The Final Frontier: The Dramatic Death of Stars and the Birth of Black Holes

Abstract

Gravity, acting unrelentingly on a star’s mass, tries to collapse it. The outward pressure generated while the star burns its nuclear fuel balances the inward force of gravitational collapse. But when a star has exhausted its store of fuel, what then? Well, gravitational collapse is inevitable. The star’s end state depends on the mass with which it started. Low-mass stars can strike a different balance between outward pressure and gravitational collapse. High-mass stars can find no such balance, so gravitational collapse continues—the star becomes a black hole.

4.1 Quiz

• 1. In 1999, NASA named their orbiting X-ray observatory Chandra in honour of Subrahmanyan Chandrasekhar. But what does the word chandra mean in Sanskrit?

  • □ Earth

  • □ Energy

  • □ Moon

  • □ Orbit

• 2. What is the mass of the most massive white dwarf yet discovered?

  • □ 1.35 times the mass of the Sun

  • □ 2.35 times the mass of the Sun

  • □ 3.35 times the mass of the Sun

  • □ 4.35 times the mass of the Sun

• 3. Which of the following statements about white dwarfs is correct?

  • □ If you add mass to a white dwarf its radius will decrease

  • □ If you add mass to a white dwarf its radius will increase

  • □ If you add mass to a white dwarf its radius will stay the same

  • □ There is no clear relationship between white dwarf mass and radius

• 4. Which is the closest white dwarf to Earth?

  • □ 40 Eridani B

  • □ Procyon B

  • □ Sirius B

  • □ Wolf 489

• 5. What is the surface temperature of Sirius B?

  • □ 25,200 K

  • □ 10,600 K

  • □ 6240 K

  • □ 2150 K

• 6. Which of the following is NOT a planetary nebula?

  • □ Ghost of Jupiter Nebula

  • □ Little Ghost Nebula

  • □ Skull Nebula

  • □ Trifid Nebula

• 7. In 1864, William Huggins observed the Cat’s Eye Nebula—a famous example of a planetary nebula. His observations led to which of the following (incorrect) hypotheses?

  • □ A new element, nebulium, must exist

  • □ A planetary nebula must have a Jupiter-sized planet at its core

  • □ A planetary nebula represents the birth pangs of a new star

  • □ Planetary nebulae lie beyond the confines of the Milky Way

• 8. In recorded history, how many supernovae may have been visible to the naked eye?

  • □ 1–3

  • □ 6–10

  • □ 15–20

  • □ More than 20

• 9. In what year did the earliest recorded supernova occur?

  • □ 4500 BCE

  • □ 4 BCE

  • □ 185 CE

  • □ 393 CE

• 10. John Aubrey, in Brief Lives, tells that Queen Elizabeth ‘sent for [Thomas Allen (an English mathematician)] to have his advice about the new star that appeared in the Swan or Cassiopeia’. By what name is that ‘new star’ commonly known?

  • □ Tycho’s Nova

  • □ Kepler’s Star

  • □ Crab Nebula

  • □ Cas A

• 11. The brightest stellar event in recorded history was a supernova, which appeared to be about 16 times brighter than Venus. Which supernova was it?

  • □ SN 393

  • □ SN 1006

  • □ SN 1054

  • □ SN 1572

• 12. ASASSN-15lh is (as of January 2024) the most inherently luminous supernova event ever observed. At its peak, how bright was it?

  • □ 57 million times brighter than the Sun

  • □ 570 million times brighter than the Sun

  • □ 5.7 billion times brighter than the Sun

  • □ 570 billion times brighter than the Sun

• 13. Astronomers believe that a Type I supernova originates from a white dwarf in a binary system. This understanding of Type I supernovae was developed in large part from observations of which event?

  • □ SN 1961V

  • □ SN 1972E

  • □ SN 1983N

  • □ SN 1987A

• 14. A plerion is a type of what?

  • □ Black hole

  • □ Magnetar

  • □ Stellar cluster

  • □ Supernova remnant

• 15. The star with the highest known surface temperature (210,000 K) is of what type?

  • □ Cepheid variable star

  • □ Red giant star

  • □ T Tauri star

  • □ Wolf–Rayet star

• 16. Astronomers detected the first gamma-ray burst in July 1967. What type of detector was involved?

  • □ A Soviet ballistic missile early-warning system

  • □ A Franco-German UV telescope

  • □ A US military satellite

  • □ A Chinese telecommunications network

• 17. What was noteworthy about the gamma-ray burst GRB 221009A, which took place two billion light-years from Earth?

  • □ It affected Earth’s upper atmosphere

  • □ It lasted seven times longer than any other burst

  • □ It repeated four days later

  • □ It was the most distant gamma-ray burst ever detected

• 18. Neutron stars are incredibly dense. How much would a sugar cube’s worth of neutron star material weigh?

  • □ About the mass of ten Sherman tanks

  • □ About the same as the mass of water in 100 Olympic-sized swimming pools

  • □ About the same as the mass of the entire human population

  • □ About half the mass of the Sun

• 19. The most massive neutron star known (as of January 2024) is PSR J0740+6620. In comparison with the Sun, how massive is this neutron star?

  • □ 0.98 ± 0.01 times the mass of the Sun

  • □ 1.41 ± 0.02 times the mass of the Sun

  • □ 2.08 ± 0.07 times the mass of the Sun

  • □ 12.1 ± 1.8 times the mass of the Sun

• 20. What was the initial nickname of the object PSR B1919 + 21?

  • □ IOU-1

  • □ LGM-1

  • □ NICE-1

  • □ XYZ-1

• 21. In 2022, the Ulster Bank of Northern Ireland released a new £50 polymer note. The image of which scientist features on the note?

  • □ Margaret Theodora Meyer

  • □ Ernst Julius Öpik

  • □ John Stewart Bell

  • □ Susan Jocelyn Bell Burnell

• 22. Who coined the term ‘pulsar’?

  • □ Susan Jocelyn Bell Burnell

  • □ Antony Hewish

  • □ Anthony Michaelis

  • □ Fred Hoyle

• 23. Apart from the Sun, which was the first astronomical object to be identified as a source of X-rays?

  • □ Aquarius X-1

  • □ Crab Nebula

  • □ Scorpius X-1

  • □ X-Centaurus

• 24. How quickly does PSR J1748–2446ad, the fastest known pulsar, rotate?

  • □ 87 times per second

  • □ 101 times per second

  • □ 332 times per second

  • □ 716 times per second

• 25. What is the period of the slowest known pulsar?

  • □ 0.92 seconds

  • □ 3.3 seconds

  • □ 23 seconds

  • □ 75.9 seconds

• 26. Magnetars are neutron stars with unusually large magnetic fields. How close would you have to be to a magnetar before its magnetic field started to disrupt your biomolecular structure?

  • □ About 10 m

  • □ About 100 m

  • □ About 10 km

  • □ About 1000 km

• 27. How long does a fast radio burst typically last?

  • □ About a nanosecond

  • □ About a millisecond

  • □ About a second

  • □ About a minute

• 28. In 2011, researchers using the radio telescope at Parkes Observatory discovered a type of signal they called a peryton. What turned out to be the cause of the perytons?

  • □ Premature opening of microwave ovens in the observatory kitchen

  • □ Identification signals from aircraft landing at a nearby airstrip

  • □ Lightning flashes

  • □ Solar flares

• 29. Kip Thorne and Anna Żytkow proposed the existence of what are now called Thorne–Żytkow objects. What is a Thorne–Żytkow object?

  • □ A neutron star at the centre of a giant star

  • □ A pair of co-orbiting black holes generating a specific gravitational wave pattern

  • □ A primordial black hole in orbit around a cosmic string

  • □ A traversable wormhole

• 30. The Hulse–Taylor binary consists of a pulsar and a neutron star emitting gravitational radiation as they orbit around a common centre of mass. The orbits are decaying. How long until final inspiral, when the two stars collide?

  • □ 3 million years

  • □ 30 million years

  • □ 300 million years

  • □ 3 billion years

• 31. GW170817 was a merger of two neutron stars. The merger is believed to have created heavy metals such as gold. How much gold did GW170817 create?

