Test Review 3

Test 3

• Test covers Chapters 9,11-14 and section 5.3• Part 1: Short questions and problems• Part 2: bonus problems, extra 30 points• Show your work everywhere• Don’t forget to prepare formula sheet• Bring your calculator• Textbook and lecture notes are not allowed

Less math, but more concepts

H-R diagram

90% of stars

Specific segments of the main sequence are occupiedby stars of a specific mass

Majority of stars are here

The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.

L ~ M3.5 only for main-sequence stars!

5.3

sunsun M

M

L

L

star mass (solar masses) time (years) Spectral type

60 3 million O3

30 11 million O7

10 32 million B4

3 370 million A5

1.5 3 billion F5

1 10 billion G2 (Sun)

0.1 1000's billions M7

Lifetime T ~ M/L ~ 1/Mp-1 = 1/M2.5 ; p ~ 3.5

M = 4M; 32

15.2

M

M

T

T sun

sun

Lifetime = Amount of hydrogen fuel

Rate of energy loss

T ~ 3x108 years

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

H-R diagram

• 90% of stars are on the main sequence and obey the mass-luminosity dependence L ~ M3.5

• Stars on the main sequence generate energy due to nuclear fusion of hydrogen

• In the end of their lives stars move to the upper right corner of the H-R diagram

Cutoff at masses > 100 M and < 0.08 M

Spectral Lines of Giants

=> Absorption lines in spectra of giants and supergiants are narrower than in main sequence stars

Pressure and density in the atmospheres of giants are lower than in main sequence stars.

=> From the line widths, we can estimate the size and luminosity of a star.

Distance estimate (spectroscopic parallax)

Luminosity classes

• Ia bright supergiant

• Ib Supergiant

• II bright giant

• III giant

• IV subgiant

• V main-sequence star

Example Luminosity Classes

• Our Sun: G2 star on the Main Sequence:

G2V

• Polaris: G2 star with Supergiant luminosity:

G2Ib

Luminosity

sT

24

m

Jstar theof areaunit fromFlux

Surface area of the star = 4R2

Luminosity, or total radiated power L = T4 4R2 J/s

Intensity, or radiation flux on the Earth:

)mJ/(s4

22 d

LI

R

d

It is convenient to compare with the Sun or any other star:

42

b

a

b

a

b

a

T

T

R

R

L

L

Surface temperature and color indices

K)(

103)nm(

6

T

Can be applied to any black-body emitter!

Binary Stars More than 50 % of all stars in our Milky Way

are not single stars, but belong to binaries:

Pairs or multiple systems of stars which

orbit their common center of mass.

If we can measure and understand their orbital

motion, we can

estimate the stellar masses.

Measuring diameters and masses

A

B

B

A

m

m

r

r

Estimating Stellar Masses

Recall Kepler’s 3rd Law:

Py2 = aAU

3

Valid for the Solar system: star with 1 solar mass in the center.

We find almost the same law for binary stars with masses MA and MB different

from 1 solar mass:

MA + MB = aAU

3 ____ Py

2

(MA and MB in units of solar masses)

Visual Binaries

The ideal case:

Both stars can be seen directly, and

their separation and relative motion can be followed directly.

Spectroscopic binaries

Stars are seen as a single point

• Spectra of both stars are distinguishable

• Sometimes spectrum of only one star is seen

STELLARWOBBLE

RECEDING:

REDDER

APPROACHING:

BLUER

The Doppler Effect The Doppler effect allows us to

measure the source’s radial velocity.

vr

/0 = vr/c

Determining the orbital period

1. Below is a radial velocity curve for a spectroscopic binary. Estimate the mass of each star if the mass of the binary system is 6 solar masses.

Time (days)

V (km/sec)r

-15

-20

-25

-10

-5Star A

Star B

MA dA = MB dB

V ~ d

2

1

A

B

A

B

B

A

V

V

d

d

M

M

6 BA MM

Over 100 extrasolar planets discovered

...BUT

THE PLANET CANNOT BE SEEN

MOTIONS OF THE STAR

BETRAYITS PRESENCE !

