Download electron degeneracy pressure and white dwarfs

Today in Astronomy 102: electron degeneracy
pressure and white dwarfs
 Detailed description of
degeneracy pressure:
confinement by quantum
repulsion (Pauli exclusion
principle) leads to additional
resistance to compression
(wave-particle duality;
Heisenberg uncertainty
principle).
 Electron degeneracy
pressure, white dwarf stars,
and the prevention of black
holes less than 1.4M .
11 October 2001
The central star in this planetary
nebula, NGC 6543, is well on its way
to becoming a white dwarf. (Hubble
Space Telescope and Chandra X-ray
Observatory/ NASA, STScI, CfA)
Astronomy 102, Fall 2001
1
Elementary particle masses
In a reference frame in which the particle is at rest,
m  9.1094  10 28 gm (electron)
m  1.6726  10 24 gm (proton)
m  1.6750  10 24 gm (neutron)
 To reveal their wave properties, electrons need to be
confined to atomic dimensions (about 10-8 cm); thus
neutrons and protons to a space a factor of about 1836
smaller (in round numbers, about 10-11 cm), that number
being the ratio of these particles’ masses to that of the
electron.
 Photons -- particles of light -- have rest mass 0. (They also
have no rest frame; they always appear to travel at the
speed of light.)
11 October 2001
Astronomy 102, Fall 2001
2
Confinement of elementary particles
Particles like electrons, protons and neutrons can be confined
to a small space by being surrounded by other particles of the
same type, very nearby.
9 electrons sharing a twodimensional region.
36 electrons sharing the same area.
Each is confined to a smaller space.
(They
don’t even
wander
into each
other’s
cell! Why
not?)
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Confinement of elementary particles (continued)
This confinement has to do not only with the electric repulsion they
may experience; there is an additional quantum-mechanical
repulsion of electrons by each other, which sets in at very small
distances, such that wave properties are displayed.
 If the separation is small enough that this quantum repulsion is
bigger than the electric repulsion, the electrons are said to be
degenerate.
 Note for those who have taken physics or chemistry before: you
may know this quantum repulsion as the Pauli exclusion
principle.
 Protons can confine each other in a similar fashion; so can
neutrons. Because electrons are less massive, though, they
become degenerate with less confinement (a space roughly 1800
times larger, as we have seen).
 Photons do not do this; the Pauli principle does not apply to
light.
11 October 2001
Astronomy 102, Fall 2001
4
Implications of confinement arising from the wave
properties of elementary particles
If one confines an electron wave to a smaller space, its
wavelength is made shorter.
Just as is the case for light, a shorter wavelength means a
larger energy for each confined electron.
11 October 2001
Astronomy 102, Fall 2001
5
Implications of confinement arising from the wave
properties of elementary particles (cont’d)
With this increase in energy, each electron exerts itself harder
on the walls of its “cell;” this is the same as an increase in
pressure.
So:
 squeeze a lot of matter from a very small space into an
even smaller space...
 electrons are more tightly confined...
 thus the electrons have more energy and exert more
pressure against their confinement.
This extra pressure from the increase in wave energy under
very tight confinement is degeneracy pressure. (Fowler,
1926)
11 October 2001
Astronomy 102, Fall 2001
6
Implications of confinement arising from the wave
properties of elementary particles (cont’d)
 Another, equivalent, way to view the wave-particle
duality-induced extra resistance to compression is to
invoke the Heisenberg uncertainty principle:
The more precisely the position of an elementary
particle is determined along some dimension, the less
precisely its momentum (mass times velocity) along
that same direction is determined.
 In other words: confining a bunch of elementary particles
each to a very small distance (thus determining each
position precisely) leads to a very large variation in their
momenta and speeds.
 Confine to smaller space => increase speed of particles on
average => increase the force they exert on their “cell
walls” (degeneracy pressure).
11 October 2001
Astronomy 102, Fall 2001
7
Electron degeneracy pressure and the prevention of
black holes
Questions:
 Most stable stars are stable because their weight is held up
by gas pressure. Do stars exist that are held up by electron
degeneracy pressure, rather than gas pressure?
• Yes: white dwarfs.
 How are such stars made?
• From normal stars at the end of life, when they have
run out of fuel, can’t generate pressure, and collapse
under their own weight.
 Can electron degeneracy pressure balance gravity for all
compact stars, preventing them from collapsing so far that
they acquire horizons and become black holes?
• Not entirely.
11 October 2001
Astronomy 102, Fall 2001
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White Dwarf Stars
White dwarfs are stars similar in mass and temperature to
normal stars, but are much fainter and much smaller - the
size of planets. Discovered in 1862, they were a hot topic in
astronomy in the 1920s. Thousands are known today.
Sirius B
Sirius A
Chandra X-ray Observatory image (NASA/CfA)
11 October 2001
Sirius, the brightest star in
the sky, has a companion
star which is a white dwarf.
They orbit each other with a
period of about 50 years.
Sirius A is vastly brighter
than Sirius B at visible
wavelengths; the contrast is
smaller in this X-ray image.
Astronomy 102, Fall 2001
9
Sirius B: a fairly typical white dwarf
 From its measured distance from Sirius A and their orbital
period (plus Newton’s laws), we know that the mass of
Sirius B is 1.00M .
 From its observed color (blue-white), we know that its
temperature is rather high: 29,200oK, compared to 5500oK
for the Sun and 10000oK for Sirius A.
 Its luminosity is only 0.003L , much less than that of
Sirius A  13L  .
 From all of this information, astronomers can work out
3
9.8

