A core-collapse supernova may leave behind a remnant, an ultracompressed ball of material called a neutron star. The processes that form neutron stars ensure that these stars are rapidly rotating and strongly magnetized at birth.
Pulsars are objects that appear to emit regular bursts of electromagnetic energy. The accepted explanation for pulsars is the lighthouse model, in which a rotating neutron star sends a beam of energy into space. If the beam sweeps past Earth, we see a pulsar. The pulse period is the rotation period of the neutron star.
A neutron star in a close binary system can draw matter from its companion, forming an accretion disk. The material in the disk heats up even before it reaches the neutron star, and the disk is usually a strong source of X-rays. As gas builds up on the star's surface it eventually becomes hot enough to fuse hydrogen. As with a nova explosion on a white dwarf, when hydrogen burning starts on a neutron star, it does so explosively. An X-ray burster results. Even more energetic are gamma-ray bursts, which may result from the violent merger of neutron stars in distant binary systems.
The rapid rotation of the inner part of the accretion disk causes the neutron star to spin faster as new gas arrives on its surface. The eventual result is a very rapidly rotating neutron star—a millisecond pulsar. Many millisecond pulsars are found in the hearts of old globular clusters. They cannot have formed recently, so they must have been spun up by interactions with other stars.
Careful analysis of the radiation received has shown that some pulsars are orbited by planet-sized objects. The origin of these "pulsar planets" is still uncertain.
The upper limit on the mass of a neutron star is about 3 solar masses. Beyond that mass the star can no longer support itself against its own gravity, and it must collapse. No known force can prevent the material from collapsing all the way to a pointlike singularity, a region of extremely high density where the known laws of physics break down. Surrounding the singularity, at a distance of a few kilometers for a solar-mass object, is a region of space from which even light cannot escape—a black hole. Astronomers believe that the most massive stars form black holes, rather than neutron stars, after they explode in a supernova.
Conditions in and near black holes cannot be described by Newtonian mechanics. A proper description involves the theories of relativity developed by Albert Einstein early in the twentieth century. Even relativity theory fails right at the singularity, however.
The "surface" of a black hole is the event horizon; its distance from a singularity is called the Schwarzschild radius. At the event horizon the escape speed equals the speed of light. Within this distance, nothing can escape. Photons passing too close to a black hole are deflected onto paths that cross the event horizon and become trapped.
Relativity theory describes gravity in terms of a warping, or bending, of space by the presence of mass. The more mass, the greater the warping. All particles—including photons—respond to that warping by moving along curved paths. A black hole is a region where the warping is so great that space folds back on itself, cutting off the interior of the hole from the rest of the universe.
To a distant observer the clock on a spaceship falling into a black hole would show time dilation—it would appear to slow down as the ship approached the event horizon. The observer would never see the ship reach the surface of the hole. At the same time, light leaving the ship would be subject to gravitational redshift as it climbed out of the hole's intense gravitational field. Light emitted just at the event horizon would be redshifted to infinite wavelength. Both phenomena are predictions of the theory of relativity. The gravitational redshifts due to both Earth and the Sun are very small but have been detected experimentally.
Once matter falls into a black hole, it can no longer communicate with the outside. However, on its way in, it can form an accretion disk and emit X-rays just as in the neutron-star case. The best candidates for black holes are binary systems in which one component is a compact X-ray source. Cygnus X-1, a well-studied X-ray source in the constellation Cygnus, is a long-standing black-hole candidate. Studies of orbital motions imply that the compact objects are too massive to be neutron stars, leaving black holes as the only alternative.
