8'8 PLANETARY NEBULAE

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8.8 PLANETARY NEBULAE
NGC 1360
Helix NGC 7293
Messier 27
Shapely 1
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PLANETARY NEBULAE
Planetary Nebulae have the appropriate
characteristics to suggest that they are created
as discussed above. They thus mark the demise of
the red giant branch for stellar masses in the
range Mo M 4Mo, and although they only last
a few tens of thousands of years Planetary
Nebulae are plentiful because there are plenty of
seed objects. Statistically there are
approximately 2 10-7 PN/Mo. To date up to about
1000 objects have been identified as Planetary
Nebulae. The wide variety of shapes we see is
partly the result of the random orientation of
hollow spheroidal shapes on the sky. On the
basis of the earlier discussion we would expect
to see

• The hot inner star which was the core of the
  red giant and is now no longer shrouded by the
  envelope. This object will be very (uv) hot (105
  K) and have about the same luminosity as the
  parent red giant.
• The outer gas cloud which was the envelope of the
  red giant should be expanding away

STATISTICAL ARGUMENTS

• At any one instant in the Galaxy we have 104
  Planetary Nebulae with a lifetime of typically
  104 years
• The number of white dwarfs in the galaxy is
  roughly 109 with a lifetime of approximately
  109 years
• Likewise the lifetime of red giants is of the
  order of 106 years and the number of red giants
  is approximately 106 within the galaxy
• On a statistical basis we can see that all three
  types of object appear to be generated at the
  same rate and this implies that on a statistical
  basis they are simply different points along the
  same evolutionary path.

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OBSERVATIONAL ASPECTS
The observational evidence bears out the idea
that Planetary Nebulae are derived from red
giants which blow off their envelope. The hot
central star has roughly the same luminosity as
red giants, but is at a much higher temperature
and therefore smaller. This object is a white
dwarf which we will later discover to be (very
often) a helium star held up by degeneracy
pressure. As predicted from the evolution of red
dwarfs. The H-R diagramme and table below give a
schematic view of the evolutionary tracks for
stars of different masses.
Track A
Red Giants
104
Track B
PN
Track C
102
Luminosity (Lo)
Main Sequence
1
White Dwarfs
10-2
300,000
100,000
30,000
10,000
3,000
Temperature (K)
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OBSERVATIONAL ASPECTS
Expanding Shell
Flourescent Line Emission
uv
Central Hot Star

• The nuclear star is a luminous high temperature
  uv emitting object
• The intense uv radiation is absorbed in the
  material of the expanding shell, causing
  ionisation of the gas
• This causes the gas to fluoresce and leads to the
  emission of discrete spectral lines.
• The lines show that the material is
  predominantly hydrogen but other materials (He,
  C, N, O) are also present
• The envelopes have typical diameters of 1015 m
• The shells are expanding with typical velocities
  of 20 km s-1 (that is fairly gently). This is
  inferred from the measurements on the wavelength
  shifts and broadening. These velocities are
  consistent with the required escape velocities
  from red giant stars.
• There is plenty of Infra-Red emission which
  strongly suggests the production of dust grains
  in the outer (cool and oscillating) envelopes of
  red giants.
• There is evidence that organic molecules are also
  present
• This is one of the sources of the ashes and dust
  which are used in the cosmic re-cycling processes.

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9. WHITE DWARFS
9.1 OBSERVATIONAL ASPECTS SIZE, MASS AND
DENSITY
The position of white dwarfs on the H-R diagramme
is as shown. However there is clearly something
very different about white dwarfs when we look at
the details
L
Main Sequence
104

• Their temperature corresponds to O, B, A main
  sequence stars
• The luminosity is a factor 10-4 lower

