Phys 171 Midterm Review

Scientific Reasoning

• A scientific hypothesis should be falsifiable.
• A successful scientific hypothesis should be consistent with previously accepted theories and or experimental results.
• A scientific hypothesis should have predictive power.
• Science prefers simpler and/or "more elegant" hypotheses.
• Experiments and observations are the final arbiter of scientific questions. A scientific theory must make successful predictions of experimental or observational results in order to be believable.

Brahe, Kepler, Galileo, and Newton

• Brahe's main contributions were his accurate observations which laid the observational foundation for Kepler's laws and his rejection of Aristotelean dogma when contradicted by observational evidence.
• Kepler's main contribution was his laws of planetary motion:
• The planets orbit in elliptical paths with the Sun located at one focus of the ellipse.
• A planet speeds up when it is closer to the Sun such that the "pie shaped" area swept out by the line connecting the planet to the Sun traces out equal areas in equal times.
• A planet which is closer to the Sun will have an orbital period that is shorter than the orbital period of a more distant planet. The orbital radius (or semi-major axis) cubed times the period squared is a constant.
• Galileo's accomplishments included:
• The "establishment" of the scientific method.
• The first use of the telescope as an astronomical instrument which led to discovery of mountains and craters on the moon, thousands of stars invisible to the naked eye, sunspots, Venus' phases, Saturn's "ears" (now known to be rings), the 4 large moons of Jupiter, the observable angular size of the planets, and the fact that the Milky Way is composed of many thousands of faint stars.
• His famous book: "A Dialog on the Two Chief World Systems, Copernican and Ptolemaic" which helped establish the correctness of the Copernican model but also led to Galileo's arrest and incarceration by the Roman Catholic Church.
• Newton is generally considered (by physicists, at least) to be the greatest scientist that ever lived. His accomplishments included:
• The theory of dynamics which explains the motion of bodies and forces.
• His universal law of gravitation which explained both terrestrial gravity and the forces affecting celestial bodies with a single theory. One puzzle was the concept of "action at a distance" that this theory required.
• The invention of the reflecting telescope.

Distance Measurements

• The size of the Earth and the distance to the moon can be determined
• The measurement of stellar parallax allowed the accurate determination of the distances to nearby stars.
• The distances of more distant objects can only be determined if we have some physical understanding of what they are.

Electromagnetic Radiation and Atoms

• electrons in atoms have decrete energy levels
• only one electron is allowed in each "state"
• spectral lines are caused by photon emission or absorption when an electron changes energy levels.
• spectral lines can be seen in emission when observing a diffuse hot gas
• spectral lines can be seen in absoption when observing a diffuse backlit gas
• a solid or dense gas radiates a continuous spectrum of radiation.
• this radiation has the "black body" form
• with _max proportional to 1/T (temperature).

Stellar and Galactic Astronomy

• The use of spectroscopy in astronomy allowed astronomers to determine both the chemical composition and the radial velocity of distant astronomical objects. Students should know how this works and they should know about the Doppler effect.
• Normal stars are supported by the pressure generated by the heat from nuclear reactions
• "Dead stars"
• electron or neutron degeneracy pressure gives a maximum mass which controls the fate of "dead stars"
• stars like the Sun die as white dwarfs supported electron degeneracy pressure.
• more massive stars explode as type II Supernovae
• Type I supernovae are caused by the explosion of white dwarfs in binary star systems.
• Cepheid variable stars were found to be a particularly useful distance indicator through the Cepheid "period-luminosity" relation.
• Using Cepheid variable stars discovered in nearby Galaxies, Hubble discovers that spiral nebulae are actually external galaxies. Spectra of these external galaxies leads Hubble to propose "Hubble's law" which states that galaxies are receding from us at a velocity proportional to their distance.

Einstein's Theory of Gravity

• the speed of light is the "universal speed limit"
• space-time diagram: only the "past" light-cone can affect a space-time point.
• General Relativity: Einstein's Theory of Gravity
• Equivalence Principle: gravity = acceleration
• curved space-time
• explains details of Mercury's orbit
• gravitational redshift
• gravitational bending of light rays
• Black Holes
• escape velocity > c (the speed of light)
• Event Horizon at the Schwartschild Radius, R_sch
• Light rays emitted from inside the event horizon cannot escape.
• Observer will have no escape once inside the event horizon.
• Observer may not notice falling through the event horizon unless the black hole is small so that tidal forces are large.
• An outside observer never sees anything move beyond the event horizon.
• singularity: a black hole's mass is all at this point
• photon sphere: at R = 1.5 R_sch light rays are bent so that photons can orbit the star in circular orbits.
• Dead stars with masses above 3M_sun are too massive to be neutron stars, and so they are expected to become black holes. (Note that stars more massive than 3M_sun do often create neutron stars because most of their mass is lost in a supernova explosion.)
• Black holes can be very bright because the matter falling into them can release a lot of energy.
• The centers of most galaxies are thought to have black holes of 10⁶ - 10⁸ M_sun.

