Final Review

Final exam

• Same structure: multiple choice and show-work parts

• Bonus questions and problems

• Covers chapters 15 through 18

• Bring your formula sheets

• Better: rewrite your formula sheet

Your final test will weigh 1.5 times more than your mid-term test

Final score = 0.22 x (test 1 + essays) + 0.22 x test 2+ 0.22 x test 3 + 0.34 x final test

Preparing to the test

• Give extra attention to the following:– Review your mistakes in your tests 1-3– Meaning and units for all formulas– Chapters 15 and 18– Homework problems – Summary and review questions in the end of

each chapter– Always try to answer the question: how do

we know?

Formulas and problems you need to refresh:

Useful units of distance:

1 AU = 1.5x1011 m (orbits of planets)

1 light-year (ly) 1016 m ~ 105 AU (interstellar distances)

1 ly = c1 year (the distance the light travels in 1 year)

Velocity of light in vacuum c = 3 108 m/s

1 parsec (pc) 3.26 ly = 3 1016 m = 206,265 AU

1 kpc = 1000 pc; 1Mpc = 106 pc

Please memorize typical scales for the astronomical objects!!

Absolute and apparent magnitude

1

212 log5.2

I

Imm For two objects 1 and 2:

5log5 dmM

Absolute magnitude M is the apparent magnitude that an object would have if it were (in our imagination) placed at a distance of 10 parsecs (which is 32.6 light years) from the Earth.

Absolute and apparent magnitude for the same object:

Here d is in parsecs!

5/)5(10 Mmd

1

212 log5.2

L

LMM

Luminosity

sT 2

4

m

J areaunit fromFlux

Surface area of the star A = 4R2

Intensity, or radiation flux on the Earth:

)mJ/(s4

22 d

LI

R

d

AT 4 Luminosity

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.

You don’t want to forget the orbital motion and Kepler’s laws!

m

M

r

v

a

F

rP

rvvPr

22

Uniform circular motion

velocityangular 2

P

r

vrva

22

velocityorbital2

2

r

GMv

r

GM

r

v

III Kepler’s law:

2

3

21 P

aMM

Note units!! M in solar masses, a in AU, P in years

For a star orbiting around the galaxy, Mstar + Mgalaxy = Mgalaxy

Temperature of black-body radiation and peak wavelength

K)(

103)nm(

6

T

Wien’s law: Note units!!

Redshift z = (Observed wavelength - Rest wavelength)

(Rest wavelength)

Doppler effect: cvc

vz

;

0

cvcv

c

v

z

radial

~;1/1

1

22

00

0

z

Cosmological redshift and the Hubble law

Hubble and Humason 1931: Vrecession = H0 R

Current value of the Hubble constant H0 70 km/s/Mpc

The universe expands!

HRv Know velocity from redshift measurements

Find the distance if H is known

Or, know velocity and distance independently

Can determine H

The Milky Way Galaxy

Rotation curve allows one to find the mass of the galaxy

Fig. 13-6a, p. 264

r

GMv

r

rGMv

)(

3

3

4~)( rrM

)(3

4rGrv

Galaxy contains 10 times more matter than we see

Matter extends beyond the visible disk!

There is much more matter than we see!

Dark matter halo (or corona)

p. 188

Age of the star clusters from turnoff points

The age of our Galaxy

13 billion years for oldest globular clusters in the halo

9-10 billion years for oldest open clusters in the disk

Disk seems to be younger than 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

The Abundance of Elements in the Universe

Logarithmic Scale

All elements heavier than He

are very rare.

Linear Scale

No elements with atomic mass 5 and 8

There were no metals in the early universe, before the first stars were born

All heavy elements were synthesized in stars

That is why metals are so rare in the universe

That is why old stars of Population II are metal poor

Younger stars of Population I contain metals synthesized by previous generations of stars

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

History of the Milky Way

The traditional theory:

Quasi-spherical gas cloud fragments into smaller pieces, forming the first, metal-poor stars (pop. II);

Rotating cloud collapses into a disk-like structure

Later populations of stars (pop. I) are restricted to the disk of the Galaxy

Fig. 12-6b, p. 234

Milky Way center in Sagittarius

Fast rotation of spiral filaments around Sgr A*

Our Galaxy Center. If one takes pictures every year it seems that some stars are moving very fast (up to 1500 kilometers per second). The fastest stars are close to the center - the position marked by the radio nucleus Sagittarius A* (cross).

Explanation: the dark mass ~ 2.6 million solar masses

Rotation curve for the Galactic Center

Is this a black hole?!

