IB DP Physics · Theme E · Nuclear and quantum physics

Fusion and stars.

E.5 · Where starlight comes from, how stars live and die, and how we measure them
Standard level and higher level

SL + HL · 6 hours E.4 Fission ↗

The journey

Four parts.

Part 01

Fusion

The energy released, and the temperature and density it needs

Part 02

The life of a star

Equilibrium, and how mass decides what happens

Part 03

The HR diagram

Its regions, and lines of constant radius

Part 04

Measuring stars

Distance by parallax, and radius from luminosity and temperature

E.5 FUSION AND STARS · SL + HL

Part 01

Fusion.

Light nuclei joining together: the energy source of the stars

E.5 FUSION AND STARS · SL + HL

Joining light nuclei

What fusion is.

DEUTERIUM AND TRITIUMPRESS → TO PLAY

Two light nuclei join to make a heavier one
The product is more tightly bound: mass is lost and energy released
Per kilogram of fuel, several times more than fission

E.5 FUSION AND STARS · SL + HL

The reaction on the slide

\({}^{2}_{1}\mathrm{H} + {}^{3}_{1}\mathrm{H} \rightarrow {}^{4}_{2}\mathrm{He} + {}^{1}_{0}\mathrm{n}\). Nucleon numbers: \(2 + 3 = 4 + 1\). Proton numbers: \(1 + 1 = 2 + 0\). It releases 17.6 MeV, most of it carried by the neutron, and it is the reaction that experimental fusion reactors on the Earth use, because it is the easiest to start.

Fission and fusion compared

Both release energy by ending with nuclei that are more tightly bound than the ones they began with. Fission gets there by splitting the heaviest nuclei; fusion by joining the lightest. One fusion releases less energy than one fission, 17.6 MeV against about 170 MeV, but the fuel is fifty times lighter, so the energy per kilogram is greater.

In the Sun the fuel is ordinary hydrogen, and it takes a chain of three steps to build one helium-4 nucleus from four protons.

\({}^{2}\mathrm{H}\) is deuterium and \({}^{3}\mathrm{H}\) is tritium: the isotopes of hydrogen with one and two neutrons.

Binding energy per nucleon

Why fusion releases energy.

The curve is steep for light nuclei
\({}^{2}\mathrm{H}\): 1.1 MeV per nucleon. \({}^{4}\mathrm{He}\): 7.1 MeV per nucleon
Fusion releases energy up to iron, at the peak

Beyond iron, fusing nuclei would take in energy.

MEASURED VALUESHORIZONTAL SCALE STRETCHED BELOW 60

E.5 FUSION AND STARS · SL + HL

The same curve as E.4, read from the other end

Fission starts at the far right and moves up towards the peak by splitting. Fusion starts at the far left and moves up towards the peak by joining. Both end closer to iron-56, the most tightly bound region.

Why the left side is so steep

The strong nuclear force acts only between neighboring nucleons. In deuterium each nucleon has one neighbor. In helium-4 each has three, all close together, so each is bound far more tightly. Helium-4 is an especially stable arrangement, which is why it stands above the line of its neighbors on the graph and why it is the particle emitted in alpha decay.

Why iron is the end

A star can only gain energy by fusing nuclei that end up higher on the curve. Once the core of a massive star has turned to iron, no further fusion can support it, and Part 2 shows what follows.

The elements heavier than iron in the Earth, and in you, were made where energy was available to be absorbed: in exploding stars and colliding neutron stars.

Worked example

Energy released in fusion.

\({}^{2}\mathrm{H}\) + \({}^{3}\mathrm{H}\) → \({}^{4}\mathrm{He}\) + n · masses / u \({}^{2}\mathrm{H}\) 2.014102 · \({}^{3}\mathrm{H}\) 3.016049 \({}^{4}\mathrm{He}\) 4.002602 · n 1.008665 Before: \(2.014102 + 3.016049 = 5.030151\) After: \(4.002602 + 1.008665 = 5.011267\) \(\Delta m = 5.011267 - 5.030151 = -0.018884 \,\mathrm{u}\) \(E_\mathrm{released} = -(-0.018884) \times 931.5 = \class{res}{17.6 \,\mathrm{MeV}}\)