The Expanding Universe

• Hubble discovered the apparent expansion of the Universe.
• velocities easy to determine
• distances difficult
• Interpretation of Hubble's result
• expansion with no center
• Cosmology with Einstein's Theory of gravity
• predicts expanding or contracting Universe
• Einstein fudged with "cosmological constant"
• homogeneous and isotropic - assumptions
• Olber's paradox

The Big Bang Theory and Its Observational Predictions

• An expanding Universe can:
• have low density, have negative spatial curvature, and expand forever.
• have critical density, have zero spatial curvature, and expand forever reaching zero expansion velocity at t = infinity.
• have high density, have positive spatial curvature, and recollapse in a "Big Crunch".
• A positive cosmological constant acts yields a repulsive gravitational force.
• An exotic form of matter known as "false vacuum" has negative pressure and provides exactly the same gravitational force as a cosmological constant.
• Recent observations of distance supernovae seem to indicate the presence of a cosmological constant (or "false vacuum" matter).
• The Supernovae of type Ia are used as distance indicators. These are white dwarfs which acrete matter from a binary companion and then explode when they reach the maximum white dwarf mass.
• Helium and trace concentrations of Deuterium and Lithium are formed in the hot Big Bang. The Big bang prediction for the Helium abundance agrees with observations.
• The formation of most elements heavier than Helium can be explained by nuclear reactions within the stars and then dispersed through supernova explosions.
• Alpher and Hermann showed that the Big Bang predicted a cosmic microwave background radiation with a temperature of 10 degrees above absolute zero or less.
• The abundance of Deuterium produced by the Big Bang tells us how many baryons there are in the Universe.

The Cosmic Microwave Background Radiation

• In 1965, a group at Princeton rediscovered Alpher and Hermann's prediction of the cosmic microwave background radiation, and began to build an experiment to search for it.
• Before they could begin their observational program, the results of Penzias and Wilson at Bell Labs came to light, and it was realized that they had discovered the cosmic microwave background radiation (CMBR). When the thermal or "blackbody" nature of the CMBR was confirmed by subsequent experiments, this came to be regarded as a strong observational confirmation of the Big Bang Theory.
• After Penzias and Wilson's discovery, the study of the CMBR emerged as a new field of astrophysics. CMBR research focused on two problems:
• The study of the CMBR spectrum. Does it follow the thermal radiation or "blackbody" curve as predicted? A small departure from the blackbody curve might indicate a significant amount of energy generation early in the history of the Universe.
• The study of anisotropy in the temperature of the CMBR across the sky. This can be caused by:
• The motion of the Earth with respect to the CMBR radiation rest frame. This yield a "dipole" anisotropy.
• Small perturbations in the density of matter across the sky. These are the seeds for the formation of galaxies, and they are easier to understand theoretically than galaxies because galaxy formation is a complicated process.
• The study of the CMBR poses a number of observational difficulties due to the fact that its temperature is so low and because the atmosphere emits and absorbs radiation at wavelengths shorter than and close to the peak of the CMBR spectrum. Because of this, CMBR experiments are done from such places as:
• mountain tops
• high altitude balloons
• cold, dry sites such as Saskatoon or the South Pole
• outer space
• The most important CMBR results since the discovery of the CMBR have come from the COBE satellite. These include:
• The very precise measurement of the CMBR spectrum by the COBE-FIRAS experiment.
• The first detection of CMBR anisotropy beyond the dipole term. This is interpreted to be a detection of primordial density perturbations of the kind that served as seeds for galaxy formation.
• The amplitude of the CMBR anisotropy detected by the COBE-DMR experiment is consistent amplitude expected to be caused by primordial "seed" density perturbations for a Universe dominated by dark matter.
• Future CMBR satellites known as MAP and PLANCK will study the CMBR anisotropy on scales as small as 0.1-0.2 degrees (compared to 7 degrees for COBE). These may allow a test of the Inflationary Universe Model.

