Hello. 
Last time, we saw how we can use, various 
relationships to measure distances to 
stars, or star clusters. 
Chief among them, relations between 
period, and luminosity for pulsating 
variables like Cepheids, but also fitting 
of the main sequence for the star 
clusters. 
These are examples of a really more 
general idea of Distance indicator 
relations. 
So let's just recap what that really 
means. 
In general, if we can find a correlation 
between some distance independent 
quantity, like color or temperature or 
stars or period was for pulsating 
variables, this correlates well with 
something that Does depend on distance 
like, stellar luminosity. 
And we can calibrate that relation, then 
we can see how stars in, other clusters, 
of different distances, fit on that 
relation. 
So now when we compare, these 
correlations inapparent quantities, say 
apparent magnitudes instead of absolute 
magnitudes, clusters of different 
distances Will systematically shift from 
each other. 
From magnitude of shift, we can find out 
what the relative distance is between 
them. 
So thus, we can use these distance to 
indicate the relations as a measure of 
relative distance. 
If we can actually calibrate them, in the 
absolute sense, then of course we can get 
the optimal distances as well. 
In this is how Cepheids work. 
So now the question is, can we do the 
same for galaxies, and the answer is yes. 
The first relation, that we'll consider, 
is so called, surface brightness 
fluctuations. 
And, conceptually, it's fairly simply. 
Suppose we are looking at some galaxies 
with a detector that has square pixels, 
like, typically they do, and each of 
these pixels would cover, a certain 
number of stars. 
Now if we, remove the galaxy twice as 
far, then there'll be, four times as many 
stars in each pixel, and that will look 
smoother. 
The mathematical reason be, behind this 
is, fairly simply. 
For elliptical galaxies and bulges, most 
of the light comes from luminous red 
giants. 
And it's the fluctuations in the numbers 
of the red giants that will determine 
what's going on here. 
So if we have, say, stars, that 
contribute most of the light and the 
average flux is f and the number of them 
in a pixel is, and the total amount of 
flux from the pixel will be n*f. 
And the variance of that will be due, 
given by the square root of the number of 
stars. 
It's simple statistics. 
So, if we move galaxy further, the number 
will, go as the distance squared, but 
flux will decline as the distance square. 
And, therefore, the variance will scale 
with the square. 
Of the distance, universally. 
And square root of that, which is rms or 
sigma as the first power. 
This is why a galaxy that's twice as far 
would appear twice as smooth. 
So now, if we actually knew what real 
luminosity is for an average star that 
corresponds to that. 
Average f, then we can determine the 
distance. 
Roughly speaking, the absolute mag, mean 
the absolute magnitude M corresponding to 
that main flux F, is the one at the top 
of the Red Giant branch, and we can try 
to calibrate that using Andromeda Galaxy 
to which we measure distance using other 
techniques. 
Using that gives the following 
calibrating relationship here. 
Note that this a purely empirically 
calibrated relationship using a galaxy to 
which we know the distance. 
However, there is likely to be some 
dependence on the actual coefficients of 
this equation, or I should say 
correlation, that will depend on things 
like the mean metallicity of stars in a 
given population, and so on. 
We also have to take care of things like 
removing the effects of foreground, 
extinction, presence of foreground stars 
in our galaxy, background galaxies behind 
the galaxy we're studying, and so on. 
Nevertheless, this is a very powerful 
method, and can be used to measure 
distances up to 100 megaparsecs using 
Hubble-Space telelscope. 
Here is a simulation of how that works. 
The 2 columns of pictures correspond to 2 
galaxies, fictitious galaxies that 1 of 
which is twice as far as the other. 
Top panels schematically show where the 
stars would be relative to the pixel 
grid. 
Now if we average the fluxes we can see 
in the grey scale images the second row 
what say. 
Picture might look like. 
Now let's add more pixels or 
alternatively we can move galaxies 
further out and so we see those grainy 
structure which is indeed what is 
observed. 
Finally, we convolve this with 
atmospheric turbulence thus seeing 
because we are not observing in a vacuum. 
Don't have to do this last step if 
observing with a space telescope. 
And the last row shows selected images of 
2 different galaxies, of 2 different 
distances from us. 
Obviously the one that's further away is 
much smoother. 
100 megaparsecs is a good distance but 
we'd like to push even further into 
what's called pure Hubble. 
So remember, the definition of Hubble's 
Law is that distance is proportional to 
the recession of velocity. 
That's due to the expansion of the 
universe. 
However, in reality, galaxies do move. 
Since the time they form, they acquired 
so called peculiar velocities because of 
the mutual gravitational attraction. 
You can imagine, that if there is a large 
concentration of mass, in some direction 
from a galaxy, it will be accelerated 
towards it. 
And over time it will develop certain 
velocity. 
So the actual observed radio velocities 
of galaxies have these two components, 
the components that's due to the 
expansion of the universe, the Hubble 
flow, and the other one which has to do 
with dynamics of large scale structure. 
Now, typically Today these peculiar 
velocities of galaxies are a few hundred 
kilometers per second. 
For example Milky Way moves 600 
kilometers per second relative to the 
cosmic micro background which is a good 
physical embodiment of the combo and 
coordinate grid, and galaxies new 
cluster. 
So, in our case, we are a member of the 
local group, and that is part of a larger 
concentration, the local super cluster. 
