Degeneracy

1
Chapter 13 The Stellar Graveyard

• Degeneracy Stars
• Brown Dwarfs
• White Dwarfs
• Neutron Stars
• Black Holes

X-ray image of supernova remnant G11.2-03, from
A.D.386.
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The Dead Stars

• The End States of Stars
• Nothing The entire star is dispersed into
  interstellar space,
• White dwarfs Remnants of low mass stars ( M lt
  1.4M?), typical size 109 cm (about the size
  of the Earth).
• Neutron stars Remnants of high-mass stars (1.4
  M? lt M lt 3 M?), ypical size 106 cm (about
  the size of a big mountain).
• Black holes Stars with mass larger than 3 M?
  will evolve into black holes.
• Supporting Mechanisms (Force Against Gravity)
• Regular stars are supported by the thermal
  pressure generated by the nuclear fusion
  processes.
• White dwarfs and brown dwarfs are both supported
  by the degenerate pressure of electrons, although
  there are different core materials.
• Neutron stars are supported by the degenerate
  pressure of neutrons.
• Black holes is the end state of a massive star in
  which gravitational contraction is so strong that
  it eventually overwhelm the degenerate pressure
  of neutrons.

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Degenerate Stars

• Three types of stars are supported by
  degenerate pressure
• Brown Dwarfs
• Supporting Mechanism Electron degenerate
  pressure
• Origin Failed stars
• Core Composition Electrons, hydrogen nuclei
• Mass Mbd lt 0.08 Msun
• White Dwarfs
• Supporting Mechanism Electron degenerate
  pressure
• Origin Remnants of low and medium mass (M lt
  8 Msun) main sequence stars.
• Core Composition Electrons, helium, carbon, and
  other heavy element nuclei (from medium-mass
  stars).
• Mass 0.08 Msun lt Mwd lt 1.4 Msun
• Neutron Stars
• Supporting Mechanism Neutron degenerate pressure
  
• Origin Remnants of high-mass (M gt 8 Msun) main
  sequence stars.
• Core Composition Neutrons
• Mass 1.4 Msun lt Mneutron lt 3 Msun

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White Dwarfs

• White dwarfs are remnants of low-mass main
  sequence stars, supported against gravity by
  electron degenerate pressure.
• Depending on its initial mass, the composition of
  the core is different.
• very-low-mass stars ? helium core
• low-mass stars ? carbon core
• Medium-mass stars ? heavier element cores
• The atomic nuclei in the white dwarfs, such as
  helium, and carbon, are bosons, not fermions. The
  exclusion principle does not apply to bosons, and
  degeneracy pressure does not arise from these
  particles. They dont help to fight against
  gravity!

White dwarf in planetary nebula
White dwarf in globular cluster M4 Each circle
marks a white dwarf.
The white dwarf companion of Sirius
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The Fate of the White Dwarfs

• The degenerate pressure DOES NOT depend on the
  temperature of the white dwarf. An isolated white
  dwarf, without further interaction with other
  stars, will slowly irradiate away its thermal
  energy into space and cool down, and eventually
  come into thermal equilibrium with the universe
  (very cold).
• Howeverif a white dwarf is not left alone, such
  as those in close binary system, the final stage
  of the evolution is not necessarily the
  equilibrium state with the universe. These white
  dwarfs still have a life after death

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The Afterlife of White Dwarfs in Close Binary
Systems

• A white dwarf in a close binary system can gain
  substantial amount of mass if the companion is a
  main-sequence or giant star, giving it a new
  life.
• In these close systems, mass from the other star
  can be transfer to the white dwarf.
• The in-falling matter forms an accretion disk
  around the white dwarf.
• Because of the strong gravity at the surface of
  the white dwarfs, the in-falling speed is very
  high!
• The friction between the gas causes the
  temperature of the accretion disk to rise,
  emitting light in the optical and UV wavelength
  ranges,
• Sometimes even X-ray!

White dwarf in X-ray from ROSAT
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Nova

• The gas (mostly hydrogen) in the accretion
  disk, obtained from the companion star, may fall
  into the white dwarf, accumulating on the
  surface, forming a shell of hydrogen gas.
• The temperature and pressure build up on the
  surface gradually, eventually reaching the
  hydrogen fusion temperature of 10 million
  degrees.
• Hydrogen shell is ignited and release large
  amount of energy ? Nova.
• This process may repeat itself, however, the
  frequency for the occurrence of nova is not
  well-established.

