welcome in the last class we discussed the
hr diagram and how we can show the position
of the star on the hr diagram and how we can
infer considerable information from this today
we are going to study the evolution of stars
and much of the knowledge about the evolution
of stars comes from solving the equations
that govern the state of a star so you take
the equations that govern the state of a star
and solve them
now this is a very tricky business and it
can be computationally very intensive because
the chemical composition changes and if you
look at the evolution of stars one also has
to deal with situations which are not always
in equilibrium so it is a very complicated
thing it can be a very complicated thing so
i will briefly give you some idea about the
evolution of a star some of the results that
people have obtained about the evolution of
a star stars form by the collapse of gas clouds
in the interstellar medium okay that is how
stars form
so you have gas clouds 
in the interstellar medium 
the is which is also referred to as the ism
and these gas clouds collapse to form stars
now when a gas cloud collapses so just imagine
this is a gas cloud and it is collapsing 
as it collapses the temperature is going to
rise and after a certain amount of collapse
the temperature in the center in some central
region is going to be adequately high is so
that the hydrogen gets ionised
so this part the temperature is adequately
high it is around 4000 so in the interior
in this part it is around 4500 kelvin it may
be more inside but this is the temperature
at which hydrogen starts to get ionized so
it may be more inside but at the surface of
this region it is around this much 4500 now
when you look at this cloud from outside so
if you are looking at this cloud from here
the neutral part is transparent
the neutral part it is largely transparent
neutral gas can only absorb at certain specific
frequencies or wavelengths which correspond
to the atomic transitions for other frequencies
or wavelengths they do not interact with the
radiation at all okay so it is largely transparent
so if you are looking from here you will not
see these neutral parts where it is cooler
when you reach the ionized part you have thomson’s
scattering and the optical depth is going
to reach unity the part of the star that you
will see is essentially this around 4500 kelvin
okay so as the gas cloud collapses during
the collapse you will see a surface of this
temperature more or less this constant temperature
the temperature inside may go up but you will
not be looking into it because it becomes
optically thick quite fast
so these are represented so the track of a
cloud that is collapsing to form a star is
represented on the hr diagram by what is known
as a hayashi track so he was the japanese
scientist who work these out so it follows
what is called the hayashi track on the hr
diagram let me draw that
so the log we are going to plot an hr diagram
in this diagram we will plot the log of the
temperature in kelvin this is going to be
in kelvin and along the yaxis we will plot
the log of the ratio of the star to the solar
luminosity luminosity to the solar luminosity
so l is the luminosity of the star and this
is the solar luminosity this is the diagram
we are going to use the hr diagram
observationally you cannot directly determine
these you would have used the colour and the
absolute magnitude but here we are going to
draw this theoretical hr diagram which is
equivalent okay and let me put the axis so
we will start from 35 and we will go up to
5 so 354 45 5 okay so this is going to be
4 okay so that is the temperature range and
the luminosity range is going to go from 2
so which is a hundred times fainter than the
sun and it is going to increase in intervals
of 2
so this is going to be 0 this will be 2 and
this will be 4 so this is where the sun would
be this is hundred times brighter than the
small luminous than the sun this is 10000
times more luminous than the sun we have seen
that it goes up to the luminosities can vary
from roughly hundred times fainter than the
sun to 8 into 10 to the power so roughly 10
to the power 5 times brighter than the sun
so it will be up will go up to five okay so
this encompasses the entire possibility roughly
okay so the hayashi track the temperature
remains a constant at a value around 4500
which corresponds to 36 in this log scale
okay so a hayashi track will look something
like this the luminosity will keep on decreasing
the temperature will not change okay a gas
cloud which is forming a star as it collapses
will move at a it will go down at a constant
temperature that is the surface that you are
looking at okay
the interior temperature may be higher so
in the hr diagram we plot the surface temperature
the effective temperature so the hayashi track
is going to come like this straight down okay
now let me draw the main sequence
the main sequence as it collapses what is