In the Sun, overall: 4 \({}^{1}\mathrm{H}\) → \({}^{4}\mathrm{He}\) \({}^{1}\mathrm{H}\) 1.007825 u \(\Delta m = 4.002602 - 4 \times 1.007825\) \(\Delta m = -0.028698 \,\mathrm{u}\) \(E_\mathrm{released} = -(-0.028698) \times 931.5 = \class{res}{26.7 \,\mathrm{MeV}}\)

E.5 FUSION AND STARS · SL + HL

The fraction of mass that disappears

For hydrogen turning into helium, 0.0287 u vanishes out of 4.0313 u: 0.71%. Seven parts in a thousand of the mass of the hydrogen is converted to energy. In fission the figure is less than one part in a thousand. That fraction, 0.007, is the number that decides how long a star can shine.

Method

It is exactly the mass-defect calculation of E.3 and E.4: add up the masses on each side and use \(\Delta m = m_\mathrm{final} - m_\mathrm{initial}\). The rest-energy change is \(\Delta E_\mathrm{rest} = \Delta m c^{2}\). The released energy is \(E_\mathrm{released} = -\Delta m c^{2}\), so multiply the negative of the signed mass change in u by 931.5 to get MeV. Including the released energy, \(\Delta E_\mathrm{total} = 0\). Keep all the decimal places until the subtraction is done.

The masses used are those of neutral atoms, which include electrons. For deuterium and tritium the electrons balance on the two sides. For four hydrogen atoms becoming one helium atom they do not: two electrons are left over, and they annihilate with the two positrons the chain produces. That energy is part of the 26.7 MeV.

Temperature and density

The conditions for fusion.

Nuclei are positive: they repel
The strong force acts only within about \(10^{-15}\) m
High temperature: enough kinetic energy to get that close
High density: enough collisions each second

TWO PROTONS APPROACHING

E.5 FUSION AND STARS · SL + HL

How hot

From B.1, the average kinetic energy of a particle is \(\frac{3}{2} k_\mathrm{B} T\), so a hotter gas has faster nuclei. Two protons approaching head-on convert their kinetic energy into electric potential energy as they close in (D.2). To get within reach of the strong force they must not be stopped and turned back first. In the core of the Sun the temperature is about fifteen million kelvin.

How dense

Even at that temperature a head-on collision at the right speed is rare, and most collisions that do happen fail to produce fusion. A high density of nuclei means an enormous number of collisions each second, so that enough of them succeed. The center of the Sun is about 150 times as dense as water, and still a gas of bare nuclei and electrons: a plasma.

Why fusion is hard on the Earth

A star holds its hot, dense fuel together by its own gravity. A laboratory has to do it with magnetic fields or lasers, at temperatures of a hundred million kelvin, far hotter than the Sun, to make up for a much lower density.

Two conditions, two reasons: high temperature to overcome the repulsion; high density for a high rate of collisions.

The story · England, the 1920s

Background story · not assessed

What keeps the Sun shining.

Photograph of Arthur Eddington — **Arthur Eddington**

The Sun has shone for thousands of millions of years. No chemical fuel, and no slow shrinking under gravity, could last so long
Four hydrogen nuclei have slightly more mass than one helium nucleus
Eddington proposed that the Sun is powered by fusing hydrogen into helium, and built the physics of stars on the idea in his book of 1926
Fusion had not yet been seen in any laboratory

A star explained from a table of atomic masses and \(E = mc^{2}\).

Photograph: Smithsonian Institution, no known restrictions.

E.5 FUSION AND STARS · SL + HL

Worked example

How long can the Sun shine.

\(L_{\odot} = 3.85 \times 10^{26} \,\mathrm{W}\) Signed rate of change of the Sun’s mass \(\dfrac{\Delta m}{\Delta t} = -\dfrac{L}{c^{2}}\) \(= -\dfrac{3.85 \times 10^{26}}{(3.00 \times 10^{8})^{2}}\) \(= \class{res}{-4.3 \times 10^{9} \,\mathrm{kg} \,\mathrm{s}^{-1}}\)

Mass \(2.0 \times 10^{30}\) kg · 10% of it fuses 0.71% of that mass becomes energy \(E = 0.10 \times 2.0 \times 10^{30} \times 0.0071 \times c^{2}\) \(E = 1.3 \times 10^{44} \,\mathrm{J}\) \(t = \dfrac{E}{L} = \dfrac{1.3 \times 10^{44}}{3.85 \times 10^{26}} = 3.3 \times 10^{17} \,\mathrm{s}\) \(t = \class{res}{1.0 \times 10^{10} \,\text{years}}\)

E.5 FUSION AND STARS · SL + HL

Four million tonnes a second

The Sun’s mass falls by \(4.3 \times 10^{9}\) kg every second, and it has done so for 4.6 thousand million years. It is so massive that in all that time it has lost less than a thousandth of its mass in this way.