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The Chandrasekhar Limit

• The newly created helium may accumulate on the
  surface of the white dwarf, increasing its mass.
  However, there is a limit on the maximum mass a
  white dwarf can have
• When we increase the mass of the white dwarf, the
  electron degeneracy pressure will increases, but
  it does not increase linearly and indefinitely,
  because the speed of the electrons cannot exceed
  the speed of the light.
• This means that there is an upper limit on the
  degenerate pressure the white dwarfs can provide,
  and an upper limit on the mass of the white
  dwarfs!
• The Chandrasekhar Limit (or the white dwarf
  limit) is the upper limit of the mass of the
  white dwarfs 1.4M?.
• At this mass, the speed of electrons in the white
  dwarf would be equal to the speed of light!
• So far, NO observed white dwarf have mass
  larger than 1.4 M?, confirming Chandrasekhars
  theory!

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White Dwarf Supernova

• Every time the hydrogen shell is ignited, the
  mass of the white dwarf may increase (or
  decrease, we dont know for sure yet).
• The mass of the white dwarf may gradually
  increase,
• At about 1 M?, the gravitation force overcomes
  the electron degenerate pressure, and the white
  dwarf collapses, increasing temperature and
  density until it reaches carbon fusion
  temperature.
• The carbon inside the white dwarfs are
  simultaneously ignited. It explodes to form a
• White dwarf supernova. (Type I).
• Nothing is left behind from a white dwarf
  supernova explosion (In contrast to a
  massive-star supernova, which would leave a
  neutron star or black hole behind). All the
  materials are dispersed into space.

White Dwarf Supernova is a very important
standard candle for measuring cosmological
distance
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White Dwarf and Massive Star Supernovae
Because the mass of white dwarfs when they
explode as supernovae is always around 1.0 M?,
its luminosity is very consistent, and can be
used as a standard candle for the measurement of
distance to distant galaxies (Chapter 15). The
amount of energy produced by white dwarf
supernovae and massive star supernovae are about
the same. But the properties of the light emitted
from these two types of supernovae are
intrinsically different, allowing us to
distinguish them from a distance.

• Massive star supernovae spectrum is rich with
  hydrogen lines (because they have a large outer
  layer of hydrogen).
• White dwarf supernovae spectra do not contain
  hydrogen line (because white dwarfs are mostly
  carbon, with only a thin shell of hydrogen).
• The light curve is different.

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Type I and II Supernovae

• Supernovae are divided into to types
  observationally according to the characteristics
  of their spectra.
• Type I Supernovae without strong hydrogen
  spectrum.
• Type I supernovae can be either white dwarf or
  massive star supernovae. They are formed from
  stars that have shed their outer hydrogen layer
  before going supernova.
• Type I supernovae are further divided into Type
  Ia, Ib, and Ic, with different light curves.
• White dwarf supernovae are Type Ia.
• Type II Supernovae with strong hydrogen lines.
• All Type II Supernovae are considered massive
  star supernovae because they have a larger outer
  hydrogen layer.

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• Degeneracy Stars
• Brown Dwarfs
• White Dwarfs
• Neutron Stars
• Black Holes

13
Neutron Stars

• The physics that accounts for the generation
  of the degeneracy pressure in a neutron star is
  identical to that of the degenerate pressure of
  the electrons in a white dwarf, since neutrons
  are fermions.
• Similar to the Chandrasekhar limit for the white
  dwarfs, there is also a upper limit on the mass
  of neutron stars, for the same physical reason.
  The degenerate pressure of the neutrons cannot
  hold off gravitational contraction forever. But
  its precise value has not been accurately
  determined theoretically yet, due to insufficient
  knowledge of nuclear physics.
• The estimated upper limit of the mass of the
  neutron star is about 3M?.
• Properties of Neutron Stars
• Size 10 km.
• Strongly magnetized 109 Gauss (average on
  Earth is about 0.5 Gauss)
• Rapidly rotating 1,000 rotation per second
• Very high temperature 1,000,000 K on the
  surface

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Neutron Star as a Giant Magnet

• If the main sequence star is a magnetic field
  star, then its magnetic fields maybe trapped in
  the neutron star as the main sequence star
  undergoes gravitational collapse.
• The magnetic fields are intensified by a
  tremendous amount, because they are concentrated
  into a much smaller space.

• The angular momentum of the main sequence star
  (or the part of it thats left) is preserved.
  Because of the neutron star is much smaller
  compared with the original main sequence star, it
  will be spinning at a much higher rotation rate
  (recall angular momentum conservation and the
  spinning ice skater).
• The axis of the magnetic fields may not be
  aligned with that of the rotation axis (just like
  the magnetic field of the Earth).

