Slide 1

Your next test will be November 14

Review session: Friday, Nov. 10, 5:30-7:30 pm

Room 205 ENPH-Teaching wing

Slide 2

Summary of Post Main-Sequence Evolution of Stars

M > 8 Msun

M < 4 Msun

Evolution of 4 - 8 Msun stars is still uncertain.

Fusion stops at formation of C,O core.

Mass loss in stellar winds may reduce them all to < 4 Msun stars.

Red dwarfs: He burning never ignites

M < 0.4 Msun

Supernova

Fusion proceeds; formation of Fe core.

Slide 3

Evidence for Stellar Evolution: Star Clusters

Stars in a star cluster all have approximately the same age!

More massive stars evolve more quickly than less massive ones.

If you put all the stars of a star cluster on a HR diagram, the most massive stars

(upper left) will be missing!

Slide 4

HR Diagram of a Star Cluster

Slide 5

Example: HR diagram of the star cluster M 55

High-mass stars evolved onto the

giant branch

Low-mass stars still on the main

sequence

Turn-off point

Slide 6

Estimating the Age of a Cluster

The lower on the MS

the turn-off point, the older the cluster.

5.2

1~M

T

Age of a cluster = lifetime of stars on the turnoff point

5.2

M

M

T

T sun

sun

Slide 7

Fate of massive stars

Slide 8

When particle velocities become relativistic, they cannot be increased anymore;

Matter becomes “softer”: P ~ 4/3

The core gives in to gravity and collapses

Pressure of degenerate gas cannot be increased indefinitely

Upper limit on white dwarf masses

Slide 9

Chandrasekhar limit: 1.4 Msun

This is because gravitational pressure increases with mass. Electron pressure should also increase, and the only way to do it is to compress the star.

For a given mass, find the radius at which equilibrium is reached

Slide 10

Death of massive stars

What happens when the core is heavier than the Chandrasekhar

limit?

Slide 11

Slide 12

Reactions proceed faster and faster, until …

Slide 13 Fig. 7-16, p. 126

Slide 14

The iron core of a giant star cannot sustain the pressure of gravity. It collapses inward in less than a second.

The shock wave blows away outer layers of a star, creating a SUPERNOVA EXPLOSION!

Slide 15

Supernova in Centaurus A

Precise mechanism – still unknown

Slide 16

Still many problems with SN scenario

• Outward shock seems to get stalled by inward falling layers

• Neutrinos carry away over 90% of the SN energy and deposit it to outer layers. However, computer models do not “explode”.

Turbulent convection seems to help the outward shock to break through the infalling layers

Slide 17

Type I and II SupernovaeCore collapse of a massive star:

Type II Supernova

If an accreting White Dwarf exceeds the Chandrasekhar mass limit, it collapses,

triggering a Type Ia Supernova.

Type I: No hydrogen lines in the spectrum

Type II: Hydrogen lines in the spectrum

Energy release due to radioactive decay of 56Ni and 56Co

Slide 18

Type Ia supernova: why there is no hydrogen?

White dwarf gains mass beyond 1.4 Msun

It collapses, and violent C-O fusion begins in the coreCarbon deflagration: a white dwarf explodes

Slide 19

Type Ib supernova: why there is no hydrogen?

Main-sequence blue giant that has lost its outer hydrogen-rich envelope

Its remnant develops an iron core and collapses

Slide 20

Chemical elements in the Universe How did the elements form?

• The Big Bang Elements Formed: H, He, (Li)

• Stellar Nucleosynthesis Elements Formed: Almost all other elements

• Supernovae Explosion Elements Formed: Heaviest elements

Slide 21

Supernova nucleosynthesis

• All elements up to A ~ 250 are synthesized!

• S-processes: “slow” synthesis of elements up to iron

• R-processes (r = rapid): rapid neutron capture

Slide 22

Atomic nucleus

n + (A,Z) ===> (A+1,Z) ===> (A+1,Z+1) + e + neutrino n + (A+1,Z+1) ===> (A+2,Z+1) ===> (A+2,Z+2) + e + neutrino And so on

Slide 23

Cosmic abundance

Earth

Slide 24

Practically all heavy elements are due to stars

We owe our very existence to SN explosions

Slide 25

Neutron Stars

The central core will collapse into a compact object of ~ a few Msun.

