Last Section of AY4
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Last quiz: Thursday, June 5
Optional Final: Week of June 9
Neutron stars, pulsars, x-ray binaries
Relativity
Black Holes
Neutron Stars
• There is a last test of SNII theory
• If the scenario is correct, there should be a VERY
dense, VERY hot ball of neutrons left behind after
the explosion.
• Supported by neutron degeneracy (although there
is an “atmosphere” of normal matter)
This is called neutron “star”
White dwarf
6000 km
Neutron Star
10 km
Neutron Stars: Predictions
• Neutron star mass: > 1.4Mo
• Neutron star radius: 10 - 80 km
• Neutron star density: 1014 grams/cm3
100 million tons/thimble (all of humanity)
• Initial Temperature: >2,000,000k
• Neutron star remnant will be spinning
rapidly and have a huge magnetic field
Neutron Star Spins
• The reason n-stars are predicted to be rapidly
spinning is another Law of Physics called
`Conservation of Angular Momentum’.
• Linear momentum is a property of a moving
object and is a vector quantity: of a moving object
to remain in motion.


• To change linear momentum p  m v
you need to exert a force on an object.

Conservation of Angular
Momentum
• Any spinning object has `angular
momentum’ which depends on how fast it is
spinning and how the object’s mass is
distributed.
• `how fast’ -> w (greek letter omega)
• `mass distribution’ -> Moment of inertia (I)


L  Iw
Conservation of Angular
Momentum
• Conservation of angular momentum means:
L initial  L final
Moment of
Inertia
Iiw i  I f w f
Ii w f

I f wi
Angular
velocity
Conservation of Angular
Momentum
• Think about those ice skaters. With arms
out, a skater has a large moment of inertia.
Pulling his/her arms in reduces the moment
of inertia.
Arms out: large I, low spin rate
Arms in: small I, high spin rate
Conservation of Angular
Momentum
• The moment of inertia for a solid sphere is:
2
2
I  MR
5
• If a sphere collapses from a radius of
7x105km to a radius of 10km, by what
factor does it’s spin rate increase?
• Conservation of angular momentum means:
Linitial  L final
Iiw i  I f w f
2
2
2
MRi w i  MR 2f w f
5
5
Ri2w i  R 2f w f
5 2