  • □ A mass of gold equivalent to 3–13 times the mass of the Earth

  • □ A mass of gold equivalent to 0.1–0.9% of the mass of the Earth

  • □ 300–800 million tons of gold

  • □ 5000–8000 tons of gold

• 32. How much power does Earth emit in gravitational waves as it orbits the Sun?

  • □ 5 mW

  • □ 200 W

  • □ 7 GW

  • □ 215 TW

• 33. What did Einstein co-patent in 1930?

  • □ A laser pointer

  • □ A microwave oven

  • □ A refrigerator

  • □ A rotary engine

• 34. Which of the following phrases was coined by John Wheeler?

  • □ To make an apple pie from scratch, you must first create the universe

  • □ Mass tells spacetime how to curve; curved space tells mass how to move

  • □ Reality is merely an illusion, albeit a very persistent one

  • □ Science without religion is lame; religion without science is blind

• 35. Of a book on relativity, published in 1923, Einstein said it was: ‘…the finest presentation of the subject in any language’. Who wrote the book?

  • □ Paul Dirac

  • □ Arthur Eddington

  • □ Hendrik Lorentz

  • □ Hermann Weyl

• 36. In 1919, observations of a solar eclipse confirmed Einstein’s general theory of relativity. The eclipse was observed from Sobral, in Brazil, and from which island?

  • □ Cape Verde

  • □ Mauritius

  • □ Principe

  • □ Zanzibar

• 37. The Global Positioning System must take account of general relativity. According to the theory, clocks orbiting 20,000 km above Earth will tick faster, relative to the ground, by how much?

  • □ 45 microseconds per day

  • □ 0.8 milliseconds per day

  • □ 247 milliseconds per day

  • □ 1.1 seconds per day

• 38. The English priest John Michel was the first person to propose the existence of what?

  • □ Black holes

  • □ Galaxies

  • □ Light pollution

  • □ Variable stars

• 39. Karl Schwarzschild was the first person to provide an exact solution of Einstein’s equations of general relativity. He died in 1916 aged 42, from what?

  • □ A sniper bullet, while serving on the Russian front in World War I

  • □ A depressive illness brought on by his struggles with the maths of general relativity

  • □ A rare autoimmune disease, called pemphigus

  • □ An infected wound, suffered while constructing trenches in Belgium in World War I

• 40. Who coined the term ‘event horizon’?

  • □ Stephen Hawking

  • □ Charles Misner

  • □ Roger Penrose

  • □ Wolfgang Rindler

• 41. Which future Nobel prize winner was an examiner on Stephen Hawking’s PhD?

  • □ Peter Higgs

  • □ Brian Josephson

  • □ Roger Penrose

  • □ Martin Ryle

• 42. An astronaut, coming too close to the singularity of a black hole, will be disrupted by gravitational tidal forces (a process called spaghettification). If an astronaut falls into a supermassive black hole, at what point will the astronaut’s body be spaghettified?

  • □ Outside the event horizon, at a distance of 10 times the Schwarzschild radius

  • □ Outside the event horizon, at a distance of 3.14 times the Schwarzschild radius

  • □ At the event horizon

  • □ Inside the event horizon

• 43. A non-rotating solar-mass black hole has a Schwarzschild radius of 2.95 km. What is the radius of the smallest possible stable circular orbit around this black hole?

  • □ 2.95 km

  • □ 8.86 km

  • □ 52,500 km

  • □ 696,340 km

• 44. Which was the first source to be widely accepted as a black hole?

  • □ 3C 371

  • □ Arp 220

  • □ Cygnus X-1

  • □ Sgr A*

• 45. What is the Hawking temperature of a solar-mass black hole?

  • □ 60 nK (0.00000006 K)

  • □ 20 mK

  • □ 0.9 K

  • □ 2.7 K

• 46. What is the evaporation time of an isolated solar-mass black hole?

  • □ 10⁹ years

  • □ 10¹² years

  • □ 10²¹ years

  • □ 10⁶⁷ years

• 47. If a primordial black hole was evaporating right now, roughly what would its mass have been shortly after the big bang?

  • □ 10² kg (roughly the mass of a person)

  • □ 10¹¹ kg (roughly the mass of Valetudo, a small moon of Jupiter)

  • □ 10²⁷ kg (roughly the mass of Jupiter)

  • □ 10³⁰ kg (roughly the mass of the Sun)

• 48. Whose discovery led to the so-called black hole information paradox?

  • □ Richard Feynman

  • □ Stephen Hawking

  • □ Roger Penrose

  • □ John Wheeler

• 49. An early resolution to the black hole information paradox, which preserves information, was shown to lead to another paradox. What is that paradox called?

  • □ Firewall paradox

  • □ Fortress paradox

  • □ Partition paradox

  • □ Roadblock paradox

• 50. Who coined the term ‘wormhole’, referring to connections between different parts of spacetime?

  • □ Ludwig Flamm

  • □ Stephen Hawking

  • □ Roger Penrose

  • □ John Wheeler

4.2 Solutions

1.	C	26.	D
2.	A	27.	B
3.	A	28.	A
4.	C	29.	A
5.	A	30.	C
6.	D	31.	A
7.	A	32.	B
8.	B	33.	C
9.	C	34.	B
10.	A	35.	B
11.	B	36.	C
12.	D	37.	A
13.	B	38.	A
14.	D	39.	C
15.	D	40.	D
16.	C	41.	C
17.	A	42.	D
18.	C	43.	B
19.	C	44.	C
20.	B	45.	A
21.	D	46.	D
22.	C	47.	B
23.	C	48.	B
24.	D	49.	A
25.	D	50.	D

4.3 Guide: The Death of Stars

Gravity, acting unrelentingly on a star’s mass, tries to collapse it. The outward pressure generated while the star burns its nuclear fuel balances the inward force of gravitational collapse. But when a star has exhausted its store of fuel, what then? Well, gravitational collapse is inevitable. The star’s end state depends on the mass with which it started. Low-mass stars can strike a different balance between outward pressure and gravitational collapse. High-mass stars can find no such balance, so gravitational collapse continues—the star becomes a black hole.

4.3.1 White Dwarfs

Low-mass, red dwarf stars burn so slowly that none has yet faced its demise. But we can observe the fate of stars with the mass of the Sun, and stars up to about eight times the mass of the Sun: they become white dwarf stars. A white dwarf represents the end stage of stellar evolution for about 97% of all stars.

• 1. The Indian–American astrophysicist Subrahmanyan Chandrasekhar—known to his colleagues simply as Chandra (a word that means ‘moon’ in Sanskrit, and is the name of the Hindu god of the Moon)—made wide-ranging contributions to science. He was one of the first scientists to combine the study of physics with astronomy to form the discipline of astrophysics. Chandra’s scientific style was unique. He would pick an area of research, work in it for several years, write a book encapsulating the field, and then move on to something else. One of his research areas was stellar structure and evolution, and this work led to the award of a Nobel prize, the second such prize in his family—his paternal uncle, Chandrasekhara Venkata Raman (1988–1970), was awarded the 1930 Nobel prize for work on the scattering of light. It was Chandra who deduced the structure of white dwarf stars.

• 2. When a star such as the Sun has no more fuel for fusion, it no longer produces an outward-pointing pressure to counterbalance the inward-pointing force of gravity. Gravity causes the star to collapse. What happens to the material of the star? Quantum theory provides the answer: no two electrons can be in the same quantum state so, as the star contracts, the lowest electron energy levels fill quickly and the remaining electrons are forced to occupy higher energy levels. The more electrons there are in a given volume, the higher the maximum energy of an occupied level. Pressure must therefore be exerted on the electrons to compress them; we encounter electron degeneracy pressure. A white dwarf star thus reaches a new stable equilibrium: electron degeneracy pressure counterbalances further gravitational collapse. This stability only exists up to a certain mass limit. Beyond a certain mass, the balance of gravitational attraction and electron degeneracy pressure breaks down: gravity wins and further collapse takes place. This mass limit, about 1.4 times the mass of the Sun, is the Chandrasekhar limit. A star can start out with more than that mass and still end up as a white dwarf, so long as it loses sufficient mass before gravitational collapse takes place. Mid-size stars do indeed shed mass, through a variety of mechanisms, in their later evolutionary stages. A stable white dwarf star, however, can never be more than about 40% more massive than our Sun. The most massive white dwarf yet discovered goes by the catchy name of ZTF J1901+1458, and it has a mass 1.35 times greater than that of the Sun. Later in this chapter we shall explore the fate of higher-mass stars.