A supernova explosion of a M > 8 Msun star blows away its outer layers.

Slide 26

Formation of Neutron Stars

Compact objects more massive than the Chandrasekhar Limit (1.4 Msun) collapse further.

Pressure becomes so high that electrons and protons combine to form stable neutrons throughout the object:

p + e- n + e

Neutron Star

neutronization

Slide 27

Neutron StarPressure of degenerate neutrons balances gravity

?? Strange matter? Quark-gluon plasma??

Neutron stars have been theoretically predicted in 30s.Landau, Oppenheimer, Zwicky, Baade

Slide 28

Fritz Zwicky 1898-1974

"spherical bastards”

Walter Baade 1893-1960

Slide 29

Predictions and discoveries

• Explained supernovae, discovered many of them, coined the term

• Predicted neutron star as a remnant of SN explosion; correctly suggested that it should be a star with the density of a nucleus resulting from gravitational collapse of a core

• Predicted that supernovae are the origin of the cosmic rays

• Discovered galaxy clusters• Predicted dark matter• Predicted gravitational lensing

Slide 30

Properties of Neutron Stars

Typical size: R ~ 10 km

Mass: M ~ 1.4 – 3 Msun

Density: ~ 1015 g/cm3

Piece of neutron star matter of the size of a sugar cube has a mass of ~ 1 billion tons!!!

Slide 31

Density ~ 1015 g/cm3; One cubic cm weighs 109 ton!

Slide 32

Degenerate gas

-The core is compressed until the inter-particle distance = de Broglie wavelength- One particle occupies finite volume in space and in momentum space- Pauli exclusion principle permits only one particle per each state

Slide 33

When particle velocities become relativistic, they cannot be increased anymore;

Matter becomes “softer”: P ~ 4/3

The core gives in to gravity and collapses

Particle density

3

3deBroglie )(

1~

h

mv

V

Nn

mneutron ~ 2000 melectron

Therefore, inter-particle distance can be 2000 times smaller and density 109 times higher!

Slide 34

Slide 35

 Pyotr Kapitza (Nobel Prize 1978) discovered that liquid helium flows without friction when cooled below 2.17 K. This phenomenon is termed superfluidity. A superfluid shows several spectacular effects. For example, superfluid helium cannot be kept in an open vessel because then the fluid creeps as a thin film up the vessel wall and over the rim.

Superfluid helium in a vessel does not rotate with it as a normal fluid does. Instead, a large number of whirlpools, called vortices, are formed.

Superfluidity

Slide 36

Isolated neutron stars are extremely hard to observe

Neutron stars have been theoretically predicted in 30s.Landau, Oppenheimer, Zwicky, Baade

Slide 37

Proper Motion of Neutron Stars

Some neutron stars are moving rapidly through interstellar space.

This might be a result of anisotropies during the supernova explosion

forming the neutron star

Slide 38

However, there are two facts that can help:

• Neutron stars should rotate extremely fast due to conservation of the angular momentum in the collapse

• They should have huge magnetic field due to conservation of the magnetic flux in the collapse

Slide 39

Conservation of angular momentum: when radius decreases, rotation velocity goes up

Slide 40

2211 RMVRMV

12

1

1

2 R

R

V

V

Slide 41

Neutron star can make over 100 rotations per second!

Conservation of magnetic field flux: BxR2 = const

Magnetic field after collapse: B ~ 1012 – 1015 Gauss !!!

Highest magnetic fields in the lab: 107 Gauss

R R

V VB B

Slide 42

Jocelyn Bell

Discovery of pulsars: Bell and Hewish, 1967

Slide 43

The enigma of pulsarsPulse repetition: from a few to 0.03 secondsPulse duration: ~ 0.001 sPeriod extremely stable: it increases by less than 1 sec in a million years

What could it be???

Only star rotation can be so stable.

However: Centrifugal acceleration < gravitational acceleration

km50~3/1

222

GM

RR

GMR

It must be a neutron star!!

Slide 44

Lighthouse Model of Pulsars

A Pulsar’s magnetic field has a dipole structure, just like Earth.

Radiation is emitted

mostly along the magnetic

poles.

