| Chapters 5 - 7 Study Guide | Chapters 12 - 14 Study Guide | Wadsworth's Home Page |
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for Michael A. Seeds, "Foundations of Astronomy" Fourth Edition (1997) Copyright © 1996 Heather L. Preston |
Study Guide for Seeds text, Chapters 8 - 11
| Chapters 5 - 7 Study Guide | Chapters 12 - 14 Study Guide | Wadsworth's Home Page |
|
for Michael A. Seeds, "Foundations of Astronomy" Fourth Edition (1997) Copyright © 1996 Heather L. Preston |
This summary goes chapter-by-chapter quickly; it does not take the place of reading the text, but provides essential and some supplemental information, summaries, and alternative explanations. Also Study Questions for Chs 8 - 11
Use the book and your notes to fill in wherever you are having trouble with understanding something -- and ask your instructor questions! :-)
Conventions: In the on-line version, I still can't do Greek letters (without cheating them in as images), so "pi" and "sigma" (the Stefan-Boltzmann constant, s in the Ch 5 - 7 study guide) are spelled out. In cases like that, multiplication of these constants is denoted by "*". Division is always denoted by "/"
Formed from the gravitational collapse of a huge cloud of gas (90% H and 10% He by number of atoms, with traces of heavier elements) and dust, the solar system started as a spinning disk of gas with one very massive lump (to be the Sun) and a couple of much less massive lumps (Jupiter and Saturn). If the mass had been distributed a little more equally, the system would have been a low- mass binary star system, but Jupiter fell short of the necessary mass to be a star by a factor of about 100.
Structure - The atmosphere of the Sun - photosphere, chromosphere, and corona - extends from the visible surface (photosphere) out to about 7 solar radii (corona). The photosphere is the opaque surface of the Sun, so it is thin -- about 500 km thick. It is the point at which the photons escaping from the center of the Sun have their last interaction with the Sun's gas (on average) and start flying freely through space. It has an absorption spectrum due to the thermal radiation coming from beneath it. The temperature at the depth to which we can see is about 6000 K. The surface looks mottled; this is called granulation. Each granule is the top end of a huge convective cell about 1500 km in diameter, where hotter gas is rising from hotter, lower regions in a single packet up toward the surface. At the edge of the granule the cooling gas is drawn back down to replace the mass displaced by the upwelling -- so the whole granule is circulating, coming up in the center and sinking at the edges. The granule is therefore brighter (hotter) in the center and darker (cooler) at the edges.
Supergranules are bunches of about 300 granules that appear to be related by being part of a more general upwelling of a large subsurface current that manifests itself on a timescale of about a day. They are about 30,000 km in diameter. Around the edges of the supergranules, spicules rise from the top of the photosphere some 12,000 km (through the chromosphere) like weeds around flagstones. Their temperature is about 10,000 K. In addition to sudying the surface motions by sensitive Doppler measurements, we can study the oscillation modes of the Sun in a process called helioseismology -- similar to the study of Earth's interior using earthquake propagation information. This can give us some idea of what's going on in the Sun's interior.
The chromosphere has an emission spectrum, which you can only see when the photosphere is blocked, such as during a total solar eclipse. Then it briefly glows bright pink, like a glowing gas-tube sign. It is a thin, hot gas, much thinner (less dense) and hotter (10,000 K at the bottom, up to 500,000 K at the top) than the photosphere. It is about 10,000 km thick, and at the top it sort of runs into the corona.
The corona is the top of a rapid temperature-rise zone where the Sun's atmosphere goes from 500,000 K up to as much as 3,500,000 K. It is extremely low-density, so that having a million-degree temperature doesn't even make it glow -- most of what you see with a coronagraph or during a total solar eclipse is normal sunlight reflecting off the dust and free electrons in the corona, so a continuum (coronal gases are moving so fast due to the high temperature that solar spectrum's absorption lines are too smeared out by Doppler shifts to see) with faint superimposed emission lines from the actual coronal gases -- highly ionized gases! The corona extends out past about 30 solar radii, but the extent seen really depends on how sensitive a detector you have and how far out of the Earth's atmosphere you put it! Remember, out at 30 solar radii you're only looking at 1 to 10 atoms per cm3.
