Final exam

• 40% - new material Ch. 15-18, 60% - previous chapters

• All - multiple choice questions • Bring green scantron form• 1/3 numerical problems, 2/3 concepts• Don’t forget to prepare formula sheets• Bring your calculator• Textbook and lecture notes are not allowed

Preparing to the test

• Pay extra attention to the following:– Your tests 1-3– Homework problems– Formula sheets indicating the meaning

and units for all formulas– Test reviews– Summary and review questions in the end of

each chapter

• Scale of different objects: planets, sun, orbits of planets, interstellar distances, Milky Way galaxy, distances between galaxies, Universe

• No need to memorize exact numbers, but try to remember the order of magnitude!

• It will help you to check whether your answers make sense

Chapters 1-3

107 mplanets

109 mSun andstars

1017 m~ 3 pcdistancebetweenstars

1021 m~ 10 kpcgalaxy

1011 m~ 1 AUSolar System

1025 m~ 500 MpcLargeststructure

1026 m~ GpcHubbleradius

Distance scale

1022 m~ 1 MpcDistancebetweengalaxies

Definitions and meaning of new units: AU, pc, kpc, Mpc

90o-L

Celestial equator

Seasons - summary1. Seasons are NOT caused by varying distances from the Earth to the Sun

2. The primary cause of seasons is the 23.5 degree tilt of the Earth's rotation axis with respect to the plane of the ecliptic.

Note: the Earth is actually closest to the Sun in January 4!

The Seasons in the Northern Hemisphere

Perihelion: 147.09 × 106 km; Aphelion: 152.10 × 106 km

Eclipses

Moon’s orbit is tilted by 5o from the ecliptic

distance)(

size)linear (rad)(

D

L

D

L 265,206rad)(265,206)arcsec(

radian = 180 degrees

DL

Convert from radian to arcseconds:

arcsec206265 arcsec3600180

deg180

rad1

1 deg = 60 arcmin = 3600 arcsec

Note units!!

Small Angle Formula

Relationship between magnitudes and intensities

Define the magnitude scale so that two stars that differ by 5 magnitudes have an intensity ratio of 100.

100;5 B

AAB I

Imm

512.2100;1 5 B

AAB I

Imm

AB mm

B

A

I

I )512.2(

)(5

2100)( 5

ABABB

A mmLogmmI

ILog

B

AAB I

ILogmm 5.2)(

Chapters 4,5

• Galileo and his discoveries

• Kepler’s laws, especially the third law

• Newton’s accomplishments

• Gravity force

• Application to the orbital motion

Remember parameters: perihelion, aphelion, semimajor axis

Elliptical orbits

a = (Rp + Ra)/2

LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes:

For the Earth P2 = 1 yr, a2 = 1 AU

)()( 31

21 AUayrP Note units!!

32

31

22

21

a

a

P

P

m

M

r

v

a

F

rP

rVVPr

22

Uniform circular motion

velocityangular 2

P

r

Va

2

velocity orbital2

2

r

GMV

r

GM

r

V

GM

r

V

rP

V

rP

32

2

222 44

;2

III Kepler’s law:

Chapters 6,7

• Telescope powers

• Different types of telescopes

• Electromagnetic spectrum

• Black body radiation

• Doppler effect

• Spectral classes of stars

Refracting/Reflecting Telescopes

Refracting Telescope:

Lens focuses light onto the focal plane

Reflecting Telescope:

Concave Mirror focuses light onto the focal

plane

Almost all modern telescopes are reflecting telescopes.

Focal length

Focal length

Telescope parameters

• Light-gathering power (ability to see faint objects)

• Resolving power (ability to see fine details)

• Magnification (least important)

Two Laws of Black Body Radiation

1. The peak of the black body spectrum shifts towards shorter wavelengths when the temperature increases. Wien’s displacement law:

max ≈ 3x106 nm / T(K)

(where T(K) is the temperature in Kelvin).

Two Laws of Black Body Radiation2. The hotter an object is, the more luminous it is.

= Stefan-Boltzmann constant

where A = surface area

L = A**T4

The Stefan-Boltzmann law:

sT 2

4

m

JFlux

Luminosity, or total radiated power:

Radiation Flux, or power emitted by unit area of a black-body emitter, is proportional to the fourth power of its surface temperature:

428

K s m

J1067.5

Shift z = (Observed wavelength - Rest wavelength)

(Rest wavelength)

Doppler effect: cVc

Vz rad

rad

;0

00

0

z

The Doppler effect: apparent change in the wavelength of radiation caused by the motion of the source

The Doppler Effect (2)

The Doppler effect allows us to measure the source’s radial velocity.

vr

The Doppler Effect (3)Take of the H (Balmer alpha) line:

0 = 656 nmAssume, we observe a star’s spectrum with the H line at = 658 nm. Then,

= 2 nm.