• 3. Electron degeneracy, which occurs at densities of around 10⁶ kg m⁻³, gives rise to an unfamiliar form of matter: degenerate matter. The properties of degenerate matter produce some peculiar phenomena. For example, its ‘squashability’ means that a high-mass white dwarf is denser than a low-mass white dwarf. If a white dwarf gets heavier, perhaps by sucking matter away from a companion star and onto itself, then it gets smaller.

• 4. The high density of white dwarf stars becomes apparent when we consider Sirius B, the companion to Sirius, the Dog Star. (Sirius B is nicknamed ‘the Pup’ for obvious reasons.) At a distance of 8.6 light-years, Sirius B is the closest white dwarf. Despite its proximity, it is not itself visible to the naked eye. Nevertheless, astronomers can measure its properties with precision. Sirius B has a mass slightly greater than the Sun packed into a sphere about the same size as Earth. This star is dense. A teaspoon of the Pup would weigh more than an elephant.

• 5. Sirius B was discovered in 1844 by Bessel, who deduced its presence by studying changes in the Dog Star’s proper motion. In 1862, the American astronomer Alvan Graham Clark (1832–1897) was the first to directly observe the companion. Further observations showed that, in addition to being dense, Sirius B is hot: its surface temperature is 25,200 K, hotter by far than the Sun’s 5770 K. Sirius B is, however, cooling. Any white dwarf, when it first forms, is exceptionally hot; but, because it no longer generates energy from nuclear reactions, a white dwarf cools as it radiates into space. In a few billion years Sirius B will be cooler than the Sun.

• 6. Consider a mid-size star—one with a mass between about 1–8 times the mass of the Sun. As we now know, its fate is to end its life as a white dwarf. But before it reaches that stage it goes through a red giant phase as it moves off the main sequence. And this can generate one of the most beautiful sights in the cosmos: a short-lived planetary nebula. (Fig. 4.1 shows an example of a planetary nebula. The term is an accident of history: these nebulous objects appeared planet-like in early telescopes. Planets are not involved, but the name stuck.) The atmosphere of a red giant disperses as an expanding shell of gas, and it leaves behind a hot core. The core emits ultraviolet radiation, which causes the shell to glow—and we see a planetary nebula. After a few tens of thousands of years the core cools, forms a white dwarf, and the glowing planetary nebula disappears from view. A planetary nebula is typically about one light-year in diameter, but there is no typical shape. Some are spherically symmetric; others, for reasons not well understood, take on exotic shapes. Those shapes give rise to some poetic names for planetary nebulae, my personal favourite being the Dandelion Puffball Nebula, also known less evocatively as NGC 6751. (Note that not all nebulae are planetary nebulae. The Trifid Nebula, for example, is not a planetary nebula, but rather a star-forming H II region.)

Fig. 4.1

An image of NGC 246, also known as the Skull Nebula, taken by ESO’s Very Large Telescope. As with other planetary nebulae, this is a glowing shell of ionised gas ejected by a central red giant star. In a few tens of millennia, the Skull Nebula will dissipate and leave behind a naked white dwarf. This planetary nebula is so-far unique: the central white dwarf has two companion stars. (Credit: ESO. CC BY 4.0 DEED)

• 7. Some planetary nebulae might appear to be substantial objects, but it turns out that in all cases the gases involved are extremely rarefied. This understanding come about rather slowly. Perhaps the first people to study planetary nebulae spectroscopically were William Huggins (1824–1910) and his wife Margaret Lindsay Huggins (1848–1915). When William analysed the spectrum of the Cat’s Eye Nebula he was puzzled: a typical stellar spectrum exhibits a continuum with dark absorption lines, but the spectrum from the nebula showed strong emission lines. The brightest emission line, at 500.7 nm, corresponded to no known element. In 1898, Margaret suggested the name ‘nebulium’ for this element. Nebulium, of course, does not exist. We now know that the observed emission line comes from doubly ionised oxygen in conditions of low density. In these circumstances one sees emission lines that are heavily suppressed in laboratory conditions. These so-called ‘forbidden lines’ are typically the brightest lines in the spectra of planetary nebulae.

4.3.2 Supernovae

The fate of most stars is to end as a white dwarf. If a star starts out with more than about eight times the mass of the Sun, however, things are different. A high-mass star cannot rid itself of enough material to get its core below the Chandrasekhar mass limit. It therefore cannot settle as a white dwarf. The result is a supernova. We consider first the supernova event itself before looking at the objects these events leave behind.

• 8. If, at the end of its life, a star’s core exceeds the Chandrasekhar mass limit then electron degeneracy pressure is insufficient to counterbalance the inexorable force of gravity. The core collapses further in an event known as a supernova. Such an event generates unimaginable numbers of neutrinos, throws vast amounts of heavy elements out into space, and produces enough radiation to be seen halfway across the universe. For a short while, a supernova can outshine an entire galaxy. (The event described above is a core-collapse, or Type II, supernova. There is another type of supernova. A Type I supernova occurs when a white dwarf is in close orbit around another star, accreting material. When the accreting material pushes the white dwarf over the Chandrasekhar limit… boom! The full supernova classification scheme is slightly more complicated, but in essence they are of Type I or Type II.) High-mass stars are relatively uncommon; accreting white dwarfs are more common but still not prevalent. So, in any particular year, our Milky Way galaxy is unlikely to host a supernova. Few, indeed, have been seen with the naked eye. It is difficult to know for certain how many supernovae have been seen by humans because some suggested candidates might have been something else, such as a comet or a classical nova. Nevertheless, most sources suggest that seven or eight supernovae have been visible to the naked eye in the historical period. With the advent of the telescope, and particularly with recent methods for identifying supernovae in distant galaxies, astronomers have been able to study supernovae almost on-demand.

• 9. The earliest recorded supernova is SN 185, which was observed by Chinese astronomers some 1800 years ago. When they described it in the Book of Later Han they called it a ‘guest star’: no light had been visible in that patch of sky and then, without warning, a star appeared like an unannounced guest. The remnant of this event can probably be associated with a shell of gas known as RCW 86. If this is correct, then the supernova was 9100 light-years away. (It is possible that people saw a supernova before 185. Indian astrophysicists have claimed that an ancient rock carving in Kashmir might represent the supernova that gave rise to the remnant HB9. The supernova could have been visible to humans living around 4500 BCE, so the interpretation is not unreasonable—but it is difficult to know for sure.)

• 10. Tycho’s Nova, or SN 1572, represents one of the most important astronomical events in history. Brahe, a truly great astronomer, who as we saw in an earlier chapter worked just before the invention of the telescope, proved the new star must be more distant than the Moon. Until then, a prevailing notion, based on Aristotelian ideas, was that the celestial sphere was unchanging, fixed, permanent—and somehow perfect. Brahe’s observations challenged that notion of the permanence of the celestial sphere and gave impetus to the Scientific Revolution.

• 11. Supernovae are inherently bright. Depending upon their distance from us, they can appear bright too. Astronomers believe SN 1006, a Type I supernova that occurred about 7200 light-years away, reached a peak visual magnitude of −7.5 when it appeared in the skies on 1 May 1006. Observers in Europe, Egypt, China, Japan, and modern-day Iraq all recorded the event, with some reports stating the star could be seen during the day. Recent work has found chemical traces, laid down in Antarctic ice, that originated when radiation from the explosion struck Earth.

