Fusion & Stars

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Fusion Reactions in Stars

Nuclear fusion is the joining of two small nuclei to produce a larger nucleus, releasing energy.
In stars, four hydrogen nuclei (protons) fuse to form one helium nucleus, releasing a huge amount of energy.
The proton-proton chain is the series of reactions that produces helium from hydrogen in stars like the Sun.
The energy released provides an outward radiation pressure that prevents the star from collapsing under gravity.
For fusion to occur, nuclei must overcome electrostatic repulsion and get close enough for the strong nuclear force to act.
This requires extremely high temperature and density, such as in the core of a star, or high-speed collisions in a particle accelerator.
On Earth, research focuses on the deuterium-tritium (D-T) reaction: ²₁H + ³₁H → ⁴₂He + ¹₀n, which releases a large amount of energy per unit mass.

Hydrogen fusion in a star

Energy Released in Fusion Reactions

When two small nuclei fuse, the resulting larger nucleus has a higher binding energy per nucleon than the original nuclei.
The mass defect (difference in mass between reactants and products) is converted into energy according to ΔE = Δmc².
For example, fusing two deuterium nuclei into helium releases about 24 MeV (28 MeV − 4 MeV).
In the proton-proton chain, the overall reaction is 4¹₁H → ⁴₂He + 2e⁺ + 2νₑ, releasing energy.
The energy released per fusion reaction can be calculated from the mass defect and used to determine the rate of hydrogen consumption in a star.

Star Formation

Stars form from a nebula — a giant cloud of hydrogen gas and dust.
Gravitational collapse causes denser clumps to form, heating up and glowing as a protostar.
When the core temperature reaches millions of kelvin, hydrogen fusion begins, producing helium and releasing energy.
The outward radiation pressure and gas pressure balance the inward gravitational force, leading to a stable main sequence star.
If outward pressure > gravitational force, the star expands; if outward pressure < gravitational force, the star contracts.
Equilibrium is maintained as long as the inward and outward forces are balanced.

Star formation stages

Life Cycle of a Star

All stars begin with the same stages: nebula → protostar → main sequence star.
A star spends most of its life on the main sequence, where it is stable due to balanced forces.
Low-mass stars (like the Sun) evolve: red giant → planetary nebula → white dwarf.
High-mass stars evolve: red supergiant → supernova → neutron star or black hole.
In red giants, helium fusion occurs in the core, and the outer layers expand and cool.
In massive stars, fusion continues up to iron, after which the core collapses and a supernova occurs.
The remnant of a supernova can be a neutron star or, if massive enough, a black hole.

Life cycle of a large mass star

The Hertzsprung–Russell (HR) Diagram

The HR diagram plots stellar luminosity (y-axis) against surface temperature (x-axis, decreasing to the right).
Most stars lie on the main sequence, a diagonal band where luminosity increases with temperature.
Red giants and red supergiants are above the main sequence, with high luminosity but cooler temperatures.
White dwarfs are below and to the left of the main sequence: hot but low luminosity.
The HR diagram only shows stars in stable phases; transitory phases are short-lived.
The Sun is a main sequence star with a luminosity of 1 (relative to itself) and a surface temperature of about 5800 K.

Emission & Absorption Spectra in Stars

A continuous spectrum contains all wavelengths and is produced by hot, dense sources like stellar cores.
An emission line spectrum consists of bright lines on a black background, produced by hot, low-pressure gases.
An absorption line spectrum consists of dark lines on a continuous spectrum, produced when white light passes through a cooler gas.
Each element produces a unique pattern of spectral lines, acting as a fingerprint for identifying elements in stars.
Stars emit an absorption spectrum because their outer atmospheres are cooler than the core.
The chemical composition of a star can be determined by comparing its absorption spectrum with known emission spectra.

Stellar Parallax

Stellar parallax is the apparent shift in position of a nearby star against distant background stars as Earth orbits the Sun.
The parallax angle p (in arcseconds) is related to distance d (in parsecs) by p = 1/d.
This method is accurate for stars up to about 100 pc away.
1 parsec is the distance at which 1 AU subtends an angle of 1 arcsecond: 1 pc ≈ 3.1 × 10¹⁶ m.
1 light-year is the distance light travels in one year: 1 ly ≈ 9.5 × 10¹⁵ m.
1 astronomical unit (AU) is the mean Earth-Sun distance: 1 AU ≈ 1.5 × 10¹¹ m.