going to happen is that the temperature in
the center and the density in the center is
then going to become adequately high that
the hydrogen starts burning the nuclear reaction
starts and it is going to form helium so that
once that will lead us to the stable age main
sequence configuration so let me also draw
the main sequence over here
the main sequence i have told you is a line
that goes like this and the temperature of
the sun we know is around 5800 kelvin so somewhere
over here so the main sequence is a line something
like this may not be exactly a straight line
but for our purposes we shall take it to be
a line something like this okay
so as the gas cloud that is collapsing approaches
the main sequence depending on the mass of
the gas cloud it will turn left and reach
the appropriate point on the hr diagram so
the more massive or the more luminous stars
will be somewhere here the less massive will
be here so depending on the mass of this cloud
it will go down like this and then it will
turn left and assume the correct position
appropriate position on the hr diagram okay
that is the stable equilibrium position so
these are the hayashi tracks
there will be different tracks for different
masses this part will be roughly the same
but it will reach a different part on the
hr diagram go and settle at a different part
in the hr diagram now stars once they are
in the hr diagram in the main sequence okay
sorry this is the main sequence so it will
approach the main sequence and go and settle
into a different point on the main sequence
once it reaches the main sequence that is
the stable hydrogen burning configurations
stars will remain there form time periods
which vary
so the least the most massive stars could
spend several millions of years there 
in that stable configuration burning hydrogen
these are the most massive stars whereas the
low mass stars could spend few tens of billions
years these are the low mass 
so the lifetime on the main sequence where
it is burning hydrogen and varies a more massive
star is more luminous that you can see from
here it burns away its hydrogen rather fast
whereas a less massive star burned its hydrogen
slowly its luminosity is low and it lives
a long life
and the lives vary from several million years
to few tens of billion years they remain on
the main sequence steadily in a steady state
okay and they burn hydrogen it is a stable
configuration that is the configuration for
which we wrote down the equations several
classes ago and that is the stable hydrogen
burning configuration
now once the star burns out all the hydrogen
it is then that the stellar evolution again
starts off okay well strictly speaking it
is not so let me make that point clear the
position that the star comes to initially
when it joins the main sequence when it goes
and reaches the main sequence is what is called
the zero age 
main sequence
this is where the core is made up of the composition
of the interstellar medium cloud from which
the star was formed now as it burns hydrogen
the composition changes it still continues
to burn hydrogen but the composition changes
and there is a small shift in the position
on the main sequence as a consequence of this
okay
and let me indicate this on the diagram so
let us for example look at the evolution of
the sun which is somewhere over here the sun
it moves slightly up as a consequence of this
okay so when it joined the main sequence it
would have been somewhere here and as a consequence
of the hydrogen burning in the center in the
core the chemical composition changes and
it moves slightly up
now the question is what happens when it has
exhausted all the hydrogen at core okay that
is the next stage so the star spends a large
part of its life there and then it has exhausted
all its hydrogen in the core and it is left
with a core that is largely inert made up
of helium
so after it has burned its hydrogen it is
left with the helium core and the helium core
is largely inert to start with okay so we
have a helium core which is largely inert
so this is the helium core in the center and
there is a shell of hydrogen being burnt still
being burnt outside it hydrogen burning is
still in progress in a shell outside it okay
so there is a shell like this where hydrogen
burning is still in progress
now as time progresses this hydrogen burning
shell it burns out more of the hydrogen into
helium so this gets bigger core gets bigger
and the shell also gets shifted to an gross
okay so this is what happens now the next
thing that happens is that this core now contracts
right there is nothing to support it there
is no energy source in the center so this
core now contracts due to gravitation and
as the core contracts it gets heated up okay
so now the core so this is the helium core
this is helium core is expanding is increasing
in size because the hydrogen shell is burning
more and more hydrogen into helium so that
keeps on increasing but this whole thing contracts
