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Supernovae, neutron stars and black holes

A star several times the Sun's mass ends as a red supergiant whose core collapses at iron, leaving a neutron star or a black hole. A type Ia supernova is a different event, a detonating white dwarf whose peak absolute magnitude near −19.3 makes it a standard candle. The Schwarzschild radius 2GM/c² fixes the event horizon.

Year 13AQA 3.9.2.6OCR A 5.5.1(f), 5.5.1(g)EDEXCEL 159, 160

The HR diagram and stellar evolution, part 2 of 2. Part 1 is The HR diagram and the life of Sun-like stars.

Builds on The HR diagram and the life of Sun-like stars and Gravitational potential.

IN THIS TOPIC

  • Describe the evolution of a massive star through the red supergiant and supernova stages to a neutron star or a black hole.
  • Explain why a type Ia supernova works as a standard candle, read its light curve, and say what standardises it.
  • Compare a gamma-ray burst's energy output with the Sun's.
  • Recall neutron star properties and use the Schwarzschild radius for black holes.

COMMON MISCONCEPTION

Every supernova is the death of a star far heavier than the Sun.

A type Ia supernova is a white dwarf of about a solar mass, pushed past the Chandrasekhar limit by accretion from a companion. That standard trigger, not a giant star, is what makes its peak luminosity standard enough to survey distances with.

Supernovae and the standard candle

A sufficiently massive star evolves from the main sequence to a red supergiant. When fusion can no longer provide enough pressure to support the core, it collapses and the outer layers are expelled in a core-collapse supernova. A star several times the Sun's mass swells past the red giant stage into a red supergiant, at the cool end of the supergiant band on the HR diagram, while its core fuses on through carbon, oxygen and silicon as far as iron, where fusion stops releasing energy and the support fails. What the collapse leaves behind is a neutron star, or a black hole if the surviving core is heavy enough.

A red supergiant's core drawn as a quarter section, six nested shells with hydrogen outermost and an iron core at the centre, each shell labelled by the nuclide its fuel makes and that nuclide's binding energy per nucleon: He-4 at 7.07 MeV, then 7.68, 7.98, 8.45 and 8.79 for Fe-56. Beside each shell a bar shows the energy that step releases per nucleon, the difference between those values. The hydrogen bar is enormous at 7.07 MeV and the rest are stubs of a half MeV or less, so the later shells buy the star ever less time. The iron core's bar runs backwards: fusing iron on to Zn-64 at 8.736 would cost energy rather than release it, because iron sits at the top of the ladder. With nothing left to burn, the support fails and the core collapses.
FIG. 1The shells in section, with what each one makes and what that step pays. Binding energy per nucleon climbs from 7.07 MeV for helium to 8.79 for iron, so the energy each shell releases shrinks the whole way: hydrogen burning gives 7.07 of the 8.79 there is, and silicon burning gives 0.34. Past iron the ladder turns down, the bar runs backwards, and the support fails.

The explosion is what an observer actually sees of all that. The star brightens with astonishing speed, its absolute magnitude dropping by ten or more on the backwards scale within weeks, and for a while it outshines its entire galaxy. Keep the core-collapse label attached: the type of supernova used to measure distances has an entirely different origin.

Some collapsing giants also fire off gamma-ray bursts, flashes of gamma radiation lasting seconds to minutes and detectable across most of the observable universe. The spec asks you to set that output beside the Sun's, so carry the two numbers. A burst radiates of order 1045 J in a few seconds, quoting the isotropic-equivalent energy, the figure computed as if the burst shone equally in every direction when in reality it is beamed. The Sun pours out 3.9 × 1026 W, so across its whole ten-billion-year life it will manage roughly 1044 J. One burst therefore radiates more in a few seconds than the Sun radiates in its whole lifetime.

A supernova light curve rising steeply for about two and a half weeks to a dashed line at absolute magnitude minus 19.3, then declining slowly over months: every type Ia peaks at nearly the same brightness, and the rate of decline corrects the rest, which is what makes it usable as a distance marker.
FIG. 2The type Ia light curve: a rise of about two and a half weeks to a peak absolute magnitude close to −19.3, then a decline over months. The peak lands near the same value every time, and the small spread that remains is corrected using the rate of decline.

