1. PSCI 1414 General Astronomy
The Deaths of Stars
Part 3: Neutron Stars and Black Holes
Alexander C. Spahn

2. Neutron Stars
neutron star: A very compact, dense star composed almost entirely of neutrons. We saw that under the very high pressures within a core-collapse supernova, a proton and an electron can combine to form a neutron (as well as a neutrino). Far from being a rare transformation, a core-collapse can convert over a solar mass of material into almost pure neutrons. Recall that white dwarfs are supported by degenerate electron pressure, and by a similar effect, neutron stars are supported against further gravitational collapse by degenerate neutron pressure.

3. Neutron Stars
Although the possibility for neutron stars to form was proposed in 1934, most scientists politely ignored the idea for years. After all, a neutron star must be a rather weird object. If brought to Earth’s surface, a single teaspoonful of neutron star matter would weigh the same as about 20 million elephants (about 100 million tons)!

4. Neutron Stars
Neutron stars can only exist for a narrow range of masses. Supported by degenerate electron pressure, stellar cores below the Chandrasekhar limit of 1.4 M⊙ form white dwarfs. Above the Chandrasekhar limit, a neutron star forms, but the maximum possible neutron star mass is only around 2 to 3 M⊙. Beyond a stellar core mass of 2 to 3 M⊙, gravity overwhelms internal pressure, and a black hole is formed. Keep in mind that these are only the masses of the stellar cores. Stars with total masses between 8 to 25 M⊙ are expected to develop neutron stars, and above 25 M⊙, black holes.

5. Neutron Stars
A couple of solar masses is a lot of material, but at extreme densities, these objects are surprisingly small. A 1.4-M⊙ neutron star would have a diameter of only 20 km (12 mi). Its surface gravity would be so strong that its escape speed would be one-half the speed of light. Even climbing a 1-millimeter “mountain” on a neutron star would require about the same energy as a person on Earth jumping straight to the Moon!

6. Pulsars
In 1967, Cambridge graduate student Jocelyn Bell and her colleagues began using an array of radio antennas to scrutinize radio emissions from the sky. While searching for something quite different, Bell noticed that the antennas had detected regular pulses of radio noise from one particular location in the sky. These radio pulses were arriving at regular intervals of seconds—much more rapid than those of any other astronomical object known at that time.

7. Pulsars
The team first wondered if they might be signals from an advanced alien civilization. That possibility had to be discarded within a few months after several more of these pulsating radio sources, which came to be called pulsars, were discovered across the sky. In all cases, the periods were extremely regular, ranging from about 0.25 second for the fastest to about 1.5 seconds for the slowest.

8. Pulsars
At the time of its discovery, the Crab pulsar was the fastest pulsar known to astronomers. Its period is seconds per rotation, which means that it flashes or rotates about 1/ = 30 times each second.

9. Pulsars
It was immediately apparent to astrophysicists that white dwarfs are too big and bulky to generate 30 signals per second; calculations demonstrated that a white dwarf could not rotate that fast without tearing itself apart! Hence, the Crab pulsar indicated that pulsars had to be much smaller and more compact than a white dwarf. We now understand that pulsars arise from rapidly rotating neutron stars, which were created in supernova explosions.

10. Pulsars
Because neutron stars are very small, they should also rotate rapidly. A typical star, such as our Sun, takes nearly a full month to rotate once about its axis. But just as a spinning ice skater speeds up when she pulls in her arms, a collapsing star also speeds up as its size shrinks.

11. Pulsars
The small size of neutron stars also leads to intense magnetic fields. The magnetic fields on neutron stars are over a million times stronger than the strongest fields produced in labs on Earth. The magnetic field of a neutron star makes it possible for the star to radiate pulses of energy toward our telescopes. As a neutron star rotates, the magnetic axis of a neutron star—the imaginary line passing through the north and south magnetic poles—is likely to be inclined at an angle to the rotation axis.

12. Pulsars
Furthermore, nearby electrically charged particles are accelerated by the magnetic fields, and this motion produces electromagnetic radiation. The result is that two narrow beams of radiation pour out of the neutron star’s north and south magnetic polar regions.