Slide 45

Slide 46

Neutron Star

(SLIDESHOW MODE ONLY)

Slide 47

General idea of a pulsar emission

Slide 48

What is pulsar radiation? Synchrotron radiation of relativistic particles!

Slide 49

Pulsars can emit also in other EM ranges: optical, X-ray, etc.

Slide 50

Slide 51

Only a small fraction of neutron stars are pulsars

• Their beams may not sweep over the Earth

• Rotation of old neutron stars slows down and the pulsar mechanism turns off (why?)

Slide 52

Pulsar Periods

Over time, pulsars lose energy and angular momentum

=> Pulsar rotation is gradually slowing down.

Slide 53

Binary Pulsars

Some pulsars form binaries with other neutron stars (or black

holes).

Radial velocities resulting from the orbital motion lengthen the pulsar period when the pulsar is moving away from Earth...

…and shorten the pulsar period when it is approaching

Earth.

Slide 54

Binary pulsars

• Test of general relativity

• Powerful X-ray sources

• The puzzle of millisecond pulsars

Slide 55

Orbital period becomes shorter: stars lose energy to gravitational radiation

Taylor and Hulse, Nobel prize 1993

Pulsar PSR1913+16: two neutron stars in a binary system

Slide 56

Slide 57

Gravitational waves: ripples of the space-time

Slide 58

This is still indirect evidence for gravitational waves

Attempts to make direct detection: project LIGO

Slide 59

Laser Interferometer Gravitational-Wave Observatory (LIGO)

Slide 60

Pulsar PlanetsSome pulsars have planets orbiting around them.

Just like in binary pulsars, this can be discovered through variations of the pulsar period.

As the planets orbit around the pulsar, they cause it to wobble around, resulting in slight changes of the observed pulsar period.

Slide 61

Fate of the collapsed core

• White dwarf if the remnant is below the Chandrasekhar limit 1.4 Msun after mass loss

• Neutron star if the core mass is less than ~ 3 solar masses

• Black hole otherwise

Slide 62

When the core is too massive, nothing can prevent collapse into a black hole

A black hole is a region of spacetime from which nothing can escape, even light (first suggested by Laplace in 1796).

Slide 63

22 2

2

1

c

GMR

R

GMmmc s

s

Derivation is wrong and picture is wrong, but the result is correct

Schwarzschild radius

Slide 64

2

2

c

GMRs

Schwarzschild radius: event horizon for a nonrotating body

To make a black hole from a body of mass M, one needs to squeeze it below its Schwarzschild’s radius

Rs

Gravitational collapse: the body squeezes below its event horizon

Slide 65

Slide 66

The event horizon is not a physical boundary but the point-of-no-return for anything that crosses it.

Slide 67

Black holes are NOT big cosmic bathtub drains!

2R

GMag

Far from a black hole R >> Rs (weak field): Newtonian gravity law holds

Approaching a black hole R ~ Rs (strong field): gravity pull runs away

RR

R

GMa

s

g

12

If our Sun collapses into a black hole, we won’t see any difference in the gravitational pull (but it will be VERY cold)

Slide 68

Black holes and other strong-field effects are described by General Theory of Relativity (A. Einstein)

Real picture

Newtonian gravity is a weak-field limit of GR

Slide 69

2005 is the World Year of PHYSICS

100th anniversary of Albert Einstein’s “miraculous year” of 1905

March 1905: the quantum nature of light

May 1905: Brownian motion shows the existence of atoms and molecules

June 1905: Special Relativity as a theory of space, time, and motion

Slide 70

General Relativity

Developed in 1907-1915 in close collaboration with mathematicians: Grossmann, Hilbert, Levi-Civita

... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now.

Marcel Grossmann David Hilbert Tullio Levi-Civita

Slide 71

The advance of the perihelion of Mercury

Slide 72

Orbits due to the force ~ 1/r2 are elliptical

Sun

MercuryPerihelion = position closest to the sun

Aphelion = position furthest away

from the sun

Perihelion: 46 million km; Aphelion: 70 million km

Slide 73

Slide 74

Mercury's perihelion precession: 5600.73 arcseconds per century

Newtonian perturbations from other planets: 5557.62 arcseconds per century

Remains unexplained: 43 arcseconds/century (Le Verrier 1855)

In reality the orbits deviate from elliptical:

Slide 75

Urbain Le Verrier 1811-1877

Predicted the presence and position of Neptunefrom irregularities in Uranus’s orbit

Neptune was found in 1846 exactly at the predicted position

In 1855 Le Verrier found that the perihelion of Mercury advanced slightly more than the Newtonian theory predicted.