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News Scientists Measured the Most Powerful
Magnet in the Universe
RELEASE 02-156http//www.gsfc.nasa.gov/topstory/
20021030strongestmag.htmlFor animation of a
magnetar, refer to http//nt.phys.gwu.edu/kovac/
magnetar SCIENTISTS MEASURE THE MOST POWERFUL
MAGNET KNOWN Scientists have identified the most
magnetic object known in the universe, the result
of the first direct measurement of a magnetic
field around a peculiar neutron star first
observed nearly 25 years ago.By following the
fate of a tiny proton whipping about at near
light speed close to the neutron star with NASA's
Rossi X-ray Explorer satellite, scientists
calculated this star's magnetic field to be up to
10 times more powerful than previously thought --
with a force strong enough to slow a steel
locomotive from as far away as the Moon.This
object, named SGR 1806-20, is one of only 10
unusual neutron stars classified as magnetars,
thousands of times more magnetic than ordinary
neutron stars and billions of times more magnetic
than the most powerful magnets built on Earth.
The strength of its magnetic field is
approximately a million billion (1015) Gauss,
according to a team led by Alaa Ibrahim, a
doctoral candidate at George Washington
University conducting research at NASA's Goddard
Space Flight Center in Greenbelt, Md
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Gyration of Charge Particles Around Magnetic
Fields
The gyration of the protons and electrons around
the magnetic field lines with speed close to the
speed of light generates gyrosynchrotron
radiation (in radio frequency).
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The Lighthouse Effect

• We do not know exactly how, but if there are
  charged particles trapped by the strong magnetic
  field of the neutron stars near the magnetic
  poles, the strong magnetic field directs the
  radiation field along the magnetic axis of the
  neutron stars.
• If the axis of the magnetic dipole is not aligned
  with the rotation axis of the neutron stars, then
  the radiation field would be sweeping through
  space, just like the light beam from a
  lighthouse.
• If the beam sweep across the Earth, we would see
  an intermittent radiation. These are referred to
  as Pulsar.
• The light beam may or may not sweep across the
  Earth ? All pulsars are neutron star, but not all
  neutron stars are pulsar.

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Neutron Stars as PulsarsThe Little Green Men?
The first pulsar was detected by Jocelyn Bell in
1967 in the constellation Cygnus. The interval
between pulses is precisely 1.337301 second.
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The Fates of Neutron Stars

• Like the white dwarfs, a neutron star will
  slowly irradiate the thermal energy into
  surrounding spaceand eventually come into
  thermal equilibrium with the cold universe.
• Also, as an isolated neutron star rotates and
  irradiates, it loses energy and angular momentum.
  Its rotation rate slowly decreasesdue to the
  conversion of rotational kinetic energy into
  radiation.
• The rotational rate of the neutron star in Crab
  Nebula was observed to be decreasing, consistent
  with theoretical expectation.
• Similar to an isolated white dwarf, the neutron
  star will eventually stop rotating, cool to the
  temperature of the surrounding universe, becomes
  inert.
• Similar to a white dwarf in a close binary
  system, a neutron star in a close binary system
  would still have a life after death.

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Neutron Star in a Close Binary System

• For a neutron star in a close binary system,
  an accretion disk similar to that formed around
  the white dwarf will be formed.
• Because the gravitational energies of the
  accretion disk around the neutron star are so
  high, the temperature of the accretion disk is
  much higher.
• X-ray binaries
• The high temperature at the inner regions of the
  accretion disk produce X-ray, some with
  luminosity as great as 105 times the luminosity
  of the Sun

Click on image to start animation
Interesting link about Neutron Star
http//sci.esa.int/
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X-ray Bursters

• Like the white dwarfs in close binary system,
  neutron stars in close binary system continue to
  draw fresh, hydrogen-rich materials from its
  companion star. Near the surface of the neutron
  star, due to the strong gravity, these materials
  accumulate in a shell only a few meters thick.
  The density and temperature of this hydrogen
  shell may be high enough to maintain a continuous
  hydrogen burning, with the helium produced by the
  fusion of hydrogen accumulating beneath the
  burning shell.
• The temperature and density of the helium shell
  may eventually be high enough for helium fusion
  to start, releasing a tremendous amount of energy
  ( 10,000 L?) ? X-ray Bursters.
• The X-ray bursters typically flare every few
  hours, with each burst lasting only a few seconds.

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What Happens Next for Neutron Stars?
The physical processes associated with novae and
X-ray bursters are strikingly similar. Therefore,
it is only natural to wonder Can neutron stars
in close binary systems continue to accumulate
mass and eventually go beyond the 3 Msun
neutron star limit and turn into black holes,
just like the process leading to the white dwarf
supernovae? Close binary white dwarfs ?
Novae ? White dwarf supernovae
Close binary neutron Stars ? X-ray bursters ?
Black holes? Quark stars? We dont know.
Unexplored subject!
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The Condition Inside a Neutron Star
We actually are not quite sure about the
condition of the matter inside a neutron star.
Theoretical investigations are still quite
preliminary, and we cannot create the same
condition in our laboratory for experimental
studies
Quark Star? Although in our current
understanding of elementary particles, protons
and neutrons are composed of even smaller
particles called the Quarks, bounded together by
the strong force, quarks cannot exist
individually. But we dont know the physics of
these elementary particles under extreme
temperature and density condition (as we imagine
must be the condition inside the neutron stars,
or black holes, or right after the Big Bang )
well enough to say if there are other forces to
resist gravity after the destruction of the
neutron stars.
Quarks Elementary particles that make up the
protons, neutrons The flavors of Quarks Up,
Down, Bottom, Strange, Charm
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Crab Pulsar From Chandra X-ray Observatory
Time-lapsed images of Crab Nebula in X-ray.
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• Degeneracy Stars
• Brown Dwarfs
• White Dwarfs
• Neutron Stars
• Black Holes
• Gamma-Ray Bursts a mystery!