We find = 0.003 = 3*10-3

Thus,

vr/c = 0.003, or

vr = 0.003*300,000 km/s = 900 km/s.The line is red shifted, so the star is receding from us with a radial velocity of 900 km/s.

Spectral Classification of Stars (2)

Mnemonics to remember the spectral sequence:

Oh Oh Only

Be Boy, Bad

A An Astronomers

Fine F Forget

Girl/Guy Grade Generally

Kiss Kills Known

Me Me Mnemonics

Sun - basic facts

• What is the Sun

• Internal structure and composition

• Source of energy

• Lifetime

• Sun’s activity and variability

Spectral class: G2Surface temperature: 5800 KLifespan: 10 billion yearsComposition by mass: ~ 71% Hydrogen, 27% Helium

Proton-proton cycle

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

Absolute magnitude

1

212 log5.2

I

Imm Recall that for two stars 1 and 2

Let star 1 be at a distance d pc and star 2 be the same star brought to the distance 10 pc.

Then 2

2

1

2

10

d

I

I 2log210logloglog 22

1

2 ddI

I

1

212 log5.2

I

Imm

5log5 dmM

Inverse:

m2 = M

5/)5(10pc)( Mmd1

212 log5.2

L

LMM

Note also:

H-R diagram

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

• Most stars are lower main sequence red dwarfs

• 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

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

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)

Deaths of stars

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

Compact Objects with Disks and Jets

White dwarfs, 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.

Continuing cycle of stellar evolution

The Structure of the Milky Way (1)

Disk

Nuclear Bulge

HaloSun

Globular Clusters

Cepheid Variables: The Period-Luminosity Relation

The variability period of a Cepheid variable is correlated with its luminosity.

=> Measuring a Cepheid’s period, we can determine its absolute magnitude!

The more luminous it is, the more slowly it pulsates.

Measuring the mass of the Galaxy

Rotation curve

Matter extends beyond the visible disk!

Dark matter halo

Two populations of stars

Walter Baade1893-1960

Ages of the stars

Their main difference is in chemical composition

Population I – metal-richPopulation II – metal-poor

Metals: all elements heavier than helium

Stellar Populations

Population I: Young stars: metal rich; located in spiral

arms and disk

Population II: Old stars: metal poor; located in the halo (globular clusters) and

nuclear bulge

The Abundance of Elements in the Universe

Logarithmic Scale

All elements heavier than He

are very rare.

Linear Scale

Classification of galaxies

Astronomers now know that the tuning fork is NOT an evolutionary sequence because each type of galaxy has very old stars. The oldest stars in any galaxy all have about the same age of around 13 billion years.

Measuring the masses of galaxies

Doppler measurements of the rotation curve + Kepler’s law

c

Vr

Supermassive black holes are found in many galaxies

Cores of some galaxies show an accretion disk with a possible black hole

Slipher 1914: found that over 90% spectra of spirals are redshifted, i.e. they are moving away from us

Hubble and Humason 1931: Vrecession = H0 d

The Hubble Law

Hubble constant H0 70 km/s/Mpc Know V -> can find d

Galaxies are quite close to each other!

Galaxy size ~ 100 kpc

Separation between neighboring galaxies ~ 1 Mpc or less

size

separation> 0.1 for galaxies

size

separation~ 10-7 for stars

Conclusion: galaxies should interact and collide very often!They collided even more often before

Active Galaxies

Galaxies with extremely violent energy release in their nuclei (pl. of nucleus).

“Active Galactic Nuclei” (= AGN)

Up to many thousand times more luminous than the entire Milky Way;

energy released within a region approx. the size of our solar system!

AGN – Active Galactic Nuclei

• Seyferts

• Radio galaxies

• Blazars

• Quasars

In the 1960s it was observed that certain objects emitting radio waves but thought to be stars had very unusual optical spectra. It was finally realized that the reason the spectra were so unusual is that the lines were Doppler shifted by a very large amount, corresponding to velocities away from us that were significant fractions of the speed of light. The reason that it took some time to come to this conclusion is that, because these objects were thought to be relatively nearby stars, no one had any reason to believe they should be receding from us at such velocities.

