White dwarfs

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Lecture 20

• White dwarfs

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Sirius B

• Wobble in the position of Sirius A led to the
  prediction of an unseen companion.

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Sirius B detection and spectroscopy

• The temperature of Sirius B is 27,000 K, almost
  three times larger than Sirius A. Surprisingly
  hot!
• Given its low luminosity, it must be very small

• Thus it has the mass of the Sun in a volume
  smaller than Earth.
• An enormous density and force of gravity.
• Estimate the central temperature and pressure

Clearly the low luminosity does not arise from
hydrogen fusion.
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Are white dwarfs white?

• White dwarfs have a range of temperatures (i.e.
  colours)

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Composition

• Heavy nuclei are pulled below the surface, while
  hydrogen rises to the top, layered above the
  helium

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Degenerate matter
Pauli exclusion principle at most one fermion
can occupy any given quantum state.
The Fermi energy is the energy that divides
occupied and unoccupied states at 0K.
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Degenerate matter

• At non-zero temperature, the degeneracy is not
  complete
• We call a gas degenerate if its average kinetic
  energy is less than the Fermi energy

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Degenerate matter

• The electron degeneracy pressure is derived from
  the Pauli exclusion principle and the Heisenberg
  uncertainty principle

(non-relativistic matter)
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Mass-Volume Relation

• Calculate the relationship between mass and
  volume for a completely degenerate star of
  constant density.

More massive stars are smaller. Electrons must
be more closely packed in more massive stars, for
degeneracy to provide sufficient pressure.
Clearly a problem here because if you keep piling
mass on its volume must go to zero. The
derivation ignored relativity, and at high enough
densities the velocities of the electrons
approach the speed of light.
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Chandrasekhar limit

• The velocities of the electrons are actually
  smaller than predicted by ignoring relativity.
  Thus they contribute less pressure the volume
  will be even smaller than predicted earlier.
• In fact, volume goes to zero for a finite mass.
• There is a maximum mass that a white dwarf can
  have.

The relativistic expression for pressure is
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Chandrasekhar limit
The relativistic expression for pressure is
This leads to the Chandrasekhar mass limit
(contains elements of quantum mechanics,
relativity, and gravity!)
A more careful calculation shows
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Break
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White dwarfs cooling
Electron conduction is very efficient, so the
interior of a white dwarf is nearly isothermal.
The luminosity, mass, and interior temperature
are related by
The cooling time can then be calculated from the
thermal energy and the luminosity.
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White dwarfs cooling

• As the white dwarf cools, the carbon (and oxygen)
  crystallize, leaving something like a huge
  diamond in the sky.

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White dwarfs star formation history

• Observations of the number of white dwarfs as a
  function of their luminosity, compared with
  theoretical models with different epochs of
  initial star formation.

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Accretion disks

• Orbital motion of the stars means mass transfer
  will form an accretion disk.

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Novae

• Accretion of fresh hydrogen builds up until a
  shell of hydrogen fusion (CNO cycle) is created.
• The sudden change in luminosity is known as a
  nova.

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Type 1a supernovae

• Type 1a supernovae arise from an accreting white
  dwarf in a close binary system.
• When the mass exceeds the Chandrasekhar limit,
  the core collapses

• These are important because they all appear to
  have the same peak luminosity (MB-19.60.2).
• Since they are so bright, they are excellent
  distance indicators for the Universe.

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Example Type 1a supernovae

• How far away can a Type 1a supernova be seen,
  using large telescopes sensitive to apparent
  magnitudes mB25 ?

At this distance, the light we are seeing was
emitted when the Universe was only a third of its
present age.
The most distant supernova ever seen, at a
distance of 12.7 Gpc the light was emitted when
The Universe was only 3.8 billion years old
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The geometry of the Universe

• It has been known since the 1930s that the
  Universe is expanding more distant galaxies are
  moving away from us more quickly.
• By comparing the distance of the supernova to
  their redshift (recession velocity) we can
  measure not only the velocity of this expansion,
  but how it has changed over time (i.e.
  acceleration of deceleration).

• But the observations of the most distant
  supernova indicate that the expansion has
  actually been accelerating!