10
km, slightly
the diameter of Sirius B; the result is
4
smaller than that of the Earth 1.3  10 km .
The mass of a star, in the size of a planet.

11 October 2001
Astronomy 102, Fall 2001

10
Mid-lecture break.
Come get your exam. Congratulations! You did very well on
it. The average score was 75.6%; the high score was 93.6%.
Number of students
25
AST 102 Exam #1
4 October 2001
20
15
10
5
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
50-55
45-50
40-45
35-40
30-35
25-30
0
Percentage score
11 October 2001
Astronomy 102, Fall 2001
11
Chandrasekhar’s theory of white dwarfs
Chandrasekhar started by combining Fowler’s theory of
degeneracy pressure with the standard theory of stellar
structure (equations expressing the balance of gravity and the
internal pressure). His initial results:
 The theory predicted sizes reasonably close to that
determined from astronomical observations of Sirius B.
 Peculiar relation between mass and size: higher-mass
degenerate stars are smaller than lower mass ones, the
opposite of what happens with normal stars.
• Reason: more mass means more pressure is required to
balance gravity, and more degeneracy pressure
requires more tightly-confined electrons (smaller star).
11 October 2001
Astronomy 102, Fall 2001
12
(cm)
Circumference
Circumference (cm)
Chandrasekhar’s first white dwarf theory
3  10
10
2  10
10
1  10
10
1997 measurements
of white dwarf
mass and
circumference by
Provencal et al.:
40 Eridani B
Sirius B
Earth’s
circumference
0
0.01
0.1
Mass  M
1

10
Mass (solar masses)
11 October 2001
Model white dwarf
40 Eri B and Astronomy
Sirius B 102, Fall 2001
Earth's circumference
13
Chandrasekhar’s theory of white dwarfs
(continued)
 For stars heavier than about a solar mass, Chandrasekhar
found from his theory that the confinement imparted so
much energy to the electrons in the
center of the star that the electron
speeds are close to the speed of light.
 Fowler’s theory of degenerate matter
did not take Einstein’s special theory
of relativity into account; therefore
Chandrasekhar had to start over, and
combine relativity and quantum
mechanics into a new theory of
relativistic degeneracy pressure.
At right: Subrahmanyan Chandrasekhar (1910–1995)
11 October 2001
Astronomy 102, Fall 2001
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Chandrasekhar’s theory of white dwarfs
(continued)
Recall that electrons, like everything else, can’t move faster
than light.
 The more massive the degenerate star, the closer the
electron speeds get to the speed of light.
 The closer the speeds get to c, the harder it is to accelerate
the electrons further.
 Thus, the electron degeneracy pressure doesn’t keep
increasing as much with tighter confinement: the electrons
reach a point where they cannot move any faster. There is
a maximum to the electron degeneracy pressure, and a
corresponding maximum weight that degeneracy
pressure can support.
 If the weight cannot be supported by electron degeneracy
pressure, the degenerate star will collapse to smaller sizes.
11 October 2001
Astronomy 102, Fall 2001
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Chandrasekhar’s relativistic white dwarf theory
(cm)
Circumference
Circumference (cm)
3  10
10
1997 measurements
of white dwarf
mass and
circumference by
Provencal et al.:
40 Eridani B
Sirius B
Maximum
mass, 1.4 M
2  10
10
1  10
10
Earth’s
circumference
0
0.01
0.1
Mass  M
1