White Dwarfs
Thus since
T
Then white dwarfs must be much smaller objects
than the main sequence stars .
i.e. about the same size as the Earth having
characteristic radii of about 5000 km.
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THE MASS OF WHITE DWARFS
Binary systems measurements taken from WD in
binary systems show that the masses are typically
of the order of a solar mass. Sirius B is a white
dwarf and was an early confirmation of the high
mass yet small size of white dwarfs.
hn dn
r dr
Red Shift of Emitted Photons
hn
r
The measurement of the red shift of photons
enables us to apply the theory of general
relativity so as to obtain the an estimate of
ratio M/R, and hence confirm the compact nature
of white dwarfs.
R
Spectral lines emitted at rest from the surface
of a gravitational object will undergo a
gravitational red shift described by the general
relativistic expression
l is the characteristic wavelength of the line
at R, and l is the apparent wavelength as seen
at infinity. Typical values of Dl / l for white
dwarfs are 10-4
Simplistically
Consider a photon at distance r which moves from
r to r dr, the frequency changes from n to n
dn
so that
giving
and hence
THE DENSITY OF WHITE DWARFS
The density is simply mass divided by volume, so
that for white dwarfs we obtain a value of
typically 108-10 kg m-3 . The average distance
between particles ( 1057 in one solar mass) is
close to the Compton wavelength
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9.2 THE STRUCTURE OF WHITE DWARFS
The only state of matter that can exist at such
high densities is one in which the material is
supported by degeneracy pressure. In the case of
white dwarfs the degeneracy pressure is derived
from the electrons. The Pauli exclusion principle
requires the electrons to inhabit higher and
higher momentum states if the stellar material
contracts in size since
If we consider a shell in momentum space between
p and p dp, then the number of electrons in the
cell at momentum p is proportional to p2. If the
corresponding spatial volume is V then the total
number of states available for electrons between
p and p dp is
The factor 2 is due to the two possible spin
orientations of the electrons.
Thus
The number of occupied electron states per
elemental volume V is
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9.3 THE PRESSURE EXERTED BY A DEGENERATE GAS
PF is the maximum momentum available for a
completely degenerate gas at the absolute zero of
temperature. (T 0). It is often called the
Fermi momentum. Thus
The pressure exerted by a gas is the momentum
transferred across the surface
Thus
i.e. the non relativistic limit.
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Integrating gives us the pressure of a degenerate
gas for the non-relativistic limit as
Taking the number density of particles as defined
above to be
we obtain
and if the star is mostly helium
then the pressure exerted by a non-relativistic
degenerate helium gas at 0 K is
The Internal Energy of the Gas
The internal energy is derived from the motions
of the individual particles
The Kinetic Energy density of a non-relativistic
gas is
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A RELATIVISTICALLY DEGENERATE GAS
We had earlier derived
For the Trans-Relativistic Zone
These intermediate cases are not suitable for
analytical solutions and in this course we will
consider only the other extreme scenario the
ultra-relativistic situation.
and
Here
so that
The pressure is
and since
Then
With
we obtain
for the pressure exerted by a highly relativistic
degenerate helium gas at 0 K
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The Internal Energy of a Highly Relativistic
Degenerate gas
Relativistically
Some key aspects of White Dwarfs may be
summarised as follows

• Most stars with masses less than 4 Mo will
  eventually finish up as white dwarfs
• The material of a white dwarf is completely
  ionised
• The degenerate material is thermally highly
  conductive
• The ions supply nearly all of the stored thermal
  energy
• In the relativistic limit the electron pressure
  is independent of the electron mass
• The degenerate electrons provide the pressure
  which supports the star
• This pressure is effectively independent of the
  temperature

