Well let us start climbing the distance 
ladder. 
First, we'll talk about, methods that we 
can find distances to nearby stars, or 
star clusters. 
The Trigonometric Parallax is the most 
basic measurement of distance in 
astronomy. 
And, hopefully all of you are, well 
famililar with it. 
It is pure geometry, and there is nothing 
uncertain about it. 
So by measuring the annual apparent 
slooshing on the sky of a star, we can 
figure out how far it is, if we know the 
distance between earth and the sun, which 
we do know, with a great percision. 
So the method by itself is safe, the 
problem is that, these are very small 
angles, and the current state of the art 
is that we can measure distances using 
parallaxes To about one kiloparsec out, 
more or less, and that's well within our 
own galaxy, never mind external galaxies. 
The next one is so-called moving cluster 
method. 
This is a statistical method. 
Stars do come in clusters, clusters move 
relative to the solar system, and stars 
have internal motions. 
Within a cluster itself. 
Since, ostensibly, all cluster stars are 
moving roughly in the same direction. 
If we look from afar, we will see them 
converging toward some distant point. 
Which is direction which they are going. 
By measuring the spread of those angles, 
we can figure out, what's the angle of 
their actual velocity vector to the line 
of sight? We also measure proper motions 
of stars in angular seconds, per unit 
time on Sky, as well as their radial 
velocities. 
We can assume that random motions within 
the cluster are same both in radial and 
tangential direction. 
So by knowing what the, what the angle is 
between The radial and tangential 
components, we can figure out what is the 
parallax to the cluster. 
Note that this makes some assumptions 
about the internal motions of the 
cluster, and it is basically a 
statistical technique. 
So the more stars you have the better, 
but cluster's have finite number of 
stars. 
By measuring distances to a number of 
stars using either one of two techniques, 
we can calibrate the Hertzsprung-Russell 
or polar magnitude diagram for stars. 
Hopefully this is something you know 
about very well. 
It is a lot of stellar luminosity versus 
temperature. 
Temperature is sometimes measured through 
color. 
The important, thing to note here, is 
that stellar luminosities are distance 
dependent according to the inverse square 
law from given measurements of apparent 
brightness. 
Temperature, measured from spectra, or 
from colors, is not. 
Star will have exact same temperature no 
matter how hot it is. 
So, once we calibrate The main sequence 
of this Hipparcos's diagram. 
If we can measure stellar colors or 
temperatures, then we can read off their 
absolute magnitudes. 
From absolute and apparent magnitudes, we 
can figure out how far they are. 
In a given cluster you may have thousands 
of stars, and therefore you can determine 
the distance very precisely, the distance 
to the cluster itself. 
Now this works fairly well for young 
stellar clusters in the galactic disc. 
However, no globular cluster is as yet 
close enough to measure parallaxes to it. 
And so something else will have to be 
done about those. 
There what we do is we measure, we use 
Field stars of the same type as those 
that live in globular clusters, the 
population 2 stars. 
HR diagram measurement is a collective 
measurement. 
But not all stars are created equal. 
Between the temperature luminosity plane, 
there is a strip within which stars are 
unstable to pulsation. 
So called instability strip. 
On the main sequence, those will be the 
Cepheids, or delta Cepheid stars. 
Among the globular clusters, horizontal 
branch, those will be RR Lyrae stars. 
There are many different kinds of 
pulsating stars. 
But those are the principal ones. 
And indeed, physics of Cepheids and RR 
Lyrae is probably the best understood of 
all It turns out that there are 
correlations between observed period, 
which again does not depend on distance, 
and luminosities of these stars, which 
do. 
Those are empirical relations and they 
can be calibrated if you had distances to 
a number of these pulsating stars using 
one of the previous techniques. 
At first, people did not know that there 
are different kinds of pulsating stars, 
they all thought it was one kind. 
And the first star that Hubble figured 
out in Andromeda was Cepheid, pulsating 
star that produced the first distance to 
another galaxy. 