13. Novae and White Dwarfs
Still other exotic phenomena occur when a stellar corpse is part of a close binary system. One example is a nova (plural novae), in which a faint star suddenly brightens by a factor of 104 to 108 over a few days or hours, reaching a peak luminosity of about 105 L⊙. By contrast, a supernova has a peak luminosity of about 109 L⊙.

14. Novae and White Dwarfs
In the 1950s, painstaking observations led to the conclusion that all novae are members of close binary systems containing a white dwarf. Gradual mass transfer from the ordinary companion star deposits fresh hydrogen onto the white dwarf.

15. Novae and White Dwarfs
Because of the white dwarf’s strong gravity, this hydrogen is compressed into a dense layer covering the hot surface of the white dwarf. As more gas is deposited and compressed, the temperature in the hydrogen layer increases. When the temperature reaches about 107 K, hydrogen fusion ignites throughout the gas layer, embroiling the white dwarf’s surface in a nuclear detonation that we see as a nova.

16. X-ray Bursters and Neutron Stars
A surface explosion similar to a nova also occurs with neutron stars. In 1975 it was discovered that some objects in the sky emit sudden, powerful bursts of X-rays. The source emits X-rays at a constant low level until suddenly, without warning, there is an abrupt increase in X-rays, followed by a gradual decline.

17. X-ray Bursters and Neutron Stars
An entire burst typically lasts for only 20 seconds, and the same object can repeat these bursts, although at irregular intervals. Sources that behave in this fashion are known as X-ray bursters. Several dozen X-ray bursters have been discovered in our Galaxy.

18. X-ray Bursters and Neutron Stars
X-ray bursters, like novae, are thought to involve close binaries whose stars are engaged in mass transfer. With a burster, however, the stellar corpse is a neutron star rather than a white dwarf. Gases escaping from the ordinary companion star fall onto the neutron star. The energy released as these gases crash down onto the neutron star’s surface produces the low-level X-rays that are continuously emitted by the burster.

19. X-ray Bursters and Neutron Stars
Most of the gas falling onto the neutron star is hydrogen, which the star’s powerful gravity compresses against its hot surface. In fact, temperatures and pressures in this accreting layer become so high that the arriving hydrogen is converted into helium by hydrogen fusion. As a result, the accreted gases develop a layered structure that covers the entire neutron star, with a few tens of centimeters of hydrogen lying atop a similar thickness of helium.

20. X-ray Bursters and Neutron Stars
When the helium layer grows to about 1 m thick, helium fusion ignites explosively and heats the neutron star’s surface to about 3 × 107 K. At this temperature the surface predominantly emits X-rays, but the emission ceases within a few seconds as the surface cools. Hence, we observe a sudden burst of X-rays only a few seconds in duration. New hydrogen then flows onto the neutron star, and the whole process starts over. Indeed, X-ray bursters typically emit a burst every few hours or days.

21. X-ray Bursters and Neutron Stars
When the helium layer grows to about 1 m thick, helium fusion ignites explosively and heats the neutron star’s surface to about 3 × 107 K. At this temperature the surface predominantly emits X-rays, but the emission ceases within a few seconds as the surface cools. Hence, we observe a sudden burst of X-rays only a few seconds in duration. New hydrogen then flows onto the neutron star, and the whole process starts over. Indeed, X-ray bursters typically emit a burst every few hours or days.

22. The Special Theory of Relativity
In 1905, Albert Einstein proposed his special theory of relativity. This theory describes how motion affects our measurements of distance and time. It is “special” in the sense of being specialized. In particular, it does not include the effects of gravity. The word “relativity” is used because one of the key ideas of the theory is that all measurements are made relative to an observer.

23. The Special Theory of Relativity
In particular, the distance between two points is not an absolute, nor is the time interval between two events. Instead, the values that you measure for these quantities depend on how you are moving, and these values are relative to you. Thus, one of Einstein’s key discoveries is that someone moving in a different way would measure different lengths for objects and different durations of events. In our everyday experience we do not notice that time and distance measurements depend on the speed between an observer and the events being observed because, as we will see, these effects are most noticeable when that speed approaches the speed of light.

