Chapter 33, Problem 33GP

(a)

To determine

Time at which the universe cooled below the threshold temperature for producing kaons.

Answer to Problem 33GP

Time at which the universe cooled below the threshold temperature for producing kaons is ${10}^{-5}\text{ s}$ .

Explanation of Solution

Given: Mass of kaons is $M\approx \text{500 MeV/}{c}^2$ .

Formula used:

Kinetic energy is calculated as

  $KE=\frac{3}{2}kT$

Calculation:

The temperature that corresponds to kinetic energy is

  $\begin{array}{c}KE=\frac{3}{2}kT\\ T=\frac{2}{3}\frac{KE}{k}\\ T\text{ =}\frac{2}{3}\frac{(500\times {10}^6\text{ eV/}{c}^2)c^2(1.60\times {10}^{-19}\text{ J/eV})}{(1.38\times {10}^{-23}\text{J/K})}\\ T\text{ = 3}\text{.86}\times {10}^{12}\text{K}\end{array}$

As shown in the figure below, time that corresponds to the temperature on the cosmological calendar

Figure 1

Time which corresponds to temperature $\text{T = 3}\text{.86}\times {10}^{12}\text{K}$ on the cosmological calendar is $\approx {10}^{-5}\text{ s}$ .

Conclusion:

Time at which the Universe cooled below the threshold temperature for producing kaons is ${10}^{-5}\text{ s}$ .

(b)

To determine

Time at which the Universe cooled below the threshold temperature for producing Y particle.

Answer to Problem 33GP

Time at which the Universe cooled below the threshold temperature for producing Y particle is ${10}^{-7}\text{ s}$ .

Explanation of Solution

Given:

Mass of Y is $\text{M}\approx 9{\text{500 MeV/c}}^2$ .

Formula used:

Kinetic energy is calculated as

  $KE=\frac{3}{2}kT$

Calculation:

The temperature that corresponds to kinetic energy is

  $\begin{array}{c}KE=\frac{3}{2}kT\\ T=\frac{2}{3}\frac{KE}{k}\\ T\text{ =}\frac{2}{3}\frac{(9500\times {10}^6{\text{ eV/c}}^2){\text{c}}^2(1.60\times {10}^{-19}\text{ J/eV})}{(1.38\times {10}^{-23}\text{J/K})}\\ \text{= 7}\text{.34}\times {10}^{13}\text{K}\end{array}$

From figure 1, time which corresponds to this temperature on the cosmological calendar is $\approx {10}^{-7}\text{ s}$ .

Conclusion:

Time at which the Universe cooled below the threshold temperature for producing Y particle is ${10}^{-7}\text{ s}$ .

(c)

To determine

Time at which the Universe cooled below the threshold temperature for producing muons.

Answer to Problem 33GP

Time at which the Universe cooled below the threshold temperature for producing muons is ${10}^{-4}\text{ s}$ .

Explanation of Solution

Given:

Mass of muons is $M\approx 1{\text{00 MeV/c}}^2$ .

Formula used:

Kinetic energy is calculated as

  $KE=\frac{3}{2}kT$

Calculation:

The temperature that corresponds to kinetic energy is

  $\begin{array}{c}KE=\frac{3}{2}kT\\ T=\frac{2}{3}\frac{KE}{k}\\ T\text{ =}\frac{2}{3}\frac{(100\times {10}^6{\text{ eV/c}}^2){\text{c}}^2(1.60\times {10}^{-19}\text{ J/eV})}{(1.38\times {10}^{-23}\text{J/K})}\\ \text{= 7}\text{.73}\times {10}^{11}\text{K}\end{array}$

From figure 1, time which corresponds to this temperature on the cosmological calendar is $\approx {10}^{-4}\text{ s}$ .

Conclusion:

Time at which the Universe cooled below the threshold temperature for producing muons is ${10}^{-4}\text{ s}$ .