Endpoints of Stellar Evolution:

White Dwarf Stars:

White dwarfs are the endpoints of evolution of stars like the Sun (near one solar mass). In fact, the mass limit for a white dwarf is the Chandresekhar Limit, 1.4 solar masses.

The radius of a white dwarf is about 10 ⁴ km, that is, planet-sized (the radius of the earth is about 6000 km). If a star were to shrink from about the size of the sun, 10 ⁶ km, down to 10,000 (a factor of 100), its volume (V=4/3 л r ³) would shrink by 100 ³ =10 ⁶. If little or no mass is lost and the volume shrinks by this factor, the density, ρ = M/V, would increase by a factor of 10 ⁶. So the density of a white dwarf star is on the order of one million grams per cubic centimeter (since the Sun’s density is about 1 g/cm³ ). The weight of one teaspoon of wd star material, at the earth’s surface, would be about one ton.

What halts the collapse of a white dwarf star? A white dwarf can be thought of as a gas of electrons (plus nuclei that we won’t worry about right now). Electrons are “fermions” which obey the Pauli Exclusion principle, which says that only one fermion can be in a given state. When the electrons fill all the lowest energy states, none can lose any energy (since all lower states are filled), so the electrons are at T (temperature)=0. They are called “degenerate.” It is also true that two fermions cannot be in the same place, so they resist being pressed closer and closer together, which results in “electron degeneracy pressure.” The collapse of a white dwarf star is halted by electron degeneracy pressure.

The more massive a wd star is, the smaller it is. Also, if a star contracts by a factor of 100, it should spin 100 ² = 10,000 times faster, by conservation of angular momentum. The sun rotates with a period of about 10 ⁶ seconds, so a wd star should rotated with a period of about 100 seconds. Its magnetic field should be stronger by the same factor.

The best known white dwarf star is Sirius B; the only one we have looked at is 40 Eridani B.

Neutron Stars and Black Holes:

In a supernova collapse and explosion, when an aging star runs out of nuclear fuel and collapses catastrophically, a “squeezed-down” remnant is left, which could be a neutron star or black hole. Essentially what happens is that the electron degeneracy pressure is overwhelmed, and the stellar remnant collapses until that collapse is halted by “neutron degeneracy pressure,” neutrons also being fermions. This happens because the electrons are used up in converting protons to neutrons: e + p à n + ν. With no electron degeneracy pressure, the stars collapses to a radius of about 10 km, where neutron degeneracy pressure halts the collapse. The star is a ball of neutrons, neutron-rich nuclei, and other exotic particles. It has collapsed from something like 10 ⁶ km to 10, a f actor of 10 ⁵. The volume decreases by (10 ⁵) ³ = 10 ¹⁵, so the density increases by a factor of 10¹⁵. Because some mass is lost and converted to energy, we say that its density is about 10¹⁴ g/cm ³. As was the case with the wd star, the neutron star should rotate very rapidly, so that if it shrinks by a factor of 10⁵, it should rotate (10⁵)² =10¹⁰ times as rapidly. If it rotated in 106 seconds, as does the sun, before collapse, it should rotate with a period of 10⁶/10¹⁰=.0001 seconds! Such rapidly rotating neutron stars can sometimes be observed as pulars, with periods as short as about .0015 seconds (1.5 milliseconds) –“millisecond pulsars.” This happens when the star’s rotation axis and magnetic axis are misaligned, and searchlight-like beams of radiation (visible, radio, x-rays), which radiate from the magnetic poles, sweep by the earth. The supernova remnant in the Crab nebula is a neutron star which we see as a pulsar.

Black Holes:

The maximum mass of a neutron star is no much more that 3 solar masses. If the supernova remnant has 5 or more solar masses, gravity will overwhelm neutron degeneracy pressure, and the star will collapse without limit, down, theoretically, to infinite density, no volume, a “singularity.” Once the surface of the star falls within the Schwarzschild radius, R_s = 2GM/c², nothing—light, particles, etc, can get out, the star is completely black. The Schwarzschild radius for the sun is about 3 km; if it were compressed to that radius it would become a black hole.

A black hole is a “space-time singularity,” it is a star which has in some sense left the universe, leaving behind only its gravitational field. Space is curved around it so strongly that light not only cannot get out, but will orbit the black hole forming a “photon sphere” around it.

How could we detect a black hole? The best chance is to find one in a binary star system, where mass from the second star, as it evolves to become a giant or supergiant, falls in on the black hole. When the matter falls into the black hole, it is compressed and heated, and emits x-rays. Such is the case with Cygnus X-1, a spectroscopic binary in Cygnus, and the first x-ray sources discovered in that constellation. It is an x-ray emitted, but only the spectrum of the companion star is seen, but that spectrum tells us that the dark star must be nearly 10 times the mass of the sun, far too massive to be a white dwarf or neutron star.

