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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.

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Final exam l.jpg
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

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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

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Chapters 1-3

  • 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

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Distance scale

107 m


109 m

Sun and


1011 m

~ 1 AU

Solar System

1021 m

~ 10 kpc


1022 m

~ 1 Mpc




1017 m

~ 3 pc




1025 m

~ 500 Mpc



1026 m

~ Gpc



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

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Celestial equator

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Seasons - summary

  • Seasons are NOT caused by varying distances from the Earth to the Sun

  • 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.

The Seasons in the Northern Hemisphere

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

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

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

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Small Angle Formula

Convert from radian to arcseconds:

  • radian = 180 degrees

1 deg = 60 arcmin = 3600 arcsec

Note units!!

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Relationship between magnitudes and intensities

Define the magnitude scale so that two stars that differ by

5 magnitudes have an intensity ratio of 100.

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Chapters 4,5

  • Galileo and his discoveries

  • Kepler’s laws, especially the third law

  • Newton’s accomplishments

  • Gravity force

  • Application to the orbital motion

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Elliptical orbits

Remember parameters: perihelion, aphelion, semimajor axis

a = (Rp + Ra)/2

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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

Note units!!

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Uniform circular motion

III Kepler’s law:

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Chapters 6,7

  • Telescope powers

  • Different types of telescopes

  • Electromagnetic spectrum

  • Black body radiation

  • Doppler effect

  • Spectral classes of stars

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Refracting/Reflecting Telescopes

Refracting Telescope: Lens focuses light onto the focal plane

Focal length

Reflecting Telescope: Concave Mirror focuses light onto the focal plane

Focal length

Almost all modern telescopes are reflecting telescopes.

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Telescope parameters

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

  • Resolving power (ability to see fine details)

  • Magnification (least important)

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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:

lmax≈ 3x106 nm / T(K)

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

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L = A*s*T4

Two Laws of Black Body Radiation

2. The hotter an object is, the more luminous it is.

The Stefan-Boltzmann law:

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

s = Stefan-Boltzmann constant

Luminosity, or total radiated power:

whereA = surface area

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(Observed wavelength - Rest wavelength)

Shift z =

(Rest wavelength)

The Doppler effect: apparent change in the wavelength

of radiation caused by the motion of the source

Doppler effect:

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The Doppler Effect (2)

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


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The Doppler Effect (3)

Take l0 of the Ha (Balmer alpha) line:

l0 = 656 nm

Assume, we observe a star’s spectrum with the Ha line at l = 658 nm. Then,

Dl = 2 nm.

We findDl/l0 = 0.003 = 3*10-3


vr/c = 0.003,


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.

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Spectral Classification of Stars (2)

Mnemonics to remember the spectral sequence:

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Sun - basic facts

  • What is the Sun

  • Internal structure and composition

  • Source of energy

  • Lifetime

  • Sun’s activity and variability

Spectral class: G2

Surface temperature: 5800 K

Lifespan: 10 billion years

Composition by mass: ~ 71% Hydrogen, 27% Helium

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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.

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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

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Energy Transport Structure one helium nucleus; 27 MeV is released.

Inner convective, outer radiative zone

Inner radiative, outer convective zone

CNO cycle dominant

PP chain dominant

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Absolute magnitude one helium nucleus; 27 MeV is released.

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.


m2 = M


Note also:

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H-R diagram one helium nucleus; 27 MeV is released.

  • 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

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

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

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Amount of hydrogen fuel spectroscopic binary systems.

Lifetime =

Rate of energy loss

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

T ~ 3x108 years

M = 4M;

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Estimating the Age of a Cluster spectroscopic binary systems.

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

The lower on the MS the turn-off point, the older the cluster.

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Measuring diameters and masses spectroscopic binary systems.

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 stellarmasses.

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Estimating Stellar Masses spectroscopic binary systems.

RecallKepler’s 3rd Law:

Py2 = aAU3

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 =


(MA and MB in units of solar masses)

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0 spectroscopic binary systems.

Deaths of stars

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Summary of Post Main-Sequence Evolution of Stars spectroscopic binary systems.


Fusion proceeds; formation of Fe core.

Evolution of 4 - 8 Msun stars is still uncertain.

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

M > 8 Msun

Fusion stops at formation of C,O core.

M < 4 Msun

Red dwarfs: He burning never ignites

M < 0.4 Msun

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  • 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

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Compact Objects with Disks and Jets white dwarf

White dwarfs, black holes and neutron stars can be part of a binary system.

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

=> Strong X-ray source!