He and others tried to explain it with a new planet Vulcan, new asteroid belt, etc.

In the eyes of all impartial men, this discovery [Neptune] will remain one of the most magnificent triumphs of theoretical astronomy …

Arago

I do not know whether M. Le Verrier is actually the most detestable man in France, but I am quite certain that he is the most detested.

A contemporary

Slide 76

Problem with Action at a Distance

Direct, instantaneous connection between cause and effect!

By the beginning of the XX century, it became clear that Newtonian gravity has other problems

m1m2

0221

21 rr

mGmFF

1F

2F

If ball 1 moves, ball 2 instantaneously feels it.

Faster than light propagation??

Slide 77

Newton’s theory is a weak-gravity limit of a more general theory: General Relativity

Even in the weak gravity of the Earth and the Sun, there are measurable deviations from Newtonian mechanics and gravitation law!

• Advance of Mercury’s perihelion

• Bending of light by the Sun’s gravity

General Relativity predicts new effects, completely absent in the Newton’s theory: black holes, event horizon, gravitational waves.

Einstein’s idea:

Slide 78

Gravity is a strange force. It has a unique property:

M

m

R

2R

mMGF

2R

MG

m

Fa

All bodies in the same point in space experience the same acceleration!

Slide 79

Acceleration of Gravity

Acceleration of gravity is independent of the mass of the falling object!

Iron ball

Wood ball

Slide 80

This means that in the freely-falling elevator cabin you don’t feel any effects of gravity! You and all objects around you experience the same acceleration.

The opposite is also true: In outer space you can imitate the effect of gravity by acceleration.

Slide 81

"You mighta seen a house fly, maybe even a superfly, but you ain't never seen a donkey fly!"

Donkey

Slide 82

In 1907, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. At this point there occurred to Einstein, described by him as the happiest thought of my life , namely that an observer who is falling from the roof of a house experiences no gravitational field. He proposed the Equivalence Principle as a consequence:-

... we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.

Equivalence Principle

Slide 83

Immediate consequences of the Equivalence Principle:

• Bending of light in the gravitational field

• Time flow and frequency of light are changed in the gravitational field

Acceleration a = gravity gc

The bulb emits flashes of light 2 times per second

An observer on the floor receives flashes faster than 2 times per second

First observed on the Earth by Pound and Rebka 1960: relative frequency shift of 10-15 over the height of 22 m.

Slide 84

Light should be bent in the gravitational field

Slide 85

If gravity can be eliminated by motion, no special force of gravity is needed!

How to explain that in the absence of any force the trajectories are not straight lines?

Because space and time are curved!

The force of gravity is actually the acceleration you feel when you move through space-time

Slide 86

Newtonian point of view

0.5 1 1.5 2 2.5 3

1

2

3

4

5

6

t

x

x = v0t

0.5 1 1.5 2 2.5 3

2468101214

t

x

x = v0t + at2/2

F = 0F = ma

Einstein’s point of view: no real gravity force

F = 0, but trajectories are curved due to curvature of space-time

v0 = const

Slide 87

M

m

R1

21

1 R

MGa

All bodies experience the same acceleration, but only in a small region of space. In another region this acceleration is different. Time flows with a different rate, and paths are bent differently in these two regions.

R2

22

2 R

MGa

Slide 88

Space-time gets curved by masses. Objects traveling in curved space-time have their paths deflected, as if a force has acted on them.

Main idea:

“Curvature” of time means that the time flows with a different rate in different points in space

"Matter tells spacetime how to bend and spacetime returns the complement by telling matter how to move."

John Wheeler

Slide 89

The shortest path between two cities is not a straight line

Shortest paths are called geodesics; they are not straight lines!

Slide 90

Slide 91

About 1912 Einstein realized that the geometry of our world should be non-Euclidean.

He consulted his friend Grossmann who was able to tell Einstein of the important developments of Riemann, Ricci and Levi-Civita.