R
7 10
9
w f  w i  
w

4.9
10
wi
 i
R
 10 
2
i
2
f
• Sun rotates at 1 rev/month. Compress it to 10km
 and conserve L, it will spin up to 1890
revolutions/second (and fly apart)
Magnetic Fields
• Magnetic field lines are also conserved.
When the core collapses, the field lines are
conserved, and the density
of the field lines goes way
up . This is the strength
of the magnetic field.
Neutron Stars
• The possibility of n-stars was discussed way
back in the 1930’s but for many decades it
was assumed they would be impossible to
detect (why?)
• But, in 1967, Jocelyn Bell and Tony Hewish
set up a rickety barbed-wire fence in the
farmland near Cambridge England to do
some routine radio observations.
LGMs
• Bell and Hewish discovered a source in Vela that
let out a pulse every 1.3 seconds. Then they
realized is was accurate to 1.337 seconds, then
1.3372866576 seconds. They soon realized that
the best clocks of the time were not accurate
enough to time the object. They called it ‘LGM’.
First Pulsar
• Bell was a graduate student at
the time. The source was
assumed to be man made, but
when no terrestrial source
could be identified, they
briefly considered an artificial
extra-terrestrial source.
• When a second source was
discovered (Cass A) they
announced the discovery as a
new phenomenon.
• The discovery led to a year of wild
speculation, but explanations involving
neutron stars quickly rose to the top.
• A pulsing source with period of 0.033
seconds was discovered in the Crab nebula.
• Big clue! Spin the Sun or Earth or a WD 30
times per second and they will be torn
apart.
• Need a small object with very large
material strength.
Pulsars
• The new objects were named ‘pulsars’ and
is was soon discovered that they were
slowly slowing down -- this provided the
answer to the mystery of why the Crab
Nebula was still glowing.
• There are now more than 1000 known
pulsars in the Galaxy.
Pulsars: The Lighthouse Model
• So, what is the pulsing
all about?
• The key is to have a
misalignment of the
nstar magnetic and
spin axes?
• What do you call a
rotating powerful
magnetic field?
Lighthouse model
• A rotating magnetic field is called a generator. The
pulsar is a dynamo which is typically about 1029
times more powerful than all the powerplants on
Earth. The huge electric field rips particles off the
surface and accelerates them out along the
magnetic field axis.
• The misalignment of the magnetic and spin axes
results in a lighthouse-like effect as the beam
sweeps past the Earth once per rotation period.
Pulsars
• The period of the Crab pulsar is decreasing by 3 x
10-8 seconds each day. The rotational energy is
therefore decreasing and the amount of the
decrease in rotation
energy is equal to
the luminosity of
nebula. Old pulsars
spin more slowly.
• There is a mysterious cutoff in pulsar
periods at 4 seconds. The Crab will slow to
this in about 10 million years. The pulsar
will turn off. Although the n-star will still be
there, it will be essentially invisible.
• Most pulsars have large space velocities.
This is thought to be due to asymetric SNII
explosions.
Pulsars
• Do all SN remnants have pulsars?
• No - some SN remnants are from SNI
• No - some rotating neutrons stars will have
beams that don’t intersect the Earth
Milli-sec Pulsars and X-ray
Binaries
• Since the first x-ray
telescopes went into
space on rockets it has
been known that there
are Luminous X-ray
stars.
• In 1982, the first of
many milli-second
pulsars was discovered
• The two phenomenon are connected.
• When a neutron stars has a close
companion, it pulls material through the L1
point. This material flies down to the
surface of the n-star and crashes onto the
surface, releasing LOTS of gravitational
potential energy. This energy comes out
mostly as x-rays and is modulated with the
n-stars spin.
Mass-transfer and N-stars
• Some of the x-ray binaries have allowed a
measurement of the neutron star mass:
In 10 of 11 cases, M=1.44Mo
This is good! Neutron stars are all supposed
to be more massive than the Chandrasekar
limit and there is even reason to expect
them to be close to this limit as that is what
initiated the core collapse in a SNII
Millisecond Pulsars
• The discovery of pulsars that were spinning more
than 100 times per second (the first was spinning
640 times per second) threw the field for a loop.
When some millisecond pulsars were discovered
in old star clusters it was even more confusing.
• Eventually it was determined that all millisecond
pulsars were in close binary systems and were
`spun up’ by accreting material.
Detecting Neutron Stars
• Detecting n-stars via their photospheric
emission is difficult.
• N-stars are VERY hot, but have a tiny
surface area so have low luminosity.
• Initial temperature may be greater than
3,000,000k so a very young n-star will emit
most of its Planck radiation in X-rays.
• First isolated n-star
observed in
photospheric light
was discovered in
1997.
• Tsurface=700,000
• Estimated age is 106
years.
• This is combined xray through visible
light image
• In 2002 there were
about 6 isolated nstars known that are
seen in the light of
their Plank
radiation.
• Most are very
nearby (<300 pc)
and traveling VERY
fast.
Puppis A remnant with 2 millionK n-star racing away at
600 km.sec. Estimated age is 6000 years.
Sun: R=105km
density=6 gram/cm3
Neutron `star’: R=20km
density=1014
Mass > 1.4Mo
White Dwarf: R=6000km
density=106
Mass < 1.4Mo
Is there a limit to neutron
degeneracy?
• Yes! Gravity wins the final battle. The
current best estimate for the maximum mass
of a neutron-degenerate star is 3Mo.
• If a neutron star exceeds this mass it will
collapse into an infinitely small volume
called a black hole.
• But, this story starts with Einstein’s theories
of special and general relativity.
Special Relativity
• Various experiments starting in the late
1800s suggested that the speed of light was
constant, independent of the motion of the
observer.
• This is very counter-intuitive.
• The spaceship traveling in the same direction of a photon
measures the photon zooming away at the speed of light
NO MATTER how fast the spaceship is traveling!
Special Relativity
• Einstein (and others before him) decided to
take the speed of light as an invariant and
not make any assumptions about the two
properties that go into determining speed:
Space and Time
Time Dilation and Length
Contraction
• The invariance of the measured speed of
light independent of the motion of the
observer can be understood if:
(1) Clocks run more slowly as speed
increases.
(2) Metersticks shrink as speed increases.
Say What?
Time Dilation
• As your speed with respect to another
observer increases, your watch runs more
slowly than the observers. This is called
`time dilation’
T
T0
1 (v /c)
2
Note, when v<<c, T=T0
Time Dilation
• As v approaches c, v/c -> 1 and the denominator
goes to zero. Dividing by zero gives infinity.
As v approaches c, time grinds to a stop!
• Q. Suppose you measure an event that lasts
for 1 second by your watch. What will your
friend in a spaceship moving at 0.98c
measure as the duration of the event?
T
T0
1 (0.98)
2
 5.02T0
• Time has been stretched by a factor of 5 for
your friend.