A type Ia supernova has a different origin. No massive star is involved at all. A white dwarf in a binary system detonates as a runaway thermonuclear explosion, and the fuel is broadly similar every time, a carbon-oxygen dwarf igniting through a limited set of routes, more than one of which is still argued over. That is why the peaks come out alike, near absolute magnitude −19.3, without being identical. Any object of known absolute magnitude is a standard candle. Measure its apparent magnitude, run the distance modulus, and out comes the distance.

Two qualifications belong with that. The trigger is not settled. A dwarf may pull matter from an ordinary companion until it approaches the limit, or two white dwarfs may spiral together and merge, or a dwarf below the limit may be set off by a detonating skin of helium, and which route dominates is still argued over. Nor are the peaks identical. They scatter by a few tenths of a magnitude, and the fix comes from the light curve itself, since the brighter explosions fade the more slowly.

The decline rate over the first fortnight or so after the peak gives the true peak brightness of that particular explosion, which is how astronomers pull the scatter down to around a tenth of a magnitude. Type Ia supernovae are therefore standardisable candles rather than perfectly identical ones, and they are bright enough to serve as distance markers across billions of light years.

The same measurements produced a further result. In the late 1990s distant type Ia supernovae came out consistently fainter than their red shifts predicted, and within the standard cosmological model that means the expansion of the universe is accelerating, driven by something now labelled dark energy. The claim leans on type Ia explosions behaving the same way across cosmic history, once corrected, and questioning that uniformity is the heart of the controversy. The candle is only as reliable as that assumption.

WORKED EXAMPLE

The candle as a ruler

A type Ia supernova peaks at apparent magnitude +16.7. Taking its peak absolute magnitude as −19.3, find the distance to its galaxy.

m − M = 16.7 − (−19.3) = 36.0.

5 log(d/10) = 36.0, so log(d/10) = 7.2 and d = 10 × 107.2 = 1.6 × 108 pc.

A hundred and sixty million parsecs, from one photometric measurement. Real work would first correct that −19.3 using the measured decline rate, which shifts the answer by a few per cent. This is how the accelerating universe was discovered, one exploding white dwarf at a time.

Neutron stars and black holes

Heavier cores leave denser remnants. Stars born with more than roughly eight solar masses, a model-dependent boundary rather than a sharp line, are the ones whose cores collapse this way. If the collapsing core is too massive to settle as a white dwarf, gravity crushes its protons and electrons together into neutrons, leaving a neutron star. Picture one or two solar masses compressed into a sphere roughly ten kilometres in radius, made almost entirely of neutrons, at the density of an atomic nucleus.

GUIDED PRACTICE

Nuclear matter by the teaspoon

A neutron star has mass 2.8 × 1030 kg, 1.4 solar masses, and radius 10 km. Find its density, and the mass of one teaspoonful, 5.0 × 10−6 m3.

Show the working

ρ = M / (4/3 πr3) = 2.8 × 1030 / (4.19 × 1012) = 6.7 × 1017 kg m−3.

One teaspoon holds 6.7 × 1017 × 5.0 × 10−6 ≈ 3 × 1012 kg, three billion tonnes.

Past even that, nothing holds. If the remnant is massive enough, its escape velocity, 2GM/r\sqrt{2GM/r} from the gravitational fields unit, reaches the speed of light at a finite radius. Inside that boundary nothing, light included, can leave, and the object is a black hole. The boundary is the event horizon, and its radius, the Schwarzschild radius, follows from a Newtonian shortcut, setting the escape velocity equal to c, a heuristic that happens to land on the answer general relativity gives:

Rs≈2GMc2R_{s} ≈ \frac{2GM}{c^{2}}ON THE AQA DATA SHEET

WORKED EXAMPLE

The Sun, hypothetically crushed

Find the Schwarzschild radius for one solar mass, 1.99 × 1030 kg.