• 12. If a supernova is distant then, no matter how luminous it is, it will appear faint. The All Sky Automated Survey for SuperNovae (ASAS-SN) detected ASASSN-15lh (also called SN 2015L) on 14 June 2015. Its peak apparent magnitude was +16.9, which is dim. But that dimness was due to its distance—3.8 billion light-years. At its peak, ASASSN-15lh shone with a brightness equivalent to all the light emitted by our Milky Way galaxy. It was 570 billion times more luminous than the Sun. If such a supernova event happened in our galaxy, it would be bright enough to cast shadows at night.

• 13. Type I supernovae (or, more precisely, a subset called Type Ia supernovae) are particularly important for astronomers. To a good approximation, these explosions of white dwarf stars all appear to be identical. This means if we can calibrate their peak luminosity then we can use them as standard candles, which lets us calculate their distance. In an earlier chapter we learned how Cepheid variable stars can be used as distance indicators. Since supernovae are much brighter than Cepheids, they give us a measuring stick that reaches deep into the cosmos. SN 1972E, the second brightest supernova of the twentieth century, which occurred in an irregular galaxy called NGC 5253, was important in this regard. Its location, in the outskirts of the host galaxy, meant astronomers could observe it with little interference from background objects. Furthermore, the position of the galaxy in the sky made it a reasonable target for observatories in both hemispheres. Astronomers observed the supernova for almost 2 years, and provided a host of data for the theoreticians to explain. Our understanding of Type I supernovae came, in large part, from these observations of SN 1972E.

• 14. A supernova ejects material at high speeds out into the interstellar medium, and the ejected material forms an expanding diffuse nebula called a supernova remnant. There are two main types of supernova remnant. A shell remnant, such as Cassiopeia A, emits radiation from the shell itself. A shock wave plows into the interstellar medium, heating it, which causes the emission of X-rays. The shock also accelerates electrons, causing the emission of synchrotron radiation. A plerion, also called a filled-centre remnant, emits radiation from within the expanding shell: the remnant star at the centre of the shell continuously supplies highly energetic electrons, which emit synchrotron radiation. The best-known example of a plerion is the Crab Nebula. See Fig. 4.2.

Fig. 4.2

A three-colour, composite image of one of the most famous objects in the sky: the Crab Nebula, also known as Messier 1. This is the remnant of the supernova that Japanese and Chinese astronomers recorded in 1054. It is about 6500 light-years distant. At the centre of this supernova remnant is a neutron star, spinning 30 times per second. (Credit: ESO. CC BY 4.0 DEED)

• 15. The more massive the star, the shorter its life; the shorter its life, the less likely we are to see it. Wolf–Rayet stars, with a mass typically 25 times that of the Sun, form an extremely rare class of star—only 220 or so are known. They were discovered by Charles Joseph Étienne Wolf (1827–1918) and Georges-Antoine-Pons Rayet (1839–1906). These French astronomers made their discovery in 1867, but it was more than a century before we gained a reasonable understanding of such high-mass stars—and even now aspects of their behaviour remain enigmatic. A Wolf–Rayet star is reaching the end of its brief life. It has used up all its hydrogen, and is frantically burning heavier elements to maintain equilibrium against the crushing force of gravity. Such a star is therefore exceptionally hot (the hottest known star WR 102, in the constellation of Sagittarius, has a surface temperature of 210,000 K). A fast-moving wind blows the outer layers of a Wolf–Rayet star into space, where the interstellar medium becomes enriched with heavier elements. This mass loss cannot prevent the fate of a Wolf–Rayet star, which is to end in violence as a Type II supernova.

• 16. The core-collapse of a young, massive star can sometimes result in an extremely powerful supernova—a hypernova. It seems that hypernovae are responsible for at least some of the most energetic events in the known universe: gamma-ray bursts (GRBs). The story behind the discovery of GRBs began in 1963, when the Partial Test Ban Treaty (PTBP) banned the testing of nuclear weapons under water, in the atmosphere, and in outer space. To monitor Soviet compliance with the treaty, the US Department of Defense put resources into Project Vela, one output being a group of satellites, orbiting high above the Van Allen radiation belt, with X-ray, neutron, and gamma-ray detectors on board. The idea was that Vela satellites would detect radiation from any nuclear bombs that the Soviet Union tested in the atmosphere or in space. Vela did not detect any Soviet violations of the PTBP, but it did detect gamma rays from space. The American astronomer Ray William Klebesadel (1932–) and colleagues found 16 bursts of gamma rays and determined that the bursts came from beyond the solar system. It took many years, however, before astronomers were confident the bursts came from distant galaxies rather than our own Milky Way galaxy. In 2003, astronomers studied the burst GRB 030329 and found its spectrum settled down to one that is characteristic of a core-collapse supernova. Much about GRBs remains mysterious but it seems certain that, at least in the case of long-duration bursts, a supernova is involved. (Short-duration GRBs, which last only a few seconds, appear to be generated by the collision of two neutron stars.)

• 17. The power of GRBs is astonishing. Consider what happened on 9 October 2022, when NASA’s Fermi and Swift satellites detected gamma rays and X-rays from GRB 221009A. This was the brightest burst ever seen, bright enough even to be seen with amateur optical telescopes. The burst was bright in part because it was inherently luminous, in part because it was near. The word ‘near’ requires some context, though: GRB 221009A, the closest burst ever detected, was two billion light-years from Earth. The radiation sweeping across Earth that October day had travelled 20 billion trillion kilometers to reach us! To give some feeling for its power, note that even from so far away the burst could be felt. Detectors in Germany and India demonstrated that the propagation of lightning strikes changed briefly when radiation from GRB 221009A hit: the gamma rays knocked electrons from their host atoms and this, for a short while, affected lightning in our planet’s atmosphere.

4.3.3 Neutron Stars and Pulsars

If a star starts life with more than about eight times the mass of the Sun then, after it has spent its nuclear fuel, electron degeneracy pressure will be insufficient to halt gravitational collapse. Rather than settling as a stable white dwarf, the star’s collapse continues.

• 18. Chandrasekhar showed that the mass of a white dwarf cannot exceed a certain limit. Above that limit, the electron degeneracy pressure that stabilises a white dwarf against gravitational collapse becomes insufficient for the job. Gravity wins. When a high-mass star has ended its life in a supernova event, leaving behind a core with a mass above the Chandrasekhar limit, then on a timescale of about a tenth of a second the core collapses. Electrons are pushed into atomic nuclei, where they interact with protons to form neutrons. At this point, the stellar remnant consists essentially entirely of neutrons. If the core is not too massive then neutron degeneracy pressure can counterbalance gravity and a new equilibrium is established. We have a neutron star. A neutron star packs a stellar mass into a sphere with a radius of about 10 km. The density of neutron star material is therefore huge: roughly 4 × 10¹⁷ kg m⁻³. To put that into context, calculate the mass of a sugar-cube-sized volume of neutron star material (the calculation is straightforward). Now make a rough estimate of the mass of the average human and multiply that by 8 billion (the number humans currently on the planet). You will see the two numbers are not too far apart. A sugar-cube-sized volume of neutron star material has roughly the same mass as the totality of human flesh and bone.

• 19. A neutron star typically has a mass of about 1.4 times that of the Sun. Neutron stars can be more massive than 1.4 solar masses but, as with white dwarfs, physics places an upper limit on their mass. Neutron stars are supported against further gravitational collapse by neutron degeneracy pressure and repulsive nuclear forces. Beyond a certain mass, however, no known force can withstand the crush of gravity and the neutron star collapses into a black hole. The precise value of the mass limit is difficult to calculate because the physics of neutron star interiors is not well understood. But the limit is thought to be somewhere around two times the mass of the Sun. The neutron star PSR J0740+6620 has about 2.08 ± 0.07 times the mass of the Sun, so it is close to the limit. It is the most massive neutron star astronomers have discovered to date. (Astronomers, incidentally, have an interesting way of determining the mass of this neutron star. The pulsar has a white dwarf companion, and the two stars orbit each other in a plane that is almost edge-on to Earth. When the pulsar passes behind the white dwarf, a relativistic effect causes a delay in the arrival time of the pulses. By measuring the delay astronomers can determine the mass of the white dwarf, and once they know the mass of one star in a binary system they can determine the mass of the other.)

