I can't say that we have a. 
Completely captured everything that could 
be said about core collapsed supernova, 
but that's all we're going to have time 
for. 
just as with planetary nebuli, you can 
talk about the light show, the planetary 
nebula. 
But then you want to know what happens to 
the star's core. 
We left our star's core collapsing. 
What happens 
In the somewhat more energetic high mass 
version of core collapse. 
We know that electron degeneracy is not 
going to rescue our collapsing iron core. 
Electrons are essentially gone, luckily 
neutrons, like electrons or fermions, 
they too are subject to the poly 
exclusion principal. 
They too exert a degeneracy pressure that 
is proportional to the 5/3rds power of 
their density. 
A little more careful analysis shows that 
because in the confines of a, the 
collapsing core, these neutrons are in 
fact relativistic than like electrons. 
There are corrections to this 5/3rds 
rule, it's more like 4/3rds. 
But there subject to the same kind of 
degeneracy pressure and it turns out that 
neutron degeneracy pressure. 
You can think of that partially as the 
repulsive core. 
It's essentially equivalent as the 
repulsive core of the strong force. 
Remember we saw that attraction down to 
an optimal distance of about one Fermi 
between nucleons. 
And then a repulsive force, when you try 
to force them too close. 
This is essentially the same because that 
was a quantum treatment as the poly 
exclusion principle. 
And the density of six to seven times 
ten^17 kilos/meter^3, neutrons become 
degenerate, and the core stops. 
If you imagine that the core had. 
Of a stellar mass, let's give it the mass 
of a heavy white dwarf. 
Then at this density, your white dwarf 
with a radius essentially that of the 
Earth, will have collapsed to something 
with the radius of a city, so now we're 
talking about. 
Essentially a great big nucleus very, 
very rich in neutrons made 
Of containing a solar mass in a ball of 
radius ten km, clearly the gravitational 
effects are humongous. 
The surface gravity if you compute it 
classically and ignore. 
And incorrectly relativistic effects will 
be on the order of ten to the eleven or 
ten to the twelve g. 
you certainly do not want to go stand on 
this object. 
the physics, as I said, is relativistic, 
but with our calculation of the 
Chandrasekharan degenerate matter. 
We can repeat the degeneracy calculation 
for neutrons. 
The Chandrasekharan and again the heavier 
neutron star is, the smaller its size. 
And the details of the calculation are 
slightly different, and not as well 
understood. 
Because we do not understand degenerate 
nuclear matter as well as we understand 
degenerate nuclear matter as well as we 
understand. 
The general electrons, but the 
Chandrasekhar mass, the maximum mass of 
the neutron star. 
Is. 
Not 1.4 solar masses, but 2.2 to 2.9 
solar masses. 
This depends a little bit on details of 
the model and more on the rate at which 
this thing rotates. 
As we will talk we expect these stellar 
remnants to be rotating very fast and 
with fast rotation you can support 
degenerate neutron star with a mass of up 
to almost three solar masses. 
What happens if the ca, collapsing core 
ends up being more massive than this is 
something that we'll leave for next week. 
So, some physics predictions follow from 
this, and I want to, pay attention to 
them. 
So I said we expect, the remnant of a 
star to be rotating very fast. 
This is reasonably clear, 
If there was any rotation at all in the 
core, then, you know, this is back to, to 
the fun experiment with me standing on 
the platform. 
When I pulled the weights in, 
my rotation sped up. 
In this case, remember something is 
contracting from the size of earth to ten 
kilometers. 
The rate of rotation would have to speed 
up. 
If you think of angular momentum as N 
times V times R. 
And angular momentum is essentially 
conserved. 
There is some angular momentum loss, but 
not. 
orders of magnitude and you remember that 
v is two, pi, r over p. 
The ignoring the constants, as we are 
doing all week. 
You find that angular momentum depends on 
the relevant quantities as n times r 
squared divided by p. 
And so we can solve for p. 
Write as m r squared and if you remember 
that, the mass of the neutron star is 
about the mass of the core that 
collapsed, but the radii are much 
smaller. 
The period of the rotation of the neutron 
star is related to the period with, of 
the rotation of the core by this, ratio. 
If you, so how fast was the. 
Stellar core rotating before it 
collapsed. 
While we can get a handle on this perhaps 
by looking at the rotation rates of 
typical white dwarfs which are, after 
all, the cores of smaller stars making 
some reasonable assumptions. 
Defined that estimate that the period 
with which a neutron star will be, might 
be as short as five milliseconds. 
This is fast rotations, indeed. 
This is something that rotates 200 times 
a second, an object with the mass of the 
sun rotating 200 times a second. 
similar considerations predict a very 
intense magnetic field. 
And the reason is that, of course, by the 
time you get to a stellar core, 
everything is ionized. 
And you remember that we discussed the 
fact that charged particles. 
interact with a magnetic field in this 
interesting hydrodynamic way the magnetic 
field effects the motion of the charged 
particles but the charged particles drag 
the magnetic field with them so if you 
have the magnetic field the of a core of 
a the core of a star as the star implodes 
it drags the star with it. 
What you have is the same magnetic field 
confined to a smaller area. 