10
Mass (solar masses)
11 October 2001
Relativistic
Non-relativistic
Astronomy 102, Fall 2001
WDs in visual binaries
16
How to make a white dwarf star
Start with a normal star like the Sun. Fusion of protons into
helium in the star’s center generates heat and pressure that
can support the weight of the star. The Sun was mostly made
of hydrogen (=1 proton + 1 electron) when it was born, and
started with enough hydrogen to last like this for about 15
billion years.
11 October 2001
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How to make a white dwarf star (continued)
When it begins to run out of hydrogen in its center, not
enough heat and pressure are generated to balance the star’s
weight, so the core of the star gradually begins to collapse.
11 October 2001
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How to make a white dwarf star (continued)
As the core collapses it gets hotter, though no extra heat has
been generated, just because it compresses. It gets so hot that
light from the core causes the outer parts of the star to
expand and get less dense, whereupon the star looks cooler
from the outside. The star is becoming a red giant.
11 October 2001
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How to make a white dwarf star (continued)
Eventually the core gets so hot that it is possible for helium to
fuse into carbon and oxygen. Extra heat and pressure are
once again generated and the core stops collapsing; it is stable
until the helium runs out, which takes a few million years.
The outer parts of the star aren’t very stable, though.
Matter
flowing away
from the star
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How to make a white dwarf star (continued)
Eventually the core is all carbon and oxygen, no additional
heat and gas pressure is generated, and the core begins
collapsing again. This time the density is so large – the
electrons so close together – that electron degeneracy
pressure begins to increase significantly as the collapse
proceeds.
Matter
flowing away
from the star
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How to make a white dwarf star (continued)
Electron degeneracy pressure eventually brings the collapse
of the core to a halt, before it gets hot enough to fuse carbon
and oxygen into magnesium and silicon. The unstable outer
parts of the star fall apart altogether; they are ejected and
ionized by light from the core, producing a planetary nebula.
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How to make a white dwarf star (continued)
The planetary nebula’s material expands away from the scene
in a few thousand years, leaving behind the hot, former core
of the star, now about the size of Earth. Its weight supported
against further collapse by electron degeneracy pressure, it
will do nothing but sit there and cool off, for eternity.
11 October 2001
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How to make a white dwarf star (continued)
When brand new, this degenerate star is quite hot and looks
white (like Sirius B) or even blue in color, leading to the name
white dwarf. The oldest “white dwarfs” in our galaxy, age
about 12 billion years, have had enough time to cool down to
temperatures in the few thousands of degrees, and thus look
red. (Despite this they are still called white dwarfs.)
11 October 2001
Astronomy 102, Fall 2001
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Back to Chandrasekhar’s theory of white dwarfs
 Important result of the theory: maximum mass for white
dwarfs, which turns out to be 1.4 M. Electron degeneracy
pressure cannot hold up a heavier mass.
 Implication: for stars with core mass less than 1.4 M, core
collapse is stopped by electron degeneracy pressure
before the horizon size is reached.
 However, for stars with cores more massive than 1.4 M,
the weight of the star overwhelms electron degeneracy
pressure, and the collapse can keep going. What can stop
heavier stars from collapsing all the way to become
black holes after they burn out?
11 October 2001
Astronomy 102, Fall 2001
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Final collapse of burned-out stars:
white dwarf or black hole?
(cm)
Circumference
Circumference (cm)
3  10
10
2  10
10
1  10
10
Sun, if it
doesn’t
lose any
mass
Small
star
0
0.01
0.1

1

Mass
(solar M
masses)
Mass
Relativistic
11 October 2001
The Sun and smaller
stars will become
Sirius
white dwarfs after
A
they burn out and,
lacking the gas
pressure generated
by their nuclear heat
source, collapse
under their weight.
?? What about Sirius A,
which weighs a good
10 deal more than the
limit?
Astronomy 102, Fall 2001
26
Chandrasekhar’s theory of white dwarfs
(continued)
Experimental confirmation of the theory: today thousands of white
dwarf stars are known. Sure enough, all stellar masses under 1.4 M
are represented, but no white dwarf heavier than this has ever been
found.
For this work, Chandrasekhar was awarded the 1983 Nobel Prize in
Physics. The NASA Chandra X-ray Observatory (CXO) is named in his
honor.
Seven white dwarfs
(circled) in a small section
of the globular cluster M4
(By H. Richer and M. Bolte.
Left: Kitt Peak National
Observatory 36”; right:
Hubble Space Telescope).
11 October 2001
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