How Long Will the Sun Shine?: Going Further 9–12

Not a fire

You learned in Grade 9 that the Sun gives out 3.8 × 10²⁶ J of energy every second Grade 9: Some facts about the Sun. Before anyone knew about nuclear fusion, the obvious guess was that the Sun was burning, like coal in a stove. Let’s test that.

Worked example

Could the Sun be a lump of coal?

The Sun’s mass is 2.0 × 10³⁰ kg, and it gives out 3.8 × 10²⁶ J every second. Burning 1 kg of coal gives out about 2.4 × 10⁷ J. If the whole Sun were coal (and the oxygen to burn it were brought in free), how long would it last?

Answer

Energy stored = mass × energy per kilogram:
$E = 2.0 \times 10^{30}\ \text{kg} \times 2.4 \times 10^{7}\ \text{J/kg} = 4.8 \times 10^{37}\ \text{J}$
Time = energy ÷ energy used per second:
$t = \frac{4.8 \times 10^{37}\ \text{J}}{3.8 \times 10^{26}\ \text{J/s}} = 1.3 \times 10^{11}\ \text{s}$
A year is about 3.16 × 10⁷ s, so t = 1.3 × 10¹¹ ÷ 3.16 × 10⁷ ≈ 4,000 years. A coal Sun lit when the pyramids of Egypt were being built, about 4,500 years ago, would already have burned out.

But the Sun is far older than 4,000 years. The oldest rocks on the Earth are about 4 billion years old, and there are fossils of living things about 3.5 billion years old. The oldest grains in meteorites, the leftovers from when the Sun and the planets formed, are 4.57 billion years old, measured by the radioactive atoms in them (How Old Is the Earth?, a later topic, shows how). So the Sun has been shining, much as it does today, for about four and a half billion years. No chemical fuel could do that.

A story from science

The Sun that was too young

In 1862 the great British physicist William Thomson, later Lord Kelvin, tried to work out the age of the Sun. Chemical burning was clearly too feeble. The best source he could think of was gravity: as the Sun slowly shrinks, its gas falls inwards, and the energy of the fall becomes heat, as a falling stone warms what it lands on. That could keep the Sun shining for a few tens of millions of years, and he concluded that it had very probably not lit the Earth for as long as 100 million years.

This troubled Charles Darwin. From how fast rain and rivers wear away rock, he had estimated that one valley in southern England had taken some 300 million years to form, and evolution needed hundreds of millions of years more. Kelvin was the most respected physicist of his day, and his numbers seemed solid. Darwin called Kelvin’s short ages one of his “sorest troubles”.

The answer came from the new physics of the atom. In 1920 the British astronomer Arthur Eddington pointed out that four hydrogen atoms weigh a little more than one helium atom, which the chemist Francis Aston had just measured, and that if stars could turn hydrogen into helium, the missing mass would keep them shining for billions of years. In 1939 the German-American physicist Hans Bethe worked out exactly how. Kelvin wasn’t foolish: he simply couldn’t have known about a source of energy that hadn’t been discovered yet.

A tank of hydrogen

You know how the Sun really works from Grade 9: in its core, hydrogen nuclei fuse, step by step, into helium Grade 9: Nuclear reaction in the Sun. The helium made is lighter than the hydrogen that went in, by about 0.7 %, and the missing mass has become energy, by E = mc² (How much energy? Mass into energy). With that, you can work out how fast the Sun uses its fuel, and how long the fuel will last.

Worked example

How much hydrogen does the Sun use?

The Sun gives out 3.8 × 10²⁶ J every second. a. How much mass does it turn into energy each second? b. If 0.7 % of the hydrogen’s mass becomes energy, how much hydrogen does it fuse each second? (c = 3.0 × 10⁸ m/s.)

