1. Lecture 20
White dwarfs

2. Sirius B
Wobble in the position of Sirius A led to the prediction of an unseen companion.
Sirius B
Sirius A
M/MSun
1.053
2.3
L/LSun
0.03
23.5

3. 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.

4. Are white dwarfs white?
White dwarfs have a range of temperatures (i.e. colours)

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

6. 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.

7. 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

8. Degenerate matter
The electron degeneracy pressure is derived from the Pauli exclusion principle and the Heisenberg uncertainty principle:
(non-relativistic matter)

9. 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 it’s volume must go to zero. The derivation ignored relativity, and at high enough densities the velocities of the electrons approach the speed of light.

10. 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:

11. 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:

12. Break

13. 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.

14. White dwarfs: cooling
As the white dwarf cools, the carbon (and oxygen) crystallize, leaving something like a huge diamond in the sky.

15. 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.

16. Accretion disks
Orbital motion of the stars means mass transfer will form an accretion disk.

17. 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.

18. 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=-19.6±0.2).
Since they are so bright, they are excellent distance indicators for the Universe.

19. Example: Type 1a supernovae
How far away can a Type 1a supernova be seen, using large telescopes sensitive to apparent magnitudes mB~25 ?
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

20. 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!