Quasars

Quasar Red Shifts

z = 0

z = 0.178

z = 0.240

z = 0.302

z = 0.389

Quasars have been detected at the highest

red shifts, up to

z ~ 6

z = /

Our old formula

/= vr/c

is only valid in the limit of low speed,

vr << c

Redshift z = (Observed wavelength - Rest wavelength)

(Rest wavelength)

00

0

z

Doppler effect: cvc

vz

;

0

How come that z > 1 ??

First, relativistic Doppler effect is described by a different formula:

1/1

1

22

cvc

v

z

radial

The redshift is due to the expansion of the Universe:

Contrary to popular belief, this is not a Doppler shift. Instead, as a light wave travels through the fabric of space, the universe expands and the light wave gets stretched and therefore redshifted.

1)(

)(

1

0 tR

tRz

However, cosmological redshift is not a Doppler effect!!!

Quasars1) Spectra contain strongly redshifted lines indicating large

cosmological distances to the objects Gravitational lensing also indicates huge distances

2) Broad emission line as in Seyferts, indicating rapid motion

3) Jets, intense radiation from radio waves to gamma-rays observed

This means that quasars are most luminous objects in the Universe!L ~ 1012 – 1014 Lsun

4) Host galaxies are found around nearby quasars

1)-5) indicate that quasars sit in the centers of galaxies, are extremely compact and super-luminous.

5) Rapid variability on the scale of days is observed

They are probably AGN!

Cosmology

Observational evidence?

Cosmology

Observation #1: universe is homogeneous and isotropic at large scales

It cannot be stationary! It should expand or contract

Observation #2: universe is expanding (Hubble)

It should have a beginning! Hot or cold??

Observation #3: Cosmic microwave background radiation

Hot Big Bang!

Fate of the universe: depends on mass distribution (or curvature)

Observation #4: Abundance of light elements

Confirms Hot Big Bang

Observation #5: density measurements

Observation #6: Fluctuations of background radiation

Universe is nearly flat; it contains dark matter and “dark energy”;It is accelerating in its expansion!

Observation #7: redshifts of distant Ia supernovae

The Early History of the UniverseElectron

Positron

Gamma-ray photon

Electrons, positrons, and gamma-rays in equilibrium between pair

production and annihilation

For reasons not completely understood, there was a very slight excess of ordinary matter over antimatter (by about 1 part in 109). This is why there was still some ordinary matter left over when all the antimatter had been annihilated. (This must be the case, otherwise you wouldn't be here!) All of the protons, neutrons, and electrons in matter today were created in the first few seconds after the Big Bang.

The Early History of the Universe (2)

Protons and neutrons form a few helium nuclei; the rest of protons

remain as hydrogen nuclei

Almost no elements heavier

than helium are produced.

25% of mass in helium 75% in hydrogen

No stable nuclei with 5 and 8 protons

RecombinationProtons and electrons recombine

to form atoms => universe becomes transparent for photons

Transition to matter dominated era

z ≈1000

The Cosmic Background RadiationAfter recombination, photons can travel freely through space.

Their wavelength is only stretched (red shifted) by cosmic expansion.

Recombination:

z = 1000; T = 3000 K

This is what we can observe today as the cosmic background radiation!

The Cosmic Background Radiation

R. Wilson & A. Penzias

The radiation from the very early phase of the universe should still be detectable today

Was, in fact, discovered in mid-1960s as the Cosmic Microwave Background:

Blackbody radiation with a temperature of T = 2.73 K

Cosmology with the Cosmic Microwave Background

If the universe were perfectly homogeneous on all scales at the time of recombination (z = 1000), then the CMB should be perfectly isotropic over the sky.

Instead, it shows small-scale fluctuations:

The universe could not have been perfectly uniform, though. The universe must have been slightly lumpy to form galaxies later on from the internal gravity of the lumps. Initial density variations had to exist in order to provide some direction to where surrounding matter could be attracted. The COBE satellite found slight variations in the brightness of the background radiation of about 1 part in 100,000. The slight variations exist because some parts of the universe were slightly denser than other parts. The slightly denser regions had more gravity and attracted more material to them while the expansion occurred. Over time, the denser regions got even denser and eventually formed galaxies about 1 billion years after the Big Bang.

Fluctuations of the CMB temperature

Evidence for the formation of galaxies and large-scale structure

Deriving geometry of the universe from microwave background radiation

The cosmic microwave background radiation can be explained only by the Big Bang theory. The background radiation is the relic of an early hot universe. The Big Bang theory's major competitor, called the Steady State theory, could not explain the background radiation, and so fell into disfavor.