Answer

a. From E = mc², m = E ÷ c²:
$m = \frac{3.8 \times 10^{26}\ \text{J}}{(3.0 \times 10^{8}\ \text{m/s})^2} = \frac{3.8 \times 10^{26}}{9.0 \times 10^{16}} = 4.2 \times 10^{9}\ \text{kg}$
That is 4.2 million tonnes of mass turned into energy every second.
b. That 4.2 × 10⁹ kg is 0.7 % of the hydrogen fused, so
$m_\text{H} = \frac{4.2 \times 10^{9}\ \text{kg}}{0.007} = 6.0 \times 10^{11}\ \text{kg}$
About 600 million tonnes of hydrogen become helium in the Sun’s core every second.

Six hundred million tonnes a second sounds as if the Sun must run out soon. But the Sun is enormous. And it can’t burn all of it: fusion happens only in the core, the innermost quarter of the Sun’s radius, where the temperature (about 15 million K) and the squeeze are great enough. The hydrogen in the outer layers never gets there. Detailed models of the Sun show that by the time the core’s hydrogen is used up, about a tenth of the Sun’s mass will have been turned into helium.

That is the Sun’s main-sequence lifetime: the time a star spends steadily fusing hydrogen in its core, the long, calm middle of its life. The Sun is 4.6 billion years into its 10 billion: roughly halfway.

Why will the Sun stop shining steadily long before all its hydrogen has been used?

Pick an answer to see why.

From the core to your skin

The energy made in the core today won’t reach you for a very long time. On its way out, it crosses the Sun’s layers (Activity 12.1 in Grade 9 shows them), and each one passes it on in its own way.

Figure 30.1 The inside of the Sun, cut in half, with the layers drawn to scale, and how long the energy spends in each. Light from the surface takes 8 minutes 20 seconds to reach the Earth, which on this scale is a dot, and would be about 50 times the width of the drawing away, far off the page to the right.

In the radiative zone, the gas is so dense that light can’t travel far. A packet of energy, a photon, goes only a tiny distance, from a fraction of a millimeter deep inside to a few millimeters near the top of the zone, before an atom or an electron absorbs it or knocks it in a new direction. Each time, it is equally likely to head back inwards as out. So the energy doesn’t stream outwards; it wanders, like a person lost in a crowd, taking a random step at a time. A path like this is called a random walk.

Send a photon out from the center. Each step goes the same distance in a random direction. Count how many steps it takes to reach the edge, then try a bigger Sun.

Distance to the edge:

Photons:

This photon: –
Average, 5 steps out: – (0 photons)
Average, 10 steps out: – (0 photons)
Average, 20 steps out: – (0 photons)

Press Send a photon. Each step is the same length, in a random direction, as a photon bounces off particles in the Sun.

The widget shows the rule for a random walk: to get R steps from where you started, you need about R² steps. Double the distance, and the walk takes four times as long. For the Sun, that makes the trip astonishingly slow.

Worked example

How long does the energy wander?

The Sun’s radius is 7.0 × 10⁸ m. Suppose a photon’s every step were 1 mm long (1.0 × 10⁻³ m), all the way out, and it moved at the speed of light, 3.0 × 10⁸ m/s, between steps. a. How many steps would it take to wander from the center to the surface? b. How long would that take?

Answer

a. The distance in steps is R = 7.0 × 10⁸ m ÷ 1.0 × 10⁻³ m = 7.0 × 10¹¹ steps. A random walk needs about R² steps:
$N = (7.0 \times 10^{11})^2 = 4.9 \times 10^{23}\ \text{steps}$
b. The total path is 4.9 × 10²³ × 1.0 × 10⁻³ m = 4.9 × 10²⁰ m. At the speed of light:
$t = \frac{4.9 \times 10^{20}\ \text{m}}{3.0 \times 10^{8}\ \text{m/s}} = 1.6 \times 10^{12}\ \text{s} \approx 52{,}000\ \text{years}$
The same distance in a straight line would take 2.3 seconds.