The observation of a large amount of energy coming from a very small volume in the center of the galaxy (about the size of the solar system) means that it is very likely that there is a black hole in the center of the galaxy, and perhaps all galaxies.

The Theory of Relativity:

The special theory of relativity (SR) is based on two postulates: 1) the speed of light is a constant, independent of the motion of the source or observer, and 2) the laws of physics are the same for all unaccelerated observers. From these postulates come the predictions of space contraction, time dilation, mass increase, relativity of simultaneity, and the equivalence of mass and energy: E=mc².

The general theory of relativity (GR) is a theory of gravity, and thus is appropriate for describing black holes as well as the universe as a whole, whose evolution is controlled (now, at any rate) by gravity. At the heart of it is the “principle of equivalence,” that says t hat a gravitational field and an acceleration (accelerated coordinate system) are equivalent. Thus gravity in not a “true” force, but the product of geometry, in this case, the geometry of space-time. Massive objects curve space and space-time, and this determines the path of light rays or particles.

The classical tests of GR are 1) the bending of starlight (now gravitational lensing), 2) gravitational redshift (the energy lost by light in climbing up out of a gravitational field), and 3) advance of the perihelion of mercury. It also predicts gravitational radiation. GR was published in 1916 and has survived until the present day as the only viable theory of gravity, but will one day have to give way to quantum gravity.

Galaxies, including the Milky Way Galaxy:

Our galaxy is the Milky Way galaxy. It is a flattened aggregation of some 200 billion stars, with a diameter of about 100,000 light years or 30,000 parsecs (30 kiloparsecs). The Sun is about 2/3 of the way out from the center in a spiral arm, which means we are about 10,000 parsecs or 30,000 light years from the center. The sun orbits the center of the galaxy every 250,000,000 years. When we look at the Milky Way in the night sky, and see a well-defined band of stars and star clouds across the sky, we are seeing exhibited the flattened shape of the galaxy and how it is tilted relative to our local horizon. The Milky Way is quite thin at the distance of the sun from the center, so we don’t see many stars if we look perpendicular to the plane of the galaxy. Our galaxy is an Sb normal spiral or SBb barred spiral galaxy. The curve of velocity versus distance from the center first increases with distance, showing that the inner part of the galaxy rotates much like it is rigid. Further out, the velocity should drop with distance, in the “Keplerian” region, very much like the planets slow down as we go out from the Sun. However the velocity curve flattens out, meaning that there is a huge amount of matter on the edge of the galaxy, which is not luminous (we can’t see it), the famous “dark matter.”

Hubble classisfication of galaxies; the “tuning fork” diagram:

See the text for the tuning fork diagram, in which elliptical, norma spiral, and barred spiral galaxies are arrayed according to their shape.

Normal spiral galaxies with loosely-wound spiral arms and very small nuclei are labeled Sc. Similar loosely wound barred spirals are called SBc. Somewhat more tightly wound spiral or barred spirals are Sb (like the Andromeda galaxy and perhaps the Milky Way) or SBb. Very tightly wound galaxies, with large nuclear bulges and spiral arms which more or less form a tight circle around the nucleus are Sa or Sba. Elliptical galaxies range, according to how flattened or eccentric they are, from E0 (spherical) to E7. There are also S0 galaxies (not to worry). More generally, galaxies are elliptical, spiral, or irregular.

Elliptical galaxies are elliptical in cross section, have no disk or spiral arms, and are very yellow (“red”) because dust and gas are mostly exhausted and star formation has ceased. Spiral galaxies have bluish spiral arms because that is where new young, hot, blue stars are forming, and yellowish nuclear bulges (older stars, Pop. II). Irregular galaxies are usually blue as well.

Galaxies are found in clusters, some of them small, like the “local group” of 25 or so galaxies dominated by the Milky Way, the Andromeda galaxy, and M33, and others with 2000 or more galaxies like the Virgo or Coma clusters. Clusters group into superclusters, so the local group and the Virgo cluster are part of the local supercluster.

The distribution of clusters and superclusters is inhomogeneous, with voids, walls, and other complicated features shown in some of the figures in the text. So the universe is only approximately smooth, or homogeneous. When the motion of galaxies in clusters is examined, it is found that not much more than a few percent of the total mass is actually luminous. Again, 90-99% of the matter is dark.

Cosmology

Cosmology is the study of the large-scale features of the universe. But the present large-scale features of the universe, which are now controlled by gravity, were determined by what happened in the early universe. There are two great pieces of observational data in cosmology: the expansion of the universe (Hubble Law, velocity-distance relation), and the cosmic microwave background radiation (CMB, 3K background).

