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.