So let's put it all together. 
We now have four main sequence stars. 
An understanding of the relation between, 
at least empirically, mass, luminosity, 
temperature, radius. 
And now, we can hand this over to the 
people who, modeled the Sun and 
understood the rate of re-fusion inside 
the Sun and predicted that neutrino flux 
and they can adapt their models to study 
other main sequence stars and try to test 
the hypothesis that all of these main 
sequence stars are basically. 
The sun at various sizes. 
So, what do we do? 
We take a ball of hydrogen and model its 
behavior. 
We include all of the properties we know 
of hydrogen, and of fusion, and of 
convection, and of radiation diffusion, 
and, of course, a star is a complicated 
system. 
But, the idea is, that a main sequence 
star is an odd ball of hydrogen which, in 
its dense hot core, is fusing hydrogen to 
helium. 
And you work on that assumption and you 
see, will you predict? 
The relation between luminosity and mass, 
between mass and radius, and mass and 
temperature, that the main sequence 
obeys. 
And if you do, then you have an 
understanding of stars. 
And essentially, the point is that the 
conditions for hydrostatic equilibrium 
together with the properties of the 
plasma of which the star is made 
determines everything. 
Starting from the mass of a star. 
And, I'd like to outline this, if this 
confuses you, skip it. 
But in the next slide I'm going to 
outline how a stellar model, proceeds in 
some rough level. 
So here's how we model a star. 
We start out with this picture here. 
Over star. 
And what do we know? 
Well I am going to assume that I know the 
temperature. 
Outside the star. 
And the pressure, the atmospheric 
pressure of the star, effectively the 
atmospheric pressure of the star is 
essentially zero but I'll start with some 
small value. 
And I'm going to work my way in and try 
to understand from equilibrium how the 
temperature, the pressure and of course 
importantly the mass density rho which 
I'm going to again say is very small on 
the outside, atmospheric, how these 
increase. 
I expect all of them to increase as I 
move in. 
So what do I know? 
A star is a complicated system but I'm 
going to imagine. 
Just for pretend, just so we get a sense 
of what's going on, that the interior of 
a star is an ideal gas. 
It's not but if it were an ideal gas, 
then ideal gases satisfy the equation 
that we wrote down, for this precise 
reason. 
Pv is NKT, the product of pressure times 
volume is the number of particles times a 
constant times the temperature. 
I have pressure in here that's good, I 
have temperature, I don't have density, 
but that's okay if these are all hydrogen 
atoms and if M is the mass of a hydrogen 
atom. 
Then since the mass divided by the volume 
is row, and since the mass in a given 
volume is the number of particles times 
the mass of an hydrogen atom. 
I can relate n and v to rho. 
And I find that, v divided n, which is 
the quantity I get that enters into here, 
is simply n, divided by rho. 
And so, putting that into that equation, 
I find that p, 
is equal to, n times rho, times kb times 
T. 
Okay this is a version of the equation of 
state that I can use, and now what do I 
do? 
Now I need to impose what this tells me 
of course, is that as pressure increases, 
temperature and or density will increase, 
now that's one thing I know, and then 
what I'm going to do is I'm going to 
imagine I know the conditions right 
outside the star, let's try to find 
conditions a little distance r inside. So 
this is the radius of the star. 
And I'm going to move a little distance r 
inside and try and see if you know what 
goes on here, how do you know what goes 
on there. 
And there's some things I know. 
I know the pressure here, p1 has to be 
bigger than p0. 
Why? 
Well because I've sunk in this layer 
underneath is holding up the weight of 
the 
Of the, the bit of the star that is 
outside so, I can write some crazy 
equation that says, P1 minus P0 times the 
area of the interface between these 
layers which is four Pi R squared is 
equal to, the. 
Mass of the gas in this region, 
which if I assume that its density is row 
zero. 
That's also four pi r2 squared little r 
times rho. 
That's the mass in there. 
And then I multiply that by g times the 
mass inside the radius R. 
So that's, in this case, the entire star 
divided by r2. 
square.d And I can, from here, cancelling 
these factors, 
I can find that the pressure changes r 
times row times gn over r squared. 
Okay, now. 
That gives me, 
real, tells me that if I know the 
pressure here, I can tell the pressure 
there. 
I want to relate, 
to figure out the density and the 
temperature in here. 
Of course, since the pressure changed, 
the, density and the temperature had to 
change. 
Because, 
The new pressure, which is higher, is 
equal to the product of the new density 
and temperature. 
