Chapters 15, 16, 17 and 18 - From Black Holes and Neutron Stars Through Peculiar Galaxies!

The first part of this guide is like a review from the previous test, but each of the topics pertains to something in the target chapters for the upcoming test, and there is new material added within the old topics. This guide does not take the place of reading the text, but provides supplemental information, summaries, and alternative explanations.

Use the book and your notes to fill in wherever you are having trouble with understanding something -- and ask your instructor questions! :-)

Chapter 15: Neutron Stars, Black Holes, etc.

Neutron Stars and Pulsars

Neutron stars are formed during the deaths of massive stars. If the mass of the core of the evolving star exceeds the Chandrasekhar limit of 1.4 solar masses, and the star is unable to rid itself of the excess mass by shedding a planetary nebula or via a supernova explosion, the core will collapse to form a neutron star. A neutron star is supported against gravity by neutron degeneracy pressure, which has much in common with the electron degeneracy pressure that supports white dwarfs. However, in neutron stars the pressure has risen high enough to effectively force the electrons and protons to combine to form neutrons, and thus star is essentially a big ball of neutrons, all in contact.

A typical neutron star has a mass of 1.4 - 3 solar masses. At the upper limit (the "Oppenheimer-Volkoff limit ") even neutron degeneracy pressure is insufficient to support the star against gravity and it will collapse into a black hole. The density of neutron stars is so high that their radii are only 10-15 kilometers!

Newly formed neutron stars have three special characteristics:

• First, they rotate very rapidly. This is because of the conservation of angular momentum. Although the core of the former star has shrunk, it must have the same angular momentum that it had before. Since angular momentum L = mvr, if r shrinks and m is constant, v must increase. Newly formed neutron stars rotate up to 100 times per second.
  
  
• Second, they have very high surface temperatures, up to a million degrees Kelvin or so. When the stellar core collapses to form the neutron star, there is a release of potential energy. This potential energy goes into heat, and the newly formed neutron star is extremely hot. However, since it is so small, it is not very bright in spite of its high temperature.
  
  
• Third, the magnetic field is extremely high.. This is due to the conservation of magnetic flux. Magnetic flux is simply the magnetic field of an object multiplied by its surface area. Stars have typical magnetic fields of 1-1000 Gauss. Neutron stars, on the other hand, have extremely high magnetic fields because, although they have very small surface areas, they must have the same magnetic flux as the star from which they are formed. This leads to neutron star magnetic fields of 10⁹ - 10¹² Gauss. Recall that magnetic fields in stars like the Sun are formed by a dynamo effect due to the differential rotation of the star. Neutron star magnetic fields are sometimes called fossil fields because they are simply remnants of the original stellar magnetic field. Since no dynamo is generating new fields, neutron star magnetic fields decay with time.

We generally see neutron stars only as pulsars, which appear to us to flash on and off, sometimes many times per second. The accepted model for this behavior is the lighthouse model.. In this model, the magnetic axis of the neutron star is not aligned with the rotation axis. This is common enough among magnetic objects in astronomy -- it is true of both the earth and the Sun, for example. The rapidly spinning magnetized neutron star behaves like a generator and makes a strong electric field. This electric field strips particles off the surface of the neutron star and accelerates them into space at the north and south magnetic poles.. On their way out, the charged particles spiral around the magnetic field lines of the neutron star, emitting synchrotron radiation along their paths. We only see those neutron stars whose north or south magnetic poles point at us at some time during their rotation, because neutron stars have such incredibly small surface areas that they don't emit much total light.

Only young neutron stars are visible as pulsars in the optical, because the wavelength of the emitted synchrotron radiation depends on the energy of the charged particles coming from the star. Young stars, with rapid rotation rates and high magnetic fields, emit rapidly moving, energetic particles. These can emit visible light when they spiral around the magnetic field lines. Older stars, with lower magnetic fields and slower rotation rates, are only visible as pulsars in the radio, since the charged particles they emit have lower energies.

