[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.