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White Star, Black Hole
Frank Hsia-San Shu
Taichung First Senior High School
26 February 2005
1905 = Einstein’s Miracle Year
Surface & Interior of Sun
Pressure at Center of Sun
• Pressure at center of Sun = weight of column of
material per unit area above center

GM Sun M Sun
14
1  2
Mean pressure ~


4

10
kg
m
s .
2
2
RSun RSun
• Central pressure 50 times larger: Pc  2 1016 kg m -1 s -2 .
Visualizing Central Pressure of Sun
Solar Plasma
electrons
positive ions (nuclei)
• A lot of space between individual particles, unlike the case if unionized atoms
were pressed up against each other at mean density of Sun, where most of
volume is occupied by the fuzzy electronic shells of the atoms.
• Electrons and nuclei therefore behave as freely moving particles, i.e., as an
ideal gas satisfying
P = nkT ,
where P = pressure, n = number density, k = Boltzmann’s constant, T =
temperature on Kelvin scale.
• For the central values, P = 2  1016 kg m1 s2 ,and
n = 1 1032 m3 ,we get T = 1.5  107 K at the center of the Sun.
3-D Random Walk of
Photons Out of Sun
Radiative Transfer in the Sun
X-rays
free flight
optical photons
random walk
photosphere
Slow photon diffusion from interior to surface is
what regulates energy leakage to the rate
aT (4RSun / 3)
26
LSun ~
 4 10 watts.
2
( RSun / ) ( / c)
4
3
Source of Energy Output of Sun
•
•
•
•
Every second, the Sun radiates more energy than lies under the sands of all Arabia.
When will the Sun (and the other stars) run out of energy?
If the Sun shone by burning fossil fuel, and it were entirely made of coal or oil, it
could last only about 10 thousand years. This is far too short – shorter than how long
humans have been on Earth (7 million years), much shorter than how long life has
existed on Earth (3.5 billion years).
More powerful energy sources:
– Chemical reaction with greatest release of energy per kg:
H+H = H 2 .
H
H
H2
For Sun, this reaction could last about 25 thousand years, still far too short.
– Nuclear reaction:
p
2
D + other things
p  p  H  D.
p
For Sun, this reaction (and others like it) could sustain Sun for 10 billion years. Long
enough.
Nuclear Energy
•
•
•
•
Net reaction: 4H→He (fusion reaction).
m  4mH  mHe  0.03mH .
Mass deficit:
E  mc 2  0.007(4mH c 2 ).
Energy release:
Implication: If Sun were 100% H, fusion reaction of H into He (“hydrogen
. 100 billion years
burning”) would provide store of energy sufficient to Llast
at present rate of solar usage:
• Since the Sun is actually only 70% H and since only inner 13% of mass
(core) is hot enough to “burn” H during the so-called main-sequence stage,
the main-sequence lifetime of the Sun = 9 billion years (see Problem Set 4).
• The Sun, with an age of 4.5 billion years, is half way through its mainsequence life. What it will turn into after another 4.5 billion years is one of
the more interesting stories of astronomy.
Need for High Temperatures
Stability of Solar Thermonuclear
Reactor
core
core
Lc
L
L  Lc
Sun expands; core cools; Lc  L.
L
Lc
Lc  L
Sun contracts; core heats; Lc  L.
At present (in equilibriu m), the Sun has a size RSun
which generates thermonuc lear power Lc in the core
that just offsets the loss L from the surface : Lc  L.
Summary of Interior of Sun
From Main Sequence to Subgiant
Horizontal Branch Star
Planetary Nebula to White Dwarf
Some Planetary Nebulae
Motion of Sirius A & B
Theory of White Dwarfs
• R. H. Fowler (1889-1944) proposes white dwarfs, made of element of
mean atomic number Z and atomic weight A, are supported against selfgravity by electron degeneracy pressure → mass-radius relationship such
that R↓as M↑:
h2
Z
R  0.114

5/3 
Gme m p  A 
5/3
M 1/ 3 .
• For M =1 M Sun , R is somewhat smaller than the radius of the Earth.
• S. Chandrasekhar (1910-1995) includes effect of special relativity, finds
R → 0 as M → finite value (Chandrasekhar’s limit).
• Controversy between Eddington and Chandrasekhar. Chandrasekhar
leaves England for United States, settles in University of Chicago. Gave
moving tribute in 1982 to commemorate 100th anniversary of
Eddington’s birth. Awarded Nobel Prize in Physics in 1984.
Difference Between Chocolate
Cakes and White Dwarfs
0.4 kg
0.8 kg
0.4 M
0.8 M
• As M↑, v→
e c, and P increases less quickly when
density ↑. Radius → 0 when M → M Chwhere
M Ch
Z
 0.20 
 A
2
 Gmp