Pulsars show two sorts of behavior which tend to support the lighthouse model. The first is that their periods, while incredibly stable, do tend to increase with time. That is, they are slowing down! The lighthouse model predicts this: the energy to accelerate the charged particles has to come from somewhere, and that somewhere is the energy of rotation of the neutron star. When we measure the energy emitted by a pulsar, and predict how fast the pulsar should be slowing down, agreement with observations is excellent. Second, pulsars show "glitches" -- sudden small increases in rotational velocity (or decreases in period, if you like). These are due to small changes in the shape of the neutron star. As the star "spins down" and its rotational rate decreases, there is less centripetal force supporting the outer layers. A "normal" star made of gas would just slowly readjust its shape, but the neutron star material is so stiff and inflexible that when adjustment does come, it is abrupt. This adjustment of the "crust" is sometimes referred to as a "starquake."

Millisecond pulsars: We expect to see the youngest and most rapidly rotating pulsars inside "new" supernova remnants. However, there is a class of very rapidly rotating pulsars (up to 642 times per second!) which we know to be old. How do we know this? These pulsars have relatively weak magnetic fields, and we can tell that they have weak magnetic fields because they are slowing down only very gradually (not much matter is being stripped off their surfaces and accelerated into space), and they are only visible in the radio. Since magnetic fields in pulsars decay with time, these pulsars must be old. But pulsars also spin down with time, yet these pulsars are rapidly rotating! The solution to this apparent quandry is that these pulsars have been "spun up" by accretion (incoming matter, bringing some angular momentum along with it!) from a companion star.

When it was first formed, the millisecond pulsar's neutron star looked like any other. It spun down with time, and its magnetic field decayed. However, when its companion binary star aged, it overflowed its Roche lobe and transferred mass to the neutron star (at this point, it may have looked like an X-ray burster). However, along with the mass came angular momentum, and the neutron star was spun up to become a rapid rotator. The magnetic field, though, was gone for good, and so the result was a rapidly-rotating neutron star with a low magnetic field. The low magnetic field means that few particles are stripped off the surface and accelerated into space, so the synchrotron radiation from them is faint, and the neutron star loses energy (as angular momentum) only slowly.

Black Holes

Black holes are regions of space in which the density of matter is so high that the fabric of space is warped (curved) back in on itself. In effect, it is a one-way door, and you can travel into a black hole but not out again. Why is this? We can use Newton's familiar equation for the escape velocity of anything from the surface of an object of mass "M":
V_esc=(2GM/R)^1/2, which means that R=2GM/(V_esc)²

and picture that as R gets smaller and smaller, V_esc gets higher and higher. Eventually at some radius (The "Schwarzschild Radius," which is 3 km for every solar-mass-worth in the star's mass, so a 3-solar-mass black hole has a 9-km Schwarzschild radius), V_esc = c (the speed of light, 3x10⁸ m/sec), and after that we can not know what state the matter might be in -- it can never again communicate with the Universe outside this radius! The star (or other object!) has become a black hole.

Since a black hole is that it is an object whose escape velocity is the speed of light, you can derive the Schwarzschild radius of a black hole:

R=2GM/c²

This works out to about 3 kilometers per solar mass, as mentioned above. (Q: If a BH the mass of the Sun is about 3 km in radius, what is the Schwarzschild radius of one with ten times the mass of the Sun?)

One error to avoid about black holes: they don't suck in everything around them like giant vacuum cleaners! From a distance, a black hole looks (from a gravitational point of view) like any other object with the same mass. If the Sun were reduced to a black hole, the earth's orbit wouldn't change (although it would become very dark outside!). Interesting effects only occur when you get close to a black hole, or when you drop some mass into it, and these are:

1. 1. Time dilation. Time runs more slowly for an observer near a black hole, as compared to an observer far away. If you fell into a black hole, the process would occur quite quickly for you, but would seem to take an infinite amount of time for your friends back on Earth.
   
   
2. 2. Gravitational redshift. The black hole is essentially a very deep and steep-sided gravity well, and we already know from General Relativity that light is bent by a gravitational field. Light travelling away from a black hole will be redshifted, because it will give up some of its energy to fight the gravitational field of the BH. Lower energy light has longer wavelength, hence the term redshift. If you shone a blue flashlight beam back at your friends as you fell into the BH, they would first see it as blue, then red, then infrared, then microwave, then radio, etc. as you fell in. When you reach the Schwarzschild radius, the redshift is infinite, which is another way of saying that the light never gets out. Note that the gravitational redshift can be observed on any massive body (the earth, the Sun, a white dwarf), but the greater the escape velocity, the greater the redshift, so it's easiest to see near a BH.
   