X

EARTH

X

JUPITER

150 000 000 km

30 km/s

450 km

9 cm/s

780 000 000 km

13 km/s

750 000 km

13 m/s

2010

2000

2005

1995

1990

2015

2020

0.002”

MOTIONS OF THE SUN VIEWED FROM A STAR 30 LIGHT YEARS AWAY

0.002’’ IS THE ANGULAR SIZE OF A MAN ON THE MOON OR A STANDARD NEWSPAPER FONT 300 KM AWAY Unobservable!

Eclipsing Binaries

From the light curve of Algol, we can infer that the system contains two stars of very different surface temperature, orbiting in a slightly inclined plane.

Example:

Algol in the constellation of Perseus

Coldest spots in the galaxy:T ~ 1-10 K

Composition:• Mainly molecular hydrogen• 1% dust

Jeans instability:

Thermal pressure cannot support the gas cloud against its self-gravity. The cloud collapses and fragments.

Shocks Triggering Star Formation

Globules = sites where stars are being born right now!

Trifid Nebula

Jeans instability:

Thermal pressure cannot support the gas cloud against its self-gravity. The cloud collapses and fragments.

Heating By Contraction

As a protostar contracts, it heats up:

Free-fall contraction→ Heating

Heating does not stop contraction because the core cools down due to radiation

Protostellar Disks and Jets – Herbig Haro Objects

Herbig Haro Object HH30

• The matter stops falling on the star• Nuclear fusion starts in the core• Planets can be formed from the remaining disk

The Source of Stellar Energy

In the sun, this happens primarily through the proton-proton (PP) chain

Recall from our discussion of the sun:

Stars produce energy by nuclear fusion of hydrogen into helium.

The CNO Cycle

In stars slightly more massive than the sun, a more powerful

energy generation mechanism than

the PP chain takes over:

The CNO Cycle.

Net result is the same: four hydrogen nuclei fuse to form one helium nucleus; 27 MeV is released.

Why p-p and CNO cycles? Why so complicated?

Because simultaneous collision of 4 protons is too improbable

Energy Transport Structure

Inner radiative, outer convective

zone

Inner convective, outer radiative

zone

CNO cycle dominant PP chain dominant

Evolution off the Main Sequence: Expansion into a Red Giant

Hydrogen in the core completely converted into He:

H burning continues in a shell around the core.

He Core + H-burning shell produce more energy than needed for pressure support

Expansion and cooling of the outer layers of the

star Red Giant

“Hydrogen burning” (i.e. fusion of H into He) ceases in the core.

Formation of degenerate core

Red Giant Evolution

4 H → He

He

H-burning shell keeps dumping He

onto the core.

He-core gets denser and hotter until the

next stage of nuclear burning can begin in

the core:

He fusion through the

“Triple-Alpha Process”

4He + 4He 8Be + 8Be + 4He 12C +

If M > 0.5 Msun, the temperature reaches 100 million K. Nuclear fusion of helium starts; carbon and oxygen are produced

Essentially all C and O in the universe are produced in this way!

p. 192

The Fate of Our Sun and the End of Earth• Sun will expand to a

Red giant in ~ 5 billion years

• Expands to ~ Earth’s orbit radius or more

• Earth will then be incinerated!

• It will be too hot for life in 200 million years

• Sun may form a planetary nebula (but uncertain)

• Sun’s C,O core will become a white dwarf

What is left?

A stellar remnant: white dwarf, composed mainly of carbon and oxygen

Sirius B is very hot: surface temperature 25,000 KYet, it is 10,000 times fainter than Sirius A

It should be very small: R ~ 2 Rearth

Its mass M ~ 1 Msun

It should be extremely dense!