• 20. If neutron stars have a radius of just 10 km or so—about the size of a city—then how can we possibly detect them? How do we know they exist? Well, a discovery by Susan Jocelyn Bell Burnell (1943–) provided the evidence. Bell (later Bell Burnell) studied for her PhD at the University of Cambridge under Antony Hewish (1924–2021), where she helped build a radio telescope, designed by Hewish, called the IPS Array. In 1967, she noticed the telescope was picking up a peculiar signal from a star. The star pulsed every 1.3373 seconds, with a pulse width of 0.04 seconds. At the time she could not account for the signal, nor for its regularity. Bell and Hewish even entertained the hypothesis that they had detected an alien beacon, and they considered naming the star LGM-1 (the initialism standing for ‘little green man’). The true explanation was not long in coming. A star typically rotates slowly; as it undergoes gravitational collapse, it rotates more quickly. The effect is due to conservation of angular momentum. (An analogy often introduced at this point is of a spinning ice skater with outstretched arms: if the skater brings her arms to her chest then she spins faster.) A neutron star, then, spins quickly. A neutron star will also possess a powerful magnetic field, billions of times more powerful than the magnetic field of our planet. All this happens within an environment rich with electrically charged particles. This combination—an intense, fast-spinning magnetic field whipping around charged particles—means a neutron star emits beams of electromagnetic radiation from the magnetic poles. If the star’s axis is oriented in a particular direction then, as the star rotates, the beam regularly sweeps across Earth. The effect is much like a lighthouse beam regularly sweeping across our field of view.

• 21. Bell was born in Northern Ireland and the Ulster Bank has commemorated her discovery by featuring her on a £50 note. She deserved more recognition, however. Hewish was awarded a share of the 1974 Nobel prize for the discovery of radio pulses from collapsed stars; Bell Burnell was overlooked. Societal attitudes to women in science were at that time disreputable. Hewish and Bell were often interviewed by journalists, and Bell later revealed how the media circus could be demeaning: Hewish would be asked about science, she would be asked about her vital statistics. Although she has been remarkably gracious about the prize, the decision by the Swedish Academy of Sciences to ignore her decisive contributions has long been a source of controversy in the scientific community.

• 22. Neither Hewish nor Bell had a gift for coming up with catchy names. They found the first four repeating radio sources but referred to these strange new stellar objects as ‘pulsating radio stars’. It was a science journalist, Anthony Michaelis (1916–2007), working at the Daily Telegraph newspaper, who shortened the term to ‘pulsar’. The first pulsar now has the official name PSR B1919+21. The letters ‘PSR’ refer to ‘pulsating source of radio’; the numbers are celestial coordinates. Pulsars discovered before 1993 typically have the letter B in their name, those discovered after have the letter J; the letters specify which coordinate system the numbers refer to. (For example, we know from its name that PSR J0740+6620, the high-mass neutron star discussed earlier, must be a pulsar discovered some time after 1993.)

• 23. Pulsars represent the main direct evidence for the existence of neutron stars, but other observations can reveal the presence of these collapsed objects. Indeed, in retrospect we can say that the first neutron star was detected in 1962 when, as we saw in an earlier chapter, a rocket was launched with an X-ray detector on board. The detector found a bright X-ray source in the constellation of Scorpius and, since it was the first such source found in the constellation, it was given the name Scorpius X-1. Apart from the Sun, Scorpius X-1 is the brightest X-ray source in our skies. We now know that the source is a 1.4 solar mass neutron star with an 0.42 solar mass star in orbit around it. The neutron star pulls material away from the smaller star and, as this material heats up in an accretion disk, X-rays are emitted.

• 24. Back to pulsars. The fastest confirmed rotation rate for a pulsar is 716 Hz; in other words, this neutron star, PSR J1748–2446ad, rotates once every 1.39 milliseconds. If the object has a typical neutron star radius, then a point on its equator must be moving at about a quarter of the speed of light! Astronomers know of about 130 millisecond pulsars. It is not entirely certain how these stars come to be spinning so quickly. The leading idea is that they are old neutron stars that have accreted material from a companion star, leading to a ‘spinning up’ of the rotation. That so many millisecond pulsars are in globular clusters, where the density of stars is high and the chance of a neutron star capturing a companion star is relatively high, lends support to this hypothesis.

• 25. At the other extreme of rotation, PSR J0901–4046, detected in 2020 by the MeerKAT radio telescope, rotates once every 76 seconds. This discovery is puzzling. As we have seen, the conservation of angular momentum means the rotational velocity of a star increases as its diameter decreases; neutron stars therefore spin rapidly. If the neutron star is a pulsar, then it will emit radiation. Freshly minted pulsars spin rapidly, but over aeons, as they lose energy via radio emission, they slow. Eventually, for some combination of rotation and deceleration rates, the pulsar shuts down: radio emission stops. The puzzle with PSR J0901–4046 is that it is still pulsing. No other pulsar has been observed with this combination of rotation and deceleration. Prior to its discovery, the slowest pulsar had a period of 23.5 seconds. PSR J0901–4046 might force astronomers to revise their ideas about pulsars.

• 26. All pulsars are neutron stars; not all neutron stars are pulsars. Another possible outcome of a supernova event is for the neutron star to become a magnetar. The distinguishing feature of a magnetar is its extreme magnetic field, which is strong enough for the object to emit gamma-rays rather than the radio waves typically emitted by a pulsar. A magnetar’s magnetic field is almost inconceivable. Earth’s magnetic field has a strength of about 1 gauss. (The gauss is an old-fashioned unit, but when talking about Earths’ magnetic field it makes sense to use it.) The Sun’s magnetic field varies with time and location but is typically of the order of 100 gauss. An MRI subjects a patient to a magnetic field of about 10,000 gauss. The strongest magnetic field generated in a laboratory is 450,000 gauss and, using currently available materials, it is difficult to get much beyond that. A neutron star can have a magnetic field as large as 1 trillion gauss. The field of a magnetar can be a thousand times larger still: a quadrillion (1000,000,000,000,000) times greater than the magnetic field in which you are currently sitting. Strange things happen to atoms, and even to space itself, when subjected to those sorts of magnetic fields. A magnetar’s field is so strong that if you got within about 1000 km it would disrupt your biomolecular structure. You would die the sort of death supervillains routinely wish on James Bond.

• 27. Magnetars might be behind one of the most puzzling astronomical discoveries of recent years. The first fast radio burst (FRB) was detected by the British astronomer Duncan Ross Lorimer (1969–) in 2007, when analysing survey data from the Parkes radio telescope. Lorimer and his student found a radio pulse lasting just a few milliseconds. Since then, radio astronomers have found hundreds of other FRBs, radio spikes that last between a fraction of a millisecond to a few milliseconds. These astrophysical sources pump out as much energy in a millisecond as the Sun radiates in a day, but because they are so distant the strength of the signal when it reaches Earth is tiny. The nature of FRBs is still up for debate, but the involvement of magnetars is plausible.

• 28. Not all short radio bursts are FRBs. In 2011, researchers at Parkes observed 16 signals that resembled FRBs. For a while, the source puzzled them. They called this type of signal a peryton, because although the signal looked natural it was artificial. (The peryton—part stag, part bird, but casting the shadow of a human—is a mythical creature dreamed up by the author Jorge-Luis Borges.) It turned out two Matsushita microwave ovens were responsible: when hungry astronomers opened the doors while the magnetrons were still active, the ovens emitted radio pulses that the telescope picked up!

• 29. Pulsars, magnetars, FRBs… neutron stars make their existence known in a variety of interesting ways. Another possibility, which is at present hypothetical, comes from the American physicist Kip Stephen Thorne (1940–). Thorne is best known for his work on gravitation, for which he shared the 2017 Nobel prize with Weiss and Barish. In 1976, however, Thorne and the Polish astrophysicist Anna Żytkow (1947–) researched a different area of astronomy. They conjectured that in dense star clusters it might sometimes happen that a neutron star collides with a red giant star. One would then have a ‘star within a star’: a red giant with a neutron star at its core. Astronomers have found candidate Thorne–Żytkow objects, but none have so far been confirmed.