Note with respect to the last bullet that the
blue curve is the momentum spectrum for 0o K. For
a non-zero temperature we may expect the curve to
look like the red dashed curve, so that the
momentum distribution and hence the pressure will
be slightly different. However the typical
temperatures of white dwarfs are 105 K and the
thermal momenta of the electrons are negligible
compared to the Fermi momenta.
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9.4 THE CHANDRASEKHAR LIMIT
There exists an upper limit for the masses of
white dwarfs. It is easy to see why if we apply a
dimensional analysis the opposing hydrostatic and
gravitational forces.
Non Relativistic Case
The pressure
so that
Highly Relativistic Case
The pressure
so that
The Gravitational Force per unit volume is
Non Relativistic Case
The two opposing forces have different
dependencies on the radius of the star. - the
star can therefore adjust its radius for
stability. If the degeneracy pressure dominates
the star will expand to equilibrium. NOTE The
smaller the radius the greater the degeneracy
force, hence
Highly Relativistic Case
Both the hydrostatic and gravitational forces
have exactly the same radius dependence. Changes
in the radius will not enable a position of
equilibrium to be attained. The two opposing
forces do, however, have a different dependence
on the mass of the star. Since the gravitational
force has a higher dependence on mass than the
hydrostatic force it necessarily follows that at
a certain mass point the gravitational force will
dominate and the object will collapse. This
critical mass is called the Chandrasekhar limit.
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AN ESTIMATE OF THE CHANDRASEKHAR MASS LIMIT
For a highly relativistic degenerate electron gas
the pressure is
where
and the hydrostatic force is
We assume that the critical mass corresponds to
the point where the hydrostatic and gravitational
forces exactly balance.
giving
so that
The dimensionless constant
can be considered the fine
structure constant of gravitation, and
characterises the strength of gravitational
interactions just as Sommerfields fine structure
constant
describes the strength of electromagnetic
interactions.
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THE SIZES OF WHITE DWARFS
It is extremely interesting to note that the
maximum mass of a degenerate star depends only on
fundamental constants.
The more massive a white dwarf is, the smaller it
is. This is readily understood in terms of the
characteristics of degeneracy pressure in order
to occupy higher Fermi momenta states so as to
provide the pressure the electrons less
volume. It is also interesting to use the
Lane-Emden equation in this context.. For an
adiabatic gas with an equation of state
and we write
where n is called the polytropic index, then it
is possible to derive the mass radius
relationship of the star to be
where x and q are dimensionless variables
related to the density and radius of the star.
Note
We are interested in the two solutions

1) g 5/3, n 3/2 giving for low density white
dwarfs (me is the mean mass per electron)
2) g 4/3, n 3 giving for the high density
case. This is the Chandrasekar mass limit which
is approached asymptotically as R--gt 0.

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9.5 THE COOLING TIMESCALE OF WHITE DWARFS
LIFETIME
There are no sources of energy in a white dwarf,
in fact it is just a hot cinder left over from
the evolution of low mass stars. However

• They have a small surface area so that the rate
  of energy loss is low µpR2
• The temperature is high and the particles contain
  a considerable amount of thermal energy

This energy resides in the nuclei, which are
completely ionised. The star is isothermal
because of the excellent conductivity of the
degenerate gas so that all the ions have the same
temperature wherever they are in the star. The
thermal energy of the white dwarf will be
Simplistically we can estimate the radiated
luminosity to be
If we define the lifetime as the period t1/2
between temperature T and temperature T/2
Then
Giving
For L 4 1022 J s-1, and T 104 K, t1/2 109
y
Although the above expression provides a
reasonable approximation it must be remembered
that in practice the surface layer of a white
dwarf will not be a simple degenerate gas. It
will have some other structure and an atmosphere
which provides the transition between the star
and the surrounding space. We will expect the
corresponding opacity to regulate the heat flow
etc...
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9.6 EVOLUTION OF WHITE DWARFS
We have seen that the cooling timescales of white
dwarfs are very long so that they will remain
visible for long periods of time due to radiation
from their surface. for a white dwarf of a single
mass we may expect its track on the H-R diagramme
to simply follow the cooling curve. However
increasing mass means smaller stars and therefore
less luminosity since
L
Main Sequence
Time
Mass
As time goes by the temperature drops.
T
9.7 ROTATION AND MAGNETIC FIELDS
It is also important to remember that white
dwarfs do not simply result as a battle between
degeneracy pressure and gravitation, other
physical forces will also come into play and
modify the final configuration.
Rotation Since all stars are rotating, then we
may expect some of this angular momentum to be
retained by the white dwarf. If all the angular
momentum (J Mr2w) were conserved, then the
rotation period will decrease by a factor
(RWD/RP)2 104
For a star like the Sun (P 30 days) this would
mean an equatorial velocity (RWD 7000 km) of
typically 150 km s-1. In practice white dwarfs
are found to rotate with velocities typically
10-20 km s-1. Even so this can have an effect on
the limiting mass. Evidently, a large fraction of
the angular momentum in the precursor red giant
star is transported outward prior to white dwarf
formation.
Magnetic Fields The conservation of magnetic
flux
means
For a main sequence star with a magnetic field
10-2 Tesla we would expect the white dwarf field
to be typically 102 Tesla if flux conservation
took place and will have a small effect on the
critical mass. Such values of surface magnetic
fields are indeed found to exist on white dwarfs.