because of gravity gravitational collapse
okay and as it contracts so this whole thing
now contracts due to gravitation 
and as it contracts due to gravitation what
happens is it gets hotter
so the core is contracting it is getting hotter
and it emits radiation also in this process
it gets hotter and it emits radiation this
is simple virial theorem okay and this as
it contracts and gets hotter the hydrogen
burning at the edge of the core also goes
up okay so the whole thing contracts and the
radiation from here this gets hotter and the
radiation from here blows 
out the outer envelope okay so the outer envelope
the outer part there is more energy being
generated
there is energy being released from the center
more energy because it contracts which if
you have in a star if you have more energy
being generated at the center we know from
virial theorem that it will expand if a star
loses energy it contracts if a star gains
energy it expands this we have learned this
in the virial theorem so when the center when
the core actually when this thing collapses
due to gravity collapses now it gets hotter
emits more radiation the entire star as a
whole will expand
so the outer part of the star expands and
it expands so much that the luminosity 
actually increases considerably the luminosity
goes up considerably due to the expansion
the temperature falls of the surface but the
radius increases dramatically 
and as a consequence what happens that the
outer part so let me draw a picture again
so the core of the star this is the core this
was my star the helium core collapses it becomes
smaller contracts gravitationally it gives
out more energy so the outer part of the star
increases in radius the rest of the star increases
in radius okay so it becomes now much larger
the temperature of these regions falls
but the total the rate increase in the radius
is so large that the fall in the temperature
is compensated by the increase in the radius
more than compensated and the luminosity goes
up and so this star now becomes what is called
a giant red giant 
red because it has become cooler and a giant
because its radius has gone up and its luminosity
also has gone up
so this now lies on a part of the hr diagram
that looks like this this this is a red giant
okay and this so the star leaves the main
sequence and becomes a red giant the inner
part of the star has collapsed the outer part
as its part has expanded and it has cooled
but it has expanded so much that its luminosity
has gone up okay so this is what is called
the giant the red giant branch hydrogen outside
this is the giant branch not just the shell
the rest of the star
the hydrogen shell is what is burning over
here but this whole thing is embedded in the
rest of the star okay that is what expands
this part just contract with collapse no it
does it is a part of the initial cloud this
is i am just drawing the core okay this is
a part of the whole star the star is some
bigger okay so we remember that in the star
in the main sequence a hydrogen burning only
occurs at the center at the core
so it is in the core only that the hydrogen
gets converted into helium okay energy is
generated only in the center of the star not
through the entire star okay so the core gets
converted to helium at the edge of this helium
core you have hydrogen and that hydrogen is
sufficiently hot so in a small shell there
is hydrogen burning and this thing keeps on
increasing in size because the hydrogen burns
and gets converted into helium
but in addition to that it collapses gravitationally
because the helium is inert there is no more
fusion going on yeah it is increasing in size
yeah no the helium core is still increasing
in size the mass of the helium is still increasing
but the radius is contracting well it could
have been but it is the mass that is increasing
basically the helium mass is increasing because
the hydrogen keeps on burning at the edge
which is what i have tried to show over here
yeah but the helium mass keeps on increasing
hydrogen burning still continues in the shell
and increases faster because the hydrogen
burn this when it contracts physically see
one is the mass the mass in the helium keeps
on increasing the hydrogen and the shell keeps
on burning so the whole thing if it contracts
the burning occurs faster the temperature
goes up the burning occurs faster so the conversion
to helium occurs faster because of the contraction
okay
that keeps on going on but the outer part
of the star because of the increased production
in energy that expands the outer part of the
star that expands and it becomes a giant so
the this part the mass is possibly still increasing
the mass is increasing oh there will be some
stage over there i do not know the exact figure
okay that will depend on the mass of the star
etc okay so it becomes a red giant now the
core keeps on contracting and as the core
contracts its temperature
so inside the red giant the core contracts
