[INAUDIBLE] One last thing to catch up on is, we talked about the beautiful
planetary nebula the sun will make. But then, what about that core?
I want to look a little more closely at that white dwarf which is, in some sense,
the end result of what the sun will be. And white dwarves have an interesting
discovery history, shortly after measuring the first successful parallax
measurement Bessel followed another star which he thought would be likely Sirius
which we know is very nearby and as he measured its proper motion he noticed
that Sirio Sirius wobbles and he immediately understood what was going on
he knew about Newtonian physics, Sirius has a binary partner.
the search for what was called the pup was on people wanted to find it it turned
out to be a very challenging prospect but in 1846
indeed the star was found and the reason it had been so difficult to detect was
because it was so much dimmer than serious, serious has a luminosity of 23
solar luminosities and serious b its partner only.
.03 solar luminosities Clark was very fortunate the year that he was making the
measurement turned out to be a year of ap astron.
Which is akillian for a binary system. It's the time when, in their elliptical
orbits, Sirius A and B were farthest apart, their period is a little over 50
years, and this will play into the history in a minute.
Measuring their orbits, we can figure out their masses.
We're good at that by now, after the experience of the previous week, and
Sirius, as we mentioned, has a mass a little over two solar masses, and, its
partner has about a mass of one solar mass, so it's a dim star.
With one solar mass, it's certainly not a main sequence star, we know that, because
a main sequence star with one solar mass would have about 30 times that
luminosity. Okay.
Well the game was on to find the spectrum of these objects.
You needed to measure them again. You needed to wait for a fast gen to come
around. And it did in 1915.
And an American astronomer named Adams managed to measure the temperature of the
two stars. And his results shocked the world,
because the temperature of Sirius had been known.
It's the 9900 degrees Kelvin that we knew.
But the temperature of the partner turned out to be not a lot less as you would
expect from the low luminosity, but in fact a lot more.
Hookie so now if you know the temperature and you know the luminosity.
Bada bing bada beam, sigma t to the fourth, we can compute the radius of
Sirius B, and the radius of Sirius B turns out to be less than a percent of
the solar radius, more impressively, less than Earth's radius by slightly.
It's the radius is less than Earth's radius.
This in an object with the size of Earth. And the mass of the sun this is
extraordinarily dense the surface gravity on serious b you just compute plug the
mass you know divided by the radius squared scaled to earth and you find that
the surface gravity on serious b is only 450,000 times the surface gravity on
earth you do not really want to go there. astronomers in the early twentieth
century scoffed they figured there's got to be a mistake.
There's no way that such an object could exist but it was there so it was there to
be studied. It's spectrum showed very broad hydrogen
absorption lines. Well if you have an atmosphere of
hydrogen in this kind of gravitational field it will certainly be very
compressed and you expect very broad absorption lines but otherwise it was
just a featureless black body continuum. you could make estimates of, given the
mass and the radius of the central density and pressure and the central
pressure is millions of times more than in the center of the sun.
The central temperature is. 70,000,000 Kelvin.
Well that tells me something. If, we know that there are hydrogen
absorption lines. We know the universe is made of hydrogen.
But there better not be that much hydrogen in this object because, if there
was, it would fuse and, and then this luminosity at these temperatures would be
way, way higher than this three one hundredths of a solar luminosity.
So, this is an hydrogen free object. Something rare.
It was poorly understood. But now we know.
The way to make a hydrogen-free object is first condense the, heavy byproducts of
fu, helium fusion into the core of a star.
You get basically an object that is almost exclusively carbon and, carbon and
oxygen. Blow away the hydrogen, envelope, and
what you're left with will be a white dwarf.
So now we know. White dwarves are the degenerate cores of
stars whose mass is not too high. Turns out that you can have a mass as
high as eight solar masses. We'll talk about what happens to more
massive stars in a few clips. There composition is indeed hydrogen
free, they're almost exclusively carbon and oxygen.
Their masses, after we measured many of them, lie in this narrow range between.4
and.7 solar masses for the most part. Given the much larger range of stellar
initial masses this gives you a sense of the mass loss that a star undergoes in
the course of its last thermal pulsing days when it leaves an exposed mass.
And. And here comes this guy trying to guy
Chandrasekhar again with another interesting discussion.
And this time, I want us to follow though with this.
So, let's understand what Chandrasekhar tells us about
About the weird behavior of the generate matter.
So what Chandrasekhar says is this. So consider an object here, which is
completely degenerate and not luminous so we can image that it's isothermal, but
temperature will play no roll. And, we'll give it a radius R and a mass
M. So, what do we know about this object?
Well, we can figure out its density. We could imagine, for the sake of
argument, that the density is uniform, although it's not going to be.