But people are confusing RR Lyrae which 
are much dimmer than surface with surface 
themselves and that confusion were 
through a an error of factor of 2 in the 
distance scale. 
Once Walter Baade understood that there 
really are two different kinds of period 
luminosity relations. 
That error was corrected. 
Let's talk about cepheids in more detail. 
Because they remain among the most 
important distance indicators altogether. 
They're young luminous stars. 
Therefore, they'll be found in star 
forming discs. 
And star forming regions, whereas Delta 
Ceti itself is relatively bright one 
within our own galaxy. 
It was Henrietta Leavitt working with 
Harold Shapley who recognized that there 
is a correlation between period and 
luminosity, as they were studying stars 
in the Magellanic Clouds, about 50 
kiloparsecs away. 
Since all of them were roughly the same 
distance. 
Apparent magnitude would correlate with 
period and then they understood that 
comparing that with nearby Cepheids they 
can find out how far the Magellanic Couds 
are. 
Surface are important because they are 
bright and so we can see them far away, 
we can find them in galaxies up to maybe 
25 mega parsecs or so. 
So we can calibrate distances to a number 
of nearby galaxies using Cepheids, which 
is not easy but it's possible. 
And then we can use distances to those 
galaxies to calibrate some other 
relations. 
It isn't all perfectly safe. 
The position of stars must depend on 
their internal composition and opacity, 
and therefore metallicity. the exact 
effects of metallicity are not firmly 
established as here. 
Moreover, there are external problems 
such as extinction. 
Cepheids are found in star-forming 
regions and they also tend to be dusty. 
So one has to make a, a correction for 
it. 
In very distant galaxies, they may be 
blended by other stars giving us wrong 
luminosity. 
Cepheids remain keystones of the distant 
scale and that also applies for the 
measurements with the Hubble space 
telescope. 
Here are some examples of Cepheid period 
luminosity relations in the modulaic 
clouds in different filters. 
The scatter is biggest in the blue band 
and the smallest in the near infra red. 
However the amplitudes are biggest in 
blue and smallest in infra red. 
It's a good idea to observe them in 
different bands so that the The effects 
of extinction can be taken out. 
Until Hipparchus satellite flew, we did 
not have parallax calibration of 
Cepheids. 
Distances to Cepheids until then were 
based on the distances through clusters 
in which they live. 
And distances through those clusters were 
by in large use, determined using the 
cluster map. 
However, with the Hipparcos, a handful of 
cepheids was within reach. 
And these are the actual calibration 
relations for distances to cepheids. 
As you can see, they're fairly noisy. 
But in any case, for the first time, they 
gave us an absolute calibration of the 
period luminosity relations for cepheid. 
This is going to get a lot better with 
the Gaia satellite, which is an astronomy 
mission which will measure Cepheids to a 
much larger number of pulsating stars 
with a much greater precision. 
The other important kind of pulsating 
stars are RR Lyrae. 
Their also named after the prototype star 
that was first recognized. 
Their at a population to stars, their not 
on the main sequence, but on the 
horizontal branch, which is the helium 
burning may sequence, and their found in 
old stellar populations, such as the 
globular clusters. 
They do have an advantage that their 
periods are short. 
So, it's much easier to observe full 
periodic Pulsating curve for an RR Lyrae 
Star than it is for a cepheid. 
Because they're dimmer, they can be 
really used, only within the local group 
of galaxies. 
But that's still useful, and it provides 
a welcome check, on the distances measure 
using Cepheids. 
Now, let's take a closer look to what 
happens when a star is pulsating. 
It's photosphere expands, but the 
temperature changes. 
As well. 
So the radius changes, the temperature 
changes therefore, luminosity must 
change. 
If we observe stars spectroscopically, we 
can observe the velocity of the photo 
sphere. 
Come storage us and go away from us. 
So we can measure stellar temperatures 
using colors or spectroscopy. 
We can measure velocity of the pulsating 
photosphere using spectroscopy and we can 
measure the changes in the apparent 
brightness. 
