Fusion and stars

• a) Stellar Stability: Radiation Pressure vs. Gravitational Collapse

• ⇒ A star stable:
• A star maintains structural stability due to a delicate balance between two opposing forces:

• This equilibrium is called hydrostatic equilibrium.
• 
• Figure 1 A stable star
• ⇒ Inward Gravitational Pressure
• – Gravity tries to compress the star’s matter toward its core.
• – It increases temperature and pressure in the core as material is pulled in.
• ⇒ Outward Radiation Pressure
• – Inside the core, nuclear fusion
• – The energy released in the form of photons creates radiation pressure, pushing outward against gravity.
• If gravity becomes stronger than radiation pressure, the star collapses. If radiation pressure becomes stronger, the star expands.
• When both are equal, the star is stable — like our Sun.

• b) Nuclear Fusion: The Star’s Energy Source

• ⇒ Fusion:
• Fusion is the process where lighter nuclei combine to form a heavier nucleus, releasing a tremendous amount of energy in the process.
• [math]H^1 + H^1 \rightarrow H^2 + e^+ + \nu_e[/math]
• Eventually:
• [math]4H^1 \rightarrow \text{He}^4 + 2e^+ + 2\nu_e + \text{energy}[/math]
• This process is known as the proton-proton chain, dominant in stars like the Sun.
• 
• Figure 2 Nuclear fusion
• ⇒ Energy from Fusion:
• – The mass of the resulting nucleus is less than the total mass of the fusing nuclei.
• – This missing mass is converted into energy, according to Einstein’s equation:
• [math]E = \Delta mc^2[/math]
• Where:
• – Δm is the mass lost,
• – c is the speed of light.
• This energy:
• – Increases the core temperature,
• – Creates radiation pressure,

• c) Conditions for Fusion in Stars (Density & Temperature)

• For nuclear fusion to occur in a star’s core, extreme conditions are required:
• ⇒ High Temperature
• At temperatures of ~10 million K or more, atomic nuclei gain enough kinetic energy to overcome electrostatic repulsion (Coulomb barrier) between positively charged protons.
• For hydrogen fusion:
• [math]\text{Minimum temperature} \approx 10^7\ \text{K}[/math]
• The higher the temperature, the faster the particles move, and the greater the likelihood they’ll collide and fuse.
• 
• Figure 3 Nuclear fusion of hydrogen into helium power star
• ⇒ High Density
• – Fusion requires frequent collisions between particles.
• – High density ensures there are many particles per unit volume, increasing the probability of collisions.
• – In the Sun’s core, density is about 150 g/cm³.
• Fusion starts when:
• – The core temperature is high enough (millions of Kelvin),
• – The core density allows for frequent enough collisions.
• These conditions are achieved through gravitational contraction early in a star’s formation.

• d) Effect of Stellar Mass on Star Evolution

• The mass of a star is the primary factor determining how it evolves, lives, and dies.
• Low-Mass Stars (e.g., the Sun)
• Core temperature is relatively lower.
• Fusion proceeds slowly → long lifespan (up to tens of billions of years).
• Evolves into:
• – Red giant (outer layers expand),
• – Planetary nebula (outer layers shed),
• – White dwarf (dense, hot remnant),
• – Eventually, cools into a black dwarf (theoretical—universe isn’t old enough for these yet).
• 
• Figure 4 Stellar Evolution
• ⇒ High-Mass Stars (> 8 solar masses)
• Core temperature and pressure are much higher.
• Fusion is much faster → shorter lifespan (millions of years).
• Can fuse elements up to iron (Fe) in the core.
• After iron forms (which doesn’t release energy through fusion), the core collapses:
• – Leads to a supernova explosion,
• – Leaves behind a neutron star or black hole depending on the remaining mass.
• Life Span vs Mass:
• Higher mass → shorter life (due to faster fusion),
• Lower mass → longer life (slower fuel consumption).

• e) The Hertzsprung–Russell (HR) Diagram

• The HR diagram is a graphical tool used by astronomers to classify stars based on their luminosity, temperature, color, and evolutionary stage.
• Axes:
• X-axis (horizontal): Surface temperature (in Kelvin), decreasing to the right.
• – Sometimes labeled with spectral classes (O, B, A, F, G, K, M).
• Y-axis (vertical): Luminosity (brightness), often in terms of Solar luminosity (L☉).
• ⇒ Main Regions of the HR Diagram:
• 1. Main Sequence:
• – Diagonal band from top left (hot, luminous) to bottom right (cool, dim).
• – Stars here are fusing hydrogen to helium in their cores.
• – Our Sun is a main sequence star (G-type).
• – Higher temperature → higher luminosity → higher mass.
• 2. Giants and Supergiants:
• – Above the main sequence.
• – Stars have large radii but cooler temperatures → high luminosity.
• – These stars are in later stages of their lives, fusing heavier elements.
• 3. White Dwarfs:
• – Below the main sequence, small and dim but hot.
• – Stellar remnants of low-mass stars like the Sun.

• 

• Figure 5 Hertzsprung-Russell Diagram

• f) Stellar Parallax: Determining Distance

• Stellar parallax is the apparent shift in position of a nearby star when observed from Earth at opposite points in its orbit (6 months apart).
• ⇒ Parallax Angle (ρ):
• – Measured in arc-seconds.
• – Small angle formed between Earth at two positions and the star.
• ⇒ Distance Formula:
• [math]d = \frac{1}{\rho}[/math]
• Where:
• – d is the distance to the star in parsecs (pc),
• – ρ is the parallax angle in arc-seconds.
• [math]1\ \text{parsec} \approx 3.26\ \text{light-years}[/math]
• 
• Figure 6 The stellar parallax – determine the distance
• ⇒ Example:
• If a star has a parallax of 0.2 arc-seconds:
• [math]d = \frac{1}{\rho} \\
  d = \frac{1}{0.2} \\
  d = 5\ \text{pc}[/math]
• This method is effective for stars up to a few hundred parsecs away.
• g) Determining Stellar Radii
• The radius of a star can be determined using the Stefan–Boltzmann Law:
• [math]L = 4 \pi R^2 \sigma T^4[/math]
• Where:
• – L is luminosity,
• – R is radius,
• – T is surface temperature in Kelvin,
• – σ is the Stefan–Boltzmann constant.
• ⇒ Solving for Radius:
• [math]R = \sqrt{\frac{L}{4 \pi \sigma T^4}}[/math]
• A star’s luminosity and surface temperature, both of which can be derived from observational data (e.g. brightness and color/spectral class).