And particularly the H.R. diagram we have
already discussed the equations that govern
a star which is burning hydrogen and these
equations yield solutions.
For the mass range 0.06 to 120 solar mass
so what happens for masses which are smaller
or larger than this. Suppose i take a star
whose mass is less than 0.06 solar masses
and apply the equations of hydrostatic equilibrium.
The mass distribution the luminosity distribution
etc and solve them it turns out that the density
and temperature at the core are not adequate
for hydrogen burning 
so if you have a mass smaller than this the
nuclear fusion process does not start up that
is the basic thing that you do not have a
star.
If you have a mass which is greater than 120,
the solar mass then what happens is that the
luminosity 
exceeds the Eddington luminosity 
and let me remind you we have discussed the
Eddington in a problem earlier and if the
luminosity exceeds the Eddington luminosity
the radiation pressure adequate to blow out
the outer layer of the star the gravitational
attraction is not sufficient to hold in balance
the radiation pressure.
So, if you solve those equations for a star
of mass greater than this the luminosity is
so high that the outer layer of the star the
star gets blown apart due to the radiation
pressure and if you do not have a stable solution
okay, so you find solutions in this mass range
now corresponding to this mass range.
The radius of the star varies between 0.1
to 150 sorry 15 times the solar radius, okay
so the point of note here is that the mass
range the ratio of the largest mass for which
you can have a hydrogen burning star to the
smallest mass is a factor of 2000 whereas
for this mass range the radius does not change
its not very sensitive to the mass the radius
changes only by a factor of 150, okay does
not increase proportionately it is a very
somewhat vague dependence.
But if you look at the luminosity so you saw
the equation and determine the luminosity
of the star solve the set of equations and
for the questions which we had discussed and
look at the luminosity of the star the luminosity
range between 0.011 to 8 *10 to the power
5 solar luminosity so the luminosity is extremely
sensitive to the mass 
and you change the mass by a small amount
the luminosity goes up enormously okay this
is a factor of 10 to the power 8 variation
in the luminosity.
So, the brightest star is 10 to the power
8 times brighter than the faintest star which
you can have provided the stars are burning
hydrogen because these are all what we have
been discussing till now are all solutions
of the four equations which we had written
down and they cover this mass range you have
solutions for the mass range which i told
you for this mass range and the corresponding
range of radii is this range in the luminosity
is given over here.
Ok now the temperatures okay so that temperature
look at these stars can be classified let
us now have a look at the classification of
this of the of the stars so the for the different
masses you get stars of different temperature
surface temperature.
And the stars can be classified based on the
temperature which is also indicated by the
spectrum, so this is called Spectrum spectral
classification, so the spectrum is a direct
indication of the temperature and the hottest
star is classified as O, the O stars are the
hottest 
and the M stars are the coolest and observationally
this manifests itself in the spectrum of the
stars so in the O stars you get helium absorption
spectra you get helium absorption lines to
observe to excite helium helium you have to
produce ultraviolet radiation.
So, these stars O stars produce copious amounts
of ultraviolet radiation which is adequate
to ionize helium, so you can observe helium
absorption lines whereas in the M stars you
get metal absorption lines, and these are
the two extremities the whole spectral classification
is as follows so the hydrogen burning stars.
Okay before i proceed any further a point
which i should have mentioned earlier.
Stars which burn hydrogen 
stars these are referred to as main sequence
stars and why it is called the main sequence
i shall explain to you shortly okay, so we
shall interchangeably we are terming them
as hydrogen burning stars our main sequence
stars and the equations which we have developed
are for these classes of this class of stars.
Now in general and stars can be classified
into the this based on their spectrum and
the spectral classification scheme is as follows
O Stars are the brightest hottest stars M
stars are the coolest stars and you have different
spectral classes in between so let me write
down the entire space spectral classification
scheme it is as follows O and there is a pneumonic
so let me tell you the pneumonic also O B
focus only on the capital letters OB A fine
is a fine girl or guy whichever you prefer
kiss me so this is the pneumonic by which
the astronomers remember the spectral classification
scheme.
And the different spectral classification
is O B A F G or it could be this K M the O
stars are the hottest and the M stars are
the coolest. Okay and this is a pneumonic
by which you can remember the spectral classification
scheme there is a further subdivision inside
this so the B stars for example, so you have
you can have a classification a sub classification,
so the subclass say is B0 to B9, okay so this
holds for all of them.