Evidence for a black hole of 2.6 million solar masses:

• Rotation curve indicating an ultra-compact object• Rapid variability• Dense stellar population• Radio jets

Radio jets but rather weak X-ray emission

Other galaxies contain much heavier black holes and stronger activity

Supermassive black holes are found in many galaxies

Matter is moving very fast close to the center and indicates a very massive and compact object

Local group: ~ 30 galaxies

Elliptical galaxies: (a) they have much more random star motion than orderly rotational

motion (star orbits are aligned in a wide range of angles and have a wide range of eccentricities);

(b) they have very little dust and gas left between the stars; (c) this means that they have no new star formation occurring now and

no hot, bright, massive stars in them (those stars are too short-lived)

(a) they have more orderly, rotational motion than random motion (the rotation refers to the disk as a whole and means that the star orbits are closely confined to a narrow range of angles and are fairly circular);

(b) they have some or a lot of gas and dust between the stars; (c) this means they can have new star formation occuring in the disk,

particularly in the spiral arms; and (d) they have a spiral structure.

Spiral Galaxies:

Irregular galaxies have no definite structure. The stars are bunched up but the patches are randomly distributed throughout the galaxy. Some irregulars have a lot of dust and gas so star formation is possible. Some are undergoing a burst of star formation now, so many H II regions are seen in them. Others have very little star formation going on in them (even some of those with a lot of gas and dust still in them).

Hubble Deep Field

Most galaxies are small

There were more spiral galaxies than now

A lot of blue, luminous stars: intensive star formation

Many galaxies had irregular shapes: just lumps of matter

Multiple nuclei

Evidence for frequent collisions!!

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

Mergers and galactic cannibalism

Giant elliptical in the cluster center gets bigger by eating neighbors

Initial proto-galaxies were probably very small: 106 solar masses

Large galaxies were produced by mergers of many small lumps

Collisions and mergers drive the evolution of galaxies

Collisions trigger intense star formation, consuming all gas in a short time

Spirals can be destroyed in the collision, producing ellipticals

Dwarf ellipticals and irregulars are produced in the collisions

Giant elliptical galaxies are produced by consuming many small galaxies

Active galaxies

Contain extremely active nuclei

AGN – Active Galactic Nuclei

AGN – Active Galactic Nuclei

• Seyferts

• Radio galaxies

• Blazars

• Quasars

What engine powers observed AGNs??? A supermassive black hole?!

Fig. 14-4, p.284

Unified model of AGNs: a massive black hole surrounded by an accretion disk

Model for Seyfert Galaxies

Accretion disk

Dense dust torus

Gas clouds

UV, X-rays

Emission lines

Supermassive black hole

Seyfert I:Seyfert I:

Strong, broad Strong, broad emission lines from emission lines from rapidly moving gas rapidly moving gas clouds near the BHclouds near the BH

Seyfert II:Seyfert II:

Weaker, narrow Weaker, narrow emission lines from emission lines from more slowly moving more slowly moving gas clouds far from gas clouds far from

the BHthe BH

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

Fig. 14-7a, p.287

3C273

Jets and host galaxies have been resolved for “nearby” quasars

Fig. 14-10, p.289

Fig. 14-10, p.289

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

indicates that quasars are very compact (less than a light-day)

5) Rapid variability on the scale of days is observed

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 and predicts the amount of baryonic matter in the universe: around 4% of the

critical density

Observation #5: density measurements: matter is 27% of the critical density

Observation #6: Fluctuations of background radiation

Universe is nearly flat; Total density is equal to the critical density

Missing 70% of the Universe??

The universe is accelerating

It is filled with “dark energy” that creates “negative pressure”

Observation #7: distant Ia SNe are dimmer than expected

Observation #8: most distant Ia SNe at z = 1.7 is brighter than expected

The universe was decelerating in the past

The Hubble Law: expansion of the universe

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.

Primordial nucleosynthesis: 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.

Three pillars of Hot Big Bang Model

Fig. 15-14, p.311

The History of the Universe

Universe expands as time passes

Un

ive

rse

coo

ls d

own

as

time

pa

sse

s

Fig. 15-9, p.304

Recombination (the universe is 300,000 yr old)

Temperature at recombination: 3000 K

Current temperature of microwave background: 2.7 K

The Cosmic Background Radiation (2)After 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!

Geometry and fate of the Universe Depends on density of matter AND energy combined

The more mass-energy 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.

The rate of expansion depends on density and pressure

flatk

openk

closedk

0:1

1:1

1:1

c

Critical parameter

G

Hc

8

3 2

Critical density:

Acceleration = )/3(3

4 22

2

cPRG

dt

Rd

Deriving geometry of the universe from density measurements

Orbital speeds of stars in galaxies

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 about 27% of the critical density.

Deriving geometry of the universe from microwave background radiation

Deriving geometry of the universe from microwave background radiation

Fig. 15-21c, p.319

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

1) The rest 70% is something else!!

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

2) This something else (“dark energy”) exerts negative pressure and causes the expansion of the universe to accelerate

Acceleration = )/3(3

4 22

2

cPRG

dt

Rd 2cP

How do we know?

Measurements of redshifts of distant Type Ia Supernovae

Supernovae are too faint

Universe is accelerating now, but it was slowing down in the past: at z ~ 1.7.Matter was dominant in the past. Dark energy is dominant now