What heats the corona to these incredible temperatures? Interactions with the Sun's magnetic field. This is also why the corona is not usually perfectly round when viewed during an eclipse, and also why at times of greater magnetic activity (sunspots, prominences, etc.) the corona appears to be much larger and fuller than at quiet-Sun times.
At some point out there, the overall high velocity of the gases in the corona becomes an overall gas flow outward from the Sun. The Sun loses about ten million tons of H, He, and dust from the outer edges of the corona each year. This is called the solar wind, and it moves past the Earth at about 400 km/sec. This mass loss is negligible compared to the mass of the Sun -- it would take 1014 years for the Sun to lose all its mass that way! (Q: Compare this number to the expected lifetime of the sun)
The Sun's magnetic field is reponsible for most of the other interesting stuff that we see. Basically, because the Sun is gaseous, the equatorial regions rotate faster than the polar regions, taking 25 days to complete a complete rotation at the equator, but up to 29 days near the poles. This is called differential rotation. Since the magnetic field lines in the Sun's surface are dragged along by the ionized parts of the gas (charged material doesn't move easily across magnetic field lines, and vice versa), the differential rotation stretches and winds up the magnetic field lines in the solar surface until they are nearly horizontal!
The increase in tension in the magnetic field leads to kinks and twists and loops of magnetic field lines popping out of the surface, and this is how sunspots are created. Where the field line pops out of the surface, there will be a sunspot, a region of lower temperature and high magnetic field with a specific polarity -- either N or S. Where the end of the loop re-enters the surface, there will be the second sunspot in the pair, which will of course have the opposite magnetic polarity. This is why sunspots usually happen in pairs -- they are the two points at the ends of a loop of magnetic field that's popping out of the solar surface.
Often when a loop of magnetic field pops out like this, some of the photosphere is lifted out with it (remember, the charged portion of the gas doesn't travel across field lines, only along them!), and this is called a solar prominence. If it is a slow-building thing, it is a quiescent prominence, and it can last weeks or even months. If it happens fast, it will probably push out farther, bursting out up to 500,000 km above the photosphere in just a few hours. This is called an eruptive prominence.
Of course, if the field has enough tension to kink, occasionally a field line will essentially "snap", slingshotting solar surface material outward at up to 1/3 the speed of light! This is called a solar flare, and it can really mess up your spacecraft electronics. The flares take only a few minutes to rise to eruption, then decline in an hour or less. A flare can release the energy equivalent of exploding more than 2 billion megatons of TNT, much of it in X-rays and UV and visible light, and the aforementioned high-speed charged particles (protons and electrons). When the particles encounter the Earth's magnetic field, they are channeled by the field lines toward the poles, where they encounter the upper atmospheric gases and set off beautiful displays of light called the auroras -- both northern and southern polar regions experience this. Flares can mess up the usual ham radio "bounce" off the Earth's ionisphere, and auroras can be accompanied by magnetic storms in which compasses and other flight instruments behave erratically.
When the field lines have been wound up to sufficient tension, they will start to reorganize themselves in a process called reconnection. Accompanying this reorganization of the solar magnetic field is a swap in the position of the Sun's N and S magnetic poles. It takes 11 years for the field to go through the tension cycle, from very little field tension and a few sunspots at high solar latitudes (35 degrees above and below the Sun's equator) to a great deal of tension, lots of sunspot groups, prominences, and flare activity, and the majority of the spots being seen closer and closer to the equator (5 degrees above and below the equator). Since it undergoes a polarity swap when it reorganizes, the complete cycle of the magnetic field (back to its original polarity) takes 22 years, while the cycle of the number of sunspots seen (related to the tension in the field) only takes 11 years. The diagram below is called the Maunder butterfly diagram, in honor of E. Walter Maunder, an English solar astronomer who published the first one in 1922. "Butterfly" is because it looks a bit like a butterfly's wings. It is a plot of sunspots at their latitudes versus the time they were observed. A good sunspot group can last months, although the average spot, larger than two Earths, only lasts a week or two.