24. The Special Theory of Relativity
Remarkably, Einstein’s theory is based on just two basic principles. The first is quite simple: Your description of physical reality is the same, regardless of the constant velocity at which you move. In other words, if you are moving in a straight line at a constant speed, you experience the same laws of physics as you would if you were moving at any other constant speed and in any other direction.

25. Special Relativity
Einstein’s second principle is much more bizarre than the first: Regardless of your speed or direction of motion, you always measure the speed of light to be the same. To see what this implies, imagine that you are in a spaceship that is moving toward a flashlight. Even if you are moving at 99% of the speed of light, you will measure the photons from the flashlight to be moving at the same speed (c = 3 × 108 m/s = 3 × 105 km/s) as if your spaceship were motionless.

26. The Special Theory of Relativity
Speed involves both distance and time. Since speed has a very different behavior in the special theory of relativity than in Newton’s physics, it follows that both space and time behave differently as well. Indeed, in relativity, time proves to be so intimately intertwined with the three dimensions of space that we regard them as a single four-dimensional entity called spacetime.

27. Length Contraction
The length you measure an object to have depends on how that object is moving; the faster it moves, the shorter its length along its direction of motion. This is called length contraction. In other words, if a train car moves past you at high speed, from your perspective on the ground you will actually measure it to be shorter than if it were at rest. However, if you are on board the railroad car and moving with it, you will measure its length to be the same as measured on the ground when it was at rest.

28. Length Contraction
If the idea of length contraction seems outrageous, it is because this effect is noticeable only at very high speeds, near the speed of light. But even the fastest spacecraft ever built by humans travels at a mere 1/25,000 of the speed of light. At this speed, a spacecraft 10 m long would be contracted in length by only 8 nm. For moving cars, trains, and airplanes, length contraction is far too small to measure.

29. Time Dilation
Nonetheless, physicists have confirmed the existence of time dilation by using an extremely accurate atomic clock carried on a jet airliner. When the airliner landed, they found that the on- board clock had actually ticked off slightly less time than an identical, stationary clock on the ground. For the passengers on board the airliner during this experiment, however, time flowed at a normal rate. You only measure a clock (or a beating heart) to be running slow if it is moving relative to you.

30. The Special Theory of Relativity
Now we see the significance of relative motion between observers. Relative motion changes what observers measure for certain quantities, such as length and time. Objects observed in motion are shorter in their direction of motion, and clocks observed in motion tick more slowly. However, relative motion does not change the speed of light (through empty space) and it does not change the laws used to describe nature.

31. Mass and Energy
The special theory of relativity also explains why it is impossible for a spaceship to move at the speed of light. If it could, then a light beam traveling in the same direction as the spaceship would appear to the ship’s crew to be stationary. But this would contradict the second of Einstein’s principles, which says that all observers, including those on a spaceship, must see light traveling at the speed of light. Therefore, it cannot be possible for a spaceship to travel at the speed of light. In fact, no object with mass can travel at the speed of light.

32. The general theory of relativity
Einstein’s special theory of relativity is a special case in the sense that it does not include the effects of gravity. Einstein’s next goal was to develop an even more comprehensive theory that includes gravity and its effects on space and time. This is Einstein’s general theory of relativity, which he published in 1915.

33. The general theory of relativity
A hallmark of gravity is that it causes the same acceleration no matter the mass of the object. For example, a baseball and a cannon ball have very different masses, but if you drop them side by side in a vacuum, they accelerate downward at exactly the same rate. To explain this observation, Einstein envisioned gravity as being caused by a curvature of space.

34. The general theory of relativity
In fact, his general theory of relativity describes gravity entirely in terms of the geometry of both space and time, that is, of spacetime. Far from a source of gravity, like a planet or a star, spacetime is “flat” and clocks tick at their normal rate. Closer to a source of gravity, however, space is curved and clocks slow down.

35. The general theory of relativity
Light rays naturally travel in straight lines. But if the space through which the rays travel is curved, as happens when light passes near the surface of a massive object like the Sun, the paths of the rays will likewise be curved. In other words, gravity should bend light rays.

36. The general theory of relativity
A gravitational lens refers to a distribution of matter (such as a cluster of galaxies) between a distant source and an observer, that is capable of bending the light from the source, as it travels towards the observer.