The intensity of a magnetic field, if you 
think about it as some number of magnetic 
lines, the intensity of a magnetic field 
can be measured by the flux of magnetic 
lines per unit area, since the area is 
shrinking by a factor of R squared, you 
expect a huge magnetic field, again 
plugging in what we know about the 
magnetic fields of white dwarfs. 
you expect magnetic fields on the order 
of a trillion times. 
The magnetic field of the Sun, there will 
be strong magnetic effects associated to 
this. 
You can make some estimate for the 
temperature. 
Remember I said that at collapse the 
temperature was billions of kelvin. 
By the time we see. 
The, core exposed and the envelope is 
gone, temperatures are expected to be 
down to a million kelvin. 
This means that Deans law tells us that 
these neutron stars will at least 
initially be radiating x-rays at three 
nanometer wave lengths. 
And given the temperature and the radius 
of ten kilometers, you can compute that 
the luminosity is about a quarter of a 
solar luminosity not visible from Earth 
but visible from x-ray telescopes. 
Okay so these are all nice stories that 
theorists told there was a theorist 
called Francis Vicky who first predicted 
that 
the outcome of a supernova explosion 
would be this nuclear density, 
Collection of neutrons, this was a very 
theoretical prediction. 
It was not clear at all. 
That such things could or should exist 
the discovery was made by Jocelyn Bell in 
1967 and Jocelyn Bell was a radio 
astronomer she was looking at 
interactions of magnetic field with with 
quesar radiation but what she found was a 
periodic radio signal with a period that 
was very precisely 1.337 seconds and so 
there was a blip in her radio signal at 
the wavelength she was looking at the 
frequency she was listening at every 1.3. 
In three, seven seconds and of course of 
the. 
Immediate interpretation was, well maybe 
some radio station from, some other 
country is broadcasting a time signal at 
some weird period. 
this was quickly excluded. 
The next idea was maybe these were 
Literally termed LGM signals. 
Lgm stood for little green men. 
This could be the long-awaited SETI 
signal of some intelligent being 
somewhere else 
Contacting us, maybe it was their version 
of the BBC. 
It turns out quickly they turn their 
telescopes another direction, they found 
other sources at exactly the same 
frequency, it was unlikely that 
civilizations many, many light years 
apart would all be addressing us on the 
same frequency, so this is a natural 
source. 
Soon many of what became, came to be 
called pulsars, these pulsing radio 
sources were found, their periods range 
between a tenth of a second and two 
seconds. 
So this is a little longer than the 
maximum. 
rotation period to be expected for 
neutron stars. 
But it's within the ballpark, and nothing 
else rotates that fast. 
And they're almost universally slowing 
down. 
So the rates of rotation were very, very 
precise. 
The repetition of a pulsar is such that 
you could it competes almost with atomic 
clocks. 
But, they very slowly and in a regular 
fashion are slowing down. 
if you estimate how long it'll be before 
a pulsar stops pulsing the rate of 
rotation is such that a lifetime of a 
pulsar would be predicted to be about ten 
million years. 
This is a musing view of the original LGM 
signal. 
It was recorded as was done in the 
sixties and. 
I can tell you in the late seventies, by 
an [xx] quarter moving on a rolling piece 
of paper and you see here the signals 
from the pulsar and a lot of radio 
interference from which it had to be 
distinguished. 
This is a clean done version of the same 
over here on the right and those blips 
are actually the regularly spaced but not 
very regularly shaped, this is in fact 
true, 
Pulses from the pulsar and so these are 
this is the discovery of a the first 
pulsar so what are these things well. 
I'm going to claim that there are neutron 
stars people try to invent all kinds of 
other models for what a pulsar might be, 
but you can imagine trying to invent the 
binary system. 
That rotates every.2 seconds. 
But a binary system orbiting every.2 
seconds would have two white dwarves 
right on top of each other and inside 
each other and would very rapidly slow 
down. 
You can try to imagine maybe a star that 
is rotating, and has some hot spot that 
emits radio waves. 
And again an order of magnitude 
calculation tells you something. 
So how fast can a star rotate without 
centrifugal forces ripping it apart? 
Well we can do the calculation, because 
we know everything we need to. 
In order for a star to rotate its a 
objects on its surface must be 
accelerating towards the middle with the 
sintripital acceleration that we computed 
long ago v squared over r that's two pie 
r divided by p squared divided by r 
because v is two pie r divided by. 
The period and R in this case is just the 
radius of the star. 
That's the radius at which, with which, 
of the circle at the equator, along which 
things move. 
So plugging that into here, I get one R 
cancels. 
And I get four pi squared R over P 
squared. 
And, why would these objects be? 
accelerating towards the center of the 
star well of course because gravity is 
accelerating them, in the case of earth 
we too are accelerating towards the 
center of earth at least at the equator 
with this acceleration but that is a far 
smaller number that the force that earth 
applies to us the gravitational 
acceleration g and so that is why even at 
the equator we don't float. 
What I'm asking is how fast does a star 
need to spin before it's equator floats? 
And if it spins any faster it's equator 
will be blown apart. 