and as the core contracts the helium core
contracts as it contracts the temperature
and density go up and finally in the temperature
and density are adequately high that helium
again nuclear burning starts helium now gets
converted to carbon and oxygen and also neon
okay so nuclear it contracts the core keeps
on contracting and it gets hot and dense enough
so that nuclear fusion starts again now burns
helium to carbon oxygen and some amount of
neon
now remember you would expect that once the
energy starts getting produced in the center
the temperature will increase if the temperature
increases the reaction rate 
will increase if the reaction rate increases
more energy will be produced if more energy
is produced in a star the star will expand
if the star expands then the temperature will
go down right and there is a negative feedback
in stars which maintains it in equilibrium
we have discussed this which prevents this
reaction to become a kind of runaway process
and cause an explosion
but there you would expect the same thing
to happen for the core also that which is
contracting if it contracts then the temperature
goes up the reaction goes up if the reaction
goes up it will again generate more energy
which will cause it to expand there will be
an equilibrium position this you expect it
to happen for the core also but what happens
is that the equation of state of the core
is different
the core is supported by degeneracy pressure
by electron degeneracy pressure what is degeneracy
pressure so if the energy levels if you have
fermions you cannot put all the fermions in
the same energy level you have to fill up
to a certain level okay and if you the pressure
that arises because you have to stack the
subsequent particles at higher and higher
energy levels there will be a minimum energy
and minimum pressure that will be there even
at low temperatures okay
so this is the pressure that supports the
core in the helium core this pressure would
be there even at 0 kelvin because just because
the system is made up of fermions you have
electrons whose pressure supports it okay
so here the pressure is not dependent on the
temperature it just depends on the density
and it scales as the density to the power
of 5/3 okay so this is the pressure that arises
just from the density
so if you increase the density the subsequent
electrons that you add will have to go to
higher energy levels and these will hire energy
electrons will contribute more to the pressure
okay so this is what is known as degeneracy
pressure so here the pressure is independent
of density so for masses of star which are
< 22 solar masses the core is supported by
degeneracy pressure
if the core is supported by degeneracy pressure
then if the star contracts and the temperature
and the reaction rate goes up the pressure
will not change because of the increase in
the energy production the energy production
increased energy production will cause the
temperature to go up
this will not affect the pressure and this
negative feedback mechanism does not work
over here so here it will keep on contracting
so you have what is called runaway process
so the helium reaction keeps on increasing
dramatically and you end up with what is called
a helium flash
the helium process burning becomes a runaway
process you end up with what is called a helium
flash over here the helium burning becomes
a runaway process 
okay after this the star goes into another
state which is called the horizontal branch
the horizontal branch is a state which is
analogous to the main sequence except that
you now have helium burning at the center
so it goes into something called the horizontal
branch like this okay it is analogous to the
main sequence except that you now have i am
just drawing what will happen to the sun other
stars will have similar evolutionary tracks
okay so you reach a state you go into an equilibrium
state where you have helium it is an equilibrium
state just like you had a hydrogen burning
equilibrium state here also you have an equilibrium
state where and the it is called the horizontal
branch
so masses more than this you do not have the
helium flash mass is less than this you have
an helium flash the reaction for mass is less
than this becomes runaway and then it goes
into an equilibrium state for mass is more
than this you do not have that runaway state
it straight away goes into an equilibrium
state
and you go it goes the star goes into what
is called a horizontal branch okay so in this
horizontal branch what happens in this part
of the stellar evolution what happens is that
helium gets burnt as i have mentioned earlier
helium gets burnt into carbon mainly carbon
and oxygen 
okay
so here you again you have a stable situation
over here so it is called the horizontal branch
something like this and there is a spread
so if i start with the solar mass there is
a spread over here it will not be exactly
if i take two stars of the same mass and start
