Problem 4 - An Introduction to Modern Astrophysics

Question

(a) Taking into consideration the Maxwell-Boltzmann velocity distribution, what temperature would be required for two protons to collide if quantum mechanical tunneling is neglected? Assume that nuclei having velocities ten times the root-mean-square (rms) value for the Maxwell-Boltzmann distribution can overcome the Coulomb barrier. Compare your answer with the estimated central temperature of the Sun. (b) Using the below equation, calculate the ratio of the number of protons having velocities ten times the rms value to those moving at the rms velocity. $$n_{v} d v=n\left(\frac{m}{2 \pi k T}\right)^{3 / 2} e^{-m v^{2} / 2 k T} 4 \pi v^{2} d v$$ (c) Assuming (incorrectly) that the Sun is pure hydrogen, estimate the number of hydrogen nuclei in the Sun. Could there be enough protons moving with a speed ten times the rms value to account for the Sun's luminosity?

Step-by-Step Solution

Answer

The temperature is too high for thermal collisions without tunneling, and quantum tunneling is essential in stars. The fraction of fast protons is extremely low and insufficient to account for the Sun's luminosity without quantum effects.

Step 1: Understanding the Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds of particles in a gas. The root-mean-square speed \(v_{rms}\) is given by \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of a particle.

Step 2: Calculate Temperature for Proton Collision

We need protons to have velocities ten times \(v_{rms}\) to overcome the Coulomb barrier. Setting \(v = 10v_{rms} = 10\sqrt{\frac{3kT}{m}}\), we equate the kinetic energy at this velocity to the barrier energy (\(E_{barrier}\)). Solving \(\frac{1}{2}m(10\sqrt{\frac{3kT}{m}})^2 = E_{barrier}\) leads to \(T = \frac{mE_{barrier}}{150k}\).

Step 3: Compare with Solar Central Temperature

The estimated central temperature of the Sun is about 15 million K. Using the calculated \(T\), we compare if it is feasible for enough protons to have the necessary velocity at the Sun's core temperature.

Step 4: Calculating the Velocity Ratio

Using the given distribution function, find \(n_v/n\) for \(v = 10v_{rms}\). Substituting \(v_{rms}\) and the expressions, the ratio is \(\frac{n_{10v_{rms}}}{n_{v_{rms}}} = e^{-495}\), since each exponent is independent of the actual speed value.

Step 5: Estimate Number of Hydrogen Nuclei

If the Sun is pure hydrogen, its mass \(M_{\odot} = 2 \times 10^{30} kg\). Number of nuclei \(N = \frac{M_{\odot}}{m_p}\), where \(m_p\) is the proton mass \(1.67 \times 10^{-27} kg\). Calculate this value.

Step 6: Calculate Feasibility based on Luminosity

Sun's luminosity \(L_{\odot} = 3.8 \times 10^{26} W\). Compare the total kinetic energy available from fast protons having speed \(10v_{rms}\) to the Sun's energy production. Use the previously estimated number of such protons to determine if they can account for \(L_{\odot}\).

Key Concepts

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution is a statistical law that describes how the speeds of particles in a gas are distributed when the particles are in thermal equilibrium. Simply put, it tells us that in a given environment, most particles will move at an average speed, while some will move much faster or slower.

Root-mean-square speed, often represented as \( v_{rms} \), is a key component in this distribution and is defined as \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of the particle. It's essentially a measure of the average speed of particles and very useful in calculating kinetic energy. Understanding this distribution helps in scenarios like estimating the temperature needed for protons to collide in nuclear fusion.

Coulomb Barrier

The Coulomb barrier is a term used in nuclear physics to describe the natural repulsion between two positively charged nuclei. Imagine trying to push two strong magnets together at their like poles—they naturally repel each other. This repulsion must be overcome for nuclear fusion to occur.

In stars like our Sun, immense pressure and temperature allow some protons to gain enough speed to overcome this barrier. This process is crucial, as it allows the fusion of hydrogen into helium, releasing a tremendous amount of energy.

In the given exercise, we consider the velocities that protons must have to breach this barrier, and this helps us appreciate the conditions necessary for fusion in stellar cores.

Root-Mean-Square Speed

Root-mean-square speed (\( v_{rms} \)) is a measure of the speed of particles in a gas that is particularly useful in understanding kinetic energy. It provides a type of "average" speed of particles when they are in thermal equilibrium. In essence, it's not just the average of how fast particles move, but a value that also accounts for variations in their individual speeds.

In the context of nuclear fusion, knowing the \( v_{rms} \) lets us predict necessary conditions like temperature for particles to acquire enough energy. In the exercise, protons need to surpass a velocity much higher than the \( v_{rms} \) to overcome the Coulomb barrier, which is crucial to initiate fusion reactions.

Solar Luminosity

Solar luminosity is a term that represents the total amount of energy emitted by the Sun every second. It is a fundamental measure for understanding a star's energy output and is denoted by \( L_{\odot} \). For the Sun, this value is approximately \( 3.8 \times 10^{26} \) watts.

In the context of nuclear fusion, solar luminosity provides insight into the efficacy of the fusion process occurring in the Sun's core. By analyzing how energy produced by fast-moving protons compares to the Sun's total energy output, we can determine if the number of protons reaching the necessary speeds is sufficient to account for the luminosity. This comparison is fundamental for validating our understanding of the conditions and reactions powering the Sun.