   
3. 3. Tidal effects. Tides are due to differential gravitational attraction. The tides that we're all used to are caused by the fact that one side of the earth is a little closer to the Moon than the other. Tidal forces near a black hole can be immense. Consider this: since a BH is so small for its mass, you can move things very close to it. If you moved the earth next to a 1 solar mass BH, one side of the earth would be only 3 km from the center of the BH, while the other would be over 12,000 km from it. Imagine the tides from this arrangement! Even six feet of body length can make a large difference between the force of gravity on your toes and that on your nose -- the closer you get, the worse the tidal pull! Unfortunately, then, you wouldn't live to shine your flashlight back at your friends as you fell into the BH, because the tidal forces would rip you apart.

Tidal forces are important for another reason. As matter gets shredded on its way into a BH, the resulting friction would heat up the inflowing gas (Have you ever broken a piece of metal by bending it back and forth until it broke off? Did you notice that it got pretty warm? Imagine how hot a steel beam would get if you ripped it apart by stretching it!), which would emit X-rays and gamma rays. The gas itself would pick up random motions that were some small fraction of the orbital speed of the gas at whatever radius you were observing, and around a BH, these orbital speeds get to be close to the speed of light! The "hotter" a gas, the faster the random motions in it, so the gas near the Schwarzschild radius is hot, hot, hot! This means that you can get energy out of a BH by throwing things into it. You can't get quite as much energy as you could if you converted that same amount of matter directly into energy (remember E = mc²), but you could theoretically get as much as half that amount (an efficiency of 50%). Efficiencies of 5% to 10% are more typical, though. This works for black holes of stellar mass, or for black holes at the centers of active galaxies.

All this means that one way to look for a black hole is to look for a strong X-ray source. If that source is in a binary system, and has a mass greater than about 3 solar masses, then it must be a BH since white dwarfs and neutron stars can't be that massive. Some good candidates for stellar-mass black holes include Cyg X-1 and LMC X-3 (see Table 15-4 in text on p. 299). For super-sized black holes, our own Galactic center qualifies as a candidate, as do the centers of "active galaxies."

For Black Hole fans: Everything I've said so far assumes that the black hole is essentially featureless, and indeed theorists like to say that "black holes have no hair." However, there are three parameters that define a black hole:

1. 1. Mass.
   
   
2. 2. Electric charge. If, for example, you fed a black hole a diet of only electrons, it would build up a negative charge, since electrons are negatively charged particles. However, this is not likely to be very important in general, since matter on the whole is electrically neutral. Most black holes are not likely to be very highly charged.
   
   
3. 3. Angular momentum. Black holes can rotate!

Non-rotating black holes (the kind we've discussed so far) are called Schwarzschild black holes, since Karl Schwarzschild first discovered the solution to Einstein's equations that describes them. Rotating black holes were "discovered" in Einstein's equations by Roy Kerr in 1963, so they are called Kerr black holes.

Kerr black holes are interesting because of the ergosphere, as region centered along the "equator" of the BH. It is possible to enter this region and escape. In fact, you can come out with more energy than you went in with! Where does the extra energy come from? From the rotational energy of the BH -- you speed up, and the rotation of the BH slows down a bit. This is another way to extract energy from black holes.

Black Hole "Evaporation"

In the late 1960s and early 1970s, Stephen Hawking showed that black holes can actually lose mass. The reason is a little subtle. The warping of space around a BH can be interpreted as storing energy in the space around the BH. The stored energy is equivalent to matter (by E=mc² again), so in effect the BH is always creating pairs of particles in the space around itself. Most of these particles are created inside the Schwarzschild radius, so nothing comes of them, but some are created outside and can escape, carrying off some of the black hole's mass with them. Since the whole process works better the more space is warped and since lower-mass black holes warp space more violently (since they're smaller), small black holes evaporate more rapidly than big ones. The radiation that such an evaporating black hole emits is sometimes called Hawking radiation.

Don't get too excited about this, though. The process is very slow. Solar-mass black holes have evaporation times many billions of times longer than the lifetime of the Universe so far. However, if the Big Bang created any small (mountain-mass) black holes, they should be ending their evaporation in a shower of gamma rays right about now. So far, though, no one has seen any...