Why only 10%

Fusion happens only in the core, where it is hot and dense enough. The hydrogen in the outer layers never gets there. When the core’s hydrogen is used up, after about ten thousand million years, the Sun leaves the main sequence. It is roughly half way through.

What the estimate rests on

A luminosity, a mass, and the 0.71% from the last worked example. Nothing else. A calculation this short settled a question that had puzzled the nineteenth century, when the best available answer, gravitational shrinking, gave the Sun only a few tens of millions of years.

One year is \(3.16 \times 10^{7}\) s.

Quick check

Fusion.

01What is nuclear fusion?

Light nuclei joining to make a heavier one

02Why do nuclei have to be moving fast to fuse?

To overcome their electric repulsion

03Why is a high density needed?

For a high rate of collisions

04A mass defect of 0.0050 u. Energy released?

\(0.0050 \times 931.5 = 4.7 \,\mathrm{MeV}\)

05Up to which element does fusion release energy?

Iron

06\({}^{2}\mathrm{H}\) + \({}^{2}\mathrm{H}\) → \({}^{3}\mathrm{He}\) + ?

A neutron

PRESS → TO REVEAL EACH ANSWER

Part 02

The life of a star.

A long balance between gravity and pressure, and what happens when it ends

E.5 FUSION AND STARS · SL + HL

Why a star keeps its size

A star in equilibrium.

Gravity pulls every part of the star inwards
Radiation pressure and gas pressure push outwards
Fusion in the core supplies the energy that keeps the pressure up

Balanced: a stable star.

INWARDS AND OUTWARDS

E.5 FUSION AND STARS · SL + HL

A balance that corrects itself

Suppose the fusion in the core speeds up a little. The core gets hotter, the pressure rises and the star expands. Expanding cools the core and thins it out, so the fusion slows again. If instead the fusion slackens, the pressure drops, gravity squeezes the core, the core heats up and the fusion recovers. The star is its own thermostat, which is how the Sun has kept almost the same output for thousands of millions of years.

What radiation pressure is

Light carries momentum, so it pushes on whatever absorbs it. Photons from the core are absorbed and re-emitted countless times on their way out, each time giving the gas an outward push. In stars like the Sun the pressure of the hot gas does most of the work; in the most massive stars the pressure of the radiation takes over.

The phrase the guide uses: an equilibrium between outward radiation pressure and inward gravitational forces.

One number sets the story

Mass decides.

More mass: stronger gravity, so a hotter, denser core
Fusion runs far faster: much greater luminosity
So the fuel is used up sooner
And the star ends differently

Mass (Sun = 1)	Luminosity (Sun = 1)	Time on the main sequence
0.5	about 0.1	\(\sim 10^{11}\) years
1	1	\(\sim 10^{10}\) years
10	about 3000	\(\sim 3 \times 10^{7}\) years

E.5 FUSION AND STARS · SL + HL

Why bigger stars die younger

It seems the wrong way round: a star with ten times the fuel should last longer. But the rate of fusion is extremely sensitive to temperature, and the extra weight of a massive star makes its core much hotter. A star of ten solar masses has ten times the fuel and burns it about three thousand times as fast. Its life is \(\frac{10}{3000}\), about a three-hundredth, of the Sun’s.

Check the table by ratio

Lifetime is proportional to fuel divided by the rate of using it: \(\frac{\text{mass}}{\text{luminosity}}\). For the half-mass star, \(\frac{0.5}{0.1} = 5\) times the Sun’s life. Such stars have not had time to die since the universe began.

Birth

Every star starts as a cloud of gas and dust that collapses under its own gravity. Gravitational potential energy becomes thermal energy, the center heats up, and when it is hot and dense enough for hydrogen to fuse, the collapse stops and a main sequence star is born.

The figures in the table are rounded orders of magnitude. What you need is the trend, and the reason for it.

