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Frank Verbunt
White dwarfs, black holes
and dark matter
Quantum mechanics and
General Relativity
Wageningen 26 March 2015
discovery white dwarfs
quantum-mechanics:
pressure
gravity and pressure:
maximum mass
discovery neutron stars
general relativity theory
general relativity tests
black holes
the Universe
dark matter & energy
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
1 / 37
The discovery of white dwarfs: Sirius B
Friedrich Wilhelm
Bessel 1844
Frank Verbunt ( Dept. Astronomy Nijmegen)
The motion of
Sirius
White dwarfs, black holes, dark matter
1844
Sirius A and B
March 26, 2015
2 / 37
Why are white dwarfs different?
From light to density
the amount of light emitted
by a star increases with the
area of its surface and with
its surface temperature
Sirius B has (almost) the
same temperature as
Sirius A
but emits only 0.0001 as
much light
hence: its
I
I
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surface area is 0.0001,
radius 0.01,
density 106
More accurately
Sirius A: a normal star
massa 2 × that of Sun
radius 1,7 × that of Sun
density 0,4 × that of Sun
(density Sun: 1.4 × water)
Sirius B: white dwarf
mass 1,0 × that of Sun
radius is 0,0084 × that of
Sun
density 1.7 × 106 that of Sun
times that of Sirius A
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
3 / 37
The simile of the Swiss village
Mayor Heisenberg
Policeman Pauli
Average price
Low-season: free
choice of rooms.
High-season: many
rooms full; average
price high
hotel rooms have
maximum 2 persons in
minimum size
each room
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
4 / 37
Quantum-mechanics: ‘rooms for velocity’
Chandrasekhar
application to white
dwarfs
Frank Verbunt ( Dept. Astronomy Nijmegen)
Heisenberg uncertainty relation, Pauli
exclusion principle
p is momentum = mass × velocity
h is side of cube ( Planck constant)
electron: 0,00072 m/s proton: 0,000 000 4 m/s
White dwarfs, black holes, dark matter
March 26, 2015
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Velocity and pressure; stellar structure
Pressure
Equilibrium in star
a particle collides: force:
mass × velocity =
momentum
gravity pulls star in
many particles: pressure
ideal gas: higher pressure
when
star settles at equilibrium
I
I
temperature higher: higher
velocities
density higher: more
particles
at very high density
presseure depends only on
density
pressure-difference pushes
out
ordinary star: pressure of
ideal gas
without source of energy star
cools and constracts
white dwarf: degenerate
pressure
size unaffected by cooling
‘degenerate pressure’
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
6 / 37
Maximum mass of white dwarf
White dwarfs of increasing mass
a low-mass white dwarf is fully
non-relativistically degenerate
(v c)
a heavier white dwarf
I
I
is smaller
has a relativistically
degenerate center (v ' c)
the heavier the white dwarf, the
(fractionally) larger the
relativistic center
Maximum mass of white dwarf
when a white dwarf mass
exceeds Mch
I
I
central pressure increases
gravity increases faster
no equilibrium between
pressure difference and
gravity possible
gravity wins: the white dwarf
collapses
hence Mch is maximum mass
until at Mch the whole dwarf is
relativistically degenerate
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
7 / 37
From neutron to the prediction of neutron stars
1932 Chadwick discovers
neutron
1934 Baade & Zwicky predict
neutron star
radius compared to that of
white dwarf scales as mass
of electron to that of neutron:
me
1
Rns
=
× 25/3 '
Rwd
mn
580
14 mile diameter
too small to detect . . .
formation = supernova
LA Times 19 jan 1934 ⇒
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
8 / 37
Crab pulsar in visible light: 30 rotations/s
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
9 / 37
Philosopher George Berkeley criticises Newton
Berkeley 1685-1753
in an empty universe rotation
and distance are not defined:
hence they do not exist. ‘no
absolute space’
example: bucket of water
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in space with stars: water
rises along edge in
rotating bucket
in empty space: water
does not rise
(Newton: water rises in
both cases: ‘absolute
space’)
picture of merry-go-round:
‘Mach principle’ inspiration
for Einstein
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
10 / 37
SRT: Special Relativity - Theory: velocity of light
low velocities
high velocities
velocity ball w.r.t. cyclist:
20 + 30 = 50 km/h
velocity light w.r.t. cyclist:
200 000 + 300 000 , 500 000 km/s
but 300 000 km/s! how can this be?
velocity = distance / time:
something wrong with distance
and/or time
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
11 / 37
General Relativity - Theory: heavy and inert mass
a heavy mass is strongly attracted by gravity, e.g. from Earth; a low
mass less so
it takes more effort to bring a heavy mass into motion than a light
mass: the inert mass of a heavy object is bigger
as a result a low-mass object falls equally fast as a heavy one: heavy
mass = inert mass. Accidentally? Newton: yes. Einstein: no!