This forms the basis of so called 
Baade-Wesselink Method. 
If the pulsating stars were perfect black 
bodies, this would be an excellent pure 
physics based method to determine 
distances to them. 
Unfortunately, real stars are not perfect 
black bodies, but they're not too far 
either. 
So, at any given time, the flux from a 
star will be it's luminosity, which is in 
itself, given by Stefan Boltzmann 
formula. 
It's proportional to the temperature to 
the fourth power. 
And to the surface area of the star, 
which is proportional to the square of 
its radius. 
And it's universally proportional to the 
square of the distance. 
We can measure those quantities all 
throughout the pulsation period. 
So temperatures are directly observable 
from photometry. 
But so are the fluxes. 
And the only remaining question is, can 
we find out the radius? We can, in a way, 
because if we integrate motion of the 
photosphere as traced by the radial 
velocity, we can find out how much the 
radius has been changing. 
So we have 3 equations in 3 unknowns and 
we can solve for that. 
Therefore, we can obtain distances purely 
from measurements. 
And assumptions about black body nature 
of stellar photospheres. 
The problem is, the stars are not perfect 
black bodies. 
And some modelling of stellar 
photospheres has to be done in order to 
actually make the method to work. 
So there is model dependence. 
And that's where the uncertainties come. 
From. 
There are a couple more statistical 
methods that are based on stellar 
indicators. 
Globular star clusters themselves have a 
distribution of luminosities. 
It turns out that their distribution 
function, the luminosity function of 
globular clusters, seems to be universal 
among galaxies for reasons that are not 
really well understood at all. 
Actually you can think of many reasons 
why this shouldn't be the case. 
But empirically, the do seem to be very 
similar. 
Thus, if we can calibrate the luminosity 
function of globular clusters in the 
milky way using local distance 
indicators, then we can apply it to 
luminosity functions of globular clusters 
in other galaxies. 
A good thing about this is the globual 
clusters are much brighter than most 
stars, and so they're easier to find and 
easier to measure. 
Now one problem is that the number of 
globular clusters vary widely among the 
galaxies. 
Elliptical galaxies, early type spirals, 
have most, late type spirals. 
Hardly have any and therefore there be a 
statistical uncertainty for those 
galaxies. 
A similar method uses luminosity function 
of planetary neubli. 
As you recall planetary nebuli 
represents. 
Stellar envelopes that have been shed by 
a star, following it's horizontal branch 
phase. 
They are illuminated and iodized by the 
incandescent core that remains. 
And most of their light emerges in 
recombination emission lines. 
A very prominent line among those, is the 
line of ionized oxygen at 5007 angstroms. 
We can measure luminositys of those lines 
alone and then we can form luminosity 
function that is distribution of 
luminosities for that emission line alone 
for planetary nebula. 
That too, turns out to be more or less 
the same for the nearby galaxies. 
Ostensibly, that reflects the way in 
which stars evolve. 
But there isn't solid strong physical 
basis. 
This is an empirical relation. 
And again It, it is statistical. 
It can work up to the distance of the 
vertigo cluster, which is not so bad, but 
not beyond. 
And finally, there is, the tip of the red 
giant branch. 
The stars cannot get more luminous than 
certain amount. 
This is related to the, so called 
Eddington Luminosity, which, hopefully, 
you have heard about, and which we can 
address later. 
So, empirically, they don't seem to get 
brighter than a certain limit. 
And, if we can observe stars in other 
galaxies, nearby galaxies like Andromeda, 
and find out what are the most luminous 
ones. 
Where does luminosity stop? Then that 
threshold can be used as standard candle. 
The advantage of this is, of course, that 
these stars are bright. 
The disadvantage is that it is not a 
terribly well-defined indicator. 
There aren't very many of those stars so 
the numerical fluctuations can affect the 
result. 
That is it about the stellar. 
Distance indicator. 
Next time we will talk about so called 
Distance Indicator Relations for 
galaxies.