Up to about eight solar masses

The life of a star like the Sun.

FOUR STAGESSIZES NOT TO SCALE

When the core runs out of hydrogen, fusion there stops, the core contracts and heats, and the outer layers swell and cool.

E.5 FUSION AND STARS · SL + HL

Red giant

With no fusion to hold it up, the helium core is squeezed by gravity and gets hotter. Hydrogen goes on fusing in a shell round it, faster than before. The extra output pushes the outer layers outwards to a hundred times their old size. Spread over that huge surface, the star is very luminous but cool, and so red. In the core, helium eventually starts fusing to carbon and oxygen.

Planetary nebula

A star of this mass never gets hot enough to fuse carbon. Its outer layers drift away into space as a glowing shell. The name is historical: in a small telescope such a shell looks like the disk of a planet.

White dwarf

What remains is the bare core: about the mass of the Sun in a body the size of the Earth. No fusion happens in it. It is hot, so it shines, and it simply cools for the rest of time.

A white dwarf is held up not by thermal pressure but by electrons resisting being squeezed into the same state. That works only up to 1.4 solar masses.

More than about eight solar masses

The life of a massive star.

FOUR STAGESSIZES NOT TO SCALE

The core is hot enough to fuse heavier and heavier nuclei, until it is iron. Then fusion can supply no more energy.

E.5 FUSION AND STARS · SL + HL

Red supergiant

Each time a fuel runs out, the core contracts, heats up and lights the next: helium to carbon, carbon to neon, and on through oxygen and silicon to iron. Each stage is shorter than the last: the final one takes about a day. The star ends with layers like an onion, the heaviest elements at the center.

Supernova

Iron is at the peak of the binding energy curve. Fusing it would absorb energy, not release it. With nothing to hold it up, the iron core collapses in less than a second. The outer layers fall in, rebound and are blown into space in an explosion that can briefly outshine a whole galaxy.

What is left

The collapsed core is a neutron star: more than the mass of the Sun packed into a ball about 20 km across, at the density of an atomic nucleus (E.1). If the core is more massive still, nothing can stop the collapse, and it becomes a black hole.

The carbon, oxygen and iron scattered by such explosions are the raw material of later stars, of planets and of living things.

The story · Harvard, 1925

Background story · not assessed

What the stars are made of.

Black and white photograph of Cecilia Payne seated at a desk with papers, in front of filing cabinets and shelves — **Cecilia Payne** at the Harvard College Observatory

She came from England to Harvard in 1923. Her thesis of 1925 used the lines in stellar spectra to work out what stars are made of
Her answer: mainly hydrogen and helium. The leading astronomer of the day called it “spurious” and urged her not to claim it
In 1929 he reached the same result himself. Her thesis was later called the most brilliant ever written in astronomy

Photograph: Smithsonian Institution, Science Service, public domain.

E.5 FUSION AND STARS · SL + HL

Quick check

The life of a star.

01Which two things balance in a stable star?

Gravity inwards; radiation (and gas) pressure outwards

02What decides how a star’s life goes?

Its mass

03Which lives longer: a star of 0.5 or of 10 solar masses?

The 0.5 solar mass star

04What does the Sun become after the red giant stage?

A planetary nebula, then a white dwarf

05Why does fusion stop at iron in a massive star?

Iron is at the peak of the binding energy curve

06What can a supernova leave behind?

A neutron star or a black hole

PRESS → TO REVEAL EACH ANSWER

Part 03

The HR diagram.

One graph on which every star has a place

E.5 FUSION AND STARS · SL + HL

Luminosity against temperature

The Hertzsprung–Russell diagram.

Luminosity upwards, in units of the Sun’s
Temperature across, decreasing to the right
Both scales are logarithmic
Stars fall into a few regions, not everywhere

EACH DOT IS A STARILLUSTRATIVE

E.5 FUSION AND STARS · SL + HL

The backwards axis

Temperature runs from hot on the left to cool on the right. This is a historical accident: the diagram was first drawn with the classes of stellar spectra along the bottom, and the classes had been lettered before anyone knew they were a temperature sequence. The convention has stuck. Blue-white stars are on the left, red stars on the right.

Why logarithmic

Stars range over a factor of ten thousand million in luminosity, from a ten-thousandth of the Sun’s to a million times it. On an ordinary scale all but the brightest would be squashed against the axis. On a logarithmic scale each step up is a factor of a hundred.