M/V ~ 106 g/cm3

V

M

All atoms are smashed and the object is supported by pressure of degenerate electrons

White dwarf should be extremely dense!

M/V ~ 106 g/cm3V

M

Strange properties of degenerate matter

• It strongly resists compression: P ~ 5/3

• Pressure does not depend on temperature

Compare with classical gas: P ~ T

Evolution of sun-like stars on H-R diagram

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.

Stars > 8 solar masses

Reactions proceed faster and faster, until Fe and Ni are synthesized

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!

Precise mechanism – still unknown

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.

• Evolution of sun-like stars: red giant, planetary nebula, white dwarf

• Evolution of massive stars: red giant or supergiant, supernova

• Three types of compact objects – stellar remnants: white dwarfs, neutron stars, black holes. Limits on their masses. Pulsars as rotating neutron stars

• Compact objects in binary systems. Accreting X-ray binaries

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

Stellar nucleosynthesis

• All elements up to Atomic mass ~ 250 u are synthesized!

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

• R-processes (r = rapid): rapid neutron capture during SN explosion; all elements heavier than iron

Continuing cycle of stellar evolution

Supernova Remnants

The Cygnus Loop

The Veil Nebula

The Crab Nebula:

Remnant of a supernova

observed in a.d. 1054

Cassiopeia A

Optical

X-rays

The Remnant of SN 1987A

Ring due to SN ejecta catching up with pre-SN stellar wind; also observable in X-rays.

Synchrotron Emission and Cosmic-Ray Acceleration

The shocks of supernova remnants

accelerate protons and electrons to extremely

high, relativistic energies.

“Cosmic Rays”

In magnetic fields, these relativistic

electrons emit

Synchrotron Radiation.

Crab nebula: the remnants of supernova 1054

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

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.

Formation of Neutron StarsCompact objects more massive than the

Chandrasekhar Limit (1.4 Msun) collapse further.

Density and T become so high that electrons and protons combine to form stable neutrons throughout the object:

p + e- n + e

Neutron Star

Properties of Neutron Stars

Typical size: R ~ 10 km

Mass: M ~ 1.4 – 3 Msun

Density: ~ 1014 g/cm3

Piece of neutron star matter of the size of a sugar cube has a mass of ~ 100 million tons!!!

• 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

2211 RMVRMV

12

1

1

2 R

R

V

V

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!!

General idea of a pulsar emission

Exact mechanism of pulsar radiation is still unknown!

2

2

c

GMRs

Schwarzschild radius: event horizon for a spherically symmetric object

Rs

Black hole: an object shrinks below its event horizon

K. Schwarzschild

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

General RelativitySee also Ch. 5 in Seeds

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

In 1672 Giovanni Cassini together with Jean Richter (1630-1696) made parallel observations of the Mars parallax in Paris (France) and Cayenne (French Guiana, N. coast of South America)

They were also able to determine the solar parallax as ~ 9 arcseconds and find the distance to the Sun (Astronomical Unit) as 140,000,000 km.Current value is 8.8 arcseconds, or 149,597,892 km.

Newton’s theory has been confirmed by increasingly precise observations …

Parallax angle A

A

BD

B

D

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 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

Newton’s theory has been confirmed by increasingly precise observations …

The advance of the perihelion of Mercury

One little speck on the brilliant face of Newton’s theory:

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.

Mercury: the closest planet to the Sun

Sun

MercuryPerihelion = position closest to the sun

Aphelion = position furthest away

from the sun

Perihelion: 46 million km; Aphelion: 70 million km

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:

This is only 12,000 km per century, or 29 km per one period!

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!

• Precession 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:

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??

Same problem existed in electromagnetic theory, but was solved by Maxwell

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!

Acceleration of Gravity

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

Iron ball

Wood ball

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.

In outer space you can imitate the effect of gravity by acceleration.

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

Donkey

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

Immediate consequences of the Equivalence Principle:

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

Bending of light in the gravitational field

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

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

Light path is bent in the accelerated frame.It should be also bent in the gravitational field!