There's another relation that I know. 
Which is the important thing that I can 
put in, is that I know the luminosity. 
There's a neck flux of energy the star is 
radiating at a luminosity L. 
And so, that means that, since the 
temperature of everything is constant. 
A net amount of energy, L, is being 
transferred through this layer at any 
given time. 
And if I know the properties of the 
layer, I can figure out how much hotter 
it has to be inside in order for an 
amount of energy L per second, a rate, 
the power of L to propagate out. 
And how this works depends on what the 
mechanisms are, if it's convection I need 
to use the convection properties of the 
fluid. 
If it's radiation diffusion the radiation 
diffusion properties of conduction, 
conductive properties. 
But if I know the luminosity I can figure 
out T1 minus T0, so this allows me, 
starting with T0 to figure out T1. 
And then I use this equation to figure 
out P1. 
And then I use this equation to figure 
out row one. 
And now I understand things one layer in 
and then I can proceed in and build the 
entire model of the star one layer at a 
time. 
If this was not illuminating to you, just 
imagine that somebody models stars. 
So what have we learned? 
Well the important thing is, looking way 
down at the middle of the star at the 
core. 
That there are, 
I talked about the PP Chain, which is how 
the sun produces energy. 
And stars with a mass of about one 
one-half solar masses or more, 
so just about the Sun. 
the core is hotter, the bigger the star 
is, because there's more pressure there's 
more layers to integrate through, and 
more mass. 
And it turns out that at higher, hydrogen 
at higher temperature can produce fusion 
in a much more efficient way through 
something called the carbon nitrogen 
oxygen or CNO chain. 
this is a process in which, the steps 
are, are many, and detailed, and we don't 
need to go into them, 
but you can read up on them. 
But basically, the presence of carbon in 
the core catalyzes fusion, in the sense 
that carbon fuses with hydrogen to 
produce nitrogen, which decays to leave 
carbon, a different isotope of carbon, 
which fuses again to produce another 
isotope of nitrogen, 
which fuses with another hydrogen atom, 
producing oxygen, 
which decays to an isotope of nitrogen, 
which fuses with yet another proton, 
emits an alpha particle, 
and returns us to the original carbon 
atom. 
so then end result is one, two, three, 
four protons go in, two week decays take 
place and a helium atom is spat out. 
The carbon is recycled in this process so 
it's not being fused into anything, but 
at higher temperature, this process goes 
much faster than the PP chain the point 
is that the rate at which this process 
goes rises much more is much more 
temperature dependent than the rate of 
the PP chain basically because to start 
it all off and its several stages in the 
middle, you need not to bring two protons 
close together, overcoming their, 
electrostatic. 
Repulsion. 
You need to bring a proton near to a 
carbon nucleus, 
which has charged six and that 
electrostatic repulsion is much stronger 
and so 
it takes higher temperatures to get the 
CNO chain started. 
This is the qualitative difference 
between the big stars and little stars. 
What are the results of this? 
Well it turns out that what happens is 
that the mechanism by which all of these 
stars, 
main sequence stars indeed can be 
explained as a ball of hydrogen, fusing 
hydrogen to helium in its core. 
Details of the model depend on the size 
of the ball of hydrogen you start with. 
Very small stars with a mass less than a 
half a solar mass. 
It turns out that there is no radiation 
zone as there was in the Sun. 
There is just a convection zone. 
The entire star convects. 
This is important for estimating their 
life time because remember in the core 
hydrogen is being converted to helium. 
Then this helium-enriched hydrogen. 
Bubbles up through the entire star. 
And new fresh hydrogen-poor, helium-poor 
hydrogen is convected down. 
So this star, when it runs out of helium 
in the core wil essentially be, when it 
runs out of hydrogen in the core, will 
essentially have converted a large 
fraction of the entire star to helium. 
And that's the way small stars work. 
Medium mass stars like the sun work we 
described. 
The interior is a radiation diffusion 
region then the outer mentor is the 
convective region where hydrogen atoms 
are less ionized, they are therefore more 
opaque and in and therefore they absorb 
the radiation heat up and then well up 
and its a hydrodynamic process of 
convection that brings the heat up to the 
surface of the star. 
In bigger stars it turns out convection 
and radiation are replaced. 
In these stars the entire, 
the temperature is so high that much of 
the interior of the star is actually 
ionized. 
But in the interior because there is a 
temperature gradient, 
the exterior is, of the core, is cool. 