The real steps are shorter deep inside the Sun, where the gas is densest, and longer near the top of the radiative zone. Careful calculations that allow for this give about 170,000 years for the energy to work its way out through the radiative zone (estimates range from tens of thousands to a few hundred thousand years, because they depend on exactly how dense and opaque the Sun’s insides are). So the sunlight warming you today set off from the core long before the first farmers planted a seed.

In the outer 30 % of the Sun, the convective zone, the gas is cool enough to be more opaque still, so radiation can hardly get through at all. Instead the gas itself carries the energy: hot gas rises, gives up its heat at the surface, cools and sinks again, as water does in a pot of boiling dal. This is much faster: the energy crosses the convective zone in weeks to months. You can see the tops of the rising columns on the surface as bright granules, typically about as wide as Nepal is long, with darker, cooler gas sinking down the narrow lanes between them.

A close-up of the Sun's surface: a pattern of bright, rounded cells like grains of rice, separated by thin dark lanes, with a scale bar — **Figure 30.2** The surface of the Sun, photographed by the Inouye Solar Telescope in Hawaii in December 2019; the bar is 10,000 km long, and the white outline in the corner is the USA, to the same scale. Each bright cell is a granule: hot gas rising, typically 1,000 to 1,500 km across. It spreads, cools and sinks back down the dark lanes. Photo: NSO/NSF/AURA, CC BY 4.0

At the photosphere, the visible surface, at about 5,800 K, the gas finally becomes thin enough for light to escape. From there it flies straight through empty space at 3.0 × 10⁸ m/s, and reaches the Earth, 1.5 × 10¹¹ m away, in 500 seconds: 8 minutes 20 seconds. When it lands on your skin, the light is absorbed and becomes heat in you.

Follow the energy all the way and you have an energy model of the kind you built in Building an energy model: mass in the core becomes the energy of fast-moving particles and gamma rays; it travels outwards as radiation, then as the motion of hot gas, then as light again; and at the end it warms the Earth, grows its plants and drives its weather. At every stage, the amount of energy stays the same.

A photon takes 8 minutes 20 seconds to cross the 150 million km to the Earth. Why does the energy take so much longer, over 100,000 years, to cross the 700,000 km inside the Sun?

Pick an answer to see why.

A message from the core, today

Since the light we see left the core so long ago, how do we know the core is still burning? Fusion has one more product that the grade books leave out: every time four protons become a helium nucleus, two neutrinos are made. A neutrino is a particle with almost no mass and no charge, which hardly ever hits anything. Neutrinos fly straight out of the Sun in about 2 seconds, and reach the Earth 8 minutes 20 seconds later.

A story from science

The tank in the gold mine

In the 1960s the American chemist Raymond Davis decided to catch the Sun’s neutrinos. He filled a tank with 615 tonnes of a dry-cleaning fluid, rich in chlorine, 1,478 m underground in the Homestake gold mine in South Dakota, so that the rock above would shield it from everything except neutrinos. Very rarely, a neutrino would turn a chlorine atom into an atom of radioactive argon. Every couple of months Davis flushed the tank and counted the argon atoms: only a handful each time, out of more than 10³⁰ atoms in the tank.

He found them, from 1968 on: proof that fusion is going on in the Sun’s core right now. But only about a third as many as the theory predicted. This solar neutrino problem puzzled physicists for thirty years. In 2001 a detector in Canada showed the answer: neutrinos come in three kinds, and on the way from the Sun many change from the kind Davis could catch into the others. The Sun was fine; the neutrinos were stranger than anyone had thought. Davis shared the Nobel Prize in 2002 with Masatoshi Koshiba, whose detector in a mine in Japan, Kamiokande, had seen the Sun’s neutrinos arrive from the Sun’s direction.

Bigger stars, shorter lives

The Sun’s 10 billion years comes from a calculation. How do we know the calculation is right? We can’t wait 10 billion years. Instead, we look at other stars, which are at every stage of their lives, and of every mass.

