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Today in Astronomy 102: neutron degeneracy
pressure and neutron stars
 The relativity of mass
 Neutron stars and the
Oppenheimer maximum
mass.
 Collapse of burned-out
stars, the formation of
neutron stars, and
supernovae.
 Pulsars are neutron stars.
 When is black-hole
The remnant of supernova 1987A in
the Large Magellanic Cloud, seen by
formation inevitable?
the Hubble Space Telescope in 1995
(NASA/STScI).
16 October 2001
Astronomy 102, Fall 2001
1
The maximum mass and the speed-of-light limit
Why is the speed of light the maximum speed that can be
reached by any moving body? (Remember, this was not part
of the two original axioms from which Einstein started…)
 Because of the relativity of mass: if a body with rest mass
m0 moves at speed V with respect to an observer, the
observer will measure a mass
m0
m
.
V2
1 2
c
for the body. (Another result of the Special Theory.)
 Note similarity to the formula for time dilation: in
particular, that the denominator approaches 0, and thus m
approaches infinity, if V approaches c.
16 October 2001
Astronomy 102, Fall 2001
2
The maximum mass and the speed-of-light limit
(continued)
So, suppose you have a body moving at very nearly the speed
of light, and you want it to exceed the speed of light. What
can you do?
 It needs to accelerate for its speed to increase.
 You need to exert a force on it in order to make it
accelerate.
 The force required is, essentially, proportional to the
product of mass and acceleration in your reference frame.
(Nonrelativistic version of this statement is Newton’s
second law: force = mass times acceleration.)
 But the mass approaches infinity as V approaches c, and
thus an infinite force is required to accelerate it further.
There’s no such thing as an infinite force, so c is the
ultimate speed limit.
16 October 2001
Astronomy 102, Fall 2001
3
Final collapse of burned-out stars
Electron degeneracy pressure can hold up a star of mass
1.4M or less against its weight, and do so indefinitely. Stellar
cores in this mass range at death become white dwarfs.
For heavier stars: gravity overwhelms electron degeneracy
pressure, and the collapse doesn’t stop with the star at planet
size.
 As the star is crushed past a circumference of 104 cm or so,
all the electrons and protons in the star are squeezed
together so closely that they rapidly combine to form
neutrons: p  e  E  n   e .
 Eventually, then, the collapse might be stopped by the
onset of neutron degeneracy pressure.
 A star whose weight is held up by neutron degeneracy
pressure is called a neutron star.
16 October 2001
Astronomy 102, Fall 2001
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Oppenheimer’s theory of neutron stars
Neutron stars were first proposed to exist, and to cause
supernovae by their formation, by Zwicky and Baade (1934).
First calculations of their sizes: Landau (1938).
 Neutron stars are analogous to white dwarfs, but the
calculations are much more difficult, since the strong
nuclear force and general relativity must be taken into
account. (For white dwarfs, special relativity suffices
because the gravity of these stars is not strong enough to
make general relativistic effects substantial.)
 As such, they may also be expected to have a maximum
mass, as white dwarfs do. For stars more massive than
this maximum, neutron degeneracy pressure will not
prevent the formation of black holes.
16 October 2001
Astronomy 102, Fall 2001
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Oppenheimer’s theory of neutron stars (cont’d)
First calculation of maximum mass: Oppenheimer and
Volkoff (1939). They got 0.7M; more recent calculations, with
improvements in the expression of the nuclear forces, give
1.5-3 M. (We will use 2M in this course.)
Albert Einstein and J.
Robert Oppenheimer at
Caltech in 1939. They
probably were, at that
moment, discussing the
prevention of black holes
by neutron-star formation.
16 October 2001
Astronomy 102, Fall 2001
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Oppenheimer’s theory of neutron stars (cont’d)
Updated calculation using 1990s-vintage inputs for the strong
nuclear force; otherwise the same as Oppenheimer and Volkoff.
Circumference (km)
(km)
Circumference
600
Maximum
mass 2M
400
200
0
0.01
16 October 2001
0.1


1
Mass
masses)
(solarM
Mass
10
Model neutron star Astronomy 102, Fall 2001
40 Eri B and Sirius B
Circumference of
Rochester (outer
loop)
7
Circumference (cm)
(cm)
Circumference
1  10
White dwarfs, neutron stars
and black hole horizons
11
WD
10
1  10
1  10
9
1  10
8
Earth
NS
BH
7
1  10
1  10
6
1  10
5
0.01
0.1
1