 c

2




3 / 2
m p  1.4 M Sun
Mass-Radius Relationship
of Cold Bodies
R (km)
Jupiter
105
10
White
Dwarfs
R  M 1/ 3
4
10
planets
moons
3
102
M Ch
R  M 1/ 3
asteroids
10
1
15
10
10
20
10
25
1030
M (kg)
Jupiter Is Close to Being
Largest Cold Object in Universe
Similarity Between Cooling Ember
and White Dwarf
Cooling
Ember
White
Dwarf
Radiates energy,
becomes cooler.
Radiates energy,
becomes cooler.
Path to Yet More Compact Objects
• If M were to exceed 1.4 solar masses, would R of a WD
really shrink to zero?
• As ρrises, electron degeneracy-energy becomes greater and
eventually can make up the mass difference between a
proton and a neutron.
• Electrons get “squashed’’ inside protons, and ions of a WD
star become converted to neutrons.
• As ρrises even more, neutrons become degenerate and are
able to exert pressure at zero T.
• The balance of neutron degeneracy pressure and selfgravity gives rise to a new state of possible equilibrium, a
neutron star.
Eta Carina – A Pre-Supernova
Supernova 1987A and its Precursor
Star in the LMC
Crab Nebula = SN1054
Crab Pulsar
Pulsars Blink Like
Rotating Lighthouses
Neutron Stars
• On average, neutrons stars are smaller than white dwarfs
by a factor mn / me  1840. Actually, by about 103.
• Therefore, mean density is higher by factor of 109 ;
i.e.,  ≈ 1018 kg m-3 instead of 109 kg m-3 .
• At such densities (nuclear densities), matter is very
incompressible, with the pressure rising rapidly as the
density increases.
• Rate of rise of pressure with density = square of speed of
sound. Speed of sound cannot exceed speed of light →
limit to how much matter, even nuclear matter, can resist
gravity → M ns  3M .
Cosmic Scale
Atoms Are Mostly Empty Space
Stellar Processing Determines Relative
Abundances of the Elements
Summary of Dying Stars
• Red Supergiant:
– Dust grains form in cool outer
atmosphere and expelled by
radiation pressure.
input of
small ISM
solids
• Planetary Nebula:
– Shell of gas and dust illuminated
and excited by hot central star,
which is exposed core of red
supergiant destined to become WD.
glowing
shell of
ionized
gas
• Supernova Remnant:
– Ejecta of processed matter from
high-mass star which impacts with ISM
and forms supernova remnant that is a
source of cosmic rays.
neutron
star or
black
hole
Idea of Black Hole
•
Consider the escape speed from the surface of a body of mass M and radius R:
m
ve  2GM / R
M
R
(See Problem Set 3.)
•
For a neutron star of mass M  1.4 M and radius R  104 m, we get
ve  1.9  108 m/s,
which is 63% the speed of light c!
The above calculation motivates us to ask a question first asked by Pierre
Laplace (1749-1827): For given M, what R would yield an escape speed equal
to c? Answer:
2GM
•
R
•
c2
.
This is the correct answer, but the reasoning is incorrect on two counts.
– First, Newtonian mechanics giving the equation for a marginally bound orbit,
E
1
GMm
mv 2 
0
2
R
for
v  ve
does not apply to photons of zero rest mass m and speed c.
– Second, Newtonian concepts about gravity do not apply to a situation where the
gravitational field of a body is so strong that even light finds it difficult to escape its
clutches.
•
Two mistakes combine to give the right looking answer! Correct derivation
based on Einstein’s theory of general relativity first given by Karl Schwarzschild
(1873-1916).
Event Horizon of Black Hole
Two Views of Gravitation
• Newton:
– Gravity is a force which pulls on all things with mass.
– Mass acts as the source that generates the force of
gravitation.
• Einstein:
– There is no such thing as the force of gravity.
Gravitation arises when spacetime has curvature;
indeed gravitation is spacetime curvature.
– Mass-energy and stress (e.g., pressure) act as the
sources that generate spacetime curvature.
Basic Postulates of
General Relativity
Bending of Light
Gravitational Bending of Light
Ant Analogy for Bending of Light
Lensing of Background Galaxies by
Galaxy Cluster
Observed Lensing of Background
Galaxies by Galaxy Cluster
Flight Circles about a Black Hole
•
•
•
•
•
•
•
•
•
RSch = 9 km for a 3 solar-mass BH.
Start with flight circle of circumference = 2π·90 km.
You deduce you’re 90 km radially from BH. Don’t jump to conclusions.
Lower yourself inward radially by 32 km.
Fly around; measure circumference = 2π·60 km. (?)
Lower yourself inward radially by another 33.75 km.
Fly around; measure circumference = 2π·30 km. (??)
Lower yourself inward radially by another 19.8 km.
You compute 32+33.75+19.8 = 85.55. Subtracted from 90, won’t this
bring you inside RSch = 9 km? (!)
• Don’t worry; lower yourself by 19.8 km as we requested.
• Fly around; measure circumference = 2π·15 km. Whew!