You met this in Grade 9: stars differ mainly because of their mass, and heavy stars burn out fast Grade 9: The color of a star. Now for the numbers. Many stars come in pairs that orbit each other, and from the size and speed of their orbits, astronomers can weigh them, using Newton’s law of gravitation. From how bright they look and how far away they are, they can work out how much energy each gives out every second, its luminosity. Put the two together, for hundreds of stars, and a pattern appears: a heavier star is far brighter. A star with twice the Sun’s mass is about 20 times as bright; a star with ten times the mass, many thousands of times.

That decides how long a star lives. A star with twice the Sun’s mass has twice the fuel, but burns it 20 times as fast, so it lasts about 2 ÷ 20 = a tenth as long as the Sun.

Slide the mass of a star. The top graph shows how bright stars of each mass are (the line is fitted to 509 stars weighed in pairs; the dots are some well-known ones). The bottom one shows how long the star can live, from its fuel and how fast it burns it.

Mass: 1.0 × the Sun’s

Brightness: 0.97 × the Sun
Life (simple estimate): 10 billion years
Compared with the Sun’s: about the same

About the Sun’s mass: about 10 billion years. The Sun is 4.6 billion years into its life.

Try it

Spica

Spica, the bright blue-white star in the constellation Virgo, is a pair of stars. The heavier, Spica A, has 10 times the Sun’s mass and is about 20,500 times as bright. Estimate its life.

Answer

$t = 10\ \text{billion years} \times \frac{10}{20{,}500} = 4.9\ \text{million years}$
About 5 million years: two thousand times shorter than the Sun’s. (Detailed models of heavy stars give lives two or three times longer than this simple estimate, because a heavy star stirs its big core and so uses a bigger share of its hydrogen: for a star of 15 Suns, about 11 million years instead of 5. Either way, its life is very short. The dots on the bottom graph of the widget are those models.)

The smallest stars, red dwarfs like Proxima Centauri, are so dim that their fuel lasts for hundreds of billions of years, even trillions: they stir their whole insides and use almost all their hydrogen. The universe is 13.8 billion years old, so no red dwarf has ever died of old age.

There is a second check, and it is beautiful. In a star cluster, a few hundred or thousand stars were born together from one cloud, so they are all the same age. In a young cluster, stars of every mass are still shining steadily. In an older one, the heaviest stars are missing from the main sequence: they have already swollen into giants or died. The older the cluster, the lower the mass at which this cut-off comes, exactly as the lifetimes predict. Clusters of every age, from a few million years to over 10 billion, fit the same pattern.

Star P has 3 times the Sun’s mass; star Q has half the Sun’s mass. Which will shine longer, and why?

Pick an answer to see why.

The Sun’s future

Now we can tell the Sun’s life story, from the calculations and from all the stars seen at each stage. Grade 9 and Grade 10 gave the outline (Death of a star and The life of a star): red giant, then a white dwarf. Here are the numbers, from a detailed model of the Sun by Klaus-Peter Schröder and Robert Smith (2008).

Figure 30.3 The Sun’s life, on a time line to scale, with how bright it is at each stage (the Sun’s brightness today = 1). The last two and a half billion years are opened out below, because so much happens so fast.

Even in its calm middle age, the Sun is not quite steady. As helium builds up in the core, the core shrinks and heats up, and fusion runs a little faster. The young Sun was only about 70 % as bright as today; it is getting about 1 % brighter every hundred million years. That sounds slow, but in about a billion years the Sun will be some 10 % brighter, enough to make the Earth hot enough for its oceans to start evaporating away. Life on Earth has a long future, but not one as long as the Sun’s.

About 5.4 billion years from now, the core’s hydrogen will be gone. Hydrogen will go on fusing in a shell around the dead helium core, and the Sun’s outer layers will swell and cool. Over the following two billion years it will grow into a red giant: at its biggest, about 7.6 billion years from now, 256 times its present radius and 2,700 times as bright, and the strong wind blowing from it will have carried away a third of its mass.