What the empty spaces mean

If any combination of temperature and luminosity were possible, the dots would be scattered evenly. They are not. The pattern is a record of how stars work: only certain combinations are stable, and stars spend most of their lives in one of them.

Drawing the temperature axis increasing to the right. In an exam sketch, label the hot end and the cool end.

Who lives where

The regions of the HR diagram.

Region	Where on the diagram	What the stars are like
Main sequence	A band from upper left to lower right	Fusing hydrogen in the core. About 90% of stars
Red giants	Above the main sequence, on the right	Cool but luminous, so very large
Supergiants	Across the top	The most luminous and the largest
White dwarfs	Lower left	Hot but faint, so very small
Instability strip	A narrow band crossing above the main sequence	Stars that pulsate, varying in brightness

E.5 FUSION AND STARS · SL + HL

The main sequence is a sequence of mass

Part 2 showed that a more massive star is hotter and far more luminous. So the heavy stars sit at the upper left of the band and the light ones at the lower right. A star stays at very nearly the same point on the band for as long as it is fusing hydrogen in its core. It does not slide along the main sequence as it ages.

Reasoning about size

A red giant has a low surface temperature, so each square meter of its surface radiates little. Yet its luminosity is high. The only way to radiate a great deal from a cool surface is to have an enormous surface. A white dwarf is the opposite case: hot, so each square meter radiates strongly, yet faint overall, so it must be tiny.

Why most stars are on the main sequence

Fusing hydrogen is by far the longest stage of a star’s life. Catch a star at a random moment and that is most likely what it is doing.

You need to know that the instability strip exists and where it is. The pulsating stars in it, including Cepheid variables, are not part of the course.

Size on the HR diagram

Lines of constant radius.

\(L = \sigma A T^{4} = 4 \pi R^{2} \sigma T^{4}\)

For one radius, \(L \propto T^{4}\): a straight line on these axes
Sloping down to the right
Larger stars: lines further up and right

THREE RADII

E.5 FUSION AND STARS · SL + HL

Where the lines come from

A star is nearly a black body, so by the Stefan–Boltzmann law of B.1 its luminosity is \(4 \pi R^{2} \sigma T^{4}\). Fix the radius and the luminosity depends only on \(T^{4}\). Halving the temperature cuts the luminosity by sixteen. On logarithmic axes a power law is a straight line, and since temperature decreases to the right, the line slopes down to the right.

Reading sizes

The white dwarfs lie along the line for a hundredth of the Sun’s radius: about the size of the Earth. The main sequence cuts across the lines at a shallow angle, from a tenth of a solar radius at the cool end to about ten at the hot end. Red giants are near a hundred solar radii, and the largest supergiants reach a thousand: larger than the orbit of Jupiter.

Moving at right angles to the lines, up and to the right, means getting bigger.

Sketching and interpreting lines of constant radius is required. Two stars on the same line have the same size, however different they look.

A life as a path

The Sun’s path on the HR diagram.

Now: on the main sequence, hardly moving
Red giant: up and to the right. Cooler, larger, more luminous
Outer layers lost: across to the left as the hot core is uncovered
White dwarf: down to the lower left, then cooling

A SKETCH OF THE PATH

E.5 FUSION AND STARS · SL + HL

The speed along the path is very uneven

The dot on the slide moves steadily, but a real star does not. The Sun will spend about ten thousand million years at its starting point, about a thousand million as a red giant, a few tens of thousands crossing to the left, and then the rest of time fading as a white dwarf. The number of stars seen in each region reflects how long stars spend there.

The path is not a journey through space

The diagram is a graph of properties. A star “moving” on it is staying where it is in the sky while its temperature and luminosity change.

A massive star starts higher up the main sequence and moves to the right across the top of the diagram, into the supergiants, before it explodes.

Each move can be explained with \(L = 4 \pi R^{2} \sigma T^{4}\). Up and right: \(R\) is growing faster than \(T^{4}\) is falling.

Quick check

The HR diagram.

01What is plotted on each axis?

Luminosity up; temperature across

02Which way does temperature increase?

To the left

03Where are white dwarfs?

Lower left: hot and faint

04A star is cool but very luminous. What does that say about its size?

It is very large

05Where on the main sequence are the most massive stars?