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
12 / 37
GRT-: simple axioms, complicated mathematics
Axioms
General relativity
SRT: velocity of light the
same for all observers at
constant velocities
Complicated mathematical
equation can only be solved in
two very simple cases
ART: velocity of light the
same for all observers at
accelerating velocities
1. spherical mass in
otherwise empty universe
(classical tests of GRT)
acceleration = gravity
Schwarzschild solved this
case analytically!
Consequence: black hole
Einstein 1915
re-writes physics
horrendously complicated
mathematics
needs help from Hilbert. . .
Frank Verbunt ( Dept. Astronomy Nijmegen)
2. homogeneous and
isotropic universe
Nowadays: fastest computers
solve other problems numerically
(e.g. two masses)
White dwarfs, black holes, dark matter
March 26, 2015
13 / 37
GRT: the field equations of Einstein 1915
distribution of mass = curvature of space-time
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
14 / 37
GRT-: differences with Newton’s gravity
Distances in GRT
Consequences: 1
in empty space the
circumference of a circle is
O = 2πr
long axis of ellipse advances
(Einstein 1915: Mercury)
near a massa the circumference
of a circle is O < 2πr
to draw this we must draw a
hollow
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
15 / 37
GRT: the classical tests
Consequences: 2
Consequences: 3
light follows curved orbit near
mass (Eddington 1919: solar
eclipse)
distances in direction of
mass are longer (Shapiro
1964: Venus 0.0002 s)
this made Einstein famous
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
16 / 37
GRT: binary neutron stars and stronger tests
neutron-star binary: orbital period 8 hr
rotation long axis ellipse: as much in one day as Mercury in a century!
orbit shrinks due to emission of gravitational radiation
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
17 / 37
GRT: binary neutron stars and stronger tests
2 effects determine masses of neutron stars; 3rd effect tests GRT
PSR1913+16
Frank Verbunt ( Dept. Astronomy Nijmegen)
PSR J0737−3039
White dwarfs, black holes, dark matter
March 26, 2015
18 / 37
GRT: binary neutron stars and stronger tests
The accurracy of the pulsar as a clock allows unprecented accurracy in
determining the orbit (period, eccentricity, masses of the neutron stars).
valid on
pulse-period
derivative
2nd derivative
orbital period
derivative
eccentricity
periastron-motion
mass pulsar
mass companion
P
P˙
¨|
|P
Pb
P˙ b
e
ω˙
M1
M2
Frank Verbunt ( Dept. Astronomy Nijmegen)
PSR1913+16
6 July 1984
0.059030002593481(7)s
8.62713(8) × 10−18 ss−1
< 2 × 10−29 s s−1
0.322997448930(4)d
−2.4184(9) × 10−12
0.6171338(4)
4.226595(5)◦ yr−1
1.4414(2)M
1.3867(2)M
PSR J0737−3039,
30 May 2004
0.022699378599624(1)s
1.75993(5) × 10−18 ss−1
0.10225156248(5)d
−1.252(17) × 10−12
0.0877775(9)
16.89947(68)◦ /yr
1.3381(7)M
1.2489(7)M
White dwarfs, black holes, dark matter
March 26, 2015
19 / 37
GRT: strong effects and the black hole
deeper depression for larger mass
when the ratio radous / mass becomes too small, the bottom drops
out: the distance to the edge (the ‘horizon’) is infinite
we call this a black hole
to understand this we consider first the Newtonian dark star
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
20 / 37
Gravitation according to Newton: the dark stsr
Michel 1784, Laplace 1795
the escape velocity vesc
depends on ratio
mass/radius
r
vesc =
Properties
a particle of light that moves up
is decelerated and falls back:
light cannot reach us
the star for us is dark:
astre occlu
2GM
R
a dark star is stable
vesc for Sun: 440 km/s
when we compress the Sun
to radius 3 km vesc is equal to
the velocity of light
Frank Verbunt ( Dept. Astronomy Nijmegen)
one can travel there . . . and
return in a finite time
clocks near the surface tick at
the same rate as clocks far away
White dwarfs, black holes, dark matter
March 26, 2015
21 / 37
Gravity according to Einstein: the black hole
classical tests
light is bent already outside the
horizon
nothing can pass the horizon
from inside, not even light
a dark star inescapably
collapses
one can travel there . . . but then
cannot return
the radius Rs of the horizon
equals the radius of the dark
star
Frank Verbunt ( Dept. Astronomy Nijmegen)
the traveller sees his clock tick
normally but for a faraway
observer the clocks slow down
near the horizon and the
traveller hovers just outside it
White dwarfs, black holes, dark matter
March 26, 2015
22 / 37
Gravity according to Einstein: the black hole
classical tests
light is bent already outside the
horizon
nothing can pass the horizon
from inside, not even light
a dark star inescapably
collapses
one can travel there . . . but then
cannot return
the traveller sees his clock tick
normally but for a faraway