If the Sun's energy output varies by more than a percent or so, it can have a serious effect
on the weather! The Maunder Minimum, a 50-year gap in the sunspot cycle, occurred in the midst
of the "little ice age" from 1430 to 1850, which may have been caused by such a decrease.
Figures for CH 8: Sun's magnetic field cycle and the Maunder "butterfly" diagram
Geometric parallax is the original way to measure distances to stars. It is measured by taking a very precise "photo" of a star's position against the background of more-distant stars, then taking another "photo" exactly the same way six months later and measuring the apparent shift in the star's position from one "photo" to the next (it's not actually a snapshot!). half of that shift, measured in seconds of arc (arc-seconds, or "), is defined as the "parallax angle" of the star. If we know the parallax angle in arc seconds, we know the distance to the star in parsecs! One parsec is defined to be the distance at which a star would have a parallax angle of one arcsecond. Any smaller shift, and the star is farther away than 1 pc. Guess what: all of them (except the Sun) are farther away, so all measured parallaxes for stars are less than one arcsecond -- really hard to measure!
1 pc = 3.26 light-years = 206,265 A.U., and the geometric parallax formula can be written two ways: either d=1/p or p=1/d.
The other method of measuring distances has to do with looking at how bright the star appears to be in our sky (by measuring its apparent magnitude), at whatever its true distance is. That apparent magnitude (for which we use the symbol mv ) is affected both by the actual luminosity of the star and by its distance from us, so it's something that we must measure (except in hypothetical problems!). Remember, since it's Hipparchus's old magnitude scale, the more negative the number, the brighter the magnitude (like rank or importance, first is 'better' than ninth).
How does the apparent magnitude tell us the distance? We need one other piece of information -- we need the intrinsic brightness of the star. Because astronomers work in "magnitudes," the way we compare stars' intrinsic brightnesses is called absolute magnitude, for which we use the symbol Mv. Absolute magnitude is defined as the apparent magnitude a star would have if it were at a distance of 10 pc from us. If we lined them all up at 10 pc, we could then compare their intrinsic brightnesses, since the distance factor would be removed. Therefore, unless the star is already at a distance of 10 pc, absolute magnitude is something that you have to be told, or that you must determine in some other way (not through simple measurement). If you are told the absolute magnitude and you measure the apparent magnitude, you can determine the distance using the distance modulus:
mv-Mv = 5log(d) - 5
where d is the distance in parsecs and mv and Mv are the apparent magnitude and the absolute magnitude. The log(d) just means that if d=10, log(d)=1 and if d=100, log(d)=2 and if d=1000, log(d)=3 and so forth; basically, it counts the power that 10 is raised to. The only part that requires a calculator is when your distance is not a direct multiple of ten! Then, in most calculators, you use the log function key.
So, with the distance modulus (mv-Mv) and the parallax formula (d=1/p), given any two values (mv, Mv or d) you should be able to find the third, and if you're given p you can find d.
Once astronomers started being able to measure distances to stars, they could begin to get some idea of what their intrinsic luminosities were, and they discovered that most stars they could see fell along a single curved line on a plot of luminosity (or absolute magnitude) versus spectral type (or temperature) -- this was the origin of the Hertzsprung-Russell diagram (see figure). The line along which most of the stars fell was called the main sequence (this line was given a Roman- numeral spectral class designation of V to distinguish it from other lines on the diagram along which other types of stars would fall). The strengths of the Balmer absorption lines of Hydrogen in a star's atmosphere are due to its surface temperature, and so determining these line strengths from stellar spectra gave astronomers a quick way of determining stellar temperatures. A G0 star would be right on the definition of a G-star's Balmer-line strengths, but a G5 would be halfway between a G star and a K star in line strengths. Our Sun is a G2 spectral type, and since it is on the main sequence (spectral class V), its complete spectral designation would be G2V.