• 30. In 1974, the American astronomers Russell Alan Hulse (1950–) and Joseph Hooton Taylor, Jr. (1941–) discovered a particularly interesting pulsar. The pulse arrival time varied in a repetitive manner. Hulse and Hooton argued this could be explained if the pulsar were in orbit with another star. The companion turned out to be a neutron star. This, then, was a binary neutron star system. Hulse and Taylor continued to make observations of the pulse timings and they concluded the pulsar orbit was decaying. The decay was in precise agreement with a prediction of general relativity, namely that such a system should lose energy through the emission of gravitational radiation. And as the system loses energy, the orbit shrinks. Observations of this system provided the first experimental verification of the existence of gravitational waves. Hulse and Taylor were awarded the 1993 Nobel prize in physics for their work. Right now, the gravitational radiation emitted by the system is about 2% of our Sun’s power output in visible light. This energy loss is causing the two stars to draw closer by a few meters every year. In about 300 million years, the two neutron stars will make a final inspiral—leading to a spectacular collision that will light up the cosmos. (Fig. 4.3 provides an impression of the inspiralling orbits in a slightly different system: a neutron star and a white dwarf.)

Fig. 4.3

Hulse and Taylor discovered a pulsar and a neutron star in orbit around each other. In 2007, astronomers using the Green Bank Telescope discovered a pulsar and a white dwarf in orbit around each other. This is an artist’s impression of the pulsar and the white dwarf. The image also provides an impression of the gravitational waves—ripples in spacetime—generated by the system. The emission of gravitational radiation is causing the orbits to decay, and eventually the neutron star and white dwarf will merge in an immense explosion. (Credit: ESO/L. Calçada. CC BY 4.0 DEED)

• 31. The Hulse–Taylor binary is not the only pair of orbiting neutron stars. And this is important because when these stars eventually inspiral the merger releases vast amounts of certain heavy elements. In 2017, LIGO observed gravitational waves from the merger of two neutron stars in an event called GW170817. Follow-up observations with traditional telescopes enabled astronomers to estimate the extent to which such mergers supply heavy elements to the interstellar medium. The GW170817 event produced a staggering amount of gold: between 3–13 Earth masses (17.7–76.7 sextillion tonnes, where a sextillion is 1000,000,000,000,000,000,000). It seems clear these mergers are the primary source of europium, gold, and various other heavy elements that we see around us today.

• 32. Just as orbiting neutron stars emit gravitational waves, so Earth emits gravitational waves as it orbits the Sun. However, whereas the gravitational waves from inspiralling neutron stars are detectable, the power of the gravitational radiation emitted by the Earth–Sun system is negligible: about 200 W, enough to let you run three light bulbs. The energy lost in gravitational radiation is about two trillion trillionths (2 × 10⁻²⁴) of Earth’s orbital kinetic energy each year. Not only are these gravitational waves too weak to be detected, the total amount of gravitational radiation emitted over the course of Earth’s existence is far too small to have altered our planet’s orbit in a detectable fashion. Gravity is a feeble force. And yet, as we have seen, gravity can crush stars into white dwarfs and neutron stars… or even into something more extreme.

4.3.4 Gravity and Curved Space

To understand what happens when gravity becomes strong enough to overwhelm the structural integrity of even a neutron star, we can no longer rely on Newton’s theory. We need Einstein’s theory of general relativity.

• 33. In 1930, the Hungarian–American physicist Leo Szilard (1898–1964) applied for a patent for an invention he had made with a colleague. Four years earlier, the two physicists had been moved by the story of the deaths of a family in Berlin, caused by a faulty refrigerator seal that had leaked toxic gases. The two men developed a three-fluid refrigerator which, because it had no moving parts, was safer and more reliable than existing appliances. Since their design was less efficient than the competition, their refrigerator was not a commercial success. This failure did not prove a hindrance to their careers, however. Three years later, Szilard came up with the idea of a nuclear chain reaction. Szilard’s co-inventor was Einstein. I mention this anecdote not because it relates in any way to astronomy, but rather to illustrate the range of Einstein’s accomplishments. In 1930 Einstein was, of course, already world famous. His annus mirabilis occurred in 1905, when he published four papers, all of which were worthy of the Nobel prize. But perhaps his greatest contribution—certainly greater than the three-fluid refrigerator—came a decade later. Einstein’s theory of general relativity is one of the high points of human thought.

• 34. Einstein published his field equations of general relativity in 1915. In one sense the equations are simple. In fact, they are so simple I can give them here: G_μυ + Λg_μυ = κT_μυ. Only a few symbols are involved. Those symbols, however, hide a tremendous amount of complication. For example, the symbols G_μυ and T_μυ terms are tensors (the Einstein tensor and the stress–energy tensor, respectively). This means the single equation shown above represents ten independent equations. When gravity is weak, the equations reduce to Newton’s equations; often, a Newtonian description is entirely sufficient for practical purposes. But in some cases gravity is strong, and then we need the field equations. The field equations can be difficult to solve; in most cases, indeed, they are impossible to solve without the aid of a computer. Fortunately, thanks to the American physicist John Archibald Wheeler (1911–2008), we can understand in general terms something about what these equations mean. Wheeler had a knack for cutting through mathematical difficulties and creating a pithy phrase to describe the physics. Noting that spacetime is a ‘thing’, an active player rather than a static background in which events happen, Wheeler argued the Einstein field equations tell us spacetime can curve; mass (and, which is the same thing, energy) causes the curvature of spacetime. The same field equations, in turn, tell us that a freely moving mass follows the curvature of space. Hence Wheeler’s maxim: mass–energy tells spacetime how to curve; curved spacetime tells mass–energy how to move. This is a simple way of appreciating what the Einstein field equations are saying. Remember, though, that Einstein’s equations make everything precise: this is not just a qualitative relationship, but a quantitative one: the field equations of general relativity make definite predictions that can be tested.

• 35. One prediction of the theory is that a large mass—the Sun, for example—will bend the path of any light beams that pass nearby. The field equations can be used to determine the amount of the deflection. Einstein calculated the deflection of light from a distant star grazing the edge of the Sun, and pointed out the effect might be detected during a solar eclipse. Sir Arthur Stanley Eddington (1882–1944)—a gifted astronomer and mathematician, and also a brilliantly clear teacher of physics (Einstein was a fan of his 1923 book Mathematical Theory of Relativity)—set out to test the prediction.

• 36. Eddington knew that, during the 1919 eclipse, the Sun would be in front of the Hyades cluster of bright stars so at totality he would be able to photograph many stars near the Sun’s disk. According to Einstein, the Sun’s gravity would cause the position of these stars to shift, with stars closest to the Sun exhibiting the largest shift. Eddington could compare the position of stars at totality with reference photographs of the same stars when the Sun was nowhere nearby, and thus compare the size of the shift with the prediction. At short notice, Eddington organised an expedition to Sobral in northern Brazil and the island of Principe off the west coast of Africa. Eddington led the Principe team; an astronomer from the Royal Greenwich Observatory, an observer carrying the impressive name Andrew Claude de la Cherois Crommelin (1865–1939), led the Sobral team. Conditions in Brazil meant Crommelin made the better observations, and these were sufficient to confirm Einstein’s predictions. The result made the front-pages of newspapers across the world. Taking his cue from the Rubaiyat of Omar Khayyam, Eddington wrote:

  Oh leave the Wise our measures to collate

  One thing at least is certain, light has weight

  One thing is certain and the rest debate

  Light rays, when near the Sun, do not go straight.