And our entire local group is falling to 
it with a speed of a few hundred 
kilometers per second. 
Since these large scale non-uniformities 
extend out to scales of maybe 100 
megaparsecs or so. 
Up until that scales, or thereabouts, 
there, there are going to be peculiar 
velocity components. 
Going to larger scales, things pretty 
much average out, and if the peculiar 
velocity is few hundred kilometers per 
second, few thousand kilometers per 
second hubble flow would make that 10% if 
we go to. 
Tens of thousands of kilometers per 
second of recession velocity. 
Then the peculiar velocities will only 
matter on a percent level. 
This is why I would like to measure 
Hubble constant to distances that are 
considerably larger than 100 megaparsecs. 
And for that, we need some very luminous 
objects. 
Galaxies and supernovae are such luminous 
objects and both have been used to push 
measurement of the Hubble constant all 
the way out to the Hubble flow. 
This is where galaxy scaling relations 
come in. 
Remember we explain how it works at the 
beginning of this clip and if we can find 
a quantity for galaxies that correlates 
with another one What the first one, with 
being distance dependant, such as 
luminosity. 
The outer one being distance independent. 
Then we can, actually do the same trick, 
as we did with cephates. 
There are 2 examples, of such, distant 
indicator relations for galaxies. 
1 for spiral galaxies, called the 
Tully-Fisher relation, and 1 for 
elliptical galaxies, called the 
fundamental plane. 
Because these distance indicator 
relations, provide a relative Distance of 
galaxies from each other, they do have to 
be calibrated locally. 
So, for example, the Tully-Fisher 
relation can be calibrated with distances 
to spirals that are now unsafe from 
Cepheids. 
Or as the Fundamental Plane relation can 
be calibrated from the distances using 
surface brightness fluctuations. 
Each of which, of course, has been 
calibrated with the lower ranks of the 
distance ladder. 
Now one thing to note here is that we 
have some idea where these relations come 
from, but their exact physical standing 
for evolution are not yet well 
understood. 
So we have to beware. 
We're using a tool whose origins and the 
right functions are not perfectly well 
understood. 
The Tully-Fisher relation, for spirals, 
is a very useful one, it, connects the 
luminosity of a galaxy, with its maximum 
rotational speed. 
When we talk about, properties of 
galaxies, later on in the class, we will 
address these relations in much greater 
detail, but, for now, just take it for 
granted, that this is indeed the case, 
and I'll show you the plot. 
The luminosity of spiral galaxies are 
well protected with rotational speeds, 
and the relation has the power of slope 
of 4, namely luminosity is proportional 
to the fourth power of the circular 
velocity. 
Roughly speaking, the power varies 
depending on the filter that is used and 
so on. 
We can also express that in terms of 
magnitudes, because log of the luminosity 
* -2.5 gives the absolute magnitude, will 
manifest itself as the width of the 
observed radio line of neutral hydrogen. 
Because as the hydrogen atoms move in the 
galaxy their Doppler shifted this way or 
that and faster the rotation speed. 
More of the Doppler Browniung rays. 
So we can establish this, this 
correlation using spiral galaxies with 
well measured distances nearby and then 
try to apply it to much more distant 
ones. 
In practice the scatter of the 
Tully-Fisher relations is typically 10 to 
20%. 
The very best that has been done is about 
9% and that's including all the 
measurement errors. 
It's actually a very good correlation 
when done well. 
There are some problems. 
The light from spiral galaxies is 
affected by the extinction in them. 
There is lots of dust that absorbs the 
blue light, or absorbs light at all 
wavelengths, but more so in the blue. 
And, the fluctuation in star formation 
can affect the luminosity of galaxy on 
short time scales. 
So here is the actual Tally-Fisher 
relation for a set of galaxies nearby, in 
3 different filters. 
There is a gradual change in slope. 
But that's okay. 
Whatever we do, we can just calibrate to 
that slope. 
But also note that the scatter improves, 
the further in the red we go, for the 
reasons I just mentioned. 
There is less effect of the extinction, 
and less susceptibility fluctuations due 
to the very luminous young stars. 
The other important correlation is so 
called fundamental plane. 
This is actually a set of bivariate 
correlations, which connect one property 
of elliptical galaxies with the 
combination of 2 others. 
But it's usually shown in this form, 
where a distance dependent quantity. 
Radius determining a certain consistent 
fashion is correlated against a 
combination of velocity dispersion. 
Which, again, is the Doppler broadening 
of spectroscopic lines in galaxy 
spectrum. 
And it means surface brightness. 
Surface brightness not depending on the 
distance, at least in Euclidean. 
Space. 
The observed scatter of the fundamental 
plane visible bands say a is about 10%. 
So it is as good as the Tully-Fisher. 
It may be even slightly lower and 
actually why is it so small is an 
interesting problem enough itself, which 
will come back to when we talk about 
properties of elliptical galaxies. 
Its zero point nowadays, tends to be 
calibrated with surface brightness 
fluctuations. 
The fundamental plane comes in another 
flavor, so called DN sigma relation, 
where DN is the diameter of a galaxy at 
which its surface brightness reaches 
certain value. 
This turns out to be a slightly oblique 
projection of the fundamental plane, but 
it basically works in the same way. 
Next time we will talk about use of 
supernova as a standard candle, a very 
popular method these days.