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Lecture 20
White dwarfs
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
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.
Are white dwarfs white?
• White dwarfs have a range of
temperatures (i.e. colours)
Composition
• Heavy nuclei are pulled below the surface, while
hydrogen rises to the top, layered above the helium
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.
2

F 
2me
 2 Z   
3  

A

m
 
H 

2/3
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
2

F 
2me
 2 Z   
3  

A

m
 
H 

2/3
Degenerate matter
The electron degeneracy pressure is derived from the Pauli
exclusion principle and the Heisenberg uncertainty principle:

xp 
2

3 
P
2 2/3
5
2

me
 Z   
 

A
m
  H 
5/3
(non-relativistic matter)
Mass-Volume Relation
Calculate the relationship between mass and volume for
a completely degenerate star of constant density.
MV  constant
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.
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:
P
3 
2 1/ 3
4
 Z   
c  

A
m
  H 
4/3
Chandrasekhar limit
The relativistic expression for pressure is:
3 
P
2 1/ 3
4
This leads to the Chandrasekhar mass limit:
M Ch 
(contains elements of quantum
mechanics, relativity, and gravity!)
A more careful calculation shows:
M Ch  1.44M Sun
 Z   
c  

A
m


H 

3 2  c 
 
8 G
3/ 2
4/3
 Z  1 
 

 A  mH 
2
Break
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:
L
0.03LSun
 M  Tc


 
7 
 M Sun  2.8 10 
7/2
The cooling time can then be calculated
from the thermal energy and the
luminosity.
White dwarfs: cooling
2.5
T 
t 
 A  T0 
 1  1.2  7 

9
T0 
 12  10 K  10 yr 
• As the white dwarf
cools, the carbon (and
oxygen) crystallize,
leaving something like a
huge diamond in the
sky.
2 / 5
2.5
T
L 
A
t 
  0 
 1  1.2  7 

9
L0 
12
10
K
10
yr
 


7 / 5
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.
Accretion disks
• Orbital motion of the stars means mass transfer will
form an accretion disk.
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.
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.
Example: Type 1a supernovae
How far away can a Type 1a supernova be seen, using large
telescopes sensitive to apparent magnitudes mB~25 ?
 d  m  M 
25  19.6  1  9.92
log   
1 
5
5
 pc 
d  8.3 109 pc
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
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!