So, there are two 9 is a subclass for each
spectral type again you have a subclass which
is labeled by the number 0 to 9 where this
is hottest hotter than this okay, so the temperature
is essentially inferred from the spectrum
straight away okay now given the same temperature
it turns out that you can have classes of
different luminosities also so to indicate
this there is another index which indicates
the luminosity.
And this runs from 1 to 5 and 51 are the giants
so the 1 are the most luminous stars 5 are
the lesser luminous star so 5 refers to the
main sequence stars 1 of us two giants which
we shall come to okay super giants giant so
this is a luminosity classification in addition
to the spectral classification you also have
a luminosity classification which tells you
the luminosity so the sun for example the
sun is classified as G. 25 because that is
the classification of the sun just have a
look where does the sun lie.
The sun is a typical star very ordinary kind
of star and it is not a very hot star neither
is it a very cold star it lies somewhere in
the middle of the spectral classification
scheme it is G star there is a sub classification
too which is somewhat towards the hotter side
amongst the G stars and it is a main sequence
star burning hydrogen which is indicated by
5, so you could have a G star which is not
on the main sequence also, okay so the G2
is an indication of the temperature.
This is an indication of the luminosity this
is how stars are classified observationally
and this is an outcome also of the equations
so the equations also if you take a mass and
solve the equations you will get it from the
surface and properties you will get the spectral
classification class and you will get luminosity
also okay another outcome.
A very interesting outcome is the convection
whether the energy transport is convective
or radiative on the diverted eighty again
we have discussed this and i have told you
that the criteria for deciding which of these
will occur is that if the temperature gradient
so if if we have radiation transport and if
that if the radiative temperature gradient
if this 
exceeds the idiomatic transport temperature
gradient then we have convection and convection
is a very efficient means of transporting
energy.
So, the temperature gradient then does not
exceed this value it remains at this value
so with a lower temperature gradient you are
able to transport more energy, okay now so
the question is when you solve these equations
the 40 questions with the extra conditions
that we have discussed it turns out that for
a lot mass low masses the energy transport
is entirely convective 
and then as you keep on increasing the mass
once you cross 0.4 times the solar mass.
So once you cross this you develop a radiative
core and you convection region gets smaller
and smaller that is outside so the envelope
is convective so the core is radiation radiative
then and is transported radiatively in the
core and the envelope it looks like this so
you have radiative transport here and then
you have convection in the outer parts just
drawn a little part of it then as you keep
on increasing the mass if you look at stars
of more and more mass the radiative core keeps
on increasing .
The sun we have seen that 70 percent from
the inside the energies transported radiatively
and 30 percent out of 30 percent energy transport
is conducted okay when you reach 1.5 solar
masses the energy transport is entirely radiated
so here the energy transport is entirely radiative
and then when you cross two solar masses the
core becomes convective the core becomes convective
once you cross two solar masses the core becomes
convective and for more masses more than this
the core becomes convective.
So here what happens is the core becomes convective
here, the convection is transported radiatively
outside for masses more than this so for masses
more than this the region where the nuclear
fusion is going on you also have convection
taking place in the same place where the energy
is generated okay, so this gives you a picture
of the solution and whether energy is transported
convectively or radiatively. now let me introduce
the HR diagram this is called the herds.
Rustle diagram referred to as the H.R diagram
the HR diagram is one of the most useful methods
for looking at stars for understanding stars
and looking at stars this in this HR diagram
what is plotted in principle is as follows
so in principle what is plotted is as follows
and you plot a log of the luminosity the luminosity
increasing this way versus the log – log
temperature so the temperature decreases in
this direction it increases this way. So,
the luminosity let me just put it here it
increases this way.
The L. L. Increases in this direction T. Decreases
in this direction because it is – log T
so a more the more loving us an object the
upper or higher up in this graph in this diagram
it will appear and the cooler an object is
the more to the right it will shift okay so
let me again make this point clear and this
side is cooler, and this side is more luminous
that is how to interpret the study in practice
you cannot observationally determine these
two things.
In practice from the observational point of
view what is plotted let me show that here
so from the observational point of view let
me put it in a different diagram.
So, from in practice what you plot 
is you plot the absolute magnitude in some
band let us say in the V band 
and it is decreasing in this direction, so
it is increasing this way right more the more
the luminosity the more negative the absolute
magnitude will become which is -2.5 log fluxes
okay, so it is decreasing in this direction
the magnitude is decreasing or the luminosity
is increasing so this these the sense over
here remains intact.