Length Contraction
• In the same way, metersticks (space)
contracts in the direction of motion.
L  L0 1 (v /c)
• But wait, there’s more!
2
Mass
• Mass grows with speed.
M
M0
1 (v /c)

2
Constant Speed of Light
• The shrinking rulers and slowing clocks
conspire to let observers in any moving
frame measure the same speed of light.
The Reason Travel to other
Galaxies will be Difficult
• The slowing clocks and increasing mass conspire
to make it impossible for objects with mass to ever
reach the speed of light.
• The increasing mass requires an ever-larger force
to accelerate to larger speed and the force needed
would become infinite. (F=ma)
• Even if you could find the force, your clock would
slow and slow and the last step would take and
infinitely long time
Is this right?
• Yes! There are many tests of Special
Relativity.
• In particle accelerators mass increase and
time dilation effects are routinely measured
• There have been tests flying very accurate
clocks in high-speed jets that show time
dilation directly.
• We might not be here if not for time dilation
in the frame of cosmic rays called muons.
General Relativity
• Einstein’s theory of General Relativity is a
theory of gravity
• The basic idea is to drop Newton’s idea of a
mysterious force between masses and
replace it with the 4-dimensional
SpaceTime Continuum
General Relativity
• In GR, mass (or energy) warps the spacetime
fabric of space.
• Orbits of planets around stars are not due to a
central force, but rather the planets are traveling in
straight lines through curved space
Imagine tossing a shotput onto your bed and rolling marbles
at different speeds and distances from the shotput. (also imagine
that you have a frictionless blanket on the bed).
The marbles that are moving slowly or close will fall down toward
the shotput. If you look from above, it will appear as if the marbles
were attracted to the shotput.
Fabric of Space
• This is a RADICALLY
different view of the
Universe and gravity
• In regions where space is not
strongly curved, GR reduces
to Newton’s law of gravity
• Einstein pointed out his new
theory would explain the
Precession of the Perihelion
of Mercury
The Deflection of Starlight
• There were several other predictions of GR, one
important one was that light rays would also
follow straight lines through curved space.
Tests of GR
• In 1919, during a total eclipse of the Sun,
the predicted deflection of starlight for stars
near to the limb of the Sun was measured
and Einstein became a household name.
• GR also predicted that time would slow in
strongly curved space. This was verified
experimentally in 1958.
Tests of GR
• There is another long
list of predictions
made by GR -- in
every case to date,
they have verified the
theory perfectly.
The Bending of Light
• The bending of light in GR leads to some
very useful and interesting phenomenon.
• One is the effect called a gravitational lens.
• The light from a distant galaxy is bent by a
large mass along the line of sight to to
Earth. If everything is lined up perfectly,
you get an “Einstein Ring”. Very useful for
identifying and measuring “Dark Matter”
Foreground
galaxy
• It is rare to have a
Background
close-to-perfect
lensed quasar
alignment. The more
common case is to
have a set of discrete
images.
On to Black Holes
• The second very interesting aspect of light
bending in General Relativity is the idea of
Black Holes. It starts with the concept of
escape velocity.
Escape Velocity
• Imagine feebly tossing a rocketship up in the air. It
falls back to Earth because its kinetic energy was
less than its gravitational potential energy.
• However, toss it with a larger and larger velocity
and it will go higher and higher before falling
back to Earth.
• There is a velocity above which it will not return
to Earth -- this is the escape velocity.
Escape Velocity
• To determine the escape velocity from Earth
you set the gravitational potential energy
equal to kinetic energy and solve for
velocity
2G M
Vescape 
R
Radius from which you want to escape
Mass of the
object from
which you
want to escape
Escape Velocity
• Note that the escape velocity doesn’t
depend on the mass of the escaping body.
• For the Earth, put in the mass and radius of
the Earth (for escape from the surface of the
Earth) and you get:
Vesc= 11 km/sec = 25,000 miles/hr
Escape Velocity
• Now suppose you shrink the Earth to 1/100
of its current radius (at constant mass).
What happens to Vesc?
Vesc
2GM