• 37. We have now encountered two predictions of general relativity: light can be deflected by mass, and space can support gravitational waves. In the next section we shall discuss another prediction: when gravity becomes strong, a black hole can form. These are all applications of the theory to situations that arise out in space. But general relativity has practical applications too. The Global Positioning System (GPS) allows a receiver to determine its location on Earth by using the time at which signals were emitted from a satellite. The time of emission is provided by an on-board atomic clock and is encoded into the signal. Then, since we know the speed of light and the position of the various satellites, it is straightforward for the GPS receiver to determine its location. It turns out that, if you want positions to be known to an accuracy of 15 m, you need to know clock times within the GPS system to an accuracy of 50 ns (the time taken for light to travel 15 m). The on-board clocks are travelling at 14,000 km/h relative to clocks on Earth and so, as we saw in the previous chapter, we must consider special relativity: the orbiting clocks tick slower by about 7 microseconds per day. But general relativity states that, because the orbiting clocks are higher in Earth’s gravity well than ground-based clocks, the on-board clocks tick faster by about 45 microseconds per day. The net effect is that, compared to a ground-based clock, a clock on a GPS satellite ticks faster by about 38 microseconds per day. The computers involved in the GPS system know how to compensate for these special and general relativistic effects. If no compensation were made, GPS would fail its navigational requirements in a couple of minutes. Although general relativity might at first glance to be an esoteric theory with no practical application, modern technology demands that we understand its effects.

4.3.5 Black Holes

Above a certain mass, a white dwarf cannot withstand the crush of gravity; it collapses into a neutron star. Above a certain mass, a neutron star cannot withstand the crush of gravity; it collapses into a … what? The fate of a high-mass star is to become a black hole.

• 38. A black hole is a region of space where the velocity required to escape the clutches of gravity exceeds the speed of light. The idea predates Einstein’s theory of general relativity. The English priest John Michel (1724–1793) made several suggestions that, given he was writing 250 years ago, were astonishingly prescient. He understood that double stars move under their mutual gravitation; he brought statistical techniques to bear on astronomical questions; he proposed the existence of seismic waves; and he understood how to make magnets. Also, in 1783, he argued that black holes—what he called ‘dark stars’—must exist. He showed that, for a given size of star, above a certain mass the escape velocity would be greater than the speed of light: the star would be ‘dark’. It was another two centuries before Michell’s ideas were taken seriously.

• 39. The basis for a general relativistic study of black holes was laid down by the German scientist Karl Schwarzschild (1873–1916). When war broke out in 1914 Schwarzschild, already an accomplished astronomer, volunteered for military service. His scientific background was put to use in Belgium (where he was in charge of a weather station) and then in France (where he calculated missile trajectories for an artillery unit). He was later transferred to Russia where, despite the war, he wrote three important physics papers. Two of these papers discussed general relativity. One of them, by giving an exact solution of Einstein’s field equations only a few months after Einstein published his theory, provided a description of the geometry of space close to a point mass—the first step towards understanding the simplest sort of black hole. Schwarzschild showed that for every non-rotating object there is a size such that, if the physical radius of the object becomes smaller than this size, light cannot escape from the object. The value of this size is now called the Schwarzschild radius and it is the effective radius of a black hole. (The Schwarzschild radius of the Sun is about 3 km; squash all the Sun’s mass into a 3 km radius and it becomes a black hole.) This was an impressive achievement, especially given that Schwarzschild was serving in the military at the time. Schwarzschild died during World War I, but he was not a direct casualty of the war. While in Russia he contracted pemphigus, a disease in which the sufferer’s immune system mistakenly identifies skin cells as being alien and launches an attack. It leads to painful blisters on the skin and mucous membranes. His early death was a loss to science.

• 40. If, following a supernova, the remaining stellar core is about two or three times the mass of the Sun (the precise figure is not yet established) then neutron degeneracy pressure will be insufficient to prevent further gravitational collapse. No other known force can halt the collapse. The star becomes a black hole (another of Wheeler’s coinages). In popular imagination a black hole is a ravenous beast that will eventually suck in everything. The Schwarzschild solution, though, tells us that far away from a black hole we notice nothing special: at a far distance the gravitational field is the same as it would be if a normal star of the same mass were in its place. Things only get interesting when you are close to the black hole. In the 1950s, the Austrian-born physicist Wolfgang Rindler (1924–2019) coined the term event horizon to describe the key feature of a black hole. The event horizon, which for a non-rotating black hole lies at the Schwarzschild radius, is a boundary beyond which events cannot affect an external observer. It is a ‘point of no return’, beyond which it becomes impossible to escape. Imagine you are an astronaut approaching a (large) black hole. As you fall past the event horizon you don’t notice anything particularly special; nothing marks the boundary. But past that boundary, light itself cannot escape—so neither you nor anything else can now affect the universe beyond the horizon. A distant observer watching this process sees something completely different: you will appear to move more slowly as you approach the horizon, with gravitational redshift causing your image to become increasingly red. Your image appears to freeze at the event horizon. Beyond the horizon lies a singularity—a place where Einstein’s theory, and our understanding of the universe, breaks down.

• 41. Is a black hole singularity inevitable? Well, yes. Roger Penrose (1931–) started out as a mathematician, but became interested in cosmology and then brought his deep mathematical insight into the field. Penrose was already an established expert in general relativity when, in 1966, he was an examiner on a PhD thesis written by Stephen William Hawking (1942–2018). The two became colleagues. In the late 1960s they proved the Penrose–Hawking singularity theorems, which showed how singularities are an inevitable feature of general relativity. Up until then, many astronomers believed there might be some ‘get out clause’ that meant black hole singularities could not form in the real world. It seems they were wrong. Penrose was awarded the Nobel prize in 2020 for ‘the discovery that black hole formation is a robust prediction of the general theory of relativity’. Kip Thorne, himself a Nobel prize winner, stated that Penrose ‘revolutionised the mathematical tools that we use to analyse the properties of spacetime’.

• 42. Black holes generate unusual effects. Imagine approaching a solar-mass black hole. If you are falling feet-first then, as you get close, your feet will feel a stronger gravitational force than your head. You will experience tidal forces, which will stretch your body into a long, thin, spaghetti-like shape. A black hole can cause the spaghettification of anything and everything—and, with a solar-mass black hole, the process happens outside the event horizon. (For a supermassive black hole, of the type that typically sits at the centre of a galaxy, the process of spaghettification happens at some point inside the event horizon. Wherever it happens, though, if you get too close to a black hole you will be spaghettified.)

• 43. Spaghettification is one unfamiliar effect of black holes. Another is gravitational time dilation: put crudely, your clock ticks more slowly, relative to a distant clock, as you approach a black hole. Yet another unfamiliar phenomenon is the existence of an innermost stable circular orbit (ISCO). If you are outside the event horizon of a black hole you can in principle escape from its clutches, but you cannot necessarily circle the hole in a stable orbit. For example, if the necessary orbital speed exceeded the speed of light then that orbit would be impossible. General relativistic effects combine to ensure that the ISCO lies outside the Schwarzschild radius. The location of the ISCO depends on the mass and spin of the black hole, and whether the velocity of the orbiting particle is in the same direction as the black hole’s spin. For the simple case of a non-rotating black hole with the mass of the Sun, though, we can run the numbers. The Schwarzschild radius of the Sun is about 3 km (2.95 km, to be precise). If Captain Kirk wanted to put the USS Enterprise in a stable circular orbit around a solar mass black hole, then he would have to make sure the starship got no closer than 8.86 km from the centre of the black hole. If the Enterprise found itself anywhere between 2.95 km and 8.86 km from the centre of the black hole then the possibility of escape would still exist, but a stable circular orbit would be impossible. (All this assumes, of course, that a Star Trek faster-than-light ‘warp speed’ is impossible.)

• 44. All of the above might seem theoretical. Is there any observational evidence for the existence of black holes? The first hints came in 1964, when instruments on board an Aerobee suborbital rocket detected a strong X-ray source in the constellation of Cygnus. The source was named, logically enough, Cygnus X-1. Subsequent observations showed Cygnus X-1 to be a binary system, in which a compact object and a blue supergiant star orbit around a common centre of mass every 5.6 days. Repeated observations established the mass of the compact object: about 15–20 times the mass of the Sun. This is far beyond the maximum allowed mass of a neutron star. Astronomers know of only object this could possibly be—the astronomical community long ago accepted that Cygnus X-1 contains a black hole. The Cygnus X-1 black hole pulls material from its companion star, and the material forms an accretion disk around it. (Fig. 4.4 is an artist’s impression of the Cygnus X-1 system.) As the material in the disk spirals down towards the event horizon it becomes extremely hot—the temperature is high enough for jets of X-rays to be emitted. It was these jets that Aerobee detected back in 1964.