Still not as hot as the center of the 
core. 
The temperature gradients means, that the 
rate at which fusion occurs changes very 
rapidly. 
This leads, to such a power, differences 
in the power output between different 
layers, 
that in fact the star becomes convected, 
so in a star the inner 
region of the star is heat tran, is 
transferred by convection. 
So this region all gets mixed up, but the 
external region is in fact 
transfers heat by radiation. 
the fact that this is what's predicted by 
the models, 
and then predictions of luminosity as 
the, as it depends on mass. 
And radius, as it depends on mass, 
and temperature as it depends on mass are 
reproduced. 
The, measurements from the main sequence. 
Tells us that our models have it right. 
So we've an understanding of the basic 
physics of how, main sequence stars work. 
They're a ball of hydrogen, fusing 
hydrogen to helium at its core and, 
transmitting the energy to the outside by 
convection and radiation diffusion, and 
then radiating that, radiating that 
energy out. 
we did, are not going to reproduce all of 
the details of understanding the nuclear 
cross sections that go into modeling. 
But I think we understand the basic 
principles, and, the, 
rate of fusion in the core, as we said 
for the sun, is in equilibrium with the 
pressure, as we saw, applied by the outer 
layers, 
and therefore the mass determines the 
rate of fusion. 
And as the star, star sits on the main 
sequence, we'll see next week what 
happens before and after. 
But today let's finish up by discussing 
what happens in the 10 billion years, 
say, for the sun that it sits on the main 
sequence. 
Remember the sun is about 4 1/2 billion 
years old, so it's about half done with 
its main sequence life. 
How does the sun today compare to the way 
it was when it was a brand new fresh zero 
age main sequence star, as we call it? 
So the star has, the Sun formed, as it 
was mostly hydrogen. 
It has a core where, hydrogen starts to 
fusing to helium. 
Over time, as fusion goes on, the 
the region, hydrogen in the core becomes 
enreached, enriched in helium. 
So, what that means is that if you have 
your star over here, and the core in 
there there is more and more Helium in 
the core and correspondingly less and 
less Hydrogen. 
This means that the probability for 
collision of two protons in order to 
initiate fusion decreases, because some 
of the time the protons are not running 
in to each other. 
They're running into helium atoms, and in 
the sum they do not have enough energy to 
actually fuse effectively with helium. 
And so the helium is inert and there's 
less and less protons running into each 
other. 
So when we predict that the rate of 
fusion would decrease in the core this 
would decrease the radiation pressure, 
because the pressure emanating from the 
core part of large particles that is 
radiation pressure or this energy. 
furthermore, and this is a tricky. 
It turns out to be the dominant effect is 
that when we particles undergo fusion 
four hydrogen atoms are converted to one 
helium atom, and if you remember our PV 
equals MKT, 
what's really going on is the number of 
particles is decreasing, 
because four protons have only made one 
alpha particle, when the partic-, the 
number of particles decreases, the 
pressure goes down. 
Now this cannot happen. 
The reason this cannot happen is because 
remember the pressure on the inside of 
the store was predicted and determined by 
the need to hold up the outer layers. 
How can the pressure decrease? 
The pressure decreases. 
The outer layers of the star will 
contract the star, the core. 
Now, this is the first time we're meeting 
this sort of a phenomenon that I call 
expansion by contraction, and it'll come 
back. 
It sounds like a weird economic theory, 
but it is a property of stars, so, I 
want to understand it this first time, 
because it will repeat, many times over 
the course of our study of stars, and 
So, what happens is the pressure has not 
decreased, instead the core contracts. 
So, the material that was previously the 
core is contracted to a smaller radius. 
This, in turn, heats it up. 
Both, because we're, converting 
gravitational potential energy. 
The entire star scrunches down. 
And so the entire star undergoes 
Kelvin-Helmholtz heating. 
And of course, this is most extreme in 
the core. 
Moreover. 
Because a thermodynamically it heats up 
the temperature in the core rises and the 
density rises. 
How long does this go on? 
Well, until fusion rates increase enough 
to hold up the rest of the star. 
So in fact, the pressure here is 
essentially equal or larger, in fact, 
because of the contraction to the 
pressure previously obtained there. 
And density and so on followed. 
In that process, something else happened. 
The region between what the court now 
occupies and where the court used to be. 
Didn't get evacuated. 
This is now occupied by an inflow of new 
hydrogen that used to reside just outside 
the court. 