Figure 30.4 The Sun at its biggest as a red giant, drawn to scale with the orbits of the inner planets as they are today (the planets themselves are drawn much bigger than to scale). On this scale, the Sun today is the tiny yellow dot in the corner, and the white dwarf it leaves at the end would be too small to see.

Mercury and Venus will be swallowed. As the Sun loses mass, its grip on the planets weakens and their orbits grow, so the Earth will move outwards; but the drag of the Sun’s swollen outer layers will probably pull it in. Schröder and Smith calculated that the Earth will most likely be swallowed too.

Then the core, at 100 million K, starts fusing helium into carbon and oxygen. That lasts only about 130 million years. In the end the Sun puffs its outer layers off as a glowing planetary nebula, and its core is left as a white dwarf: just over half the Sun’s present mass, packed into a ball about the size of the Earth, cooling for ever. That is the life span of a star like the Sun: about 10 billion years of steady shining, a little over 2 billion years of swelling and dying, and then a long, slow fading.

A restless Sun

From far away the Sun looks perfectly steady. Close up, it is anything but. Look at it with a safely filtered telescope and you will usually see dark spots on it.

The disc of the Sun, yellow-orange, with a row of dark spots near the middle and a few smaller ones scattered elsewhere — **Figure 30.5** The Sun with sunspots, photographed by NASA’s Solar Dynamics Observatory on 14 January 2013. The biggest spots here are several times wider than the Earth. Photo: NASA/SDO/HMI, CC BY 4.0

Sunspots are places where the Sun’s magnetic field is very strong, thousands of times stronger than the Earth’s. The field holds back the rising hot gas, so the spot is cooler, about 3,800 K instead of 5,800 K. It is still hotter than any furnace on the Earth, and glows brightly; it only looks dark next to the even brighter surface around it. Spots last from a few days to a few months.

From 1610 Galileo and others, with the first telescopes, watched sunspots move across the Sun from day to day, and realized that the Sun turns, about once every 25 days at its equator. In 1843 the German amateur astronomer Heinrich Schwabe, who had looked at the Sun on every clear day for 17 years (hoping to spot a planet passing in front of it), noticed that the number of spots rose and fell, over about ten years. Astronomers have counted sunspots ever since, and with older observations, the yearly counts go back to 1700. Now you can find the pattern yourself.

These are the numbers of sunspots seen each year since 1749. Drag the two yellow markers onto peaks far apart; the widget counts the cycles between them. Then try two neighboring peaks.

Data: NOAA Space Weather Prediction Center.

From the peak of: 1761 (cycle 1)
To the peak of: 2024 (cycle 25)
Cycles between: 24 in 263.3 years
Average cycle: 11.0 years

Over many cycles the average settles near 11 years, although single cycles vary a lot.

The number of spots goes up and down in the sunspot cycle, or solar cycle, on average about 11 years long. Measured from one minimum to the next, single cycles have lasted from 9 to 14 years (from peak to peak, from 7 to 17), and some peaks are three times as high as others. The cycles are numbered from the one that peaked in 1761. Cycle 25 reached its peak in October 2024; the spots are now (as of 2026) becoming fewer, towards the next minimum around 2030.

Cycle 25 reached its peak in late 2024. Around when would you expect the peak of cycle 26?

Pick an answer to see why.

Does the Sun’s brightness change?

Satellites have measured the Sun’s total output since 1978. At the top of the Earth’s air, about 1,361 W reach each square meter facing the Sun, and over a sunspot cycle that goes up and down by only about 0.1 %, about 1 W/m². Surprisingly, the Sun is slightly brighter when it has more spots: the dark spots come with even larger bright patches around them.