Okay so the luminosity is observationally
quantified to the absolute magnitude of the
logarithm of the luminosity and the temperature
as i have told you is observationally quantified
to the color, so you could use the B.- V color
for example okay could use U-B color and these
are the two things that are plotted along
the two different axis so let us say B-V so
this is a color magnitude diagram it is a
magnitude color like the magnitude diagram
so bear in mind the color.
Let us look at the color if the the larger
the value of the color. it it means if this
value is larger it means that the blue is
fainter than the visual band visual band is
at a lower frequency higher larger wave length,
so a larger wave length having larger radiation
tells us that it is cooler so the more i go
in this direction the larger the value of
the color the cooler the stars okay the more
i go in this direction the less the more luminous
the stars.
Now a point which i should make that early
on when this diagram this diagram was introduced
in the beginning of the in the end of the
nineteenth century or beginning of the twentieth
century when this diagram was introduced this
people had no method of determining distances
to stars except for parallax even now there
are no diet very few direct methods but the
parallaxes were measured only for very nearby
stars so there was no way you could determine
the absolute magnitude what you had to do
was.
So you do not know what this number is for
a star you do not know the luminosity but
what you could do was you could look at stars
which appear in the cluster so it could be
a globular cluster or it could be an open
cluster a cluster but a cluster and so that
the distance to the cluster is approximately
the same from us so if you look at a collection
of stars to which the distance is approximately
the same then the distance introduces a factor
which is the same for all the stars.
Okay so up to a multiplicative constant we
know the luminosity the multiplicative constant
distance affects all the stars equally or
if you look think of it in terms of the absolute
magnitude what you could measure was the apparent
magnitude but the apparent magnitude differs
from the absolute magnitude/5 log the distance
right and the distance was the same for all
the stars we are looking at them in a cluster
or in or in some kind of a cluster in some
kind of a group which is physically associated
with one another.
Okay so this is what people looked at and
what people found was that the stars the bulk
of the stars lie along a band which runs diagonally
across this diagram so the bulk of the stars
it was found lie in a band which runs diagonally
roughly like this across the diagram okay
and if you had a different distance all that
would do is it would shift this and dieting
there because the distance would shift the
sun. So, the curve would go up and down for
different clusters different groups of stars.
But it is found that for all of them the bulk
of the stars lie on band like this and this
is called the main sequence 
okay 
now later so this explain so what it tells
us.
It tells us that there is a tight correlation
it tells us that there is a tight correlation
between the temperature and luminosity 
of a star and understanding this was a very
big issue what gives rise to this tight correlation
between the temperature and luminosity of
a star and bulk of the stars turned out line
of band like this. There are stars elsewhere
also okay, but this was a very dominant feature,
and this was called the main sequence it was
later realized that these are the stars which
are burning hydrogen.
These are the hydrogen burning stars which
lie on the main sequence all the hydrogen
burning stars in the HR diagram they lie on
a band like this called the main sequence
okay and it was later realized also that this
there are we have discussed the equations
there are fundamentally questions governing
this that has been discussed but it is not
clear why they should lie on a straight line
like this we shall see later that there are
simple scaling laws.
Which give us some idea why the hydrogen burning
stars should lie on a region like that okay
this is something that we shall discuss later.
But things have changed since then and there
was a satellite called HIPPARCOS 
this satellite was launched with the explicit
aim of measuring parallaxes okay 
to nearby stars parallax distances to nearby
stars and it can this satellite could has
measured parallaxes to an accuracy of one
million arcsecond 
okay and it has measured legs distances to
around one hundred thousand stars in the solar
neighborhood so let me show you a HR diagram
made with the HIPPARCOS data it has around
17000 within 300 light years.
And focus on this diagram if we just draw
your attention to this diagram this HR diagram
so this is the HR diagram which has been measured
by the heat of the stars, so this shows you
all the stars that have been whose paralyze
distances have been measured using HIPPARCOS
it has got 17000 stars that cannot all the
star 17000 stars within 300 light years the
luminosities are what have been measured.
So, this is the absolute magnitude of what
has been measured using the HIPPARCOS data
the HIPPARCOS satellite measured the distances
okay which goes into determining the absolute
magnitude apparent this is the difference
of the apparent magnitudes in the B band and
the V band and this has been measured from
ground based observations okay HIPPARCOS those
basically give the distances so notice that
the bulk of the stars lie on a band that runs
from left to right across the HR diagram.