This new influx of hydrogen is heated and 
compressed. 
It's now closer to the center of the star 
than it was before and it, in turn, 
begins to undergo pressure. 
So there are two processes going on. 
One is the pressure in the core is 
maintained by compressing the core. 
This in turn heats the core. 
Increasing the rate of fusion, both by 
increasing the temperature and the 
density. 
And further more, the amount of fusing 
material grows, because some of the 
hydrogen that was too cold, and not 
pressurized enough to fuse is now 
contracting to the point where it begins 
to fuse. 
The net result is that the core heat, the 
fusing region grows, and the total power 
production of the star increases. 
When the core contracts, the luminosity 
increases. 
The increased power output from the core, 
in turn, applies extra radiation pressure 
and extra thermodynamic pressure to the 
rest of the star, and the star balloons 
out a little bit. 
So over the course of its main sequence 
life time, as it runs out slowly of 
helium, hydrogen in the core, the stars 
core contracts and heats, 
and creates more hydrogen into the core, 
resulting in the fact that the outer 
envelope of the star grows, and 
luminosity increases. 
This is a very important fact. 
So, if we apply to the sun, what does it 
tell us? 
It tells us that the sun is now about 25% 
brighter than when it formed 
And the this is halfway through its main 
sequence lifetime. 
the sun is now brighter than when it 
formed 4 billion years ago. 
The sun was a rather dim star compared to 
what it is now. 
this meant that the earth was presumably 
cooler. 
There are all kinds of complications 
trying to understand how this is 
consistent with what we find on earth but 
it is certainly what our stellar models 
predict. 
Moreover, we can run our simulations and 
the core of the sun is now 60% helium. 
When I said it's depleted in hydrogen, I 
meant that of course, there are regions 
outside the core that are beginning to 
fuse as new clean hydrogen, untainted by 
helium, is 
compressed to the densities and 
temperatures, where it too can begin to 
fuse, which is why the sun has a larger 
luminosity now. 
It's going to continue to brighten over 
the next 5 billion years, making Earth 
warmer and warmer, and you can ask global 
warming, carbon dioxide, will the sun do 
us in? 
And yes, the sun will probably make Earth 
uninhabitable, and depending on your 
estimates, somewhere between 1 and 3 
billion years. 
So, if we managed to control our carbon 
emissions for that long, the sun will in 
any event do life on Earth. 
this in fact is a little bit imprecise 
because once you get into billion year 
periods there are other factors that come 
into account. 
The sun loses mass and as it expands, the 
outer layers are now farther and farther 
from the center of the sun. 
Their escape velocity decreases. 
S the solar wind increases. 
At, in it's last couple a billion years 
the sun's lo, mass loss rate increases 
significantly so that the earth will be 
orbiting a star that is less and less 
massive. 
Which means that with it's current energy 
it will probably be propelled to a higher 
orbit. 
Once you allow orbits to change you have 
to also remember that orbits in the solar 
system, I said are stable to hundreds of 
millions, maybe a billion year out. 
But we do not have really reliable 
predictions that maybe in a few billion 
years there won't be some orbital 
residents that will completely throw the 
orbits out of kilter, and so a billion 
years out is a long time. 
We don't know exactly what will happen, 
but at some point in the next three 
billion years, Earth will probably be a 
problem. 
If we survived that long we have to deal 
with this. 
Another busy week, but I think very 
productive. 
What have you learned by looking at the 
stars? 
For 90% of them, we have a pretty good 
physics understanding of how they work 
and it matches all the data, and this 
came from decades of painstaking 
observations, measuring the positions of 
stars over the decades to do astrometric 
measurements of binaries and proper 
motion. 
Measuring the parallax angles of 
thousands of stars in competing their 
distance carefully, measuring thousands 
and thousands of spectra in classifying 
them and understanding the patterns. 
A lot of really, really careful 
observation, and then detailed in the 
miracle modelling with the advent of high 
speed computing and putting all that 
together with experiments under that to 
determine the properties of there is 
nuclear processes, we actually understand 
a lot about how stars work. 
That's good for the 85% that are on the 
main sequence, what about the rest? 
Not all stars are main sequence stars. 
We need to understand where the rest come 
from. 
A related question it turns out will be, 
well what was the sun like before it 
became a main sequence star? 
And, even more excitingly, what happens 
when the hydrogen runs out, and it can no 
longer be a main sequence star. 
And that is what we're going to start 
studying next.