Determination of Stellar Radii

The radius of a star can be found by combining Wien's displacement law and the Stefan-Boltzmann law.
Wien's displacement law: λmax T = 2.9 × 10⁻³ m K, gives the surface temperature from the peak wavelength.
The Stefan-Boltzmann law: L = 4πr²σT⁴, relates luminosity, radius, and temperature.
If luminosity and temperature are known, the radius can be calculated.
For example, Betelgeuse has a radius about 1000 times that of the Sun.

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Übungsfragen

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1.Which of the following is the correct definition of nuclear fusion?
Easy
- AThe joining of two small nuclei to produce a larger nucleus
- BThe splitting of a large nucleus into two smaller nuclei
- CThe joining of two large nuclei to produce a smaller nucleus
- DThe emission of an α particle from a nucleus
2.Which of the following is the correct definition of nuclear fission?
Easy
- AThe joining of two small nuclei to produce a larger nucleus
- BThe splitting of a large nucleus into two smaller nuclei
- CThe joining of two large nuclei to produce a smaller nucleus
- DThe emission of a β particle from a nucleus
3.In a main sequence star, the inward gravitational force is balanced by the outward radiation pressure.
Easy
True or false?
4.Which force must be overcome for two nuclei to fuse?
Easy
- AGravitational force
- BElectrostatic repulsion between protons
- CStrong nuclear force
- DWeak nuclear force
5.What is the name of the force that acts when nuclei get close enough to fuse?
Easy
- AElectrostatic force
- BGravitational force
- CStrong nuclear force
- DWeak nuclear force
6.What are the two factors that the power output of a star depends on according to the Stefan-Boltzmann law?
Medium
- ASurface temperature and radius
- BLuminosity and distance
- CMass and age
- DColour and chemical composition
7.What is the definition of a parsec?
Medium
- AThe distance travelled by light in one year
- BThe mean distance from the centre of the Earth to the centre of the Sun
- CThe distance at which the parallax angle is 1 arcsecond
- DThe distance at which the parallax angle is 1°
8.A light-year is a unit of time.
Easy
True or false?

