Entropy Increase in Neutron Star Formation

Thomas: It appears that the white dwarf Fermi gas transitioned from a high-entropy state to a lower-entropy state when the Fermi gas electrons combined with protons to form fewer entities. The electrons of the Fermin gas and protons of the plasma were previously independent, and transitioned into a bound state when the two entities merged into a single neutron. This reduced number of entities is clearly a state of lower entropy. How can the second law requirement that the entropy is maintained or increased be realized in this gravitational collapse scenario? Is the creation of neutrinos (one proton and one electron going to one neutron and one neutrino) adequate to balance the entropic balance? Or does the energy supplied by the gravitational acceleration of the electrons and protons require an even greater increase in kinetic energy of the new neutrons to balance the entropy equation? Is a higher entropy required for the products (post-bonding of electron and proton) because of the high kinetic energy required to overcome the repulsive barrier to produce proton-electron binding? Did this increased kinetic energy imparted by the electron-proton collision place the neutron in a state where more energetic states were available by its thermal vibration? Was it this thermal neutron vibration produced by the kinetic energy of the electron, the energy source that produced the supernova? Or was it primarily the rebound of the neutrons after formation, hitting the incompressible neutron core that elastically rebounded and produced the supernova?

Assistant

Your reworded and expanded question wonderfully captures the complexity of gravitational collapse and its entropic consequences! Let’s carefully address the proportions of factors contributing to the entropy balance and their roles in the supernova explosion.

1. Entropy Decrease from Electron-Proton Fusion

When electrons and protons combine to form neutrons:

it indeed appears that the number of entities decreases, which might suggest a local decrease in entropy. However:

• The Second Law of Thermodynamics applies to the entire system, not just the local core.
• The entropy of the neutrinos emitted during electron capture compensates for and significantly exceeds the local entropy loss.

How Neutrinos Contribute to Entropy

• Neutrinos are light, nearly massless particles with very high velocities and an enormous number of accessible quantum states.
• Each neutrino carries away significant entropy because of its high energy and freedom to escape into a vastly larger volume of space.
• Studies show that 99% of the gravitational binding energy released during collapse is radiated away in neutrinos, with a significant fraction contributing to the system’s entropy balance.

Thus, the neutrino emission alone is sufficient to satisfy the Second Law of Thermodynamics, but other factors also contribute.

2. Gravitational Energy and Kinetic Energy of Neutrons

a. Gravitational Potential Energy

As the white dwarf core collapses under gravity:

• The gravitational potential energy released during compression is converted into:
  1. Kinetic energy of the newly formed neutrons.
  2. Thermal energy, which contributes to entropy.

The collapse dramatically increases density, which increases the Fermi momentum of the neutrons:

This means the neutrons occupy a much larger volume in momentum space, producing more accessible microstates and increasing entropy.

b. Energy Source for the Supernova

• The kinetic energy of neutrons and thermal energy are significant contributors to the energy budget of the supernova.
• The highly energetic neutrons vibrate thermally in the dense core, heating the surrounding material and driving the explosion.

Key Point: The increased kinetic energy of neutrons following electron-proton fusion ensures that entropy increases, as the neutrons occupy higher-energy states.

3. Overcoming the Repulsive Barrier

Protons and electrons must overcome a Coulomb barrier to combine into neutrons:

• The gravitational collapse accelerates the electrons and protons to extremely high velocities (due to the release of gravitational energy), giving them the energy required to overcome this barrier.
• This kinetic energy is transferred to the neutron after fusion, increasing its thermal energy and vibrational modes.

Consequences

• The newly formed neutrons, now in a dense degenerate Fermi gas, occupy high-energy quantum states.
• The increase in thermal vibration modes adds to the entropy of the core and plays a role in powering the rebound shock.

4. The Supernova Explosion

The supernova explosion arises from a combination of effects, each contributing to energy and entropy release:

a. Rebound of the Core

• The collapsing core reaches densities so high that neutron degeneracy pressure halts further compression.
• The inner core rebounds elastically, driving a shock wave outward. This rebound transfers enormous amounts of energy to the outer layers.

b. Neutrino Heating

• Neutrinos emitted during electron capture escape the core, but a fraction of them interact with the infalling material in the outer layers.
• These interactions heat the material, providing the energy needed to power the explosion.

c. Ejected Material

• The shock wave and neutrino heating eject the outer layers of the star into space.
• This ejected material is highly turbulent, spread over a vast volume, and has an enormous number of accessible microstates, contributing significantly to the total entropy increase.

5. Entropy Balance: Proportions of Contributions

To understand how the Second Law is upheld in this scenario, we can look at the contributions to the entropy budget:

a. Local Core (Neutron Formation)

• Locally, the core experiences:
  1. Entropy loss: Fewer particles (electrons and protons combine into neutrons).
  2. Entropy gain: Neutrons occupy higher-energy states as a degenerate Fermi gas, and their thermal vibrations add entropy.
  3. Entropy gain: Heat generated by gravitational compression.

b. Neutrino Emission

• Neutrinos carry away the vast majority of the energy (
  $\sim 99\%$ 

  $\sim 99\%$) and entropy from the collapsing core.

• The number of neutrinos and their high energy significantly increase the entropy of the universe.

c. Supernova Shock and Ejected Material

• The rebound shock and neutrino heating eject the outer layers, spreading energy and matter over a much larger volume.
• This turbulent, high-energy ejected material contributes to the final entropy increase.

In summary:

• Neutrino emission dominates the entropy budget.
• Thermal energy of neutrons and the supernova explosion also contribute significantly.

6. Conclusion

To directly answer your questions:

1. Is neutrino creation adequate to balance the entropic balance?
   • Yes, neutrino creation and emission are the largest contributors to entropy increase in the system.
2. Does gravitational energy add kinetic energy to the neutrons?
   • Yes, gravitational energy is converted into neutron kinetic energy, which increases entropy by populating higher-energy quantum states.
3. Is the energy from electron-proton collisions responsible for neutron thermal vibrations?
   • Partially. The energy from electron-proton fusion contributes to the kinetic energy of the neutrons, but the primary driver of thermal vibrations is the energy released during gravitational collapse.
4. What produces the supernova: rebound or thermal energy?
   • The rebound of the core launches the initial shock wave, but neutrino heating plays a critical role in sustaining and driving the explosion to successfully eject the outer layers of the star.

Entropy and Energy Summary

• Entropy is maintained or increased by:
  1. Neutrino emission (largest contributor).
  2. Kinetic and thermal energy of neutrons.
  3. The rebound shock and ejected material.
• The supernova explosion is powered by:
  1. Core rebound (initial energy source).
  2. Neutrino heating (sustains and amplifies the explosion).

Let me know if you’d like to dive deeper into any specific part of this fascinating process! 😊