Over centuries, the cycles change too. From 1645 to 1715 almost no sunspots were seen at all, a quiet spell called the Maunder minimum. It came during one of the coldest parts of the Little Ice Age, when glaciers grew in the Alps and rivers in Europe froze over in winter. A quieter, slightly dimmer Sun probably played a part, alongside large volcanic eruptions, which scientists think mattered more; how much each contributed is still debated. You read in Grade 10 that changes in the Sun can affect the climate Grade 10: Causes of climate change; but the Sun has not got brighter since satellites started measuring it, so it can’t be causing today’s warming.

Activity

At home: see sunspots safely

Never look at the Sun, not with your eyes, sunglasses, binoculars or a camera, and never look into the mirror while it reflects the Sun: either can blind you in seconds. This way you look only at a picture of the Sun on a wall.

You need: a small flat mirror (a make-up mirror), a sheet of paper, tape, scissors, a sunny window or balcony, and a shaded wall at least 10 m away from it (a room or corridor with the curtains drawn works well).

Cut a hole about 1 cm across in the paper, and tape the paper over the mirror so that only the hole shows.
Put the mirror in the sunshine and turn it so that it reflects a spot of sunlight onto the shaded wall. Keep your eyes on the wall, not on the mirror. Don’t shine it into anyone’s eyes, or at a road.
The spot on the wall is not a picture of the hole: it is a picture of the Sun, made by the small hole like a pinhole camera. Make the room as dark as you can.
Tape a sheet of paper on the wall where the image falls, and trace the Sun’s edge and any dark spots. Do it again at the same time each day for a week.

What you should see

The Sun’s disc appears as a round patch about 9 cm across for every 10 m from the mirror to the wall (the Sun is about half a degree wide: 10 m × 0.0093 = 9.3 cm). If there are big sunspots, they show as small gray smudges, and over the days they move across the disc, all together, as the Sun turns; a spot takes about two weeks to cross from one edge to the other. Near a solar maximum you will often see some; near a minimum, the Sun may be spotless for weeks.

Space weather

Around sunspots, the twisted magnetic fields store huge amounts of energy, and sometimes they snap into a new shape and let it go. A solar flare is a sudden flash, from a few minutes to hours long, brightest in X-rays. A coronal mass ejection (CME) is a cloud of about a billion tonnes of the Sun’s outer gas, with its magnetic field, blown into space at hundreds to a few thousand kilometers per second. Both are far more common near the peak of the sunspot cycle.

A huge loop of glowing orange gas bursting out from the edge of the Sun into space, with a tiny picture of the Earth added beside it for size — **Figure 30.6** A coronal mass ejection leaving the Sun on 31 August 2012, photographed by NASA’s Solar Dynamics Observatory in ultraviolet light. The Earth has been added, to scale, for size. This one wasn’t aimed at the Earth, but three days later it gave the Earth’s magnetic field a glancing blow, and auroras followed. Photo: NASA Goddard Space Flight Center, CC BY 2.0

The X-rays from a flare reach the Earth in 8 minutes 20 seconds, and can upset radio signals on the side of the Earth facing the Sun. A CME arrives one to three days later. When it hits the Earth’s magnetic field, it shakes it, causing a geomagnetic storm. The Sun’s changing activity and its effects on the Earth are called space weather, and like the weather, it is now forecast. Spacecraft between the Earth and the Sun watch for storms coming, among them India’s Aditya-L1, which has watched the Sun from 1.5 million km away since January 2024.

A storm’s most beautiful effect is the aurora: the charged particles stream down along the Earth’s magnetic field near the poles and make the air glow, green and red, as a gas glows in a tube. Its harmful effects come from the changing magnetic field, which drives electric currents through anything long that conducts: power lines, pipelines and old telegraph wires. Satellites suffer too.