Mehr Fragen

9.Which of the following correctly describes the proton-proton chain?
Medium
- AFour hydrogen nuclei fuse to produce one helium nucleus
- BTwo helium nuclei fuse to produce one hydrogen nucleus
- COne hydrogen nucleus splits to produce two helium nuclei
- DFour helium nuclei fuse to produce one hydrogen nucleus
10.What are the conditions necessary for nuclear fusion to occur in stars?
Medium
- ALow temperature and low pressure
- BHigh temperature and high pressure
- CHigh temperature and low pressure
- DLow temperature and high pressure
11.In the deuterium-tritium fusion reaction, what are the products?
Medium
- AHelium-4 and a neutron
- BHelium-3 and a proton
- CTwo helium-4 nuclei
- DHelium-4 and a positron
12.What is the main sequence of the Hertzsprung-Russell diagram?
Medium
- AA region where stars are very luminous and cool
- BA diagonal band where most stars are found, with luminosity increasing with temperature
- CA region of hot, low-luminosity stars
- DA region of cool, high-luminosity stars
13.Why do hot, low-pressure gases produce an emission line spectrum?
Hard
- AElectrons transition between discrete energy levels, emitting photons of specific wavelengths
- BPhotons of all wavelengths are emitted due to close interactions between atoms
- CPhotons are absorbed at specific wavelengths, creating dark lines
- DThe gas is so hot that it emits a continuous spectrum
14.What causes absorption lines in a star's spectrum?
Hard
- APhotons are emitted by the star's core at specific wavelengths
- BPhotons are absorbed by cooler gas in the star's outer atmosphere at specific wavelengths
- CThe star's surface is too hot to emit a continuous spectrum
- DThe star's magnetic field splits spectral lines
15.How can the radius of a star be determined from its luminosity and surface temperature?
Hard
- AUsing Wien's displacement law
- BUsing the Stefan-Boltzmann law
- CUsing the inverse square law of flux
- DUsing the parallax equation
16.Which of the following statements about the Hertzsprung-Russell diagram are correct? (select all that apply)
Medium
- ALuminosity is plotted on the y-axis
- BTemperature increases from left to right
- CMost stars are found on the main sequence
- DWhite dwarfs are hot but low luminosity
- ERed giants are hot and high luminosity
17.Which of the following are stages in the life cycle of a low-mass star like the Sun? (select all that apply)
Hard
- ANebula
- BProtostar
- CRed supergiant
- DWhite dwarf
- ESupernova
18.White dwarf stars are hot but have low luminosity.
Medium
True or false?
19.The strong nuclear force has a very long range.
Medium
True or false?
20.The Sun is currently on the main sequence.
Easy
True or false?
21.In a fusion reaction, the total mass of the products is greater than the total mass of the reactants.
Medium
True or false?
22.A star with a higher surface temperature will have its peak emission at a shorter wavelength.
Medium
True or false?
23.Match each term with its correct definition.
Medium
- Luminosity
- Apparent brightness
- Stellar parallax
- Parsec
- The apparent shifting of a nearby star against distant stars as Earth orbits the Sun
- The total power output of a star
- The distance at which a star would have a parallax angle of 1 arcsecond
- The intensity of light received from a star per unit area
24.Match each stage of stellar evolution with its correct description.
Medium
- Nebula
- Protostar
- Main sequence
- Red giant
- A cloud of gas and dust from which stars form
- A collapsing cloud that heats up and glows before fusion begins
- A stable star where hydrogen fusion occurs in the core
- An expanding, cooling star after core hydrogen is exhausted
25.Place the stages in the life cycle of a low-mass star in the correct order, starting from the beginning.
Medium
- Nebula
- Protostar
- Main sequence
- Red giant
- Planetary nebula
- White dwarf
26.Place the stages in the life cycle of a massive star in the correct order, starting from the beginning.
Hard
- Nebula
- Protostar
- Main sequence
- Red supergiant
- Supernova
- Neutron star or black hole
27.Compared to the Sun, this star is:
Medium
- ALarger and cooler
- BSmaller and hotter
- CLarger and hotter
- DSmaller and cooler
28.Which star has the highest surface temperature?
Medium
- AAljanah
- BDeneb
- CFawaris
- DSadr
29.Which star has the lowest surface temperature?
Medium
- AAljanah
- BDeneb
- CFawaris
- DSadr
30.Which star has the largest radius?
Medium
- AAljanah
- BDeneb
- CFawaris
- DSadr
31.Which of the following statements about the proton-proton chain are correct? (select all that apply)
Medium
- AIt produces a helium nucleus from the fusion of four protons
- BIt occurs in the cores of stars
- CIt involves the fusion of deuterium and tritium nuclei
- DIt releases energy due to mass defect
- EIt requires extremely low temperatures
32.Which of the following are conditions necessary for nuclear fusion to occur in stars? (select all that apply)
Medium
- AVery high temperature
- BVery high pressure and density
- CLow kinetic energy of nuclei
- DPresence of a strong magnetic field
- ENuclei must overcome electrostatic repulsion
33.Which of the following correctly describes the energy released in the fusion of two deuterium nuclei into a helium nucleus?
Medium
- AThe energy released is the difference in binding energy between the products and reactants
- BThe energy released is the sum of the binding energies of the reactants
- CThe energy released is equal to the binding energy of the helium nucleus