The Dark Matter Mystery

• There is strong evidence for the existence of dark matter in
• Galaxies
• The rotation curves of spiral galaxies tend to remain flat (v(r) = cont.) as far from the center of galaxies as they can be measured. This means that the mass of the galaxy must be dominated by a dark component that extends well beyond the visible extent of the galaxy.
• The observed orbital motion of small satellite galaxies about larger galaxies like the Milky Way also indicates that the galaxies must have a substantial mass that is not in the form of stars.
• Clusters of Galaxies
• The observed orbital motion of the cluster galaxies in the cluster indicates that the cluster must have more mass than the stars in the galaxies can provide.
• Background galaxies are gravitationally lensed by the cluster and are bent into thin arcs. This requires the clusters to have a substantial dark mass.
• There are also theoretical arguments for dark matter:
• Theoretical galaxy formation scenarios only seem to work if there is a substantial amount of non-baryonic dark matter in the Universe.
• The theory of Big Bang nucleosynthesis can account for the production of Helium and Deuterium (heavy Hydrogen) if the total density of of baryons is much less than the critical density but larger than the observed density of stars. This suggests that some baryonic dark matter must exist, but that there may be some non-baryonic matter as well.
• Theoretical arguments such as the Inflationary Universe model imply that the Universe must have the critical density which implies that there must be a great deal of dark matter.
• Candidates to comprise some or all of the dark matter include:
• Cold Dark Matter (particles with small velocities)
• Weakly Interacting Massive Particle (or WIMPs). These are predicted by some particle physics theories.
• Axions. Another particle dark matter candidate.
• Hot Dark Matter (particles with large velocities)
• Neutrinos with a mass of about 10^-8 of a proton.
• MAssive Compact Halo Objects (or MACHOs). These are objects with masses between that of the moon and about 100 times the mass of the Sun. Prime candidates include planetary mass objects or stars without enough mass to burn their nuclear fuel (these are brown dwarfs), the remnants of dead stars, or black holes. These can be detected by gravitational microlensing.
• Dark Matter Searches:
• WIMPs can be detected by looking for the effects of rare collisions of WIMPs with solid detectors as the WIMPs pass through the detector on their orbit through the Galaxy.
• MACHOs can be detected via gravitational microlensing of stars in external galaxies such as the Large Magellanic Cloud. This is the only experiment that has detected any dark matter at present. It is also the research project of you esteemed professor, so it would be good for you to know about it!!

Beyond the Big Bang Theory

• The Inflationary Universe Model. This model postulates an early period of accelerating expansion due to the repulsive gravitational force of a "false vacuum" state which may have existed in the early universe. This model seems to explain:
• The "horizon problem": why does the CMBR temperature appear to be the same across the sky to 1 part in 100,000 when regions separated by more than 2 degrees on the sky have not been able to communicate with each other since the Big Bang?
• The "flatness/oldness problem": why did the Universe start out close enough to the critical density to expand for 15 billion years rather than re-collapsing or expanding to very low density in less than a second?
• The origin of the primordial density perturbations. Inflation provides a way to generate the right kind of perturbations.
• The Inflationary Universe Model scheme for producing the primordial density perturbations may be tested by the upcoming CMBR anisotropy satellites: MAP and PLANCK.
• Origin of the Universe theories. There are a number of theories which seek to explain the origin of the Universe. They all use the inflation idea, and it is not clear that any of them can be tested observationally. These include:
• The creation of the Universe from Nothing. The Universe (of 0 energy) is created by a quantum process from a state of "nothing".
• The Eternally Inflating Universe. The Universe continues forever in an inflationary expansion, but every once in a while, a piece of the Universe stops expanding. These regions become ordinary "big bang" universes like ours.
• Chaotic Inflation. This is a variation of the inflation model. It holds that the Universe starts in a "chaotic" state, but that some small fraction of the Universe will happen, by chance, to be dominated by the energy density of the "false vacuum". These regions will start inflating and eventually generate a Universe like the one we live in.
(These have been covered very superficially, so any question on these would be quite easy.)

Possible Calculation Problems

• The equation describing the relationship between brightness, luminosity, and distance: 
• The 1/d² dependence of brightness on distance, d, is the crucial feature.
• The equation for the Doppler shift of light given in the class notes for Thursday, Feb. 25.
• The Cosmological Redshift Formula: 
• The Temperature-expansion relation: