A Major League Puzzle

Pre-Game Analysis

Foreword: Spring Training

Chapter 1: The Playing Field

Chapter 2: The Gamma-Ray Burst Baseball Card

Chapter 3: Watching The Game

Chapter 4 - Pre-Game Analysis

Chapter 5: The First Pitch
Chapter 6: What Game Are We Playing?
Chapter 7: Extra Innings
Glossary

Nevertheless, there are some conclusions that can be drawn from the statistics of the game that most everyone can agree upon. The Chicago Cubs probably will not win the World Series next year. Ozzie Smith will be in the baseball hall-of-fame some day. The agreed-upon facts of baseball, however, are more of a global nature than they are specific. Perhaps this is the price of consensus.

Like good baseball fans, scientists studying the gamma-ray bursts compile their own statistics, perform their own analyses, and make predictions on the outcome of future observations. In this chapter, we examine some of the statistics of the gamma-ray burst game and explore the pre-game predictions of scientists prior to the launch of the BATSE instrument in April 1991.

The Angular Distribution of Bursts

Perhaps the simplest statistics that scientists use to study the bursts are those related to their angular distribution. The angular distribution of gamma-ray bursts provides the information necessary to answer the question "Where on the sky do the bursts occur?". The angular distribution of the bursts is the batting average of the gamma-ray bursts - the first statistic any astronomer seems to be interested in - and it can provide important clues to the origin of these events. One obtains the burst angular distribution by simply placing a dot on a map of the sky corresponding to the location of each detected gamma-ray burst.

Most objects we observe in the sky are not distributed uniformly. Stars, planets, and other celestial objects are not simply scattered randomly across the entire sky, but instead are distributed in recognizable patterns or in preferred directions. The nine planets, for example all orbit the Sun in approximately in the same plane called the Ecliptic. Since the Earth is also a member of this Ecliptic plane, as viewed from Earth, the planets are all found within a narrow band of the sky. We never, for example, observe any of the other eight planets near the north celestial pole. As another example, consider our own Milky Way Galaxy. As we discussed previously, the Milky Way is a collection of stars, gas, and dust that is roughly circular yet also thin and flat, shaped like a pizza or a frisbee. Our solar system sits inside this thin disk far away from the center. From our vantage point, therefore, the other stars in the Milky Way appear as a narrow band of light stretching across the sky.

Distribution of Globular Clusters around the Galactic Center

As a third example, consider the group of objects known as Globular Clusters; huge spherical collections of more than a million stars which are tightly grouped around the center of the Milky Way. The angular distribution of 113 Globular Clusters is presented in the picture above. Because we are not near the center of the Milky Way, the Globular Clusters are seen in a preferred direction. This is evidenced by their tendency to "clump" near the center of the diagram, which is the direction of the Galactic center. As we look toward the center of the Galaxy (near the center of the figure), we see many of these objects, while if we look away from the Galactic center (towards the edges of the diagram), few can be observed because of their concentration about the Galactic center.

The non-uniformity of the angular distribution of objects on the sky lends insight into their nature, just as a right-handed batter who gets a majority of hits to left-field consistently might lead one to believe that the batter's nature is to pull the ball. If the bursts were found, for example, to be distributed in the sky just like the stars in the band of the Milky Way Galaxy, we would conclude that the objects generating these bursts must also be members of the Milky Way. We would then be dealing with a Galactic phenomenon, rather than some type of event which occurred outside of our Galaxy. Furthermore, since we know the approximate dimensions of our Galaxy, a distance scale and energy output for the bursts would also be established, based on the observation of a non-uniform angular distribution.

We have discussed previously the numerous spacecraft and experiments which were built prior to BATSE during the first 15 years of observations of the gamma-ray burst phenomenon. With their limited sensitivity, these instruments were only capable of detecting the brightest (and therefore probably nearby) gamma-ray bursts. The bright gamma-ray bursts observed by these experiments possessed an angular distribution that was isotropic. Unlike most objects in the sky, these detected bursts showed no preferred direction, and no tendency to align themselves with the Milky Way. Like a good poker player who shows no facial expression to hint at the cards he is holding in secret, the bright gamma-ray bursts were uncooperative in displaying any feature in the angular distribution which might give a hint as to their origin.

The Intensity Distribution of Bursts

Fortunately for scientists, the angular distribution of the bursts is not the only possible clue to their origin. The brightness distribution can also give insight into the nature of the events by telling scientists how the burst sources are distributed radially in space as one goes farther and farther from the Earth. To construct a brightness distribution, the astronomer simply counts the number of bursts that are observed at each given brightness.

The brightness of a gamma-ray burst is not solely a property of the burst itself, but also depends on how far away the burst is. When watching a baseball game from the box-seats, a ringing double into the right-field corner might produce a loud crack off the bat. Someone in the upper-deck, however, might not hear it at all because they are sitting quite far from the batter. The loudness of a sound, just like the brightness of a burst, depends on how far away one is from the source of the noise.

Mathematically, the brightness scales with the inverse of the square of the distance. As one moves twice as far away from a source, its brightness decreases by a factor of four. If you move five times as far away, the brightness decreases by a factor of 25, and so on. As an example of different sources with similar observed brightnesses, consider the difference between day and night baseball. The Sun and the lights atop the ballpark both illuminate the playing field to roughly the same brightness, although the Sun intrinsically emits much more light (it is said to be more luminous) than the bulbs at the top of the stadium. Even though the Sun emits vastly larger quantities of radiation than the lights atop the ballpark, the difference in distance makes the brightness of the two light sources approximately the same.

Unfortunately, gamma-ray bursts have distances which are unknown. We consequently do not know their luminosity, i.e. how much gamma-radiation the source is emitting, either. Therefore, a burst that appears bright to us at the Earth could either be intrinsically weak but relatively nearby, or could also be extremely strong yet farther away.

Despite this ambiguity, astronomers can learn something about how the bursts are distributed in space by looking at their brightness distribution. In the special case where the number density of burst sources (the number of sources per unit volume of space) is constant throughout all of observed space, the brightness distribution will obey a special mathematical relationship. This result is very important in the study of many objects, not just gamma-ray bursts. In this special case where the distribution is said to be homogeneous, the number of bursts observed with a brightness larger than some value P is proportional to P raised to the -3/2 power. A graph of the number of bursts above a given brightness N(>P) plotted against P will display this relationship if the number density of sources is constant. Astronomers search for this characteristic -3/2 relationship to see if the burst sources are homogeneously distributed throughout all of observed space.

To see how this special relationship is obtained, imagine yourself standing in the center of the Houston Astrodome, uniformly filled with 100 Watt light bulbs. Now, imagine that only the bulbs that are located less than some distance d away from you are actually lit, and all the rest are dark. The number of bulbs that you can see is simply proportional to the volume of space occupied by the illuminated bulbs. This volume is proportional to d-cubed. If the volume of space that included lit bulbs were increased by a factor of eight (by doubling the value of d), you would see eight times as many bulbs. The number of bulbs observed is therefore proportional to d-cubed. Now, we have discussed the fact that the brightness of an object is proportional to the inverse square of the distance between the observer and the object. Consequently, those light bulbs that are at the distance d have a brightness P that is proportional to the reciprocal of d-squared. All other bulbs are brighter because they are closer to you. We can re-write this simple relationship to state that the distance d is proportional to the reciprocal of the square-root of P. The previous relationship showed that the number of bulbs which are observed with a brightness greater than P is proportional to d-cubed. When combined with our second result, we see that the number of sources with brightness greater than P is in turn proportional to P raised to the -3/2 power.

When astronomers examined the brightness distribution of bursts observed, they found that this special -3/2 law was indeed obeyed, at least for the strong bursts that could be detected by instrumentation prior to BATSE. It was apparent that these strong bursts were uniformly distributed throughout the volume of space in which bursts could be seen by the instrumentation of limited sensitivity.

The Gamma-Ray Burst Spectral Lines

Spectroscopy, the act of splitting up observed radiation into its various energy components, is a widely applied technique in astronomy. Although it has a rather technical name, we have all seen examples of passing white light through a prism to split the light into its constituent colors. This is spectroscopy in its simplest form. A rainbow is perhaps the most common naturally occurring spectrum in visible light. Just as the splitting of white light with a prism shows that it is made up of many colors, the spectra of gamma-ray bursts can also provide the astronomer with information that can be used to investigate the nature of the events.

In several gamma-ray bursts observed with the GINGA and KONUS experiments, the spectrum showed peculiar dips or notches at evenly spaced intervals. The optical analog would be given by missing or depleted colors in a spectrum obtained by passing visible light through a prism. These notches in the gamma-ray spectra may possibly have produced when particular gamma-ray energies were filtered out of the original spectrum by some intervening mechanism. Some scientists believed that these gamma-ray energies were "picked out" and absorbed by electrons spiraling in a strong magnetic field. These particular energies where the absorption occurred are said to be in resonance with the electrons.

The concept of resonance is a familiar one to anyone who has pushed a child on a swing. The only way to get the child to swing higher and faster is to push at just the right interval, or to be in resonance with the frequency of the swinging child. If you try to push at the wrong time, the child will not swing higher (and you might hurt yourself!). Electrons in a magnetic field move in a circular trajectory around the magnetic field line, and are like a child on a swing in the sense that they circle the field line at a particular frequency. When photons with a resonant frequency strike the electrons, these photons can be absorbed, causing the electrons to reach a higher energy level, just like the child on the swing absorbs your push to swing higher. When the photons are absorbed, they are no longer present in the spectrum, and we may observe dips or notches at these energies where the photons were absorbed.

Calculations show that in order for this resonant activity to take place, producing absorption features at the observed energies, one requires a very large magnetic field, approximately one trillion times the magnitude of the Earth's magnetic field. This strong magnetic field would do a lot more to your compass needle than make it point north! Magnetic fields of this strength are believed to be found in only one type of astrophysical body, an object known as a neutron star. Neutron stars are extremely small objects, astrophysically speaking, with typical diameters of 10-15 miles. They are the remnants of old, massive stars that have used up all of their life-sustaining fuel and have ended their lives in a violent explosion called a supernova. Despite their small size, they are extremely massive, with as much material inside a neutron star as one finds in the Sun. The density of a neutron star is therefore almost incomprehensible.

Neutron stars, like the young, massive stars which lead to their creation, are known to populate the disk of our own Milky Way Galaxy. Therefore, if these resonant absorption features, or dips, in the spectra of some gamma-ray bursts are indicative of a neutron star origin for these bursts, this evidence would further indicate that gamma-ray bursts are a phenomenon produced in the disk of our own Milky Way.

Time-Profile Arguments

We have looked previously at the time profiles of several gamma-ray bursts, and noted that the brightness of these events can change rapidly in a very short period of time. Scientists can look at these time profiles to constrain the size of the region in which the bursts are created. How can temporal variations in brightness lead us to constraints on the size of the region? Let's consider the following analogy.

The typical batter can make it from home to first base in something like 5 seconds. If we knew nothing about the dimensions of the baseball diamond, we could use this fact and a few assumptions to roughly determine the extent of the infield. Everyone who has played baseball knows that you hit the ball, then run to first base. If we are so unfortunate as to hit a typical ground ball, it will be fielded and thrown to first, beating us there by a second or two and we'll be out.

A ground ball hit at a speed of nearly 50 miles per hour (we're real sluggers) has to leave the bat, reach the shortstop, be fielded (taking about 1 second), then be thrown to first all in about 4 seconds to record an out. Accounting for the 1 second it takes to field the ball and get it out of your glove before throwing the ball, there are at most 3 seconds in which the ball is traveling approximately 50 miles per hour (about 73 feet per second). At this rate of speed, the ball cannot cover more than about 220 feet in three seconds; 110 feet on its journey to the shortstop, then another 110 feet over to first base. We have just estimated the size of the diamond, which actually has 90 feet between bases. Not bad!

A similar analysis is performed on the time profiles of gamma-ray bursts. We have observed that the brightness of the bursts can vary on timescales of nearly 1/1000 of a second. The mechanism that initiates the burst emission and then turns it off (in the previous case the ground ball to the shortstop and the subsequent throw to first base) cannot travel any faster than the speed of light, nearly 186,000 miles per second. As with our estimation of the size of the baseball infield, by multiplying the time-scale of the variation (how long it takes to leave home plate and get to first base) by the speed of the information conveying the on/off signal (our ground ball to the shortstop), we observe that the region of space that is emitting the burst cannot be larger than about 186 miles or 300 kilometers in extent. This is roughly the size of New Jersey. Normal stars and other planets are many times too large to fit inside such a small region. On the other hand, neutron stars fit easily into this rather small volume of space, and are therefore possible candidate objects for events with rapid timescale variations in emission.

The Galactic Neutron Star Paradigm

We have amassed some information regarding the gamma-ray bursts that, like the statistics of baseball, can now be used to make some predictions. We have seen that both the claimed resonance dips in some of the observed spectra as well as the timing arguments both are consistent with the possibility that bursts originate on or near the exotic objects called neutron stars. We also know that the neutron stars populate the disk of the Milky Way Galaxy. On the basis of this information, one may surmise that gamma-ray bursts are a Galactic phenomenon, taking place on or near neutron stars in the disk or our own Milky Way Galaxy.

Given the supposition that bursts in fact do arise on or near Galactic neutron stars, a question arises regarding their observed angular and brightness distributions. As members of the frisbee-like portion of our Galaxy, these bursts should have a distribution on the sky that is similar to the stars we observe stretched out in the narrow band of the Milky Way. However, the observed angular distribution of the bursts is isotropic. There is not a concentration of sources in this plane. The observed angular isotropy of the bursts seems contrary to the notion that the events are distributed in the plane of the Galaxy.

Another question concerns the brightness distribution. In the disk itself, the number density of the sources may be constant, but above and below the disk there are virtually no sources at all. Clearly a distribution of objects in a disk does not therefore represent a uniform density throughout all space. The brightness distribution of objects distributed in the Galactic disk should therefore not obey our special mathematical relationship that we discussed earlier. However, the gamma-ray bursts seemingly do obey this relationship and are consistent with a uniform density throughout the volume of space that is being observed.

We have a paradox. On one hand, some observational evidence strongly suggests that the gamma-ray bursts must somehow be associated with Galactic neutron stars. On the other hand, the angular and brightness distributions of these same bursts is very different from what one would expect if the sources were distributed in the plane of the Galactic disk. How can this paradox be resolved?

The answer lies in examination of how far out the detector can see. There is a threshold below which bursts cannot be detected because they are too faint. To explore the concept of a threshold, consider the following example. If I turn a 100 Watt light bulb on and hold it 3 feet in front of your face, you will have no problem seeing the bulb. As I walk farther and farther away from you, the bulb appears fainter and fainter, obeying the inverse square law that was discussed previously. If I continue to hold the illuminated bulb while walking still farther away, the brightness of the bulb will eventually be so low that your eyes will fail to detect it. It will simply be too faint. Your eyes, just like the detectors used to observe gamma-ray bursts, have a threshold below which sources of light (or gamma-rays) cannot be detected. Sources at large distances from the detector therefore cannot be observed because they are too faint. Because the exact energy output of the bursts is not known, we cannot assign a numerical value to this threshold distance. However, we do know that at some distance, bursts will be too faint to be observed by our detector. Consequently, we have no knowledge of what the angular or intensity distributions of these faint, distant sources look like.

The gamma-ray instrumentation prior to BATSE, as we have already noted, was limited in its sensitivity and could not see out to large distances. One does not have to place a gamma-ray burst very far away before these detectors can no longer detect the incoming radiation. Our paradox is resolved by asserting that these early gamma-ray burst detectors cannot see out to the edge of the source distribution at the edge of the disk of the Galaxy. Bursts occurring at distances larger than the thickness of the Galactic disk are too faint to be observed by these detectors. The shape of the burst distribution therefore cannot be determined from their observations. The situation is analogous to standing in the middle of a forest. By only looking at the nearby trees around you, you have no chance of determining the extent and shape of the forest that surrounds you. The extent of a distribution of objects cannot be determined without being able to detect the edges of the distribution. Therefore, if one could build a more sensitive burst detector, capable of seeing to large enough distances, the edge of the gamma-ray burst distribution could be detected and its structure could be determined.

The Predictions for BATSE

Armed with the evidence supporting a Galactic disk origin for bursts and the theory that if one could build a detector that was many times more sensitive than previous experiments, the edge of the Galactic disk distribution of bursts could be observed, scientists eagerly awaited the first results of BATSE.

BATSE provides a sensitivity that is ten times greater than any previous gamma-ray burst experiment. As BATSE detected weaker and weaker bursts, scientists around the world believed that the edge of the Galactic disk would in fact be observed, thereby allowing a determination of the shape and extent of the distribution. This edge would manifest itself in two ways.

First, in the regions of space above and below the Galactic disk, there should be no sources. This marks a deviation from a uniform density throughout the volume of space being observed and, consequently, our previous mathematical relationship in the brightness distribution should be violated. There should be fewer weak bursts observed than there would be if a constant number density of burst sources existed throughout all of the newly-increased volume of space observed. This can bee seen in the inhomogeneous brightness distribution above, where the stair-stepped curve bends away from the -3/2 logarithmic slope line.

Angular Distribution of Open Clusters in the Galactic disk

Second, because instrumentation is now capable of detecting bursts at distances beyond the height of the Galactic disk, there should also be a preferred direction, or anisotropy, in the bursts' angular distribution. It was expected that the gamma-ray bursts, especially the weak ones at large distances, would show an alignment with the band of the Milky Way on the sky, similar to the figure at left, above, where the angular distribution for open star clusters is shown. These objects clearly are concentrated towards the Galactic Plane, represented by the center horizontal grid line in the figure.

If BATSE were to make these observations, they would serve as a confirmation of the Galactic disk population and neutron star origin for the gamma-ray bursts, thereby validating nearly 15 years of scientific speculation regarding these events.

It is difficult to convey the high level of certainty among the gamma-ray burst scientists regarding these expected observations of BATSE. While several scientists adopted a wait-and-see attitude, the expectation of a confirmation of the Galactic disk origin for gamma-ray bursts was extremely prevalent. In fact, less than 60 days after the launch of BATSE, the results of a study on pre-BATSE gamma-ray bursts detected on Venera 13, Venera 14, and Phobos were published in the scientific periodical Nature. The authors of this study stated that upon further analysis of data they had already collected from several pre-BATSE experiments, evidence had already been found implicating the Galactic disk (and therefore neutron stars) as the home of gamma-ray bursts. This particular team of scientists claimed to have found the answer and that BATSE would later prove them to be correct in their analysis.

"Our analysis ... suggests that GRBs are associated with the Galactic plane. ......We expect the BATSE experiment, which has high sensitivity and localization capability, on the Gamma Ray Observatory to provide further information, and the concentration of sources towards the Galactic plane reported here should become apparent after several months of operation." (J. L. Atteia et al., Nature, 351, 296, 1991)

SUPPLEMENT IV.

The Angular Distribution

The mapping of the positions of objects, regardless of whether the objects are gamma-ray bursts or not, requires some type of coordinate system. In geography, we frequently use the directions north, south, east, and west to discern the relative orientation of places to each another. St. Louis is southwest of Chicago, east of Kansas City, and north of Memphis. Astronomers use a variety of different coordinate systems to describe the positions of objects in the sky relative to each other. These systems usually treat the sky as a sphere, or a dome on which all the objects we observe are painted, ignoring the radial coordinate or the distance to the objects. Points in the sky then can be uniquely determined by two numbers, a general azimuth and a general elevation. We are familiar with such spherical coordinate systems in the context of the surface of the Earth. Any point on the Earth can be uniquely described with two numbers; a longitude, which is the relative angle between some point and the prime meridian, and a latitude, which determines the angle between the specified point and the equator.

One particularly useful coordinate system that astronomers use is the system of Galactic Coordinates. Unlike the normal longitude and latitude we use here on the surface of the Earth, which are fixed by certain points on the Earth such as the equator and the north pole, the Galactic coordinate system is fixed on the sky.

In this system, we use the terms Galactic latitude and Galactic longitude to uniquely specify the position of objects in the sky. We have observed that the Galaxy is shaped like a pizza, with our solar system located about 2/3 of the way out from the center in the plane of the disk.

These figures will help explain the coordinate system. Our Galactic coordinate system is defined so that the center of the Galaxy has the coordinates 0 degrees Galactic longitude (commonly abbreviated l) and 0 degrees Galactic latitude (commonly abbreviated b). Indeed, any object that lies exactly in the plane of the Milky Way, such as many of the objects in the Galactic disk, will have Galactic latitudes b=0. The direction from the Earth that is directly upward and perpendicular to the disk of the Galaxy is called the North Galactic Pole, and has a Galactic latitude of 90 degrees. The Galactic latitude of some object is therefore simply the number of degrees between the object and the plane of the Milky Way Galaxy.

This coordinate system is very useful for measuring the celestial distribution of various objects and determining the structure of the distribution relative to other objects in the Milky Way. For example, if we were to measure the Galactic latitude of all the stars we can observe, we would find that these values are clustered around b=0. Clearly, this is an expected result, because a majority of stars are located in the disk of the Milky Way, which, by definition has b=0. As another example, we mentioned the objects known as globular clusters. If we measure the Galactic coordinates of these objects, we find that the values of Galactic longitude for these objects are almost always between +/- 90 degrees. This occurs because most of the globular clusters are tightly packed around the center of the Galaxy. Consequently, we observe that these objects are located preferentially in the direction of the Galactic center at l = 0. Very few globular clusters have Galactic longitudes between 90 and 270 degrees, as that would place them on the opposite side of the sky from the Galactic center as viewed from Earth.

Because of the ability for this coordinate system to readily show the association of a distribution of objects with the overall structure of the Milky Way, the celestial positions of gamma-ray bursts are usually plotted in these coordinates. A burst is detected, its particular l and b values are determined, and a dot is placed on a map of the sky at the position of the detected burst. In this manner, an overall sky distribution is built up burst-by-burst.

When a large number of bursts have been detected, two prominent statistical tests are performed to determine if there is any deviation from isotropy or non-randomness in the accumulated distribution of burst positions. If the gamma-ray bursts have some distribution related to the overall structure of the Milky Way, their positions on the map should not be random.

The first test that astronomers use is called the cos(theta) test. In this test, the average of the cosine of the angle theta between the burst positions and the Galactic center is obtained. If the distribution is isotropic, the expected value of this statistic is zero. On the other hand, if a value of cos(theta) significantly greater than zero is obtained, this is an indication that bursts are preferentially located in the direction of the Galactic center, similar to the distribution of the globular clusters. A value of cos(theta) less than zero tells you that the objects are preferentially distributed away from the Galactic center. In mathematical language, this test searches for a dipole-moment in the direction of the Galactic center.

The second test is called the sin^2 b test, where b is the Galactic latitude of the burst. This test searches for an association between the distribution of the objects in question and the plane of the Milky Way Galaxy, mathematically described as a quadrupole moment in the direction of the Galactic plane. If the distribution of objects is isotropic and not associated with the plane, the expected value of the sin^2 b statistic is equal to 1/3. A value of sin^2 b that is significantly less than 1/3 would indicate to the observer that the objects are distributed preferentially in the direction of the Galactic plane. If this were found in the gamma-ray burst distribution, one would conclude that these events were likely members of our own Galactic disk. A value significantly larger than 1/3 would indicate that the burst positions had a tendency to avoid the plane of the Galaxy.

When these two statistical tests were applied to the gamma-ray burst distribution of those events detected by instrumentation prior to BATSE, the values of cos(theta)and sin^2 b were found to be consistent with 0 and 1/3, respectively. There is no indication of any anisotropy in the angular distribution of these detected events.

Brightness Distribution

The brightness distribution of the gamma-ray bursts is somewhat more complicated than the angular distribution. It is actually a convolution of two distributions, the luminosity functionphi(L) and the radial distribution of burst sources n(r). The luminosity function describes the relative amounts of gamma-ray bursts with different luminosities L. The radial distribution describes the number density of bursts at each distance r. Neither distribution is known for the gamma-ray bursts that we observe. What can be observed is the number of detected bursts N(>P) with brightness exceeding some value P. Mathematically, this quantity is related to n(r) and the luminosity function through the expression

It is important to notice that the integration over r is not from zero to some fixed value of r, but instead is integrated up to the distance at which a burst with luminosity L at some distance r produces a brightness P in the detector. This accounts for the fact that bursts of different luminosities can be observed out to different distances. As a simple illustration, consider a 100 Watt light bulb and a 10,000 Watt bulb. Because it intrinsically puts out much more light, the 10,000 Watt bulb is visible at distances much larger than those where the 100 Watt bulb can no longer be seen. Different luminosities are visible to different distances. This fact can complicate the analysis in many cases, and if not handled properly, can lead to erroneous conclusions regarding the observed N(>P) distribution.

To obtain our special mathematical relationship, consider the case where the distribution of sources n(r) is really a constant, let's call it Ao so that

If one performs the simple integration over r in the previous equation for this special case, one obtains

immediately displaying our special result that

This result is independent of the form of the burst luminosity function.

As we have observed, gamma-ray bursts detected with instrumentation prior to BATSE displayed this -3/2 behavior, indicating that their radial distribution was consistent with a constant number density.

Foreword: Spring Training

Chapter 1: The Playing Field

Chapter 2: The Gamma-Ray Burst Baseball Card

Chapter 3: Watching The Game

Chapter 4 - Pre-Game Analysis

Chapter 5: The First Pitch
Chapter 6: What Game Are We Playing?
Chapter 7: Extra Innings
Glossary

• NASA Home Page
• MSFC Home Page
• Space Sciences Laboratory Home Page
• BATSE Home Page