Fig. 4.4

An artist’s impression of Cygnus X-1, a system that is about 10,000 light-years from Earth. A dense object, with about 15 times the mass of the Sun, is pulling material away from a companion star. The companion is a blue supergiant, with a surface temperature of 31,000 K. The supergiant might seem to be extreme enough, but as the material falls onto the dense object it gets heated to even higher temperatures and emits jets of X-rays and gamma rays. The dense object is only a few kilometers in diameter: it can only be a black hole. (Credit: NASA, ESA, Martin Kornmesser (ESA/Hubble). CC BY 4.0 DEED)

• 45. Astrophysical black holes are messy places. They suck in material and spew jets of high-energy particles and radiation into the cosmos. The black holes we observe are difficult to understand in all their complicated detail. Theoreticians, however, need to understand black holes because these are places where the twin pillars of modern physics—quantum theory and general relativity—must both be applied. This is exciting for theorists because the two theories appear to be incompatible. Theorists interested in combining general relativity with quantum mechanics use mathematics to study idealized objects—black holes, yes, but ‘pure’ entities that sit aside from the messy reality of the universe we observe through our telescopes. Whether these idealized black holes have anything to do with a real-world black hole such as in Cygnus X-1 is not yet known. Stephen Hawking was one of the greatest explorers of ‘theoretical’ black hole physics. He showed how quantum effects mean black holes must emit radiation and therefore they must have a temperature. Hawking’s beautiful equation for black hole temperature (T = ℏc³/8πGMk_B), is engraved upon his gravestone in Westminster Abbey. His equation links various disparate concepts: ℏ brings in quantum mechanics; c is the speed of light; 8π arises because of the spherical nature of a black hole; G is Newton’s constant, which brings in gravity; M is the mass of the black hole; and k_B is the Boltzmann constant, which is important in thermodynamics. So can we test this theoretical prediction by observing an astrophysical black hole? Well, plug in the numbers for a solar-mass black hole and you find that such an object would have a temperature of just 60 nK. This is so cold that, if such a black hole were floating out in the depths of space, it would absorb energy from the cosmic microwave background (which is at the far higher temperature of 2.7 K). A higher-mass black hole would have an even lower Hawking temperature. For a black hole even to be at a temperature of just 2.7 K, and thus be in equilibrium with the cosmic microwave background, it would need to be a lunar-mass object. There seems to be little chance of observing the Hawking temperature of an astrophysical black hole.

• 46. If, as Hawking argued, black holes emit radiation then they must be losing mass. Over time, therefore, a black hole will get smaller—and eventually it will evaporate. The lifespan of a black hole scales as the cube of its initial mass (so a black hole that starts with twice the mass of the Sun, for example, will endure eight times as long as one with the same mass as the Sun). How long will a solar-mass black hole endure? Well, if the black hole is out in the empty depths of space, and so does not add to its mass by accreting additional material, then it will last for about 10⁶⁷ years before it evaporates. This timescale is unimaginably longer than the current age of the universe.

• 47. In the present-day universe, supernova events create black holes that are a few solar masses in size. We can never see these black holes evaporate because the evaporation timescale is too long. But in the early universe the conditions might have right to create black holes of any mass. A primordial black hole could potentially be much less massive than a typical star. A low-mass black hole, of the order of 10¹¹ kg, created when the universe was young, would only now be evaporating. Hawking radiation from a black hole is emitted at a rate that is inversely proportional to the black hole mass so, as a black hole gets less massive, the Hawking radiation increases. This leads to a runaway effect: at the end of its life, a black hole emits a burst of radiation. So, if the big bang created lots of low-mass primordial black holes, we might hope to see explosions popping off in space. Astronomers have looked for signs of such explosions. The Fermi space telescope, in particular, has searched for gamma-rays from evaporating black holes. So far, astronomers have found no evidence for the existence of primordial black holes.

• 48. Hawking’s discovery of black hole evaporation generated a paradox. On the one hand, black holes are described by general relativity—a classical theory, which considers gravity to be an effect of the warping of spacetime. The theory says a black hole is surrounded by an event horizon, but nothing special marks the boundary in space. If you cross it then it is only when you try to leave that you realise you have passed a point of no return. So: information does not leave the black hole. Quantum mechanics, on the other hand, is not a classical theory. Quantum mechanics deals with probabilities: it can give us the probability that a particular event will occur. The quantum approach only makes sense if the probabilities of all possible events add up to one. This fact means information can never be lost in quantum theory and information can never be copied. (If it could, then the complete set of probabilities for a given process could add up to less than or to more than one, which makes no sense.) But if a black hole evaporates, where does all the information inside it go? If the information simply disappears along with the black hole then a basic principle of quantum theory has been violated. If the information is contained within the emitted Hawking radiation, then there must be two copies of the information: one copy inside (since nothing can escape) and one outside. But cloning information this way also violates a basic principle of quantum theory. It seems something has to give: either quantum theory or general relativity is incomplete. Almost half a century after Hawking’s initial discovery, the black hole information paradox has spawned a cottage industry—theorists have dreamed up a plethora of proposals to address the issue. So far, the paradox remains unresolved. Indeed, in some ways the paradox has worsened.

• 49. In 1992, the American physicist Leonard Susskind (1940–) and colleagues proposed a solution to the black hole information paradox that had the merit of saving quantum theory from being violated. They argued observers outside the black hole see information gather at the event horizon, which is broadcast through Hawking radiation. Observers on the other side of the event horizon see the information located inside the black hole. Since the two classes of observer cannot communicate, there is no paradox. For this solution to work, physics in the three-dimensional space inside the black hole, where gravity is important, must be equivalent to physics in the absence of gravity on the two-dimensional surface just above the event horizon. This holographic principle—that a three-dimensional space with gravity is equivalent to a two-dimensional space without gravity—sounds bizarre. But physicists have confidence in the holographic principle, at least in certain circumstances. So maybe the holographic principle works in this case? Well, a quartet of physicists—Ahmed Almheiri (1986–), Donald Marolf (1971–), Joseph Gerald Polchinski Jr. (1954–2018), and James Kenneth Sully (1983–)—showed this approach leads to its own paradox. When the black hole has evaporated halfway, there is no longer enough information at the horizon for holography to describe the black hole’s interior. Observers passing the event horizon would meet not a harmless point of no return but a blazing firewall that would burn them to a crisp. But fundamental principles of general relativity tell us there should be nothing special about the event horizon; there should be no firewall to mark its presence. The black hole information paradox was replaced by the firewall paradox. Neither paradox has been resolved to everyone’s satisfaction.

• 50. To conclude our look at black holes, let’s consider one exciting possibility: perhaps black holes allow us to travel to distant parts of the universe almost instantaneously. The Austrian physicist Ludwig Flamm (1885–1964) first described solutions to the equations of general relativity that can lead to connections between different regions of spacetime. Two decades after Flamm’s work, Einstein and his colleague Nathan Rosen (1909–1995) discussed how general relativity permits the existence of ‘bridges’ between two separate points in spacetime—two different locations, two different points in time, or both. The term ‘Einstein–Rosen bridge’ is not particularly snappy, however. John Wheeler, with his knack of coining a memorable word or phrase, came up with the term ‘wormholes’ to describe the bridge. In a paper with his colleague Charles William Misner (1932–), Wheeler depicted a wormhole as a throat connecting an opening in one part of spacetime with an opening elsewhere. In theory, if one could traverse the throat of a wormhole, one could move between distant regions of the universe in an instant. In practice, it is not known whether wormholes exist. Even if they do, it might not be possible to traverse them. Nevertheless, wormholes are yet another fascinating aspect of general relativity.