37. The general theory of relativity

38. The general theory of relativity

39. The general theory of relativity
In the general theory of relativity, a massive object such as Earth warps time as well as space. Einstein predicted that clocks on the ground floor of a building should tick slightly more slowly than clocks on the top floor, which are farther from Earth.

40. The Formation of Black Holes
Perhaps the most dramatic prediction of the general theory of relativity concerns what happens when a large amount of matter is concentrated in a small volume. We have seen that if a dying star is not too massive, it ends up as a white dwarf star. If the dying star is more massive than the Chandrasekhar limit of about 1.4 M⊙, it cannot exist as a stable white dwarf star and, instead, shrinks down to form a neutron star. But if the dying star has more mass than the maximum permissible for a neutron star, about 2 to 3 M⊙, not even the internal pressure of neutrons (the degenerate neutron pressure) can hold the star up against its own gravity, and the core collapses.

41. The Formation of Black Holes
As the star’s core becomes compressed to enormous densities, the strength of gravity at the surface of this rapidly shrinking sphere also increases dramatically.

42. The Formation of Black Holes
According to the general theory of relativity, the space immediately surrounding the core becomes so highly curved that it closes on itself.

43. The Formation of Black Holes
Photons flying outward at an angle from such an object’s surface would arc back inward.

44. The Formation of Black Holes
An object with an escape speed exceeding the speed of light is a black hole. Surrounding the black hole, where the escape speed from the hole just equals the speed of light, is the event horizon. You can think of the event horizon as the “surface” of the black hole, although it is just a spherical boundary and not a physical surface. Once a massive dying star collapses to within its event horizon, it disappears permanently from the universe. The term “event horizon” is quite appropriate, because this surface is like a horizon beyond which we cannot see any events.

45. Inside a Black Hole
The distance from the center of a nonrotating black hole to its event horizon is called the Schwarzschild radius (denoted RSch), after the German physicist Karl Schwarzschild who first determined its properties. This radius only on the black hole’s mass. The more massive the black hole, the larger its event horizon.

46. Inside a Black Hole
Once an object (such as a stellar core) has contracted enough to form an event horizon, nothing can prevent the matter’s further collapse. Equations describing the matter within the event horizon indicate that the object’s entire mass is crushed nearly to a single point, known as the singularity, at the center of the black hole.

47. Falling into a black hole
Imagine that you are on board a spaceship at a safe distance from a 5-M⊙ black hole. A distance of 1000 Schwarzschild radii, or 15,000 km, would suffice. You now release a space probe with a video camera and let it fall into the black hole. What will you see on the video as the probe falls?

48. Falling into a black hole
Initially, at 1000 Schwarzschild radii from the black hole, the video camera would send back a rather normal view of space. But as the probe approaches the black hole, the bending of light by the black hole becomes more pronounced. Light rays passing close to the back hole are deflected so much that background stars are severely distorted.

49. Falling into a black hole
How will the probe itself look from your vantage point on the spaceship at a safe distance? To discuss changes in the probe, let’s assume it emits blue light, and it has a clock that was initially synchronized with a clock on your spaceship. You might expect that as the probe falls, its speed should continue to increase. This expectation is true up to a point.

50. Falling into a black hole
As the probe approaches the event horizon, where the black hole’s gravity is extremely strong, the gravitational slowing of time becomes so pronounced that the probe will appear to slow down! From your point of view on the spaceship, the probe takes an infinite amount of time to reach the event horizon, where it will appear to remain suspended for all eternity.

51. Falling into a black hole
Near the event horizon, the strength of the black hole’s gravity increases dramatically as the probe moves just a short distance closer to the hole. In fact, the side of the probe nearest the black hole feels a much stronger gravitational pull than the side opposite the hole. These tidal forces are like those that the Moon exerts on Earth, but are tremendously stronger.

52. Falling into a black hole
Furthermore, the sides of the probe are pulled together, since the hole’s gravity makes them fall in straight lines aimed at the center of the hole. The net effect is that the probe will be stretched out along the line pointing toward the hole, and squeezed together along the perpendicular directions. The stretching—sometimes called spaghettification—is so great that it can rip even the strongest materials apart. The end.

53. For next time…
Study!!