Rs = 2GM/c2 = (2 × 6.67 × 10−11 × 1.99 × 1030) / (3.0 × 108)2 = 3.0 km.

The Sun will never collapse this far, but the number sets the scale: to make any mass a black hole, squeeze it inside its Schwarzschild radius, about three kilometres per solar mass.

A straight cyan line through the origin on axes of Schwarzschild radius against mass, near 3 km per solar mass. An amber point works the 6 solar mass remnant at 17.7 km. A cyan point marks a real neutron star, 10 km across at 1.4 solar masses, sitting above the line and so outside its 4.1 km horizon; a note records that the Sun, at 696 000 km, is nowhere near its own 3 km.
FIG. 3Rs = 2GM/c2 drawn out: a straight line through the origin at about 3 km per solar mass, so doubling the mass doubles the horizon. A 6 solar mass remnant is worked on the line at 17.7 km. A real neutron star, 10 km across at 1.4 solar masses, sits a factor of about two and a half outside its own horizon, and the Sun sits hundreds of thousands of times outside its 3 km.

INDEPENDENT PRACTICE

The monster at the centre

Observations of stars orbiting the centre of our galaxy reveal a supermassive black hole of about 4 × 106 solar masses. Find its Schwarzschild radius.

Show the working

Rs scales in direct proportion to M, so Rs = 3.0 km × 4 × 106 = 1.2 × 1010 m.

About seventeen times the radius of the Sun, for four million solar masses. Supermassive black holes like this one appear to sit at the centre of most large galaxies, ours included.

ASSESSMENT FOCUS

  • The heavy-star path is the other half of the same question, so name each stage. Main sequence, red supergiant, core-collapse supernova, then a neutron star, or a black hole if the remnant core is heavy enough.
  • Type Ia as a standard candle is a three-step argument, and all three steps score. A carbon-oxygen white dwarf detonating in a narrow range of conditions, so a peak absolute magnitude near −19.3, standardised by the light curve, so apparent magnitude plus distance modulus gives the distance. A type Ia is not a core-collapse event, so keep massive stars out of that answer.
  • If a question asks how good the candle is, state the limitation. The peaks scatter by a few tenths of a magnitude, and the decline rate of the light curve is used to correct each one, so type Ia supernovae are standardisable rather than identical.
  • For gamma-ray bursts, quote the comparison rather than waving at it. Around 1045 J in seconds against the Sun's 1044 J over its whole lifetime.
  • Rs ≈ 2GM/c2 questions are usually substitution plus commentary. Quote the radius, then say what the event horizon is, the boundary from inside which not even light escapes.

CHECK YOURSELF

A supernova leaves a remnant core of mass 4.0 × 1030 kg, about 2.0 solar masses. If it settles as a neutron star of radius 11 km, find its density. If it instead collapses past its event horizon, find its Schwarzschild radius. Comment on each number.

Show a hint

Density needs a sphere's volume; the Schwarzschild radius is 2GM/c² straight in.

Show the answer

V = (4/3)πr3 = (4/3)π × (1.1 × 104)3 = 5.6 × 1012 m3, so ρ = M/V = 4.0 × 1030 / (5.6 × 1012) ≈ 7 × 1017 kg m−3, the density of an atomic nucleus.

Rs ≈ 2GM/c2 = (2 × 6.67 × 10−11 × 4.0 × 1030) / (3.0 × 108)2 = 5.9 km.

Both answers pack Sun-scale masses into city-scale sizes. The neutron star holds two solar masses at nuclear density, and squeezing that same core inside six kilometres would leave an object nothing can escape, light included.

Heavier stars die as core-collapse supernovae and leave a neutron star or a black hole.

A type Ia is a detonating white dwarf, standardised by its rate of decline.

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  • Describe the evolution of a massive star through the red supergiant and supernova stages to a neutron star or a black hole.
  • Explain why a type Ia supernova works as a standard candle, read its light curve, and say what standardises it.
  • Compare a gamma-ray burst's energy output with the Sun's.
  • Recall neutron star properties and use the Schwarzschild radius for black holes.

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