Upper left

06Two stars lie on the same line of constant radius. What do they share?

The same radius

PRESS → TO REVEAL EACH ANSWER

Part 04

Measuring stars.

How far away, how bright, how hot and how big, from a point of light

E.5 FUSION AND STARS · SL + HL

Distance by geometry

Stellar parallax.

\(d \,\text{(parsec)} = \dfrac{1}{p \,\text{(arc-second)}}\)

A nearby star shifts against distant ones as the Earth orbits
\(p\): half the total shift in six months
Further away: a smaller angle

NOT TO SCALE

E.5 FUSION AND STARS · SL + HL

Try it

Hold up a finger at arm’s length and look at it with one eye, then the other. It jumps against the background. Bring it closer and it jumps further. Your eyes are a few centimeters apart; for a star, the two viewpoints are opposite sides of the Earth’s orbit, 300 million kilometers apart.

Why the equation is so simple

In the long thin triangle of the diagram, \(\tan p = \frac{1 \,\mathrm{AU}}{d}\), and for such tiny angles \(\tan p = p\) in radians. The parsec is defined to make the constants disappear: it is the distance at which the parallax is exactly one arc-second. So a parallax of 0.5 arc-second means 2 pc, and 0.1 arc-second means 10 pc.

How small the angles are

An arc-second is \(\frac{1}{3600}\) of a degree: the width of a small coin seen from four kilometers. No star has a parallax as large as one arc-second. The first to be measured was 61 Cygni, by Friedrich Bessel in 1838. From the ground the method reaches a few hundred parsecs; the Gaia spacecraft, launched in 2013, measures angles of tens of millionths of an arc-second.

Using the whole shift. The parallax angle is half the shift between two sightings six months apart.

AU, light years and parsecs

Units of distance.

Unit	Value
Astronomical unit, AU: the Earth–Sun distance	\(1.50 \times 10^{11} \,\mathrm{m}\)
Light year, ly: the distance light travels in a year	\(9.46 \times 10^{15} \,\mathrm{m}\)
Parsec, pc: parallax of one arc-second	3.26 ly

The nearest star: \(p = 0.769\) arc-second \(d = \dfrac{1}{p} = \dfrac{1}{0.769} = \class{res}{1.30 \,\mathrm{pc}}\) \(d = 1.30 \times 3.26 = \class{res}{4.24 \,\mathrm{ly}}\) \(d = 4.24 \times 9.46 \times 10^{15} = \class{res}{4.0 \times 10^{16} \,\mathrm{m}}\) \(d = \dfrac{4.0 \times 10^{16}}{1.50 \times 10^{11}} = \class{res}{2.7 \times 10^{5} \,\mathrm{AU}}\)

E.5 FUSION AND STARS · SL + HL

Which unit for what

The astronomical unit suits a planetary system: Neptune is 30 AU from the Sun. The light year is the unit of popular writing, and says how old the light is that you are seeing. The parsec is the working unit of astronomers, because it comes straight out of the parallax measurement.

The emptiness of space

The nearest star, Proxima Centauri, is 270 000 times further from us than the Sun is. If the Sun were a grapefruit in London, the Earth would be a grain of sand eleven meters away, and the nearest star another grapefruit in Moscow.

One parsec in meters

\(3.26 \times 9.46 \times 10^{15} = 3.08 \times 10^{16}\) m. All three conversions are in the data booklet, and converting between AU, light years and parsecs is required.

Treating a light year as a time. It is a distance.

Worked example

The radius of a star.

Sirius: \(L = 9.8 \times 10^{27} \,\mathrm{W}\) · \(T = 9940 \,\mathrm{K}\) \(L = 4 \pi R^{2} \sigma T^{4}\) \(R = \sqrt{\dfrac{L}{4 \pi \sigma T^{4}}}\) \(R = \sqrt{\dfrac{9.8 \times 10^{27}}{4 \pi \times 5.67 \times 10^{-8} \times 9940^{4}}}\) \(R = \class{res}{1.2 \times 10^{9} \,\mathrm{m}}\)

The same, as a ratio to the Sun (5800 K) \(\dfrac{R}{R_{\odot}} = \sqrt{\dfrac{L}{L_{\odot}}} \left(\dfrac{T_{\odot}}{T}\right)^{2}\) \(= \sqrt{25.4} \times \left(\dfrac{5800}{9940}\right)^{2}\) \(= \class{res}{1.7}\)

E.5 FUSION AND STARS · SL + HL

Why a ratio is easier

Divide \(L = 4 \pi R^{2} \sigma T^{4}\) for the star by the same equation for the Sun and the constants cancel: \(\frac{L}{L_{\odot}} = \left(\frac{R}{R_{\odot}}\right)^{2} \left(\frac{T}{T_{\odot}}\right)^{4}\). Luminosities are usually given in units of the Sun’s, so this form avoids large powers of ten altogether.