So that is obtained by setting the 
centrifugal acceleration equal to the 
gravitational acceleration of the star. 
You can solve this with a period. 
And here's the result you get and you. 
Sticking in a period of around a second, 
you real, you quickly realize that a 
white dwarf, the. 
Most massive object known to exist at the 
time, would blow itself apart if it tried 
to spin that fast. 
And remember we calculated it for the 
Earth about two hours is 
the 
Wrote the, the, the period with which it 
could spin before it blew itself apart. 
Remember lower satellites orbit every 
hour and a half. 
That's about the period. 
notice that this is determined completely 
by the ratio R cubed divided by M, or M 
over R cubed, which is the density. 
So we need something much denser. 
Barometrically denser than a white dwarf 
and only neutron stars turn out to be, an 
object that can exist that is dense 
enough to survive. 
Rotating this fast and then the mechanism 
is that the emission is somehow aligned 
to the star's magnetic access, axis, and 
that perhaps, remember, neutron stars are 
endowed we expect with very powerful 
magnetic fields. 
And like the Earth's magnetic field, the 
neutron star's magnetic field need not be 
aligned to its rotation axis. 
So if it is emitting light along the 
magnetic axis and. 
The magnetic axis and its axis of 
rotation, in this picture over here on 
the right, is vertical, then over a 
rotation the magnetic axis sweeps out a 
cone in space. 
And if Earth happens to lie somewhere 
along this cone, then the axis is pointed 
at us, we see a blip of radiation in this 
case radio radiation, the detection soon 
thereafter of the crab pulsar with both a 
period much shorter than a second. 
And its know relation with a super nova 
remnant nailed. 
Both the pulsars both neutron stars as 
the source pulsars and. 
Neutron stars as supernova remants and so 
how does this work. 
Now modeling a pulsar is a difficult 
exercise. 
It's not completely understood what goes 
on, and the physics that goes into it is 
a little bit beyond what we have. 
But the rough idea is that you have this 
star which is rotating very rapidly, it's 
got a very intense magnetic field. 
That means that at any point there's a 
rapidly changing and very intense 
magnetic field. 
The magnetic field, which changes, 
remember are friend Faraday, a changing 
magnetic field creates an intense 
magnetic field, this is intense enough to 
lift charged particles. 
A neutron stars mostly neutrons, but it 
does have certainly in its upper layers 
proton and heavy nuclei and certainly 
electrons. 
They are lifted. 
In fact the, electric field is so 
intense, that it can almost create, pairs 
of electrons and positrons out of the 
vacuum. 
And so it can pair create its own 
magnetic electrically charged particles. 
These have been accelerated by the 
changing magnetic field to vary 
relevantistic velocities. 
And they wind around the magnetic field 
lines as such particles are supposed to 
do. 
They wind around the, the magnetic field 
lines and are dragged around. 
So there's this magnetosphere of charged 
particles that are dragged around by the 
rotation of the pulsar. 
They wind around the magnetic field 
lines. 
This accelerated motion around the 
magnetic field lines emits a 
characteristic 
Of. 
Curvature radiation which is related to 
synchrotron radiation. 
This a mix preferentially because of the 
relativistic speeds along the direction 
of the magnetic axis. 
And so we produce this sort of lighthouse 
effect. 
the source of the energy for all of this 
emission is the rotation, and this is why 
the pulsars are observed to slow their 
rotation speeds. 
In fact, this at the end will answer the 
question I posed earlier. 
Why is the crab nebula still glowing a 
thousand years after the supernova light 
pulse as past through. 
In fact the energy source for all of the 
glow of n one is the magnetic field 
generated by the pulsar in the middle in 
fact. 
What can computer, or estimate, the 
kinetic energy. 
Included in that rotation, one knows the 
rate at which the crab pulsar is slowing, 
one can figure out how much energy that 
means its losing and compare that our 
luminosity of the crab nebula, its a firm 
calculation and the results, to within 
our uncertainties completely agree so the 
entire luminosity of a, crab nebula is 
powered by the slowing down of the 
rotating neutron star that is in the 
middle of it. 
since Bells discover, I said that, these 
particles should emit, synchrotron or 
Curvature radiation at all wavelengths. 
And indeed, pulsars have been observed in 
all bands, radio waves, infrared, 
ultraviolet, X-rays, visible light. 
The Crab Pulsar can be seen in visible 
light though it's a tricky exercise to 
find it. 
But pulsars do admit at all bends neutron 
stars are extremely exciting 
Systems you can imagine that just like 
white dwarfs when you put neutron stars 
in a binary system, you can have exciting 
mass transfer. 
The physics that describes all of these 
phenomena is relavalistic. 
And the details are going to be a little 
bit beyond what we can do. 
if we have time next week we might return 
armed with some understanding of 
relativity. 
And we'll certainly meet neutron stars 
and pulsars in the sequel but I couldn't 
not mentioning. 
Not mention them as the most exciting 
almost, second most exciting end product 
of. 
Stellar revolution. 
Of course the most exciting. 
End product of star evolution is what 
happens if neutron degeneracy is overcome 
in the collapse of the core and that will 
be the topic for next week.