Z/A = 0.5 white dwarf
Rochester
10
Mass
(solar M
masses)
Mass
16 October 2001
Neutron star Astronomy 102, Fall 2001
Rochester's circumference
8
Implications of neutron stars: (Type II) supernovae
 After the electron degeneracy pressure is overpowered, and the
electrons and protons combine to form neutrons, the star is free
to collapse under its weight. Nothing can slow down this
collapse until the neutrons are close enough together for
degeneracy to set in.
• Requires confinement to space a factor of order 1000 smaller
than for electron degeneracy pressure.
 This collapse takes very little time, and the collapsing material
is moving very fast when neutron degeneracy pressure takes
over.
 A neutron-degeneracy-pressure supported core can form from
the inner part of the collapsing material.
 The outer, collapsing material that didn’t make it into the
neutron core proceeds to bounce off this core, rebounding into
the rest of the star and exploding with great violence.
16 October 2001
Astronomy 102, Fall 2001
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A supernova forms from a dead, massive star
(not drawn to scale)
2 years
Star: 6 M, 107 km
circumference
Core: 1.4 M, 105 km
circumference
16 October 2001
Core: 104 km circumference. Electrons and
protons begin combining
to form neutrons.
Astronomy 102, Fall 2001
10
A supernova forms from a dead, massive star
(continued)
1.2
seconds
Core: 104 km circumference.
Electrons and protons begin
combining to form
neutrons.
16 October 2001
Core: 70 km circumference,
neutron degeneracy pressure
sets in.
Astronomy 102, Fall 2001
11
A supernova forms from a dead, massive star
(continued)
Core: 70 km circumference,
neutron degeneracy pressure
sets in. This makes the core
very stiff.
Outside of core: still
collapsing, moving inwards
at about 1010 cm/s. (Near
light speed!) Bounces off
stiff core.
16 October 2001
Astronomy 102, Fall 2001
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A supernova forms from a dead, massive star
(continued)
A few
seconds
Core: Still 70 km
circumference, it is now
stable.
Outside of core: the
rebounding outer-star
material explodes the rest
of the star. Energy comes
from bounce, and from
gravitational energy of
core.
16 October 2001
Astronomy 102, Fall 2001
13
About
a day
A supernova
forms from a
dead, massive
star
(continued)
Neutron
star
16 October 2001
Expanding
supernova
shell. Very,
very bright for
about a month
after explosion
(can outshine
rest of galaxy!).
Astronomy 102, Fall 2001
14
Mid-lecture break
 Be on the lookout, on your email, for a notice of the
availability of Homework Set #4 on WeBWorK.
Image of Supernova
1994D in the galaxy
NGC 4526, taken a few
weeks after it was first
discovered in March
1994, by the High-z
Supernova Search
Team, with the NASA
Hubble Space
Telescope.
Supernova
16 October 2001
Astronomy 102, Fall 2001
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Supernova 1987A
Before: the Tarantula
Nebula in the Large
Magellanic Cloud, in 1984.
The star that exploded is
indicated by the white
arrow.
After: the same field, two
weeks after the supernova
went off. It was still easily
visible to the naked eye.
Images by David Malin,
Anglo-Australian
Observatory.
16 October 2001
Astronomy 102, Fall 2001
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The appearance of a supernova as time passes
Here is an animated view of the first month after the
explosion of a supernova, courtesy of UC Berkeley’s
Supernova Cosmology Project (Perlmutter, Nugent, Conley,
Nugent).
Click on image
to see movie.
16 October 2001
Astronomy 102, Fall 2001
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Neutron stars, supernovae and pulsars
Many hundreds of neutron stars are known today; they
appear mostly as pulsars: pulsating, starlike sources of radio
and visible light, discovered in 1967 by Jocelyn Bell.
 The Oppenheimer-Volkoff theory of neutron stars, and
their maximum mass, has been confirmed in all essentials.
 Theory and experiment on nuclear matter at high density
done largely through the US and USSR nuclear weapons
development programs.
 Astronomers have only been able to measure the masses
of a handful of neutron stars; they all turn out to be
around 1.4-1.5M, comfortably less than 2M.
16 October 2001
Astronomy 102, Fall 2001
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Neutron stars, supernovae and pulsars (continued)
 The Zwicky theory of supernova explosion by neutronstar formation has basically been confirmed.
 Many pulsars are seen to be associated with supernova
remnants.
• Notably the Crab pulsar and nebula, in Taurus,
remnants of Supernova 1054.
 Not all live stars in the neutron-star mass range will
become neutron stars; many eject a large fraction of their
mass in their final stages of life and make the white dwarf
cutoff.
• In their final “mass loss” stage, light from the stellar
core lights up the ejected material, producing a
planetary nebula.
16 October 2001
Astronomy 102, Fall 2001
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Pulsar
16 October 2001
Astronomy 102, Fall 2001
20
A neutron star
observed directly
On
Off
16 October 2001
The neutron star at the
center of the Crab
Nebula, the remnant
of the supernova
visible in the year
1054. It is seen as a
pulsar in these images
taken 0.03 second
apart. (This image’s
orientation is rotated
counterclockwise
about 100o from the
previous one.)
Astronomy 102, Fall 2001
21
1  10
10
1  10
9
1  10
8
1  10
7
1  10
6
1  10
5
WD
Sun
Procyon
Sirius A
11
Circumference (cm)
(cm)
Circumference
1  10
Final collapse of burned-out stars:
white dwarf, neutron star, or black hole?
NS
BH
0.01
0.1
1


Z = 0.5 white dwarf
Mass
(solar masses)
Mass
M
16 October 2001
Neutron star
Black hole
Astronomy 102, Fall 2001
10
If these stars do
not eject mass
while in their
death throes, their
fates are as
follows: the Sun
will become a
white dwarf,
Procyon will
become a neutron
star, and Sirius A
will become a
black hole.
22
Summary: status of the Schwarzschild singularity
and black holes
 Electron and neutron degeneracy pressure can prevent the
formation of black holes from dead stars, but only for
masses below about 2M.
 Stars with masses in excess of this must eject material
during their final stages of life if they are to become white
dwarfs or neutron stars. (Most do.)
 For masses larger than this, no force known to science
exists that would prevent the collapse from proceeding to
the formation of a black hole. For very heavy stars, black
hole formation is probably compulsory. (Einstein’s
objections are overruled.)
16 October 2001
Astronomy 102, Fall 2001
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