As you know, once the temperature is known, the energy being radiated per square meter
of surface area is given by the Stefan-Boltzmann equation, E = sigma*T4 (see previous study
guide!), and if we want to get the total luminosity (Joules/sec) being given off by the star as a
whole, we must multiply the energy per square meter by the total number of square meters of the
star's surface. Fortunately, stars are spherical, so this is easy: the surface area of a sphere is
4*pi*R2 so the total luminosity is: L= (4*pi*R2)(sigma*T4) where R is the radius of the star
and T is its surface temperature and everything else is a physical constant that normally we'd look up,
but since we're comparing stars to stars, we can eliminate the constants by taking a ratio. Thus, we choose to describe the star's radius in terms of solar radii, the temperature in terms of solar temperature, and the luminosity in terms of solar luminosities, and we come out with this nice simplification of the Stefan-Boltzmann law:
L*/Lo=((4*pi*R*2)(sigma*T*4))/((4*pi*Ro2)(sigma*To4))=(R*2/Ro2)x(T*4/To4)
which is just (L*/Lo)=(R*/Ro)2x(T*/To)4 == Luminosity of a star in units of Solar Luminosities.
This is called the Luminosity Equation.
For example, if a star has twice the sun's temperature and three times its radius, then its luminosity in terms of solar luminosities is:
(L*/Lo) = (3Ro/Ro)2x(2To/To)4 = (3)2 x (2)4 = 9 x 16 = 144 which means that the star is 144 times as luminous as the Sun. If it were twice the Sun's radius and the same temperature as the Sun, then it would only be four times as luminous, since the temperature ratio would be 1! (Q: If it were the same radius as the Sun but twice as hot, what would its luminosity be, in solar luminosities?)
Thus, when we look at the Hertzsprung-Russell diagram of the various spectral classes, we can see that the big difference in brightness between the G2V class and the G2III class cannot be due to temperature, since the temperatures are the same -- it must be due to the stars' having different radii. The brighter G2III star must be a giant compared to the G2V star to be five magnitudes brighter -- and so it is, class II stars are called "giants." Likewise, the class Ia and Ib stars must be REALLY humongous, so these are called "supergiants." Finally, the really dim guys down at the lower left of the diagram are very hot, so they ought to be bright per square meter; therefore we KNOW they are small. They are called "white dwarfs."
If we can determine a star's spectral type and class by analyzing its spectrum, then we can place it approximately on the Hertzsprung-Russell diagram. If we can figure out where it should be on the H-R diagram, we can read across to the right-hand axis and find out what its absolute magnitude is (approximately).
At that point, if we have measured its apparent magnitude (mv) and we have determined its approximate absolute magnitude (Mv), we can use the distance modulus equation to get the distance to the star! Since this concept allowed astronomers to find distances in a new way, where before that the only way to measure distance was by measuring the parallax angle, this new method was dubbed "spectroscopic parallax."
Figures: Determining Geometric Parallax, and the Hertzsprung-Russell diagram
We know how to find the temperatures of stars, and by analogy with the Sun we can estimate their radii. The main property of the stars that we have not yet learned how to measure is the mass. For the Sun, fairly easy! But for other stars, we must use binary systems -- so we see them interacting with other massive bodies.
