How does the mean lifetime of a deuteron (d) in the core of the Sun compare with the mean lifetime of a proton (p) in the core of the Sun?
A) Considering the cross-sections for the two reaction steps, why does one of these two steps determines the overall rate of reaction.
B) Equating the rates of the two reaction steps, and expressing them in terms of the particle number densities and reaction cross-sections, what determines the ratio of the density of deuterons to the density of protons in the Sun's core?
1) the mean life time for d is around 1 second and for p its billions of years so thats a pretty simple answer.
a) You can consider the S-factor in both reactions and compare these to find that the probability of reaction is lower for one or the other. Alternatively you can consider how slow the p-p reaction is and consider this a limiting factor.
Step 1: Comparing the mean lifetimes
The mean lifetime of a deuteron (d) in the core of the Sun is significantly longer than the mean lifetime of a proton (p). This is because the deuteron is a more stable particle compared to the proton.
Step 2: Determining the overall rate of reaction
A) One of the reaction steps determines the overall rate of reaction due to the cross-sections of the two steps. The cross-section is a measure of the probability of a reaction occurring when particles collide.
In the core of the Sun, the first reaction step involves the fusion of two protons (p + p), which results in the formation of a deuteron (d) and a positron (e+) along with the release of a neutrino (v). The cross-section for this step is relatively small.
The second reaction step involves the fusion of a deuteron (d) with another proton (p), resulting in the formation of a helium-3 nucleus (He-3) and releasing a gamma-ray photon (γ). The cross-section for this step is much higher compared to the first step.
Therefore, even though the cross-section for the first reaction step is small, it is the determining factor for the overall rate of reaction because it takes place at a much slower rate than the second step.
Step 3: Determining the ratio of the density of deuterons to the density of protons
B) By equating the rates of the two reaction steps and expressing them in terms of particle number densities and reaction cross-sections, we can determine the ratio of the density of deuterons to the density of protons in the Sun's core.
Let n_p and n_d be the number densities of protons and deuterons, respectively. The reaction rates can be expressed as:
Rate of 1st step: R1 = n_p^2 * σ1
Rate of 2nd step: R2 = n_d * n_p * σ2
Since the rates of the two steps are equal (R1 = R2), we can equate them:
n_p^2 * σ1 = n_d * n_p * σ2
Simplifying the equation, we get:
n_d / n_p = σ1 / σ2
So, the ratio of the density of deuterons to the density of protons in the Sun's core is determined by the ratio of the cross-sections for the two reaction steps (σ1 / σ2).
To compare the mean lifetimes of a deuteron (d) and a proton (p) in the core of the Sun, we need to understand the reactions involving these particles.
A) The mean lifetime of a deuteron (d) in the core of the Sun is determined by the rate of the reaction where a deuteron is formed:
p + p → d + e+ + νe
This reaction occurs through the weak force, which has a relatively small cross-section. On the other hand, the mean lifetime of a proton (p) is determined by the rate of the reaction where a proton is converted into a neutron:
p → n + e+ + νe
This reaction occurs through the weak force as well, but it has a larger cross-section compared to the deuteron formation reaction.
Since the deuteron formation reaction has a smaller cross-section, it determines the overall rate of the reaction. This means that the deuteron formation reaction occurs less frequently than the proton-to-neutron conversion reaction, leading to a longer mean lifetime for the deuteron compared to the proton.
B) In order to determine the ratio of the density of deuterons to the density of protons in the Sun's core, we need to equate the rates of the two reaction steps and express them in terms of particle number densities and reaction cross-sections.
Let's define the following variables:
- n_d: Density of deuterons
- n_p: Density of protons
- σ_d: Cross-section for deuteron formation reaction
- σ_p: Cross-section for proton-to-neutron conversion reaction
- R_d: Rate of the deuteron formation reaction
- R_p: Rate of the proton-to-neutron conversion reaction
The rate of the deuteron formation reaction (R_d) is given by:
R_d = n_p * n_p * σ_d
The rate of the proton-to-neutron conversion reaction (R_p) is given by:
R_p = n_p * σ_p
By equating the rates of the two reaction steps, we can solve for the ratio of the density of deuterons to the density of protons:
R_d = R_p
n_p * n_p * σ_d = n_p * σ_p
Simplifying the equation:
n_d / n_p = σ_p / σ_d
Therefore, the ratio of the density of deuterons (n_d) to the density of protons (n_p) in the Sun's core is determined by the ratio of the cross-section for the proton-to-neutron conversion reaction (σ_p) to the cross-section for the deuteron formation reaction (σ_d).