Lecture 13: Energy Generation in Stars

Key Ideas:

• Energy generation in stars:
  • Nuclear Fusion in the core.
  • Controlled by a Hydrostatic "thermostat".

Putting Stars Together

• Law of Gravity
• Equation of State ("gas law")
• Principle of Hydrostatic Equilibrium
• Source of Energy (e.g., Nuclear Fusion)
• Way to transport that energy to the surface.

Hydrostatic Equilibrium

• If Pressure dominates, the star expands.
• If Gravity dominates, the star contracts.

Sets up a Core-Envelope Structure:

• Hot, dense, compact core.
• Cooler, low-density, extended envelope.

Energy Generation

To stay hot stars must make up for the energy lost by shining.

Energy sources available:

• Gravitational Contraction (Kelvin-Helmholtz)
• Nuclear Fusion in the hot core.

Main-Sequence Stars

Generate energy by fusion of 4 ¹H into 1 ⁴He.

There are two nuclear reaction paths by which a star might accomplish this fusion:

Proton-Proton Chain:

• Relies on proton-proton reactions
• Efficient at low core Temperature (T_C<18M K)

CNO Cycle:

• Carbon acts as a catalyst
• Efficient at high core Temperature (T_C>18M K)

Proton-Proton Chain:

Note that you use 6 protons in all, and end up with 1 ⁴He nucleus and 2 protons at the end, for a net conversion of 4 protons into 1 Helium, with the release of energy as gamma-ray photons, neutrinos, and postirons.

CNO Cycle:

Note that you start with one ¹²C nucleus in step one, and add 4 protons during steps 1, 3, 4, and 6, ending up with the ¹²C nucleus back at the end with the ⁴He nucleus.

The result is a net conversion of 4 protons into 1 Helium nucleus, with a release of energy in the form of gamma-ray photons, neutrinos, and positrons.

Because ¹²C is not consumed by this process (it goes in & comes out at the end), we say that it acts as a catalyst for the nuclear reaction.

Because Carbon and Nitrogen have 6 and 7 protons, respectively, in order to overcome the repulsion of all these positive charges the protons must be moving fairly fast. This is why the CNO cycle occurs at higher temperatures than the P-P chain.

Controlled Nuclear Fusion

• Higher Core Temperature = More Fusion
• Energy Generation Rate of PP Chain ~ T⁴
• Energy Generation Rate of CNO Cycle ~ T¹⁸

BUT,

• More fusion makes the core hotter,
• Hotter core leads to even more fusion, …

So why doesn't it runaway and blow up like a Hydrogen Bomb?

Hydrostatic Thermostat

• core heats up.
• pressure increases & core expands.
• expansion cools core, slowing fusion reactions.

If fusion reactions run too slow:

• core cools down.
• pressure drops & gravity contracts the core.
• contraction heats core, increasing fusion.

Summary:

• Nuclear Fusion in the core.
• Controlled by a Hydrostatic "thermostat".