G.F.B. Riemann(1826-1866)

When Planck visited Einstein in 1913 and Einstein told him the present state of his theories Planck said:

As an older friend I must advise you against it for in the first place you

will not succeed, and even if you succeed no one will believe you.

Slide 92

Several versions of Einstein’s GR in 1913-1914 were wrong.

Only in November 1915, after correspondence with Levi-Civita and Hilbert, Einstein published a paper with correct equations.

Hilbert also published correct equations, in fact 5 days earlier than Einstein.

On the 18th November Einstein made a discovery about which he wrote For a few days I was beside myself with joyous excitement . He explained the advance of the perihelion of Mercury with his theory.

Slide 93

Bending of light

Slide 94

Two British expeditions in 1919 confirmed Einstein’s prediction.

The shift was about 2 seconds of arc, as predicted

Slide 95

Gravitational lensing

Slide 96

Gallery of lenses (Hubble Space Telescope)

Slide 97

Riemann geometry of spacetime

G.F.B. Riemann(1826-1866) 22222

22 1)sin(

1dt

r

Rddr

rR

drds s

s

Schwarzschild metric:

Metric: interval ds between two close pointsas measured by a local observer

(r,,,t) are global coordinates and time

Slide 98

Low density star

High density star

Slide 99

The curvature of a 2D slice of a spherically symmetric black hole

Curvature becomes infinite as we approach the singularity r =0

Slide 100

The curvature and photon path in a 2D slice of a rotating black hole

Note the dragging of the coordinate grid (“frame dragging”)

Slide 101

Gravitational bending of light paths around a black hole

Slide 102

Gravitational bending of light near a rotating black hole

Note the frame dragging effect (spacetime gets into rotation)

Slide 103

Approaching a black hole

Circling around a black hole

Slide 104

Time dilatation

rR

tt

s

o

1

t

t0

As measured by a distant observer, clocks slow down when approaching a black hole

22 1 dtr

Rds s

Slide 105

Frequency of light is shifted in the accelerated frame.It should be also shifted in the gravitational field!

H

t = 0, V = 0

H

t = H/c, V = aH/c

Acceleration a

Doppler effect:

2c

aH

c

V

Light is emitted from the nose

Light reaches floor

Slide 106

Frequency = 1

Period of oscillations

Increase in time intervals means decrease in frequency :Gravitational redshift!

r

Rs 10

Slide 107

Gravitational redshiftPhotons always travel at the speed of light, but they lose energy when travelling out of a gravitational field and appear to be redder to an external observer. The stronger the gravitational field, the more energy the photons lose because of this gravitational redshift. The extreme case is a black hole where photons from within a certain radius lose all their energy.

Gravitational redshift is absent in the Newtonian mechanics. It is a general relativity effect.

r

Rs 10

Slide 108

Tidal forces and contraction of space squeeze and stretch the astronaut. Lateral pressure is 100 atm at a distance of 100 Rs from the event horizon

Slide 109

What would happen IF we could observe directly the collapsing stellar core:

• Photon energies decrease due to a gravitational redshift

• Luminosity decreases due to light bending• The star becomes dark within a free-fall time

of order R/c• However, from our point of view the collapse

slows down to a complete freeze as the star surface approaches the event horizon – time dilatation!

Slide 110

What is at singularity??

Slide 111

“Black Holes Have No Hair”

Matter forming a black hole is losing almost all of its properties.

Black Holes are completely determined by 3 quantities:

Mass

Angular Momentum

Electric Charge

Slide 112

Singularities Clothed and Naked

The singularity is the point of infinite density thought to exist at the center of a black hole. We have no way of understanding what would happen in the vicinity of a singularity, since in essence nature divides our equations by zero at such a point. There is an hypothesis, called the "Law of Cosmic Censorship" that all singularities in the Universe are contained inside event horizons and therefore are in principle not observable (because no information about the singularity can make it past the event horizon to the outside world). However, this is an hypothesis, not proven, so it is conceivable that so-called "Naked Singularities" might exist, not clothed by an event horizon.

Slide 113

Slide 114

Collapse of a scalar field: naked singularity!Choptuik 1990s

Hawking conceded defeat in 1997