- DThe energy released is zero because mass is conserved
34.A star has a luminosity of 4.5 × 10²⁷ W and an apparent brightness of 2.6 nW m^-2. What is its distance from Earth?
Hard
- A1.2 × 10¹⁸ m
- B3.7 × 10¹⁷ m
- C2.1 × 10¹⁷ m
- D5.5 × 10¹⁸ m
35.A main sequence star has a surface temperature of 8000 K. A second main sequence star has a surface temperature of 4000 K. Assuming both stars have the same radius, what is the ratio of the luminosity of the hotter star to the luminosity of the cooler star?
Medium
- A2
- B4
- C8
- D16
36.Which of the following statements about the proton-proton chain in the Sun are correct? (select all that apply)
Hard
- AIt converts four hydrogen nuclei into one helium nucleus.
- BIt produces positrons and neutrinos.
- CIt releases energy because the mass of the helium nucleus is less than the total mass of the four protons.
- DIt requires temperatures of about 1000 K.
- EIt is the only fusion process that occurs in all stars.
37.A star has a peak emission wavelength of 290 nm. Using Wien's displacement law (λ_max T = 2.9 × 10^-3 m K), what is its surface temperature?
Medium
- A1000 K
- B10000 K
- C100000 K
- D100 K
38.An astronomer measures a star's parallax angle as 0.001 arcseconds, the smallest angle measurable with a particular telescope. What is the maximum distance this telescope can measure, in parsecs?
Medium
- A1000 pc
- B0.001 pc
- C1 pc
- D100 pc
39.Which of the following statements about the experimental determination of a star's radius are correct? (select all that apply)
Hard
- AWien's displacement law is used to find the star's surface temperature from its peak wavelength.
- BThe Stefan-Boltzmann law L = 4πr²σT⁴ is used to find the radius once luminosity and temperature are known.
- CThe inverse square law b = L/(4πd²) can be used to find luminosity if the apparent brightness and distance are known.
- DThe star's radius can be found directly from its parallax angle alone.
- EThe radius is calculated by dividing the luminosity by the surface temperature.
40.In an absorption spectrum, the dark lines appear at exactly the same wavelengths as the bright lines in the emission spectrum of the same element.
Medium
True or false?
41.A star has a parallax angle of 0.1 arcseconds. Calculate its distance in parsecs.
Medium
42.A star has a surface temperature of 5800 K. Using Wien's displacement law, calculate its peak wavelength in nm. (Wien's constant = 2.9 × 10⁻³ m K)
Medium
43.Complete the sentence about the proton-proton chain.
Easy
In the proton-proton chain, four ____ nuclei fuse to produce one ____ nucleus.
44.Complete the sentence about the conditions for fusion.
Medium
For nuclear fusion to occur, nuclei must overcome the ____ repulsion and get close enough for the ____ nuclear force to act.
45.Complete the sentence about stellar spectra.
Medium
Hot, dense sources produce a ____ spectrum, while hot, low-pressure gases produce an ____ spectrum.
46.What is the name of the process by which four hydrogen nuclei fuse to form a helium nucleus in stars?
Easy
47.What is the name of the unit of distance defined as the distance travelled by light in one year?
Easy
48.Calculate the radius of the star in metres.
Medium
49.A star has a surface temperature of 4700 K. Using Wien's displacement law, calculate its peak wavelength in nm. (Wien's constant = 2.9 × 10^-3 m K)
Medium
50.A star has a luminosity of 6.0 × 10²⁸ W and an apparent brightness of 3.0 × 10^-10 W m^-2. Calculate its distance from Earth in metres. Use the inverse square law: b = L / (4πd²). Give your answer to 2 significant figures.
Medium
51.Complete the sentence about the equilibrium of a main sequence star.
Medium
In a stable main sequence star, the outward ____ pressure balances the inward gravitational force.
52.A star has a parallax angle of 0.05 arcseconds. Calculate its distance from Earth in parsecs.
Medium
53.Explain how a main sequence star can remain in equilibrium.
Hard
54.Explain why very high temperatures and densities are needed to bring about nuclear fusion in stars.
Hard
55.Describe how the chemical composition of a star can be determined from its spectrum.
Hard
56.Explain how the chemical composition of a star can be determined from its spectrum.
Hard
57.Outline how the parallax angle of a nearby star is determined using stellar parallax observations.
Hard
58.A star has a parallax angle of 0.1 arcseconds. Calculate its distance in parsecs.
Medium
- A9 pc
- B10 pc
- C11 pc
- D12 pc
59.A star has a surface temperature of 5800 K. Using Wien's displacement law, calculate its peak wavelength in nm. (Wien's constant = 2.9 × 10⁻³ m K)
Medium
- A600 nm
- B450 nm
- C550 nm
- D500 nm
60.Calculate the radius of the star in metres.
Medium
- A630000000 m
- B700000000 m
- C770000000 m
- D840000000 m
61.A star has a surface temperature of 4700 K. Using Wien's displacement law, calculate its peak wavelength in nm. (Wien's constant = 2.9 × 10^-3 m K)
Medium
- A555 nm
- B617 nm
- C679 nm
- D741 nm
62.A star has a luminosity of 6.0 × 10²⁸ W and an apparent brightness of 3.0 × 10^-10 W m^-2. Calculate its distance from Earth in metres. Use the inverse square law: b = L / (4πd²). Give your answer to 2 significant figures.
Medium
- A3600000000000000000 m
- B4400000000000000000 m
- C4000000000000000000 m
- D4800000000000000000 m
63.A star has a parallax angle of 0.05 arcseconds. Calculate its distance from Earth in parsecs.
Medium
- A18 pc
- B24 pc
- C22 pc
- D20 pc

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