1. The density of a neutron star is comparable to the density of an atomic nucleus. (Hint)
2. As a result of their high masses and small sizes, neutron stars have only weak gravitational pulls at their surfaces. (Hint)
3. Newly formed neutron stars have extremely strong magnetic fields. (Hint)
4. A millisecond pulsar is actually a very old neutron star that has been recently spun up by interaction with a neighbor. (Hint)
5. Millisecond pulsars are found only in globular clusters. (Hint)
6. Planet-sized bodies will never be found around a pulsar, because the supernova that formed the pulsar would have destroyed any planets in the system. (Hint)
7. Nothing can travel faster than the speed of light. (Hint)
8. All things, except light, are attracted by gravity. (Hint)
9. A black hole is an object whose escape speed equals or exceeds the speed of light. (Hint)
10. Although visible light cannot escape from a black hole, high-energy radiation, like gamma rays, can escape. (Hint)
11. It is possible for a beam of light to go into orbit around a black hole. (Hint)
12. If you could touch it, the surface of a black hole, the event horizon, would be very hard. (Hint)
13. The laws of physics break down near the center of a black hole. (Hint)
14. X-rays are emitted by matter accreting onto a stellar-mass black hole. (Hint)
15. Thousands of black holes have now been identified. (Hint)
1. No remnant remains after the explosion of a _____ supernova. (Hint)
2. A typical neutron star is _____ km in diameter. (Hint)
3. Neutron stars may be characterized as having _____ rates of rotation and _____ magnetic fields. (Hint)
4. Pulsars were discovered through observations in the _____ part of the electromagnetic spectrum. (Hint)
5. Typical pulsar periods range from _____ to _____. (Give numbers and units.) (Hint)
6. The pulse period of pulsar radiation tells us the _____ of the neutron star emitting the radiation. (Hint)
7. All millisecond pulsars are now, or once were, members of _____ star systems. (Hint)
8. X-ray bursters result from accretion of material from a binary companion onto a _____ star. (Hint)
9. According to general relativity, space is warped, or curved, by _____. (Hint)
10. If the Sun were magically to turn into a black hole of the same mass, Earth's orbit would _____. (Hint)
11. The radius of the event horizon of a black hole is called the _____. (Hint)
12. The region of extremely high density at the center of a black hole is called a _____. (Hint)
13. Photons _____ energy as they escape from a gravitational field. (Hint)
14. Black holes of stellar origin are believed to have been discovered in several _____ systems. (Hint)
15. Scientists infer that the energy-emitting region in Cygnus X-1 must be very small because of the rapid _____ of the radiation received. (Hint)
1. How does the way in which a neutron star forms determine some of its most basic properties? (Hint)
2. What would happen to a person standing on the surface of a neutron star? (Hint)
3. Why aren't all neutron stars seen as pulsars? (Hint)
4. What are X-ray bursters? (Hint)
5. Describe the line of reasoning that led astronomers to conclude that gamma-ray bursts are very distant and very energetic. (Hint)
6. What is the favored explanation for the rapid spin rates of millisecond pulsars? (Hint)
7. Why did astronomers not expect to find a pulsar with a planetary system? (Hint)
8. What does it mean to say that the measured speed of a light beam is independent of the motion of the observer? (Hint)
9. Use your knowledge of escape speed to explain why black holes are said to be "black." (Hint)
10. Why is it so difficult to test the predictions of general relativity? Describe two tests of the theory. (Hint)
11. What would happen to someone falling into a black hole? (Hint)
12. What is the principle of cosmic censorship? Do you think it is a sound scientific principle? (Hint)
13. What makes Cygnus X-1 a good black-hole candidate? (Hint)
14. Imagine that you had the ability to travel at will through the Galaxy. Explain why you would discover many more neutron stars than those known to observers on Earth. Where would you be most likely to find these objects? (Hint)
15. Do you think that planet-sized objects discovered in orbit around a pulsar should be called planets? Why or why not? (Hint)
1. The angular momentum of a solid body (see More Precisely 15-1) is proportional to its angular velocity times the square of its radius. Using the law of conservation of angular momentum, estimate how fast a collapsed stellar core would spin if its initial spin rate was 1 revolution per day and its radius decreased from 10,000 km to 10 km. (Hint)
