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.