And this is the main sequence this so all
these are all stars which burn hydrogen 
and these are the low temperature cool stars
relatively cool stars as you move to the left
you go to hotter and hotter stars bluer that
you go blue and you basically as you move
left the color gets less so you are going
towards bluer stars which are hotter and you
find that the hotter stars are also more luminous
and the hottest stars that you have the most
in this.
These are all the stars in the solar neighborhood
within 300 hundred is that all stars within
the solar neighborhood within 300 light years
there shows you the stars on the main sequence
okay now and these are not the only stars
that you have there other stars also okay
okay before we move on let me show you another
HR diagram.
This shows you another HR diagram here the
HR diagram is off a globular cluster a globular
cluster is a collection of stars is a distinct
collection of stars and now here the distances
to the individual stars are not known but
it does not matter because they are all of
those roughly the same distance so i can interpret
the apparent magnitude itself as an indicator
of the luminosity the distance has the same
effect on all these stars now here again notice
that you have the main sequence over here
this is the main sequence.
But it extends to a much smaller distance
along the HR diagram and notice that there
are deviations.
So, by as compared these two so right over
here the main sequence kind of stops and you
find stars which are off the main sequence
to this turning off from the main sequence
is a very important is a very important effect
and we shall discuss this when we did discuss
the determination of this is age cosmological
ages the age of the universe this is something
that is very important for that okay we shall
come back to this point later.
When we discussed the evolution of stars okay
now let us go back to our discussion of stars
so i have told you how stars can be classified
using represented and classify using the HR
diagram okay.
Now given for any star given the luminosity
and the radius you can define the effective
temperature and for given a star if you know
its luminosity and if you know its radius
you can define what is called the effective
temperature and we have already discussed
this for the sun for the sun the effective
temperature. comes out to be 5800 kelvin okay
now if you compare this with the actual temperature
of the sun at the photosphere so for the sun
this is the effective temperature.
We have done this exercise now for the sun
if you solve the model the solar the Steller
model which we have discussed and if you determine
the actual temperature at the photosphere
discovers out to be 6500 kelvin photospheres
remember is the region where the optical depth
becomes one that is where the radiation originates
from that is the part of the sun that you
actually see the temperature there is higher
than the effective temperature.
And this happens because some of the radiation
it passes through cooler material it gets
absorbed in the process typically the effective
temperature is lower than the temperature
at the surface that you are looking at for
stars okay, but it gives you some idea of
the temperature of the surface so one could
define this effective temperature for other
stars also 
and this is a very useful thing suppose i
find a star, let me discuss in a different
sheet suppose i find a star.
I find two stars at the same temperature 
but two stars at the same temperature one
of them L1 has a luminosity which is much
less than the luminosity of the second star
then you would conclude that the the radius
of the second star is much smaller than the
radius of the first sorry the radius of the
first star is much smaller than the radius
of the second star.
Okay so if i have stars and i plot them on
the main sequence the main sequence along
the X Axis i have the temperature along the
Y Axis i have the luminosity so i just tried
to visualize so this is the star on the main
sequence let us see and i have another star
which has the same temperature as the main
sequence let us say a very hot star it is
quite very hot, a very hot star in which part
of the diagram would it appear it would appear
to the left.
So hot star would appear somewhere over here
to the left now there is one star which lies
on the main sequence so there is a main sequence
star over here and there is another star which
is much fainter so what would you conclude
you would conclude that the second star which
is much fainter has a much smaller radius
okay so it does turn out that you do get some
stars like this over here in this part of
the HR diagram of the main sequence so these
stars are called White Dwarfs.
So now you start to appreciate the utility
of this HR diagram you take any star and put
it on the H.R diagram if you know it is a
distance or if you know it is in a group then
you compare it with the other stars you do
it just by placing it on the H.R diagram okay
and if it lies somewhere and much below the
main sequence but it is hot then it is white
dwarf it is an extremely hot star but of extremely
compact size such stars are not hydrogen burning
stars they are what are called white dogs.
There are different kinds of stars not like
the ones that we have been discussing they
do not burn hydrogen at the center to start
right all the stars that burn hydrogen are
over here similarly let us look at the stars
over here here these stars are the hot or
they are cold they are quite cold but they
have extremely high luminosities these stars
so you have a kind of star which is cold.