2. What would your mass be if you were composed entirely of neutron-star material, of density 3 
1017 kg/m3? (Assume that your average density is 1000 kg/m3.) Compare this with the mass of (a) the Moon. (b) a typical 1-km asteroid. (Hint)
3. Calculate the surface gravity and escape speed of 1.4—solar mass neutron star with a radius of 10 km. (Hint)
4. Use the radius—luminosity—temperature relation to calculate the luminosity of a 10-km-radius neutron star for temperatures of 105 K, 107 K, and 109 K. What do you conclude about the visibility of neutron stars? Could the brightest of them be plotted on our H—R diagram? (Hint)
5. A gamma-ray detector of area 0.5 m2 observing a gamma-ray burst records photons having total energy 10-8 joules. If the burst occurred 1000 Mpc away, calculate the total amount of energy it released (assuming that the energy was emitted isotropically). How would this figure change if the burst occurred 10,000 pc away instead, in the halo of our Galaxy? What if it occurred within the Oort cloud of our own solar system, at a distance of 50,000 A.U.? (Hint)
6. A 10-km-radius neutron star is spinning 1000 times per second. Calculate the speed of a point on its equator, and compare it with the speed of light. (Consider the equator as the circumference of a circle, and recall that circumference = 2
r.) Also calculate the orbital speed of a particle in a circular orbit just above the neutron star's surface. (Hint)
7. Supermassive black holes are believed to exist in the centers of some galaxies. What would be the Schwarzschild radii of black holes of 1 million and 1 billion solar masses, respectively? How does the first black hole compare in size with the Sun? How does the second compare in size with the solar system? (Hint)
8. Calculate the tidal force on a 2-m-tall human falling feet first into a 1—solar mass black hole—that is, compute the difference in the forces acting on his head and his feet just before his feet cross the event horizon. Repeat the calculation for a 1—million solar mass black hole and for a 1 billion—solar mass black hole. Compare these forces with the person's weight on Earth of 600 N. (See More Precisely 2-2.) (Hint)
9. What are the mass and Schwarzschild radius of a black hole having a blackbody temperature equal to that of the Sun—6000 K? (See More Precisely 22-3.) What is the black hole's luminosity? (Hint)
10. Using the data given in the text (assume the upper limit on the stated range for the black-hole mass), calculate the orbital separation of Cyg X-1. If the companion star's radius is 20 million km, verify (approximately) that the black hole's tidal field is sufficient to draw matter from the companion's surface. (Hint)
1. Many amateur astronomers enjoy turning their telescopes on the ninth-magnitude companion to Cygnus X-1, the sky's most famous black-hole candidate. Because none of us can see in X-rays, no sign of anything unusual can be seen. Still, it's fun to gaze toward this region of the heavens and contemplate Cygnus X-1's powerful energy emission and strange properties. Even without a telescope, it is easy to locate the region of the heavens where Cygnus X-1 resides. The constellation Cygnus contains a recognizable star pattern, or asterism, in the shape of a large cross. This asterism is called the Northern Cross. The star in the center of the crossbar is called Sadr. The star at the bottom of the cross is called Albireo. Approximately midway along an imaginary line between Sadr and Albireo lies the star Eta Cygni. Cygnus X-1 is located slightly less than 0.5° from this star. With or without a telescope, sketch what you see.
2. Set up a demonstration of the densities of various astronomical objects—an interstellar cloud, a star, a terrestrial planet, a white dwarf, and a neutron star. Select a common object that is easily held in your hand, something that would be familiar to anyone—an apple, for example. For the lowest densities, calculate how large a volume would contain the object's equivalent mass. For high densities, calculate how many of the objects would have to be fit into a standard volume, such as 1 cm3. This volume is better for this project than 1 m3 because most people do not appreciate how large a volume 1 m3 is. Present your demonstration to your class or to some other group of students. Tell them about each astronomical object and how it comes by its density.
