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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~ 1 AU
~ 10 kpc
~ 1 Mpc
~ 3 pc
~ 500 Mpc
Definitions and meaning of new units: AU, pc, kpc, Mpc
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
Small Angle Formula
Convert from radian to arcseconds:
1 deg = 60 arcmin = 3600 arcsec
Define the magnitude scale so that two stars that differ by
5 magnitudes have an intensity ratio of 100.
Remember parameters: perihelion, aphelion, semimajor axis
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
Uniform circular motion
III Kepler’s law:
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.
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).
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
Shift z =
The Doppler effect: apparent change in the wavelength
of radiation caused by the motion of the source
The Doppler effect allows us to measure the source’s radial velocity.
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.
Mnemonics to remember the spectral sequence:
Spectral class: G2
Surface temperature: 5800 K
Lifespan: 10 billion years
Composition by mass: ~ 71% Hydrogen, 27% Helium
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
Inner convective, outer radiative zone
Inner radiative, outer convective zone
CNO cycle dominant
PP chain dominant
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
The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.
L ~ M3.5only for main-sequence stars!
Rate of energy loss
Lifetime T ~ M/L ~ 1/Mp-1 = 1/M2.5 ; p ~ 3.5
T ~ 3x108 years
M = 4M;
Age of a cluster = lifetime of stars on the turnoff point
The lower on the MS the turn-off point, the older the cluster.
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)
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
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.
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!
Two populations of stars
Their main difference is in chemical composition
Population I – metal-rich
Population II – metal-poor
Metals: all elements heavier than helium
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
All elements heavier than He are very rare.
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.
Doppler measurements of the rotation curve + Kepler’s law
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
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
Conclusion: galaxies should interact and collide very often!
They collided even more often before
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!
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 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
Redshift z =
How come that z > 1 ??
First, relativistic Doppler effect is described by a different formula:
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.
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!
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!
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!
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.
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.
Protons and electrons recombine to form atoms => universe becomes transparent for photons
Transition to matter dominated era
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!
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
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:
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.
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.
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.
r < rc => universe will expand forever
Maximum age of the universe:
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!
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.
Absorption lines from hydrogen in quasar spectra tells us that there is a lot of material between us and the quasars.
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.
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!
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?
Linde, Vilenkin, …
Planck density 1094 g/cm3
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