• All flight circles are in a single plane. Clearly, presence of a 3 solarmass point-mass at center has warped our usual (Euclidean) sense of
geometry.
Black Holes Are
Punctures in Fabric of Spacetime
Behavior Near Event Horizon
Reversal of Space and Time Across
Event Horizon of a Black Hole
• Outside event horizon, by exerting enough force on the rope, I can
hold your position stationary with respect to center of BH. But there is
nothing I can do to stop the forward progression of time for you (or,
for that matter, for myself).
• As I lower you toward event horizon, your perception of stars begin to
change and blur. Are you getting a sinking feeling?
• When you get close enough to the event horizon, no rope – no matter
how strong – can stand the strain. It will snap and break, and you will
begin an inexorable fall toward the black hole.
• For you, it takes only a few milliseconds for you to reach and cross
the event horizon of the BH. But for me, it seems that you formally
take an eternity to reach the event horizon.
• In other words, as you draw near to the event horizon, there is nothing
I can do to stop your forward progression through space. But for me,
time seems to have stopped moving for you! In some sense, for me on
the outside, time and space seem to reverse roles as you approach the
event horizon. When you cross it, you will reach a different space and
time than the one that we on the outside occupy. In a certain sense,
BHs may be portals to other spacetimes and other universes!
Speculation 1-- Wormholes:
Shortcuts through Space?
y
x
Speculation 2 -- Wormholes:
Machines through Time?
x
t
Detectability of Black Holes
• Since black holes allow nothing to escape from their
“surfaces,” not even photons, how can we detect them or
verify their existence?
• One technique examines the radiation from matter drawn
from a closely orbiting star before this material falls into
the black hole in binary X-ray sources. Radiation is
associated with an accretion disk surrounding the black
hole.
• Famous example -- Cygnus X-1: Inference for compact
(unseen) companion in excess of 3 solar masses orbiting a
normal star.
• Data accumulated over the past several years show the
somewhat surprising result that most stellar-mass black
holes found in this manner have a fairly narrow range of
masses of around 7 solar masses.
Cygnus X-1 =
Interacting Binary Star
Final Stellar States in War between
Gravity &Thermodynamics
White Dwarf
Truce mediated by quantum
behavior of electrons
Neutron Star
Truce mediated by quantum
behavior of neutrons
Black Hole
Final victory for gravity over
thermodynamics
Nothing Left
Final victory of thermodynamics
over gravity (only true equilibrium?)
Speculation 3: Evaporation of BHs?
• Currently popular theoretical view: Proton is a long-lived, but ultimately
unstable particle, which will decay into positron plus other particles in
some 1032years or so.
• If this speculation is correct, then ultimately all the protons (and
neutrons, which will decay into protons if the latter disappear) in WDs
and NSs will turn into positrons that will annihilate with the electrons
present in these compact objects. The resulting photons (and other
massless particles) will then escape at the speed of light and disappear
into the universe.
• Even BHs may ultimately evaporate completely away if Stephen
Hawking (1943-) is correct. According to Hawking, (nonrotating) BHs
of a mass M have a nonzero surface temperature T given by the formula:
c 3
kT 
.
8GM
True End State of Universe?
• Today, BHs preferentially pull in matter and become more massive.
But eventually (and this may take a long, long time), when all the
neighboring matter is gone and the night sky has become even darker
than it is now, BHs will begin to slowly lose mass and energy back into
the universe.
• Since T is inversely proportional to M, this process will accelerate with
time as M becomes smaller and smaller, until the BH disappears in a
final outburst of light. Need superstring theory to understand this last
event.
• The whole story of stellar evolution is to turn normal stars into more
and more compact objects, ultimately producing WDs, NSs, or BHs.
However, if current theoretical ideas are correct that most forms of
matter are ultimately unstable toward decay into lighter particles, with
even black holes not immune from the disease, then the end game of
the evolution of the universe will see a redispersal of the
gravitationally bound “final states” of stellar evolution.
• Is the grand scheme of the cosmos then merely a mechanism to turn
matter into energy? This would be an ironic end if current ideas are
also correct, that all matter in the universe originally began in the big
bang as pure energy.
Thank you, everyone!
Happy New Year!