And so, if the density were uniform, then the density would be m4*pi))/(3*r^3).
divided by pi over 3 r cubed. Now, that's not true, because the density
is not uniform. But, I want to compute some kind of
characteristic density. So I'm going to say that it's going to be
some number, which I shall call A, say, times m divided by r cubed.
A will incorporate 4s and pis. And the relative geometry of the
situation in two-fifths and whatever else, some number of order one, times
m/r^3. What this tells me is that if this object
is held up by electron degeneracy pressure, then that pressure, Pe, is
given by k. -Fer electrons times rho to the 5/3, and
I plug this expression into this, and I get a new constant - eight to the 5/3
times, times k, which I might as well call c.
And then I get. M over R cubed.
Raised to the five thirds power. So the electron degeneracy pressure is
dependent on mass and radius in this form, that's n to the five thirds divided
by r to the fifth if you write it out explicitly.
That's worth writing out cleanly. Let me write it out cleanly.
So, for some number c the electron degeneracy pressure is given by this.
Now, what is that statement? The statement is that this object is
supposed to hold itself up against gravity.
that means when you draw my object again since I erased it at the edge of the
object pressure is zero its touching space and so.
I can ask how much pressure do you have to have in the center in order for the,
pre, the thing for that to be enough to hold that object up.
In other words, what is the gravitational pressure?
And just dimensional analysis tells us about how to figure out the effective
contribution to pressure by gravity. Because, well of course it'll depend on
G. Right and we can make an acceleration out
of G and N and R which are the only parameters we have by taking G M over R
square. And we can turn that into a force by
multiplying by the only other thing we have which is N.
So, gm squared over r squared is a characteristic force of gravity.
We want the pressure, so that needs to be the divided by area so I multiply by the
only relevant thing that could be an area and this is completely wrong.
this is not the pressure due to gravity. But if put some number here which I will
call c prime then it is and exactly what depends where exactly your measuring the
gravitational pressure due to gravity and what exactly the distribution of mass is
it so on but some version c prime times g m squared over r to the fourth.
Is always a good estimate for the gravitational pressure.
Or, if you want, the difference in pressure from here to here, which is the
amount basically the, by which the pressure in the center has to be higher
than the pressure at the edge to hold the thing up.
So let's take note of this gravitational pressure calculation that we did.
I just copied it and rated, wrote it out cleanly.
And now Chandrasekhar notes, set one equal to the other up for the case of an
objects that's held up by a. Degenerate matter and what do we do we
set this equal to that and I find that c primed times g times m squared over r to
the fourth. Is equal to C times N to the five thirds
divided by R cube to the five thirds is just R to the fifth.
Okay so. I get some cancellations.
I have all kinds of constants I don't know what to do with.
But these R's go away. And four of those R's go away.
And five thirds is less than two by a third.
So I can get rid of this. And raise n to the one third.
So, multiplying by r here I find that n to the one third times r and then
dividing by this constant is some number c over c prime g and then it's easier to
take the cube of all of this which is n r cubed.
Is this whole constant, some number cubed.
Notice that this constant depends a little bit on the geometry but it doesn't
change too much. Well, that's very reasonable.
The mass of an object times the cube of its radius, essentially times its volume
is a constant. That is the weirdest thing ever.
Let's write that out and think what that means.
That means that if you have two white dwarfs and one is more massive.
The one with the bigger end has a smaller radius.
Degenerate matter works in a very, very weird way.
The more massive a white dwarf is, the more it has to compress in order to hold
itself up against gravity. And therefore the smaller it is.
This already hints at something very bizarre as you make the white dwarf more
and more massive and. Become smaller and smaller.
The electrons are pushed to higher and higher energy levels, and eventually
relativity comes into this, that is next weeks stuff.
But when you make the correction for relativity, you find that eventually.
If you make the mass too big the radius becomes zero.
the white dwarf cannot exist with a mass bigger than what is called the
Chandrasekhar limit. And the Chandrasekhar limit is about 1.44
solar masses. You cannot have a white dwarf with mass
bigger than that, A degenerate object held up by electron degeneracy cannot
have a mass bigger than that. Our observations are consistent with
this, here is the weird mass radius relationship.
For White dwarfs for degenerate matter would
be computed as the non-relativistic blue curve.
You notice that the larger the mass, the smaller the radius.
And that's pretty weird. the relativistic corrections become
important at large masses and small radii and they show you that at the
Chandrasekhar limit, that's the end, you cannot pack more than 1.44 solar masses
into a chunk of degenerate matter, something bad will happen.
What will happen, we'll see soon.