Chapter 16:The Milky Way

Size

Sir William Herschel and his sister Caroline, working in the late 1700s and early 1800s, were the first to systematically try to study the size of the Milky Way. They made two assumptions:

1. 1. The Sun is located inside a large cloud of stars
   
   
2. 2. We can see to the edge of this cloud in all directions

If these assumptions are true, then the distance to the edge of the cloud is largest in those directions in which we see the most stars. The Herschels counted stars in 683 patches of the sky and announced that:1) the Milky Way galaxy was disk-shaped and that: 2) we lived near the center. WRONG! The Herschels didn't know the distances to any of the stars they counted, so they couldn't say how big the galaxy was. However, in the early years of this century, Jacobus Kapteyn repeated the Herschel's work using the then-new distance measurements to the stars and decided that the disk was about 10 kiloparsecs (10,000 parsecs) in diameter (off by a factor of two) and about 2 kiloparsecs (kpc) thick (too thick).

The problem with all of this work is that assumption (2) is wrong. Intervening gas and dust clouds limit our view in some directions, so the Herschels and Kapteyn were wrong about the size of the galaxy and our location in it wrong. The correct situation was determined by Harlow Shapley in the 1920s.

Shapley noticed something interesting. He found that open clusters were scattered all over the Milky Way, but globular clusters were nearly all in one direction, that of Sagittarius. He then assumed that the globular clusters were centered around the center of the galaxy, which told him right away that we were off to one side of the center.

To get the distance to the center of the galaxy, Shapley had to find the distances to the globular clusters, which he did using Cepheid variables. Recall that Henrietta Leavitt had determined, from looking at Cepheids in the Magellanic clouds, that there was a period-luminosity relationship. However, she did not know the proper absolute magnitudes to calibrate her relationship, since there are no nearby Cepheids. Shapley observed that faster-moving stars (those with large proper motions) are, on average, closer than slower-moving ones. He used nearby stars to derive a relationship (valid for large groups of stars rather than for single objects) between proper motion and distance, and applied this relationship to 11 Cepheids. Lo and behold -- he was able to replace Leavitt's apparent magnitude scale with an absolute magnitude scale, and thus give astronomers their most important distance indicator (RR Lyrae variables were a "bonus" calibration since they inhabited the same clusters as the Cepheids). After all, if you know how bright a Cepheid actually is (based only on measurement of its period), and you know how bright it appears to be, you can easily figure out how far away it is. (And you know you will be asked to do that! ;-)

Shapley derived distances for all of the globular clusters he could see, and determined that the galaxy was about 100 kpc in diameter. As it turns out, this number is still too large, because Shapley (once again) did not take interstellar extinction into account. Since he thought that all of the dimming he observed in his stars was due to distance, even though some of it was actually due to intervening dust and gas, all of his distances were too large. The accepted size of the galaxy is now about 30 kpc, and the Sun lies about 8.5 kpc from the center, so about 3/5 - 2/3 of the way out along the disk, on the edge of a spiral arm segment called the Orion arm. The disk is about 1 kpc thick.

Rotation of the galaxy

If the galaxy weren't rotating, it would collapse into a ball. We can measure the speed of the rotation by looking at stars near the Sun, and such stars orbit the center of the galaxy at about 220 km/sec, thus taking about 240-250 million years to complete a "galactic year." It is important to keep in mind that the galaxy does not rotate like a solid body -- instead, each star follows its own orbit around the center. This means that stars near the center orbit more rapidly (in general) than those near the edge, a phenomenon known as differential rotation.

If most of the mass in the galaxy were located in the center, then we'd expect the orbital velocity of stars to steadily decrease as we move outwards. (A plot of rotational velocity versus distance from the center is known as a rotation curve) This does not happen, however, and that means that there is a lot of mass in the outer regions of the galaxy, and in the galactic halo (see below). Since we don't see this mass, it isn't in the form of bright stars and H II regions, but is more probably in the form of brown dwarfs -- sort of low-mass "failed stars."

We can use the orbital velocity of stars near the Sun to derive the mass in the galaxy that is "inside" the Sun's orbit, using Kepler's third law, and we get about 10¹¹ solar masses. The true total mass of the galaxy is somewhere between twice and ten times this amount.

The above comments apply to the disk of the galaxy (see below). Halo stars don't move in flattened orbits, but instead have highly inclined orbits that take them far out of the plane of the galaxy. Usually the part of their orbits that takes them through the plane of the galaxy is the part that is closest to the center, so they are rapidly moving at this time, and are known as high velocity stars.

Structure of the galaxy

The following picture is a good representation of the structure of the galaxy.

Notice that here you are looking at the galaxy from slighly "above" the plane of the disk, so that the central bulge, composed of many, many stars doesn't appear as "bulgy" as it would if the galaxy were seen exactly edge-on! Seen edge-on, the bulge is thicker than the disk by quite a bit.

The disk component of the galaxy is the flattened part, and contains nearly all of the free gas and dust. This medium is mostly hydrogen and helium, with a few percent heavier elements (which came from supernova remnants). Much is thinly spread, but there are also giant molecular clouds within which star formation can take place. When star formation is well along in these clouds, the new hot stars will light the gas up as H II regions, within which hydrogen is ionized. Note that since most of the the free gas and dust is in the disk of the galaxy, this is where most of the star formation takes place as well.

The thickness of the disk depends on how you measure it. If you look only at H II regions and O and B stars, the disk is less than 100 pc thick, while if you look at lower main sequence stars it can be as much as 1000 pc (1 kpc) thick. The difference arises because all stars are born in the H II regions, which are concentrated near the central plane of the disk, but only the low-mass ones live long enough for their slight random up-or-down velocities to move them any distance away from the plane of the galaxy.

The disk also contains a weak magnetic field, which traps high-energy particles (mostly from supernova explosions) and keeps them "in orbit" around the galaxy. These particles are the cosmic rays.

If you were to look at the galaxy from the outside, the most obvious feature would be the spiral arms, which are long spiral-shaped patterns of young stars and H II regions.The stars are mostly grouped into two types of star clusters:

1. Associations -- small groupings of 10-100 stars that are only very weakly gravitationally bound, so that they don't stay together very long. Too wimpy (mass-wise) to be called open clusters.
   
   
2. Open Clusters -- groups of 100 to 1000 stars in a region about 25 pc in diameter. These are strongly enough bound to one another gravitationally that they will stay together for tens of millions of years.

The spiral arms themselves are not solid -- if they were, the differential rotation of the galaxy would tear them apart (stretch them out like spaghetti in a few galactic rotations!). In fact, they are regions of star formation, and we can trace them by looking for young objects -- young enough that they will still be near their region of formation. We also want to use bright objects, so that we can see them from a long way off. Such objects are known as spiral tracers, and good examples in the optical are O/B associations and H II regions.

Using O/B associations (small groups of hot young O and B stars) to trace nearby spiral arms reveals three short arm segments nearby -- we are on the edge of one, called the Orion arm. However, optical wavelengths of light are easily stopped by gas and dust clouds, so these associations are not ideal to use if you want to see for large distances across the galaxy. For that, we use radio wavelengths, particularly the 21 centimeter "spin-flip" line of H I (neutral hydrogen) gas (remember 21-cm radiation?).

Using the 21-cm radiation to look at the structure of the galaxy has another advantage: we can use the Doppler shift. Since each orbital radius around the center of the galaxy has a characteristic orbital speed, we can separate out gas at different distances from the center, and produce a map that looks like Figure 15-18 (p. 338). Note that, since the Doppler Effect applies only to radial velocities, and not to transverse (perpendicular to our line of sight), we can't disentangle the different masses of gas roughly along a line of sight from us to the galactic center (see the cone-shaped region in Figure 15-18). The result of this work is the conclusion that, while we live in a spiral galaxy, it is not a simple two-armed spiral. Instead, we appear to live in a spiral composed of many short arm segments. Some astronomers even think that our galaxy has a central bar! You should also be aware that observations of molecules like CO (found in giant molecular clouds) roughly reproduce the spiral picture given by the 21-cm. radiation.

There are two models for the formation of spiral arms. The first is the density wave theory. In this model, the spiral arms are dynamically stable regions of compression that move slowly around the galaxy. By dynamically stable, we mean that they retain the same general appearance over time even though their components are always changing. (Think of a traffic jam at a toll booth. The jam stays around for a long time even though the specific cars that make up the jam may be slowly moving through it.) The moving compression, or density, wave moves at slower-than-orbital velocity, so gas clouds catch up to it as they orbit the galaxy. The resulting compression of the gas clouds triggers the star formation characteristic of spiral arms. Would you expect to see O/B associations at the leading edge (front) of a spiral arm, or at the trailing edge (back)?

