brightness to go up by square, and so,

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radius will be proportional to, inversely
proportional to square root of surface

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brightness, which is now too shallow for
Kormendy relation.

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So, it's in between.
And that suggests that both dissipative

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collapse and dissipationless merging play
a role in formation of elliptical galaxies

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and that's almost certainly true.
Now, the resolution of the second

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parameter problem was discovery of the
so-called fundamental plane, a family of

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correlations for elliptical galaxies that
unify any number of their properties into

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two-dimensional scaling relations.
Often expressed between radius, velocity

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dispersion, and surface brightness, but
you can use luminosity in place of radius

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as well.
And so, here, the four different

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projections, the Kormendy relation, the
upper left.

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And you, and you see the little arrow bar
symbol in the corner and that tells you

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what the errors are, so quickly there is
extra scattered involved, the

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Faber-Jackson relation on the right, top
right.

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The plot of velocity dispersion, which is
like kinetic temperature of stars versus

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surface brightness, like projecting
density in the lower left, which is the

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coordinates as the cooling diagram for
galaxies.

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And there, we see absolutely no
correlation whatsoever.

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Naively, you might expect that galaxies
with higher kinetic temperatures would

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also need to have higher densities, but
that's simply not the case.

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This set of plots immediately tells you
that there are at least two independent

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coordinates here, which is why there is no
correlation in the bottom left.

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But, since there is some correlation
chances are that this is a flattened

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distribution in multidimensional space.
And indeed, if you combine quantities in

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such a way to look at this new
two-dimensional correlation edge on, it

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00:09:12,692 --> 00:09:17,060
looks essentially perfect within
measurement errors, that is the

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fundamental plane.
It is plane, because there are two

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00:09:20,904 --> 00:09:25,608
independent variables and it's
fundamental, because it connects

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fundamental properties of ellipticals.
Well, can we understand where these

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correlations come from?
Yeah, up, up to some point.

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So we can always start with the Virial
Theorem that always has to apply, and we

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can then substitute mass for luminosity
using mass-to-light ratios.

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Now, here is the crucial part.
If all galaxies were homologous, had just

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same kind of structure, only scaled
versions of each other, then we can relate

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the important three-dimensional values of,
say mean radius and mean kinetic energy,

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00:10:04,161 --> 00:10:09,746
appearing mass to the observed lines to
some constant of proportionality K.

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00:10:09,746 --> 00:10:15,006
And we can account for density structure
again with some proportionality constant,

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because we just assume they all have same
shape.

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Then we can simply, from the Virial
Theorem, come up with the expression that

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00:10:23,570 --> 00:10:28,211
radius should scale as measure of velocity
to the second power.

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00:10:28,211 --> 00:10:34,388
That could be velocity dispersion, surface
brightness to the minus 1 power, and mass

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00:10:34,388 --> 00:10:39,759
light ratio to the minus 1 power.
Likewise, we can deduce that luminosity

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will scales of fourth power of velocity or
velocity dispersion, just like in

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00:10:45,391 --> 00:10:51,463
Tully-Fisher or, or Faber-Jackson, minus 1
power of surface brightness and minus

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00:10:51,463 --> 00:10:56,855
second power of mass-to-light ratio.
So you can see that, in Tully-Fisher

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00:10:56,855 --> 00:11:02,582
relation, that suggest that surface
brightness and mass-to-light ratio somehow

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00:11:02,582 --> 00:11:05,944
together play the role of the second
parameter.

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00:11:05,944 --> 00:11:10,925
For Faber-Jackon relation, the same.
For Kormendy relation as its velocity

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00:11:10,925 --> 00:11:15,548
dispersion and mass-to-light ratio play
the role of second parameter.

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So this is what Virial Theorem plus
homology imply.

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00:11:18,980 --> 00:11:25,016
Any deviations of observed correlations
from these, are telling you something

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00:11:25,016 --> 00:11:29,572
about their assumptions.
One of them or another is wrong.

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00:11:29,572 --> 00:11:34,755
We will talk more about this in the next
module.