Expansion of the universe: in the 1920s astronomers, especially Hubble found that most galaxies are moving away from us, and when their distances were measured using the Cepheid variable yardstick, with velocities which increased with distance. This, the velocity-distance relation, v=Hd, implies that the universe is expanding. Why doesn’t this mean that the Earth is in the center of the universe? Because what is happening is that the space between t he galaxies is expanding; every observer would see the same thing. We are not at the center and there is no center. Consider an expanding balloon with spots painted on it, being blown up. The spots all recede from each other, but there is no “center”. The center of the balloon is not part of the space (the surface of the balloon).

The Hubble constant, H, determines how fast the universe is expanding. H is somewhere between 50 and 100 km per second per megaparsec, so we take it to be 75: H=75 km per second per megaparsec. That means that if the Virgo cluster is about 20 Megaparsecs away, its members should be receding from us at 20x75=1500 km/.sec. A galaxy at a distance of 1 billion parsecs (1000 Megaparsecs) would be receding at 1000x75=75,000 km/sec, ¼ of the speed of light.

The theory which, at least to this point, describes the expansion of the universe is general relativity. It allows for the possibility that the universe is closed (enough mass), open (too little mass to close it), or flat. It predicts a universe t hat expands without limit, getting thinner and thinner until the “lights go out,” or one that expands to a maximum and then recontracts. The discovery that the expansion of the universe is accelerating instead of slowing down is the most remarkable discovery in cosmology since the CMB was discovered in the mid-60s.

As the early universe cooled, it eventually reached a temperature which allowed neutral hydrogen atoms to form and for the universe, which had been an opaque plasma (charged gas), to become transparent. The temperature was a few thousand K, below 10⁴ K. The CMB originates at this time, when the universe was about 300,000 years or over 10¹² seconds old.

When the universe was about 1 second old, the temperature was about 10¹⁰ K and the energy was too high for deuterium nuclei to stay together. As the temperature dropped, deuterium nuclei began to form, starting the process of producing helium. Thus the 25% contribution of helium to the universe dates from this time.

The Weinberg-Salam transition, when the weak force split from the electromagnetic force, occurred at about 10^-12 seconds, and a temperature of 10¹⁶ K. Still earlier, and hotter, was the time when the “GUT transition” occurred, as the strong force split off from the electroweak. This is also the era in which inflation occurred.

Inflation was introduced to solve 1) the so-called “flatness problem,” which is expressed in the question, why is the universe still so nearly flat that we can’t tell whether it is positively or negatively curved, after 13 billion years? And 2) the “horizon problem,” which arises because directions in the sky only 2^o apart could never have been “causally connected” (in contact with each other) and yet are at almost exactly the same temperature. Etc. The idea is that the universe was very tiny bubble of space and time, every point in contact with every other, which was inflated by a factor of as much as 10⁶⁰, forming the early universe and eventually the universe we know. Because it inflated by so much, it is now almost exactly flat. Thus, if inflation is right, the universe is flat. And if it is, about only about 4% of the mass is familiar to us, luminous, maybe 23% is dark matter, and the rest, 73%, is dark energy, energy associated with repulsive gravity.

Finally, we note that matter is made of fermions, which interact through fields carried by bosons. Example, electrons interact by exchanging photons. The weak force is carried by the W+/- and Z particles, the strong force between quarks is carried by the gluon. There are 6 flavors of quarks, up (u), down (d), strange (s), charmed (c), top (t), and bottom (b). There are 6 corresponding leptons, the electron, muon, and tau, plus neutrinos associated with each.

THIS IS THE END OF THE MATERIAL FOR THE THIRD EXAM

The Solar System

The planets are divided into the terrestrial planets: Mercury, Venus, Earth, and Mars; and the Jovian or giant gas planets: Jupiter, Saturn, Uranus, and Neptune. This division is based on the following properties:

terrestrial planets:

1) proximity to sun (source of all or most other properties)

2) high density (around 5 g/cm³)

3) small size

4) solid surfaces

5) slow rotation

6) thin atmospheres of CO₂ and N₂ (exception being Venus)

7) few moons

The properties of the Jovian planets are essentially the opposite of these, with a density of 1-2 g/cm3, and thick atmospheres of hydrogen, helium, methane, and ammonia. The terrestrial planets formed nearest the sun, where the temperatures were higher, and were unable to retain the light gases that eventually formed the gas planets.

The Moon

The moon averages about 240,000 miles from the earth (229-252,000) and has a diameter of about 2160 miles. It has a density comparable to the earth’s crust and mantle, and this leads one to believe t hat it was formed when a Mars-sized object struck the earth in the early days of the solar system, ejecting the moon-sized mass.