from here it will not be exactly the same
they will not lie at exactly the same point
in the horizontal branch this is because in
this giant phase the stars lose mass as winds
stellar winds okay mass is ejected and there
will be a spread in values so if you look
at the whole collection of stars there will
be a big horizontal branch possibly over here
okay
it is another it is an equilibrium state the
horizontal branch is an equilibrium state
just like the main sequence where the star
is supported by helium burning which in the
main sequence was supported by hydrogen burning
you now have a star supported by helium burning
okay so now you have this helium burning star
in this horizontal branch and at the when
all the helium is burnt out it remains there
for some time
and then when all the helium is burnt out
what happens is that you have a core which
again is inert and is made up of carbon and
oxygen 
outside this code you have a shell where helium
burning is still going on and beyond this
you have a shell of helium and beyond that
you have a shell where hydrogen burning is
still going on okay this is the structure
of the core of the star and outside that you
possibly still have hydrogen left so this
is what happens at the end of the horizontal
branch
now for stars which are < 8 solar masses if
your star is < 8 solar masses the sun for
example 
this is the end of nuclear burning no more
nuclear burning proceeds beyond this so for
stars less than this this is the end of nuclear
burning
and what happens over here is that this core
will now collapse it will due to gravity 
and again as it collapses due to gravity it
will get heated up by straightforward virial
theorem and it will emit more and more energy
and this energy which is the increased source
of energy will cause the outer parts of star
to become bigger
and you will go into a second giant phase
called the so there will be a second giant
phase called the asymptotic giant branch or
the second giant asymptotic giant branch
and 
then the energy from the interior on the core
is adequate to eject the envelope so the core
keeps on collapses becomes smaller and smaller
and it ejects the envelope forming what is
called a planetary nebula it is a very beautiful
objects seen on the sky 
the interior the core heats up tremendously
core gets heated tremendously 
and the radiation the photons from the core
fall on the envelope and cause fluorescence
they cause fluorescence in the gas in the
envelope
so the track in the hr diagram looks something
like this this will go quite fast so okay
and then it will go into a state over here
slow evolution which is the white dwarf the
core so what you have left after the envelope
is blown out is a core the core is a white
dwarf
the core remains it is essentially made up
of carbon and oxygen this is a white dwarf
it is extremely hot because of the gravitational
collapse and its track is basically determined
it loses energy l is = 4 pi r square sigma
t to the power 4 so it loses energy and due
to the radiation from the surface given by
this it is extremely hot but it is very compact
and as a consequence of the loss of energy
it moves along a track like this these are
the white dwarfs
so you will see the white dwarfs in this part
of the hr diagram they are very hot stars
but they are extremely low luminosity and
these are the cores which have been left behind
after stars like the sun or < 8 solar mass
have exhausted all their fuel gone through
all these phases and then have become planetary
nebula and finally they end up as white dwarfs
now what happens to stars which are more massive
than eight solar masses now in these stars
you let me go back and do the picture what
happens to these stars in these stars you
have a core of carbon and oxygen now these
stars when they collapse when the core collapses
when the core is contracting again the temperature
and the density 
are adequately high
so you can again start nuclear fusion in these
smaller stars you do not the temperature the
core of the star once you reach carbon and
oxygen when it collapses the density and the
temperature do not reach values adequately
high for nuclear fusion to occur further here
this is not true the temperature and density
is adequately high for further nuclear fusion
to occur so more nuclear reactions occur
so you have nuclear reactions going on forming
heavier and heavier elements finally you reach
iron iron is the most stable nuclei iron nuclei
are the most stable so the binding energy
per nucleon is maximum in iron these are the
most stable nuclei and in thermal equilibrium
you cannot form nuke elements which are heavier
than iron the whole synthesis of elements
stops at iron
so you can form elements which are less heavy
than iron but finally you reach iron and you
cannot form anything heavier than that okay
so you end up with iron so at the end of nuclear
fusion you have a core which is made up of
iron you have burned everything and finally