When	What happened
1 September 1859	The Carrington event. Richard Carrington, drawing sunspots in England, saw two bright white flashes on the Sun. Seventeen hours later the biggest storm on record hit: telegraph wires sparked and some telegraph operators got shocks, and auroras were seen as far south as Cuba and Hawaii.
13 March 1989	A storm drove currents through the power lines of Quebec, in Canada. In 90 seconds the whole grid shut down: six million people were without electricity for nine hours, on a cold March night.
February 2022	A modest storm warmed and puffed up the Earth’s upper air, and the extra drag brought down about 40 of 49 Starlink satellites just launched.
10–11 May 2024	The strongest storm for twenty years. Auroras were seen across Europe and North America, and cameras at the Indian Astronomical Observatory at Hanle, in Ladakh, in the Himalaya north-west of Nepal, recorded a red aurora glowing over the mountains.

An old pen drawing of a large, irregular group of sunspots, with the places of two bright flashes marked by letters — **Figure 30.7** Richard Carrington’s own drawing of the sunspot group of 1 September 1859. The letters mark where he saw the two bright flashes of the flare appear (A and B) and where they faded away (C and D), five minutes later. Photo: Richard Christopher Carrington, Public domain

A night sky with a purple glow rising in a broad pillar above dark trees and a house — **Figure 30.8** The aurora of 10 May 2024, seen from near London, far south of where auroras are usually seen. The same storm was photographed from Ladakh. Photo: The wub, CC BY-SA 4.0

Light from a flare reaches the Earth in about 8 minutes, but the storm from its CME arrives a day or more later. Why?

Pick an answer to see why.

In short

The Sun has shone for about 4.6 billion years (from the ages of meteorites). No chemical fuel or slow shrinking could have powered it that long; only nuclear fusion can.
The Sun turns 4.2 million tonnes of mass into energy every second (E = mc²), fusing about 600 million tonnes of hydrogen into helium. Only the core’s hydrogen, about a tenth of its mass, can be used: enough for about 10 billion years. The Sun is roughly halfway.
Energy made in the core wanders through the dense radiative zone in a random walk for over 100,000 years, is carried by rising hot gas through the convective zone in weeks to months, then flies as light to the Earth in 8 minutes 20 seconds. Neutrinos from fusion reach us directly, proving the core is burning now.
A star’s brightness rises much faster than its mass (twice the mass, about 20 times as bright), so heavy stars live short lives and light ones very long ones. Stars of every mass and age, alone, in pairs and in clusters, fit this model.
In about 1 billion years the Sun will be 10 % brighter; in about 5.4 billion its core hydrogen runs out; it swells into a red giant 256 times its present size, then ends as a white dwarf.
The Sun is variable: sunspots rise and fall in an 11-year cycle; flares and coronal mass ejections cause space weather, with auroras, failing satellites and power cuts. Its total output changes only by about 0.1 % over a cycle.

Questions

Which of these stars will have the longest life?

Pick an answer to see why.

a.

A star has 4 times the Sun’s mass and is about 300 times as bright. Using the Sun’s life as 10 billion years, estimate its life. Explain why it is so much shorter than the Sun’s, although the star has more fuel.

Answer

$t = 10\ \text{billion years} \times \frac{4}{300} = 0.13\ \text{billion years}$
About 130 million years. The star has 4 times the Sun’s fuel, but it uses it 300 times as fast: the much greater weight of its outer layers squeezes its core harder and makes it hotter, so fusion there runs far faster. Fuel ÷ rate = 4 ÷ 300 of the Sun’s life, about a seventy-fifth.

b.

Describe the journey of the energy given out by one fusion reaction in the Sun’s core until it warms your hand. For each stage, say how the energy is carried and roughly how long it takes.

Answer

In the core: a little mass disappears in fusion and becomes energy (E = mc²), as the speed of the particles made and as gamma rays.
Radiative zone: the energy is carried by radiation, absorbed and given out again in random directions countless times, a random walk; this takes over 100,000 years (about 170,000 by careful estimates). On the way the radiation changes from gamma rays to X-rays, ultraviolet and visible light.
Convective zone: the energy is carried by the motion of hot gas, rising to the surface and sinking when cooled (convection), in weeks to months.
Photosphere to Earth: it leaves as light (and infrared and ultraviolet) and crosses space in a straight line at 3.0 × 10⁸ m/s, taking 8 minutes 20 seconds.
Your hand: the light is absorbed and becomes thermal energy, the motion of the particles in your skin.

c.