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

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The Structure of the Milky Way (1) white dwarf


Nuclear Bulge



Globular Clusters

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Cepheid Variables: white dwarfThe Period-Luminosity Relation

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

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

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

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Measuring the mass of the Galaxy white dwarf

Rotation curve

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Dark matter halo white dwarf

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Ages of the stars white dwarf

Two populations of stars

Walter Baade


Their main difference is in chemical composition

Population I – metal-rich

Population II – metal-poor

Metals: all elements heavier than helium

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Stellar Populations white dwarf

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

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The Abundance of Elements in the Universe white dwarf

All elements heavier than He are very rare.

Logarithmic Scale

Linear Scale

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Classification of galaxies white dwarf

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.

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Measuring the masses of galaxies white dwarf

Doppler measurements of the rotation curve + Kepler’s law

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The Hubble Law 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

Hubble constant H0 70 km/s/Mpc

Know V -> can find d

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size possible black hole


> 0.1

~ 10-7



Galaxies are quite close to each other!

Galaxy size ~ 100 kpc

Separation between neighboring galaxies ~ 1 Mpc or less

for galaxies

for stars

Conclusion: galaxies should interact and collide very often!

They collided even more often before

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Active Galaxies possible black hole

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!

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AGN – Active Galactic Nuclei possible black hole

  • Seyferts

  • Radio galaxies

  • Blazars

  • Quasars

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Quasars possible black hole

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.

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Quasar Red Shifts possible black hole

Quasars have been detected at the highest red shifts, up to

z ~ 6

z = 0

z = 0.178

z = Dl/l0

z = 0.240

Our old formula

Dl/l0 = vr/c

is only valid in the limit of low speed, vr << c

z = 0.302

z = 0.389

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(Observed wavelength - Rest wavelength) possible black hole

Redshift z =

(Rest wavelength)

Doppler effect:

How come that z > 1 ??

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

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However, cosmological redshift is not a Doppler effect!!! possible black hole

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.

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Quasars possible black hole

  • Spectra contain strongly redshifted lines indicating large cosmological distances to the objects

  • Gravitational lensing also indicates huge distances

This means that quasars are most luminous objects in the Universe!

L ~ 1012 – 1014 Lsun

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

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

4) Host galaxies are found around nearby quasars

5) Rapid variability on the scale of days is observed

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

They are probably AGN!

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Cosmology possible black hole

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Observational evidence? possible black hole

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Cosmology possible black hole

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!

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Observation #4: Abundance of light elements possible black hole

Confirms Hot Big Bang

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

Observation #5: density measurements

Observation #6: Fluctuations of background radiation

Observation #7: redshifts of distant Ia supernovae

Universe is nearly flat; it contains dark matter and “dark energy”;

It is accelerating in its expansion!

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The Early History of the Universe possible black hole



Gamma-ray photon

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

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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.

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The Early History of the Universe (2) slight excess of ordinary matter over antimatter (by about 1 part in 10

25% of mass in helium 75% in hydrogen

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

No stable nuclei with 5 and 8 protons

Almost no elements heavier than helium are produced.

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Recombination slight excess of ordinary matter over antimatter (by about 1 part in 10

Protons and electrons recombine to form atoms => universe becomes transparent for photons

z ≈1000

Transition to matter dominated era

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The Cosmic Background Radiation slight excess of ordinary matter over antimatter (by about 1 part in 10

After recombination, photons can travel freely through space.

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


z = 1000; T = 3000 K

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

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The Cosmic Background Radiation slight excess of ordinary matter over antimatter (by about 1 part in 10

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

R. Wilson & A. Penzias

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

Blackbody radiation with a temperature of T = 2.73 K

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Cosmology with the Cosmic Microwave Background slight excess of ordinary matter over antimatter (by about 1 part in 10

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:

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Evidence for the formation of galaxies and large-scale structure

Fluctuations of the CMB temperature

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.

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Observations are consistent with Hot Big Bang Model 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.

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Fate of the Universe radiation

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.

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Deceleration of the Universe? radiation

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

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

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

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Model Universes radiation

r < rc => universe will expand forever

Maximum age of the universe:

~ 1/H0

r = rc => Flat Universe

Size scale of the Universe

r > rc => 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!

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Faint gas shells around ellipticals radiation

Ellipticals 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 cluster

Galaxy cluster members are moving too fast to be gravitationally bound unless there is unseen mass.

Hot gas in clusters

The 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 spectra

Absorption lines from hydrogen in quasar spectra tells us that there is a lot of material between us and the quasars.

Gravitational Lensing

Gravitational 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.

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The case of a missing Universe radiation

Observations suggest that the universe is flat:  = 1

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

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

The rest 70% is something else!!

This something else is termed “dark energy”

It causes the universe to accelerate in its expansion!

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Do we live in a radiationspecial 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

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Anthropic Principle radiation

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?

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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, …

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Landscape of the multiverse in the place we live

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