The moon rotates and revolves with the same period, 27-1/3 days, so that it keeps (almost) the same face to the earth; it is tidally locked-on to the earth in a 1:1 resonance. Because the moon’s orbit departs considerably from a circle (5.5% either way), the moon moves faster in its orbit near perigee and slowest at apogee. When it is revolving fastest, the revolution gets ahead of the rotation, and when it is moving slowest, the rotation gets ahead of the revolution. This is because the spin rate of the moon is constant. So we see a little around both the E and W sides of the moon, and can see 59% of its surface from earth. But the far side of the moon gets just as much light during a month as the near side. There is no permanent “dark side of the moon”

Both the moon and the sun cause tides on the earth’s oceans, but the moon is the major influence because being nearer, the difference between its attraction on the far vs. the near side of the earth is greater than that of the sun, which is far away and exerts nearly the same force on each side of the earth. The moon, however, raises high tides on both the side of the earth facing it and on the side facing away. The same is true of the Sun, so when they are in line, either at new or full moon, we have high high tides and low low tides. These are called spring tides. At first and last quarter the effects of the sun and moon cancel. These are neap tides.

Surface features of the moon:

1) craters–300,000 with diameters of 1 km or more; almost all impact craters. Typically they show a central peak, ejecta blanket, secondary craters, and a ray system. They are named after scientists and philosopers from antiquity. A crater like Copernicus, 56 mi in diameter, is about 2 miles deep.

2) maria (pl.; mare, sing.)–the so-called “seas” on the moon. They are large, more or less circular impact basins, which have filled with molten rock (“lava-filled basins”) which welled up from below. They have diameters up to several hundred miles. Mare Crisium, Mare Fecunditatis, Mare Tranquillitatus, Mare Serenitatus, Mare Imbrium, etc. The far side of the moon, while covered with craters, has few maria.

3) mountains–usually the walls or ramparts of large basins. For example, the lunar alps and appenines are walls of Mare Imbrium.

4) rays–radiating streaks from impact craters, consisting of light material thrown several hundred miles in the impact.

5) rilles–sinuous canyon-like crevasses which are thought to be collapsed lava tubes.

The major planets:

(My Very Excellent Mother Just Served Us Nine Pizzas)

Mercury—nearest the sun, hottest/second hottest (daytime side), heavily cratered, no atmosphere, rotates 3 times while orbiting the sun twice (3:2 resonance).

Venus—nearest the earth in size and mass, comes closest to the earth, day longer than year, dense CO₂ atmosphere resulting in runaway greenhouse effect, brightest, most highly reflective, large volcanoes on surface, some impact craters but fairly young surface, retrograde rotation, synodic period of 584 days, most nearly circular orbit.

Earth—

Mars—about half the size of the earth, comes close every 2+ years, surface covered by reddish (oxidized) sands, large shield volcanoes, considerable cratering, water apparently once flowed on surface, two polar caps of frozen CO₂ (dry ice) and water, large canyons (Valles Marineris), two small moons, least dense of the terrestrial planets….

Jupiter—largest planet, most rapidly rotating, most moons, mostly hydrogen and helium, with dense atmosphere overlying liquid and solid metallic hydrogen interior, some methane and ammonia the chemistry of which gives rise to colors, including the equatorial belts, great red spot, etc., strong magnetic field, thin ring, Galilean moons Io, Europa, Ganymede, and Callisto. Io has erupting sulfur volcanoes, Europa seems to have a smooth frozen crust over a salty ocean, Ganymede is largest moon in solar system, Callisto is darkest, most heavily cratered. Io and Europa about size of moon, Ganymede and Callisto about the size of Mercury. More massive than all the other planets combined.

Saturn—second largest planet, spectacular ring system, many moons. Rings made of a myriad of small particles: rocks and ices; rings only a few km thick, largest moon Titan is only moon in the solar system with a dense atmosphere (N₂, methane). Least dense planet. Second most rapidly rotating.

Uranus and Neptune—about —4X the size of the earth, greenish (Uranus) to bluish (Neptune), due mainly to methane, Neptune blue with “great blue spot”, Uranus’ spin axis is nearly in the plane of its orbit Both have rings. Uranus barely visible to naked eye. Uranus’ moon Miranda seems cemented together from disparate pieces. Uranus discovered in 1781 and Neptune in 1845/6.

Pluto—most distant, coldest, probably should be considered a “Kuiper belt object” rather than 9^th major planet, retrograde rotation with period of 6.3d, one moon (largest in comparison to primary), most eccentric orbit, most highly inclined orbit, nearer the sun t han Neptune 20 years out of 248 year period (1979-99).

We will skip meteoroids, asteroids, and comets(?).