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• Black holes

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What is a Black Hole?

• From the textbook
• The black in the name black hole comes from
  the fact that nothingnot even lightcan escape
  from a black hole. The escape velocity of any
  object depends on the strength of its gravity,
  which depends on its mass and size Section 4.4.
  Making an object of a particular mass more
  compact makes its gravity stronger and hence
  raises its escape velocity. A black hole is so
  compact that it has an escape velocity greater
  than the speed of light. Because nothing can
  travel faster than the speed of light, neither
  light nor anything else can escape from within a
  black hole.

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Key Concepts Relating to Black Holes

1. Escape Velocity
2. What is escape velocity?
3. How does escape velocity depends on the mass and
   size of the star?
4. Speed of Light
5. Speed of light is finite.
6. Speed of light is the maximum speed that any
   object can achieve.
7. Photons has no mass, but its path is affected by
   gravity.

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The Escape Velocity

• Recall that the escape velocity on the surface
  of Earth is about 11 km/sec. It is the minimum
  velocity an object on the surface of Earth need
  to have for it to overcome the gravitation pull
  of the Earth to go the gravity-free space.
• The Escape Velocity on the surface of a body
  depends only on its mass and size. For a
  gravitational body with mass M and radius R, the
  escape velocity on its surface is
• v2escape 2 GM/R
• where G is the gravitational constant.
• For a neutron star with the mass of the Sun (
  300,000 Mearth) with the size of 10 km ( 1/1000
  of Rearth),
• vescape 250,000 km/sec!
• Recall that the speed of light c in vacuum is
  measured to be 300,000 km/sec, which is the
  ultimate speed limit of the universe according to
  Einsteins Special Theory of Relativity

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Early Idea of Black Holes

• Pierre Laplace in the 19th century (before
  Einsteins General Theory of Relativity was
  derived) first postulated that if an object can
  be made compact and dense enough so that the
  escape velocity on the surface of this object is
  greater than the speed of light, then even light
  cannot escape the gravitational pull of such a
  dense and compact gravitational body.
• In Laplace's time, photons were considered
  ordinary particles with very small finite mass.
  Therefore, gravity of such a compact and dense
  object should be able to trap the photons in its
  gravitational fieldthis is the reasoning that
  leads to the idea of a Black Holes.
• We know this idea is erroneous today, because
  photons are mass-less, and dont interact with
  gravitational field like ordinary particles. That
  is, Newtons formula for the gravitational force
• F G M1 M2 / R2
• does not apply to photons. If the mass of a
  photon is zero, then F 0.

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• So, how do we trap photons?
• Einsteins General Theory of Relativity

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Gravitational Distortion of Spacetime

• In classical physics, the universe is
  composed of a three-dimensional space, and a one
  dimensional time. Space and time as separate and
  independent dimensions. The three-dimensional
  space moves in the time dimension.
• In Einsteins General Theory of Relativity, space
  and time are considered inseparableand gravity
  arises from the curvature of the spacetime
  continuum.
• Both light and matter follow the same path in
  spacetime
• Therefore, in region of very strong gravity, the
  distortion of spacetime is so great that the path
  of both light and matter curves back inside

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Curvature
Large Curvature
Small Curvature
Zero Curvature
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Curvature of Spacetime Around Black Hole

• A black hole in the two-dimensional analogy is a
  bottom-less piteverything fall in if you get too
  close, and nothing comes out once they are innot
  even light!

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Bending of Light Path Around Black Holes
At a distance of about 1.5 Rsch of a black hole,
spacetime is distorted so much that photons
emitted from the back of your head actually go
around the black hole and come back to you.
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The Size of the Black Holes

• The size of an black hole depends only on its
  massit is derived in General Relativity.
  However, we can estimate the size of the black
  holes by the radius of a object at which its
  escape velocity equals to the speed of light
• Rsch 2GM/c2
• Rsch is call the Schwarzschild radius.
• The Radius of an object with mass of 1 M? is 3
  km.
• Event Horizon
• The event horizon is essentially the boundary of
  the black hole. It is equal to the Schwarzschild
  radius of the object. Nothing inside the event
  horizon can escape to the outside of the black
  hole.