So, these are cool, so you have stars which
are at relatively low temperature 
but high luminosity 
these stars are what are called giants and
super giants 
okay giants and super giants, so these are
stars which are extremely not very hot stars,
but they are extremely luminous, so the radius
of the temperature is low, but the radius
is large basically are is large these again
are not main sequence stars okay these giant
stars they have a large radius.
And because of that their surface acceleration
surface acceleration or surface gravity which
is G* mass of the star / R square this is
relatively low, so these are not extremely
massive stars they are stars of optical mass
which have to be going like the sun or maybe
somewhere around there which have the radius
have increased so the surface acceleration
has fallen and as a consequence of that what
happens is that the density of the the acceleration
is low.
And as a consequence of that the density two
things happen the density is small at the
surface and this is obvious because the R
has gone up the mass is fix and also the surface
pressure is low so at the surface the density
and pressure are both small and this manifest
itself in who is in the spectrum of the star,
so the first manifestation of the density
is that the density remember is there SAHA
ionic equation of any species and, so it affects
the ionization of the species.
That is one effect and so the density comes
into the SAHA ionization equation and it affects
the ionization state, so this affects the
ionization state that if you go back to your
SAHA ionization equation you will see that
it depends the ionization state depends on
the density and the change in the density
affects the ionization states the pressure
difference the smaller pressure manifests
itself in the line width of the lines so if
you look at the spectral lines.
From the surface the line width reflects the
pressure and the more the pressure you have
the broader lines due to pressure broadening
so for these giants you will have narrower
lines okay so the line width get narrower
because pressure is low and from the sky looking
at the spectrum 
it is possible to classify a star as a giant
etc okay so there is a classification scheme
which I had already mentioned let me tell
you that.
So classes of the class 5 are the main sequence
class 4 these are the sub giants 
class 3 these are the giants class two these
are the bright giants 
and one these are the super giants so the
main sequence are the ones which burn hydrogen
so let us go back to our HR diagram and take
a look so these are the stars that burn hydrogen
here if the local neighborhood you also find
the giant branch over here there is another
branch over here called the horizontal branch
which is kind of overlapping with a giant
branch here.
Okay which is more clearly visible over here
so if you look at the globular cluster this
is a particular globular cluster M3 okay so
the globular cluster is called M Messier on
the Messier catalog this is the main sequence
here you have the giant branching these are
all giant stars off the main sequence and
here you have these stars which lie on what
is called the horizontal branch they are the
horizontal branch now if you take a look at
this you will see that there is a very interesting
thing that the brighter part of the main sequence
is missing the brighter part of the main sequence
is missing.
Whereas you have this very extended giants
okay this indicates that there is some relation
seems to indicate that there is some relation
between these and if you look at different
globular clusters you will then see that the
turn off occurs at different places okay so
the main sequence is where these stars first
lead a large part of their life when they
are formed when stars are first form they
are formed in the main sequence they lead
a large part of their life burning hydrogen
once they finish burning their hydrogen.
Then they depart from the main sequence and
you then they become these other kinds of
stars the giant super giants they go into
the horizontal branch some of them become
wide over here etc and that is a very interesting
topic the life story of evolution of stars
and that is something that we shall take up
in the coming class in todays lecture what
we learned is that you have solutions for
stable solutions and of the equations that
govern a star for only a finite mass range
below that mass the stars do not the pressure
the core.
The temperature and density at the core are
not adequate for fusion above the mass range
the Eddington luminosity is exceeded the luminosity
exceeds the Eddington luminosity and the star
blows up away parts of the stars the luminosity
is so high so there is a finite mass range
over the mass range the range of radius is
not very large but the luminosities vary enormously
so if i go from here to here the luminosities
have changed by 10 to the power 8 or something
like that okay the radius has not changed
very much.
The temperature also has not changed by an
enormous amount the colors do not change all
that very much okay they do change the temperature
does change and may be of order or somewhere
of that order but not not like the luminosity
the luminosity is the thing that really increases
drastically and that is quite clear it should
be so.
Because the luminosity scales T to the power
4 temperature increase and by sum amount will
be a much more increase in the luminosity
so an a combined increase in the radius and
temperature will give rise to a large increase
in luminosity okay and then i discussed how
from the observational point of view how the
stars are classified you have a spectral classification
and you can get real understanding of the
stars if you look trace them on the HR diagram
so let me stop here we shall resume on this
tomorrow in the next class.