The amount of activity (active galaxies, quasars, collisions) was greater in the past than now. This shows that the universe does evolve (change) with time. The Steady State theory says that the universe should remain the same with time, so once again, it does not work.

The number of quasars drops off for very large redshifts (redshifts greater than about 50% of the speed of light). The Hubble Law says that these are for large look-back times. This observation is taken to mean that the universe was not old enough to produce quasars at those large redshifts. The universe did have a beginning.

The abundance of hydrogen, helium, deuterium, lithium agrees with that predicted by the Big Bang theory. The abundances are checked from the spectra of the the oldest stars and gas clouds which are made from unprocessed, primitive material. They have the predicted relative abundances.

Observations are consistent with Hot Big Bang Model

Fate of the Universe Depends on mass-energy density (Curvature of Space)

The more mass there is, the more gravity there is to slow down the expansion. Is there enough gravity to halt the expansion and recollapse the universe or not? If there is enough matter (gravity) to recollapse the universe, the universe is ``closed''. In the examples of curved space above, a closed universe would be shaped like a four-dimensional sphere (finite, but unbounded). Space curves back on itself and time has a beginning and an end. If there is not enough matter, the universe will keep expanding forever. Such a universe is ``open''. In the examples of curved space, an open universe would be shaped like a four-dimensional saddle (infinite and unbounded). Space curves away from itself and time has no end.

Deceleration of the Universe?

• Fate of the universe depends on the matter density in the universe.

• Expansion of the universe should be slowed down by mutual gravitational attraction of the galaxies.

• Define “critical density”, c, which is just enough to slow the cosmic expansion to a halt at infinity.

Model Universes

Siz

e s

cale

of t

he

Un

iver

se

Time

< c => universe will expand forever

> c => Universe will collapse back

If the density of matter equaled the critical density, then the curvature of space-time by the matter would be just sufficient to make the geometry of the universe flat!

= c => Flat UniverseMaximum age of the universe:

~ 1/H0

Faint gas shells around ellipticalsEllipticals have faint gas shells that need massive ``dark'' haloes to contain them. The gas particles are moving too quickly (they are too hot) for the gravity of the visible matter to hang onto it.

Motion of galaxies in a clusterGalaxy cluster members are moving too fast to be gravitationally bound unless there is unseen mass.

Hot gas in clustersThe existence of HOT (i.e., fast moving) gas in galaxy clusters. To keep the gas bound to the cluster, there needs to be extra unseen mass.

Quasar spectraAbsorption lines from hydrogen in quasar spectra tells us that there is a lot of material between us and the quasars. Gravitational LensingGravitational lensing of the light from distant galaxies and quasars by closer galaxies or galaxy clusters enables us to calculate the amount of mass in the closer galaxy or galaxy cluster from the amount of bending of the light. The derived mass is greater than the amount of mass in the visible matter.

Current tallies of the total mass of the universe (visible and dark matter) indicate that all matter constitutes only 27% of the critical density.

The case of a missing Universe

Dark matter accounts for only 27% of the total mass-energy density: DM = 0.27

Observations suggest that the universe is flat: = 1

The rest 70% is something else!!

Visible matter accounts for ~ 4% of the total mass-energy density: v = 0.04

This something else is termed “dark energy”

It causes the universe to accelerate in its expansion!

Supernovae are too faint

Accelerating now, but decelerating in the past?!

Do we live in a special universe??

• Change of physical constants by a very small amount would render impossible the life in the universe as we know it

• Adding or subtracting just one spatial dimension would make the formation of planets and atoms impossible

• Life as we know it needs a universe which is large enough, flat, homogeneous, and isotropic

Anthropic Principle

We observe the universe to be as it is because only in such a universe could observers like ourselves exist.

That is, selection effects would say that it is only in universes where the conditions are right for life (thus pre-selecting certain universe) is it possible for the questions of specialness to be posed.

This is a solution, but can we do better?

History of science teaches us that there is nothing special in the place we live

• Our local country is nothing special (ancient travelers)• Planet Earth is nothing special (Copernicus)• Milky Way galaxy is nothing special (Hubble)• Our part of the Universe is nothing special

– Self-reproducing Universe– Eternal Big Bang and ensemble of universes

Linde, Vilenkin, …

Landscape of the multiverse

Planck scale:

Planck Length

Planck Mass

Planck density 1094 g/cm3

Eternal multiverse;

Individual universes are being continuously “inflated” from a space-time “foam”.

Some of these universities can harbor life as we know it; others don’t.

A large fraction of universes CAN harbor life