you end up with an iron core and there will
be an envelope of other elements in the region
where these iron the reaction to iron could
not have formed
this is the most stable nucleus so it is a
big all the elements that we see around us
every element here present on earth present
anywhere heavier than hydrogen helium and
maybe some small amounts of lithium etc they
were all produced in stars by these nuclear
fusion processes that we discussed okay but
you can only produce up to iron by these processes
so it was a big problem how do you produce
elements which are beyond iron for example
lead and many other uranium all these very
heavier atoms nuclei okay you cannot produce
them in any equilibrium thermal equilibrium
process that in thermal equilibrium you will
go to the most stable state which is iron
so once you form the iron core there you nuclear
reaction you cannot have fusion any more nuclear
fusion will not get you any further
you cannot get any more stable nuclei now
once you form this iron core this core now
collapses gravity due to gravity right so
it collapses it will become smaller and smaller
and in the process it will get hotter also
now when this collapse occurs this entire
collapse occurs in a few seconds so this whole
thing occurs in a few seconds beyond iron
once you reach iron the whole thing has burned
out burnt and reached iron
there is no more nuclear fusion going on at
the center then the whole thing collapse collapses
due to gravity there is no energy source over
here it collapses gravitationally and it gets
heated up the whole thing occurs in a few
seconds it collapses rapidly okay and when
it collapses rapidly there is enormous heat
generated
and here what happens is that the iron again
goes up breaks up into helium and in the process
it absorbs this is an endothermic reaction
iron is the most stable nuclei i have already
told you that so in the process it absorbs
energy 
despite all of this the inner part gets tremendously
heated and it generates it radiates and there
is also neutrinos there is a so in this collapse
there is the tremendous nuclear reaction where
it gets converted into helium and there are
neutrinos okay
so this is what is which are emitted there
is tremendous amount of radiation also emitted
and you have a supernovae the envelope is
blown out 
the interior collapses and the envelope energy
is generated which causes the envelope to
blow out taking away large parts of the elements
including the iron and all else away out of
it okay there are large neutron fluxes during
this 
and the neutron impinges on the iron to produce
the heavier elements
and all of these are ejected into the interstellar
medium which then forms the material on which
the next generation of star formation occurs
the core the core of this whole thing collapses
to form either a neutron star depending on
the mass 
or a black hole 
and masses more than 25 solar masses it is
believed and go straight away and form a black
hole
so in a neutron star what happens is that
all of the iron gets converted into helium
and finally everything gets converted into
neutrons okay a black hole we shall something
if we get time we shall discuss it but main
point is that the entire thing collapses to
get smaller and smaller and very high dense
object
before we finish i should point out and you
have a supernovae the outer envelope gets
blown out its luminosity goes up tremendously
because of the increase in the radius and
you will see a very bright object in the sky
and you have what is called a supernovae now
the entire thing occurs in a few seconds and
a full computational simulation so you want
to calculate everything on a computer you
put all the equations that we have been discussing
in a computer and you want to solve it
the computational power to do this does not
exist as yet okay so there are simplifications
which people have done and these are the some
of the i have told you about some of the results
which people have obtained using these okay
because things are changing tremendously fast
over a few seconds you have to solve the entire
set of fluid equations the transfer of radiation
there are enormous neutrino flux also there
is an enormous neutrino flux etc okay
so let me just summarise briefly in today's
class i have tried to give you a picture of
what how these stars evolve and stars they
form from gas clouds they evolve along the
hayashi track they reach the main sequence
and after the main sequence they go into the
giant branch then they go into the horizontal
branch
then you have a second asymptotic giant branch
and after that the evolution is highly dependent
on the mass of the star below eight solar
masses you have the white dwarfs being formed
above eight solar masses the star explodes
in a it burns the core burns you get elements
beyond carbon and oxygen and finally it explodes
in a supernovae and depending on the mass
again you either end up with a neutron star
or a black hole