There are two problems with spiral density wave theory. The first is that density waves are not very stable -- they should die out in about a billion years or so. That means that something must keep regenerating them. The answer to this problem seems to be that the galaxy, like a string on a musical instrument, is unstable to certain disturbances, so minor fluctuations such as supernova explosions or another galaxy passing might suffice to reenergize the density wave. A more serious problem with density wave models is that they are only able to reproduce two-armed spiral galaxies -- the so-called grand design spiral galaxies. What about the multi-armed (flocculent) galaxies like our own?

The model we need for flocculent galaxies is called self-sustaining star formation. Consider a big cloud condensing to form stars. Once one star forms, it will try to blow a bubble in the cloud, causing other parts of the cloud to compress, and thus new stars to form. This means that stars will form in large groups. Over a relatively short time (tens of millions of years), the group of stars, and the remains of the cloud (now an H II region) will be stretched out into a little arc by the differential rotation of the galaxy.The result will be a galaxy full of short arcs, like our own.

Probably the true model is a combination of the two outlined above and the final appearance of a spiral galaxy may simply depend on whether or not the density wave is successfully triggered in the first place, i.e. whether there are enough supernovae in the right place at the right time, or whether or not another galaxy has passed close by in the recent past. Those galaxies which have had spiral density waves triggered will be the two-armed grand design models, while the rest (dominated by self-sustaining star formation) will be the flocculent type.

The spherical component of the galaxy comprises the nuclear bulge at the center and the halo around the outside.

The nuclear bulge is the cloud of stars at the center of the galaxy. Here things are very crowded (stars are spaced only a few thousand AU apart), and the gas and dust between us and the core make things very hard to see in the optical. However, we have used radio and infrared observations to see the center, and the picture is very interesting.

Infrared observations show us that there is a significant amount of dust in the core, and the stars are packed closely enough together to make the dust glow at IR wavelengths. Where there's dust, there's gas. This means that some star formation is probably going on in the core, unlike the halo.

The nature of the Core...First of all, in the radio, the galactic center contains the brightest source in the sky, Sagittarius A. Observations suggest that the center is surrounded by at least two concentric rings of material, which are expanding outwards at speeds of 1-200 km/sec. There is a central cavity, about 3 parsecs in diameter, which is empty of stars, but contains a large amount of mass. We know this because gas clouds close to the center are still rotating rapidly around the center, and Kepler's third law tells us that this means that there is still a lot of mass inside their orbit -- about 3 million solar masses worth. In fact, radio observations show only a small object, less than 20 AU in diameter, at the center, so current thinking is that the central object must be a massive black hole.

Some additional evidence that the central object is a black hole includes:

1. Observations that matter is slowly spiralling into the central object, at the rate of a few tenths of a solar mass per year.
   
   
2. A weak jet extends southward (Galactic South) from the central object. Jets are usually created when accretion is happening, and we would expect a massive central black hole to be accreting.
   
   
3. Sporadic electron-positron radiation is seen at an energy of 511,000 electron volts -- a signature of black hole evaporation, or Hawking radiation.

The halo of the galaxy contains a thin scattering of stars and globular clusters, with nearly no free gas or dust and consequently no star formation. Examination of the globular cluster turn-off points tells us that they are 10-15 billion years old, the oldest objects in the galaxy. Observations of the rotation curve (see above) of the galaxy imply that there is an extended halo, the galactic corona, which could extend out as far as 100 kpc and contain 10¹² solar masses.

Origin of the Milky Way

In the 1940s and 1950s, astronomers came to the realization that there are two broad types, or populations, of stars in the galaxy. First, there are those in the disk, like the Sun. We call these Population I (Pop I) stars, or somtimes disk population stars. (It rhymes: "Pop I, like the Sun"). Stars of the other type are located in the halo and central bulge, and are called Population II (Pop II) or sometimes halo population stars.

The two populations are physically similar, with one main exception: Pop II stars contain nearly no elements heavier than helium (astronomers call such elements metals). Pop I stars, as you know, do have "metals." There are other differences between the two populations -- Pop II stars are all older, lower main sequence stars with non-circular orbits, while Pop I stars include young stars with nice circular orbits around the galactic center.

Comments/email:hpreston@valdosta.peachnet.edu
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