Problem 30 - An Introduction to Modern Astrophysics

Question

One factor that contributed to the cessation of the reactions that formed neutrons at roughly \(10^{10} \mathrm{K}[\mathrm{Eqs} .(\quad 93-\quad 95)]\) was the annihilation of electron-positron pairs that occurred at that time. When the temperature became too low, the electron-positron pairs could not be replaced by pair production. (This removed the supply of electrons that could combine with protons to form neutrons.) By setting the characteristic thermal energy of a photon, \(k T,\) equal to the rest energy of an electron-positron pair, estimate the temperature below which an annihilated pair will not readily be replaced. $$\begin{aligned} n & \rightleftharpoons p^{+}+e^{-}+\bar{v}_{e} \\ n+e^{+} & \rightleftharpoons p^{+}+\bar{v}_{e} \\ n+v_{e} & \rightleftharpoons p^{+}+e^{-}. \end{aligned}$$

Step-by-Step Solution

Answer

The temperature below which electron-positron pairs won't be readily replaced is approximately \(1.186 \times 10^{10} \text{ K}\).

Step 1: Understand the Problem

We need to estimate the temperature below which electron-positron pairs cannot be replaced by pair production. We're given that the characteristic thermal energy of a photon, \( kT \), should equal the rest energy of an electron-positron pair for replacement to occur.

Step 2: Identify the Key Formula

The key relationship here is equating thermal energy to rest energy, i.e., \( kT = 2m_ec^2 \), where \( m_e \) is the rest mass of an electron and \( c \) is the speed of light.

Step 3: Plug in Known Values

We know the rest mass energy of an electron is approximately \( m_ec^2 = 0.511 \text{ MeV} \). Therefore, the rest energy of an electron-positron pair is \( 2m_ec^2 = 1.022 \text{ MeV} \).

Step 4: Solve for Temperature \( T \)

Using the formula \( kT = 2m_ec^2 \), with \( k \) as the Boltzmann's constant (\( 8.617 imes 10^{-11} \text{ MeV/K} \)), solve for \( T \). It becomes \( T = \frac{1.022 \text{ MeV}}{8.617 imes 10^{-11} \text{ MeV/K}} \).

Step 5: Calculate the Temperature

Performing the calculation, \( T \approx 1.186 \times 10^{10} \text{ K} \). This temperature is the threshold below which electron-positron pairs cannot readily be replaced by pair production.

Key Concepts

Electron-Positron Annihilation

When particles like electrons and their antimatter counterparts, positrons, come into contact, an interesting phenomenon called **annihilation** occurs. In this process, the electron and positron destroy each other, converting their mass into energy in the form of photons. This is in accordance with Einstein’s famous equation, \( E=mc^2 \), which tells us that mass can be converted into energy.
The significance of electron-positron annihilation in the early universe was profound. It played a critical role in shaping the balance between matter and energy. During the early stages of the universe, temperatures were incredibly high. Thus, electrons and positrons were plentiful and frequently annihilated each other. However, as the universe expanded and cooled, this process couldn't keep up because the thermal energy available became insufficient to create new electron-positron pairs from photons.

• This makes annihilation a self-limiting process.
• It's critical in determining when specific particles may cease to exist.
• Helps define the conditions necessary for particle formation.

Pair Production

Pair production is a fascinating process where energy is converted into mass, creating a particle and its antiparticle pair, such as an electron and a positron. This phenomenon is the reverse of annihilation and is highly dependent on the energy of the photons involved.
For pair production to occur, the available photon energy must be at least equal to the total rest mass energy of the two particles being created. Mathematically, this is represented as \( kT = 2m_e c^2 \), aligning with the energy conditions mentioned in the exercise. Here, \( m_e \) is the rest mass of an electron, and \( c \) represents the speed of light.

• When temperatures are high, pair production occurs frequently.
• As the universe cools, photons have less energy, reducing pair production rates.
• This is why at low temperatures, production of particle pairs from energy becomes rare.

The balancing act between pair production and annihilation helps define the eras in the evolution of the universe.

Thermal Energy

In the context of the universe's evolution, thermal energy plays a pivotal role. **Thermal energy** is essentially the energy possessed by an object due to the motion of its particles, and it is directly related to the temperature of the system.
For the early universe, thermal energy was immensely high, a critical condition for various nuclear and atomic interactions to occur. One way to quantify thermal energy is through the expression \( kT \), where \( k \) is Boltzmann's constant, and \( T \) is the temperature. This expression provides a measure of the average kinetic energy per particle.

• High thermal energy contributed to early universe reactions.
• It allowed processes like pair production to occur.
• As the universe cools, the thermal energy diminishes, affecting these interactions.

Understanding thermal energy helps in grasping why certain reactions ceased as the universe expanded and cooled, marking transition points like Big Bang nucleosynthesis.