Now that we know how to find stellar masses (at least of binary systems!), we can write in the masses beside the various points representing the specific stars to which they belong on the Hertzsprung-Russell diagram. We already know that the range of surface temperatures is around 2500 K for late M stars up to 100,000 K for extremely hot O stars. Now we discover that the range of stellar masses is really not that large -- from 0.08 solar masses at the lowest end of the main sequence to 80 or more at the high end (the book says 50, but there is evidence for even more massive stars). We compare this to our earlier observation that the range of stellar luminosities along the main sequence of the H-R diagram is from 10-6 times solar luminosity to 106 times solar. This implies a range of stellar main-sequence lifetimes from a few hundreds of thousands of years (highest mass stars) to more than 100 billion years at the low-mass end. For comparison, the Universe is less than 15 billion years old.
The interstellar medium is the gas (mostly Hydrogen) and dust (carbon, silicates, etc.) between the stars. Its density is typically between 1 and 10 atoms/cm3, but in dense molecular clouds it can rise to 100 or more.
Any gas cloud you can see is called a nebula (means "cloud" in Latin). The kind that have an emission spectrum are called emission nebulae -- they occur when there is a source of ionizing radiation (usually a hot star or cluster of hot stars with a lot of UV radiation) near or in a gas cloud. Many of the H atoms are excited or ionized (see previous study guide), and when the electrons cascade back down to ground level they emit photons at the exact energies and wavelengths corresponding to the energy level separations in the atoms.
If a LOT of UV is being produced so that a lot of the H is ionized, we call that an H II region (when used beside a chemical symbol for an element, such as H, He, Ca, Fe, Roman numerals signify the number of electrons stripped from the atom, plus one. So H II has had its only electron removed, and Fe IX has had eight electrons removed.). H II regions are often found around clusters of young, high-mass stars. If there's not a lot of UV, but there's a wispy blue nebula surrounding a star or group of stars, the dust in the cloud is reflecting the starlight. This is called a reflection nebula. A dark nebula is a fairly dense cloud of dusty gas, dusty enough that we can't see through it, and not excited enough to radiate visible light. However, it can be detected in radio. Here's how:
Here is the hydrogen atom diagram (again). It is basically just an electron "orbiting" a proton in our simplified picture. The electron has a negative charge and the proton has a positive charge. They also both have "spin." If the electron is spinning in the same sense as the proton, it's at a very slightly higher energy than if it were spinning in the opposite sense, and so, if the H atom is left alone long enough, everything tends to find its lowest energy state, and eventually the electron will flip its spin. This tiny drop to a lower energy requires that the electon emit a very low-energy photon, a 21-cm radio photon. So, 21-cm radiation is the signature of cold clouds of hydrogen gas. If they were warm clouds, the atoms would knock together more frequently and disturb the "settling" of the electron!
When a shock front, such as would be caused by the explosion of a supernova nearby, encounters a fairly dense molecular cloud, the cloud will begin to fragment and condense into a cluster of protostars. As these clumps in the gas cloud contract, the pressure and temperature of the gas increase. Once the temperature is a few hundred degrees Kelvin, the protostar is glowing with infrared radiation from the gravitational contraction of the clump. It is only this heat of contraction that makes the protostar glow, but it will continue to heat up as it contracts, eventually glowing brightly in the visible from the heat of its own contraction.
Ultimately, the temperature at the core of the contracting protostar will rise above 10 million degrees K. At that point, the star will start fusing H into He in its core, and it can truly be said to be a star. Its main-sequence lifetime has begun.