Sirius has a companion

Sirius B is even hotter than Sirius, about 25 000 K, but has only 0.026 of the Sun’s luminosity. The same working gives \(\sqrt{0.026} \times \left(\frac{5800}{25000}\right)^{2} = 0.009\): a radius under a hundredth of the Sun’s, about the size of the Earth. It is a white dwarf, one of the first found. Two stars side by side in the sky, the fainter one a thousand times dimmer but hotter, can differ like this only in size.

Raising the temperature to the fourth power but forgetting that the radius is squared, so a square root is needed at the end.

Putting it together

Everything from a point of light.

Step 1 · Parallax

Distance

\(d = \frac{1}{p}\)

Step 2 · Brightness

Luminosity

\(L = 4 \pi d^{2} b\)
from B.1

Step 3 · Spectrum

Temperature and composition

Wien’s law for the peak; the lines for the elements

Step 4 · Stefan–Boltzmann

Radius

\(R = \sqrt{\frac{L}{4 \pi \sigma T^{4}}}\)

With \(L\) and \(T\) the star has a place on the HR diagram, and from that its stage of life.

E.5 FUSION AND STARS · SL + HL

What is actually measured

A star is too far away to show a disk in most telescopes. All that arrives is light. From it astronomers measure three things: the direction it comes from, which over a year gives the parallax; how much power arrives per square meter, the apparent brightness; and how that power is spread across the wavelengths, the spectrum. Everything on the slide follows from those three.

Temperature and composition from the spectrum

The wavelength at which the spectrum peaks gives the surface temperature by Wien’s law, \(\lambda_\mathrm{max} T = 2.9 \times 10^{-3}\) m K. The dark lines crossing it are the fingerprints of the elements in the star’s outer layers (E.1): this is what Cecilia Payne read.

A chain is as strong as its first link

An error in the distance is squared in the luminosity and carries through to the radius. This is why precise parallaxes matter so much.

A typical long question walks down this chain. Recognize which step each part is asking for.

Quick check

Measuring stars.

01A star has a parallax of 0.20 arc-second. Distance?

5.0 pc

02Convert 10 pc to light years.

32.6 ly

03Why does parallax fail for very distant stars?

The angle is too small to measure

04A star has the Sun’s temperature and 100 times its luminosity. Radius?

10 solar radii

05What two things does a star’s spectrum give?

Surface temperature and composition

06Which law gives the temperature from the peak wavelength?

Wien’s displacement law

PRESS → TO REVEAL EACH ANSWER

Summary

Fusion and stars on one slide.

Fusion

Light nuclei join

\(E_\mathrm{released} = -\Delta m \, c^{2}\)
Needs high temperature and high density

A stable star

Gravity = pressure

Inward gravity against outward radiation pressure

Mass decides

Heavier: brighter, shorter

White dwarf, or supernova and neutron star or black hole

HR diagram

\(L\) up, \(T\) falling to the right

Main sequence, red giants, supergiants, white dwarfs, instability strip

Distance

\(d = \frac{1}{p}\)

Parsecs and arc-seconds
1 pc = 3.26 ly

Radius

\(L = 4 \pi R^{2} \sigma T^{4}\)

Lines of constant radius slope down to the right

E.5 FUSION AND STARS · SL + HL

Summary

Key words.

FusionLight nuclei joining to make a heavier one

Radiation pressureThe outward push of light on matter

Main sequenceStars fusing hydrogen in their cores

Red giantCool, luminous and very large

White dwarfHot, faint and very small: a dead core

SupernovaThe explosion of a massive star

HR diagramLuminosity against surface temperature

ParallaxApparent shift of a nearby star as the Earth orbits

ParsecDistance for a parallax of one arc-second

E.5 FUSION AND STARS · SL + HL