List three pieces of evidence that the Sun is powered by nuclear fusion, and will last about 10 billion years in all.

Answer

Any three of:

The Sun’s age: meteorites and Earth rocks show the Sun has shone for about 4.6 billion years. Chemical burning would last only thousands of years, and shrinking under gravity only tens of millions; only fusion gives out enough energy.
Neutrinos: detectors deep underground catch the neutrinos that fusion makes, arriving from the Sun’s direction at the rate fusion predicts (once their changing type is allowed for). They show fusion is happening in the core now.
Other stars: weighing stars in pairs shows that brightness rises steeply with mass. The lifetimes this gives, fuel ÷ rate, fit what is seen: heavy, bright blue stars are found only where stars are still forming, and in star clusters the heaviest stars have gone first, at the masses the model predicts for each cluster’s age.
The numbers fit: the Sun’s output and E = mc² give its fuel use, about 600 million tonnes of hydrogen per second, and the core’s fuel lasts about 10 billion years: consistent with its age of 4.6 billion.

d.

A sunspot is at about 3,800 K; the rest of the surface is at about 5,800 K. The energy a hot surface gives out per square meter is in proportion to its temperature (in kelvin) to the power 4. a. What fraction of the energy per square meter of the surroundings does a sunspot give out? b. Why does it look dark, although it is hotter than molten iron?

Answer

a. $\left( \frac{3800}{5800} \right)^4 = (0.655)^4 = 0.18$
A sunspot gives out only about a fifth as much energy per square meter.
b. It is still very bright, and would glow dazzlingly on its own. But in a photograph exposed for the brilliant surface around it, something five times dimmer looks dark. Our eyes and cameras judge brightness by comparison.

e.

Use the sunspot widget, or the peak years: cycle 1 peaked in 1761 and cycle 25 in 2024. a. Work out the average length of a cycle. b. Why can’t you use this to say exactly when the cycle after next will peak?

Answer

a.From the peak of cycle 1 to the peak of cycle 25 is 24 cycles, over 2024 − 1761 = 263 years: 263 ÷ 24 = 11.0 years per cycle.
b.It is only an average. Single cycles have lasted from about 9 to 14 years from minimum to minimum, and from 7 to 17 from peak to peak, and some cycles are much stronger than others. The Sun’s magnetic field, which drives the cycle, is not a clock; the best forecasts look at the field near the Sun’s poles at each minimum.

f.

In about 7.6 billion years the Sun will be a red giant with a radius 256 times its present one, 696,000 km. The Earth is 150 million km from the Sun today. How big will the red giant be, compared with the Earth’s present orbit? What does this suggest will happen to the Earth?

Answer

Radius = 256 × 696,000 km = 1.78 × 10⁸ km = 178 million km. That is more than the Earth’s present distance of 150 million km (1.2 times as far). So the Sun’s surface would reach beyond the Earth’s present orbit. As the Sun loses mass the Earth’s orbit will grow a little, but the drag of the Sun’s outer layers will probably pull the Earth in: it will most likely be swallowed and destroyed. (Long before then, in about a billion years, the brightening Sun will already have made the Earth too hot for oceans.)

Where to go next

How old is the Earth? The 4.6 billion years in this topic comes from radioactive clocks in meteorites, Moon rocks and tiny crystals. How they are read.
Where did the atoms in your body come from? The Sun turns hydrogen into helium; bigger stars go much further. How the carbon, oxygen and gold around you were made in stars.
How do we know the universe is expanding? The stars live and die inside a universe that has its own history. The evidence that it began 13.8 billion years ago.