The process of fusing four H nuclei to create one He nucleus (plus energy) is the energy source for all main-sequence stars. But there are two main ways of accomplishing this fusion, and they occur at different rates depending on the core temperature. For low-mass stars (below 1.1 solar mass) the core temperature never gets very far above 10 million degrees K, so the dominant H fusion mechanism is the proton-proton chain. This consists of building He from H by direct H-H fusion (one proton at a time). For higher-mass stars the core temperature is above 15 million degrees K, and the CNO cycle begins to dominate. This occurs when H fuses with Carbon, which essentially acts as a rendezvous point for the H atoms. Three more H atoms join the fused atom, producing Nitrogen and then Oxygen before the final fusion produces a newly-made He nucleus plus the original C atom all over again. In all of this activity, the net result is that four H nuclei are joined to form one He nucleus, plus energy. If you subtract the mass of a He nucleus from the combined masses of four H nuclei, you have a little mass left over. That is the mass that was converted directly into energy. The amount of energy is given by the famous equation: E = mc2 where m is the mass difference between the four H nuclei and the He nucleus, c is the speed of light (3x108m/sec). In fusion to create a single He nucleus, the energy released is 4.3x10-12 J. It takes the gamma-ray photons from fusion 106 years to reach the Sun's surface as visible light.
Neutrinos are very low-mass, high-speed (near lightspeed) particles that are created as a by-product of fusion. Scientists expect a certain very specific number to be created per second in the Sun's core, and these travel out of the Sun almost as if it weren't there -- neutrinos have a near-zero cross-section, so they don't interact with any other matter very often. The mystery about this is that the actual number of neutrinos detected (the expected number has been calculated and checked over and over) is about 1/3 the number expected. So what's going on? Possibly the Sun is experiencing some kind of fusion lull lately, the full effects of which will reach us a long time from now when the rest of the photons start to reach the Sun's surface. Or possibly neutrinos naturally oscillate between three basic types, and only one type is detectable by the existing experiments. A final alternative has some as-yet undiscovered exotic particles in the solar interior, helping to hold the core up against gravity. Only the first two have really strong followings, and the oscillating-types theory is the current favorite. Evidence is still accumulating.
Energy Transport from the core (the site of fusion) to the star's surface can occur by any of three basic means: conduction (unlikely in main sequence stars, gas densities aren't generally high enough), radiation (where the photons carry the energy directly), and convection ("boiling" motion with hot gas packets rising, a cooler gas sheath flowing downward around the outer edge of the convective cell to be warmed in turn...). In general, the most efficient process is radiation, but there are two situations in which the free flight of photons across the "radiating surface" just doesn't happen fast enough and convection begins. The first such situation is when the fusion zone is extremely small and thus the radiating surface of that small zone is physically too small to make radiation as efficient as convection. The second is when the opacity in the gas increases dramatically (this usually occurs as the temperature of the gas decreases -- so is often found in lower-mass stars). Armed with this knowledge, let's examine the internal structure of the stars:
Stability: what keeps the star in equilibrium for its entire main-sequence lifetime? Well, a small increase or decrease in core temperature will have a large effect on the fusion rate, and the energy from fusion is what heats the star up enough to support itself against its own self-gravitation. If I "nudged" the core into a slight compression, the pressure in the core would rise, as would the temperature (due to the compression), and the fusion rate would then increase fairly dramatically! All the additional energy from the increased rate of fusion would heat up the core even more, and the pressure outward would increase, so the core would expand until it reached a point at which the pressure inward was just enough to generate the fusion rate needed to push back against that inward pressure. A main-sequence star is in constant balance in this way, expansions cooling off the core and gravity pushing the star back until the rate of fusion is just enough to balance the star again. This mechanism is called the Nuclear Thermostat (AKA Pressure-Temperature thermostat).
1. Identify the following:
2. What characteristic of the Sun (mass, temperature, magnetic field, chemical composition) has a direct bearing on the vast majority of the preceding topics?
3. Identify the following:
4. A particular star of spectral type M0 III has an apparent magnitude of 5. What is its absolute magnitude? What is its distance? What would its parallax angle be? What is the name applied to stars of this type?
5. Identify the following:
6. What properties of stars can the foregoing help us to determine, if any? If none for a particular type, say that.
7. Identify the following:
8. Describe the energy transport mechanisms for an 8-solar-mass star in its different regions (core, envelope). Do the same for a 0.08-solar-mass star.