R
1
Vesc 
R
As R goes up, Vesc goes down
As R goes down, Vesc goes up
Don’t forget the square root
For this case, Vesc increases by 10x
Escape Velocity
• Reduce the radius of the Earth to 1cm and
Vesc=c (speed of light)
• In this new theory of Gravity, where photons are
affected by gravity, if the escape velocity equals or
exceeds the speed of light, that object can no
longer be observed. This is a Black Hole
Black Holes
• The critical radius for which an object of a
particular mass has an escape velocity of `c’
is called the Schwarzschild Radius.
• This is also called
• the `Event Horizon’.
Schwarzschild Radius
• You can easily calculate the Schwarzschild
radius for any mass by setting Vesc=c
Vesc
2MG
c 
Rs
2MG
2MG
c 
 Rs  2
Rs
c
2
• Every object has a radius at which it
becomes a Black Hole

Black Holes
• But, it is VERY, VERY difficult to compress
an object to its Schwarzschild radius.
• For the Sun, you would have to somehow
overcome thermal pressure, then edegeneracy, then neutron degeneracy. We
know of no `cosmic vice’ that can do that.
Black Holes
• But, go back to a neutron star and we are
building a pretty big vice. Thermal pressure
has already been overcome as has edegeneracy pressure.
• There is a limit to the pressure that can be
generated by neutron degeneracy. Its hard to
calculate, but is probably between 2Mo and
3Mo
Black Holes
• Think about the n-star core of a SNII explosion. If
say 1.6Mo of material falls back, the core will
exceed the neutron degeneracy limit and undergo
collapse to zero volume (what?) zero volume.
Black Holes
• What is left behind?
– The gravitationally force (i.e. a warp in
spacetime) including a `singularity’ at the
center of the warp
– An Event Horizon with radius given by
RSch=8.9km
Hawking
radiation
Black Hole FAQs
• What would happen if the Sun collapsed
into a Black Hole, would the Earth be
dragged in?
• No, the gravitational force at the distance
of the Earth would not change.
• Is the Event Horizon a physical boundary?
No, it is simply the distance from the center
where the escape velocity of `c’.
• What happens if a Black Hole absorbs some
mass?
As M increases, the Schwarzschild radius
also increases.
• Is there any reason to believe that Black
Holes exist?
You Bet!
This would be great. But not too likely…
Black Hole Evidence
• The best stellar-mass cases are binary x-ray
sources.
Cygnus X-1 is a good
example.
Black Hole Evidence
• Cyg X-1 is a bright x-ray source. Look there
in the visual part of the spectrum, we see a
30Mo blue main-sequence star which is a
spectroscopic binary with a period of 5.6
days.
• The companion has a mass of between 5
and 10Mo. What is it?
Cygnus X-1
• There is no sign of the companion at any
wavelength so what is it?
1) A red giant would be easily seen
2) A main-sequence star would be seen
with a little effort
3) Can’t be a WD because M>1.4Mo
4) Can’t be a n-star because M>3Mo
Cygnus X-1
• By elimination, we are left
with a black hole
• The x-rays back this up. In
an accreting WD we see
UV radiation, in an n-star
we see `soft’ x-rays, in
Cyg X-1 we see `hard’ xrays because the accreting
material falls into a deeper
potential well.
Stellar-mass Black Holes
• We now have a few dozen excellent stellar-mass
black hole candidates and few people doubt that
such objects exist.
• There was a `microlensing’ event in 1996 that was
ascribed to a blackhole gravitationally lensing a
background star.
• There are various claims that x-ray transients are
black holes accreting little bits of stuff.
Supermassive Black Holes
• Since the early 1960s extraordinarily
energetic objects called `qso’s or `quasars’
have been identified a large distances and
lookback times.
• The only explanation astronomers could
come up with for their energy source was
accreting mass onto a large (>105Mo) black
hole.
Supermassive Black Holes
• QSOs had large radio
jets emitted at
enormous velocities.
• Eventually it became
clear that QSOs were
all located in the cores
of galaxies and nearby
counterparts were
identified.
• Cen A radio jets
• The nearby systems
allowed observations
much closer to the
central engine and
over time the evidence
for the black holes has
become more direct
The Galactic Center
• After years of speculation about a possible
supermassive black hole in the center of the
Milky Way, work at Keck by Andrea Ghez
at UCLA demonstrated convincingly in
1999 that we have a 2 million solar mass
black hole at the center of the Galaxy.
Supermassive Black Hole in the
Galaxy
• 2002 observations pretty much cinch the
case for a 2.6 million solar mass black hole
in the center of the galaxy.
• See the movie!