observer the clocks slow down
near the horizon and the
traveller hovers just outside it
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
22 / 37
GRT: the universe
The Universe
‘Copernican principle’: the Sun
has no special position in the
Universe
‘extended Copernican principle’:
there are no special positions in
the Universe
homogeneous: mass-density
the same everywhere
isotropic: the same in all
directions
complicated equations
becomes very simple
Einstein 1915
Solution for homogeneous
Universe
expands or shrinks
but stars do not move from
us
the Universe is static
mathematical trick:
‘cosmological constant’
1921: Friedman: static, but
not stable
Einstein ignores this
2nd-order differential
equation: 2 constants
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
23 / 37
Nebulae beyond the stars: M81 en M82
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
24 / 37
Spiral galaxies
en
els
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
25 / 37
Vesto Slipher discovers that galaxies move away from us
first measurement 1912: Andromeda nebula moves towards us:
vr = −300 km/s. Most other galaxies move away: V.M. Slipher , 1917,
Proceedings of the American Philosophical Society, vol. 56, p.403-409
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
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The expanding Universe
Einstein & Lemaître ±1933
Lemaître
1925 Einstein static but not
stable
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the Universe expands
according to GRT
and measured by Slipher
velocity proportional to
distance
Einstein not impressed. . .
discovery later claimed by
Hubble
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
27 / 37
The expanding Universe: the Big Bang
Lemaître
George Gamow 1904-1968
the Universe expands
was smaller in the past
its temperature was higher
its density was higher
The Universe started as an
exploding primordial atom
the inital Universe was hot
and dense
therefore: nuclear fusion; all
elements made in first 3
minutes
Alpher, Bethe, Gamow 1948
as the Universe expands it
becomes transparent
radiation and matter cool
independently
radiation now 5 á 50 K
(in fact: 3 K)
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
28 / 37
Cluster Abell 1185
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
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Dark matter in clusters of galaxies
Zwicky 1933/37: Coma cluster
velocities v ' 700 km/s
size R = 0.8 Mpc
Coma cluster of galaxies
mass in galaxies too small to
confine cluster: most mass
invisible
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
30 / 37
Dark matter in clusters of galaxies
Confirmation 1
Cluster of galaxies
curved stripes in galaxy
clusters
gravity lense: mass in cluster
deforms and amplifies image
of faraway galaxy
from this derive mass in
cluster
agrees with kinematic mass
Confirmation 2
hot X-ray emitting gas
confinement requires large
mass
agrees with kinematic mass
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
31 / 37
3 K background radiation
Discovery
COBE / WMAP
from all directions
3 K + dipole=direction own
motion
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
32 / 37
3 K background radiation: fluctuations: 9 yr WMAP
5
variation across sky <
∼ 1 : 10 : how does one side know about the other?
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
33 / 37
3 K background radiation: fluctuations: 4 yr Planck
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
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Dark matter and dark energy
Het standaard-model
Dark matter fraction
solar system: negligible
10 pc (globular cluster):
negligible
103 pc (solar environment):
30 %
104 pc (galaxy): 64-84 %
107 pc (cluster of galaxies):
70 %
Dark matter nature
not baryonic
primordial black holes (?)
new type of particles (?)
Frank Verbunt ( Dept. Astronomy Nijmegen)
voor elk deeltje een
supersymmetrisch deeltje
White dwarfs, black holes, dark matter
March 26, 2015
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Dark energy
Discovery of dark energy
white dwarf collapse starts
fusion
supernova 1994D
leading to explosion =
supernova Ia
all equally bright
brightness known: distance
known
faraway supernovae indicate
push: dark energy
also indicated by details of
cosmic background radiation
Frank Verbunt ( Dept. Astronomy Nijmegen)
White dwarfs, black holes, dark matter
March 26, 2015
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Ordinary matter, dark matter and dark energy
WMAP and Planck
at largest scales: 109 pc
dark energy 72.1±2.5%
dark matter 23.3±2.3%
baryons 4.6±0.2%
Frank Verbunt ( Dept. Astronomy Nijmegen)
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March 26, 2015
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