Light path is bent in the accelerated frame.It should be also bent in the gravitational field!

t = 0, V = 0t = x/c, y2-y1 = at2/2

y2 –y1= a(x)2/2

Acceleration a

Parabola:

2/21 axyy

Light is emitted from the left wall

Light reachesthe right wall

x

y1

y1

y2

x

y

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!

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

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.

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

The shortest path between two cities is not a straight line

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

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.

First test of General Relativity: precession of perihelion for Mercury (43 arcsec per century)

Planet Observed excess Predictedprecession

Mercury 43.11+-0.45 43.03

Venus 8.4+-0.48 8.6

Earth 5.0+-1.2 3.8

Bending of light

Two British expeditions in 1919 confirmed Einstein’s prediction.

The shift was about 2 seconds of arc, as predicted

Gravitational lensing

Gallery of lenses (Hubble Space Telescope)

t

t0

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

Time dilatationt > t0

Frequency = 1

Period of oscillations

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

R

Rs 10

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

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

How to observe a stellar remnant if it does not emit radiation?

• Isolated black hole has almost no chance to be seen• Gravitational lensing is possible but very improbable• Isolated neutron star can be detected as a pulsar, or if it is

very close and hot• Isolated white dwarf can be seen if it is close enough and hot• Good news: most stars are in binary systems

– We can detect radiation from matter accreting onto a compact object. Remember, however, this is only an indirect indicator of a black hole

– We can determine the mass of an unseen companion. If it is much larger than 3 Msun – this is likely a BH. If it is between 1.4 and 3 Msun – this is likely a neutron star.

;2

3

21 P

aMM

a – in AUP – in yearsM1+M2 – in solar masses

Binary systems

If we can calculate the total mass and measure the mass of a normal star independently, we can find the mass of an unseen companion

Accretion from stellar windAccretion through Roche lobe outflow

Two mechanisms of mass transfer in a binary system

Initial ring of gas spreads into the disk due to diffusion.

To be able to accrete on the star, matter should lose angular momentum as a result of viscous friction

Friction leads to heating of the disk and intense radiation!!

White Dwarfs in Binary Systems

Binary consisting of WD + MS or Red Giant star => WD accretes matter from the companion

Angular momentum conservation => accreted matter forms a disk, called accretion disk.

Matter in the accretion disk heats up to ~ 1 million K => X-ray emission => “X-ray binary”.

T ~ 106 K

X-ray emission

Nova Explosions

Nova Cygni 1975

Hydrogen accreted through the accretion

disk accumulates on the surface of the WD

Very hot, dense layer of non-fusing hydrogen

on the WD surface

Explosive onset of H fusion

Nova explosion

Compact Objects with Disks and Jets

Black holes and neutron stars can be part of a binary system.

=> Strong X-ray source!

Matter gets pulled off from the companion star, forming an accretion

disk.

Infalling matter heats up to billions K. Accretion is a very efficient process of

energy release.

Zoo of accreting binaries:

• Novae

• X-ray pulsars

• Millisecond pulsars

• High-mass X-ray binaries: Cygnus X-1

• Low-mass X-ray binaries

• X-ray Novae

X-ray pulsar: an accreting neutron star

Compare with a radio pulsar

Main feature: strong magnetic field ~ 1012-1015 G

X-ray emission from hot accreting plasma

Measurement of binary system parameters gave M ~ 7 Msun

Low-Mass X-ray binary: accretion through Roche-lobe overflow

Low-mass X-ray binaries are best candidates because the mass of a red dwarf is much less than a black-hole mass

212

3

21 ; MMP

aMM

Black Hole X-Ray Binaries

Strong X-ray sources

Rapidly, erratically variable (with flickering on time scales of less than a second)

Sometimes: Quasi-periodic oscillations (QPOs)

Sometimes: Radio-emitting jets

Accretion disks around black holes