Chapter 25, Problem 21Q

(a)

To determine

The age of the universe for a Hubble constant of $50\text{km/s/Mpc}$.

Answer to Problem 21Q

Solution:

20 billion years.

Explanation of Solution

Given data:

The Hubble constant is $50\text{km/s/Mpc}$.

Formula used:

The relation between the age of the universe and the Hubble constant is given by the following expression,

$T_0=\frac{1}{H_0}$

Here, $T_0$ is the age of the universe and $H_0$ is the Hubble constant.

Explanation:

Recall the relation between the age of the universe and the Hubble constant.

$T_0=\frac{1}{H_0}$

Substitute $50\text{km/s/Mpc}$ for $H_0$,

$\begin{array}{c}{T}_0=\frac{1}{50\text{ km/s/Mpc}}\\ =\frac{1\text{ Mpc}}{50\text{ km/s}}\left(\frac{3.09\times {10}^{19}\text{ km}}{1\text{ Mpc}}\right)\\ =6.18\times {10}^{17}\text{ s}\left(\frac{1\text{ ly}}{3.156\times {10}^7\text{ s}}\right)\\ =1.96\times {10}^{10}\text{ yr}\end{array}$

Upon further solving,

$T_0\simeq 20\text{ billion years}$.

Conclusion:

Therefore, the age of the universe is 20 billion years.

(b)

To determine

The age of the universe for a Hubble constant of $75\text{km/s/Mpc}$.

Answer to Problem 21Q

Solution:

13 billion years.

Explanation of Solution

Given data:

The Hubble constant is $75\text{km/s/Mpc}$.

Formula used:

The relation between the age of the universe and the Hubble constant is given by the following expression,

$T_0=\frac{1}{H_0}$

Here, $T_0$ is the age of the universe and $H_0$ is the Hubble constant.

Explanation:

Recall the relation between the age of the universe and the Hubble constant.

$T_0=\frac{1}{H_0}$

Substitute $75\text{km/s/Mpc}$ for $H_0$,

$\begin{array}{c}{T}_0=\frac{1}{\text{75 km/s/Mpc}}\\ =\frac{1\text{ Mpc}}{\text{75 km/s}}\left(\frac{3.09\times {10}^{19}\text{ km}}{1\text{ Mpc}}\right)\\ =6.18\times {10}^{17}\text{ s}\left(\frac{1\text{ ly}}{3.156\times {10}^7\text{ s}}\right)\\ =13\times {10}^9\text{ year}\end{array}$

Upon further solving,

$T_0=13\text{ billion year}$.

Conclusion:

Therefore, the age of the universe is 13 billion years.

(c)

To determine

The age of the universe for a Hubble constant of $100\text{km/s/Mpc}$. The explanation for the use of the ages of the globular clusters for the maximum value of the Hubble constant.

Answer to Problem 21Q

Solution:

10 billion years.

Explanation of Solution

Given data:

The Hubble constant is $100\text{km/s/Mpc}$.

Formula used:

The relation between the age of the universe and the Hubble constant is given by the following expression,

$T_0=\frac{1}{H_0}$

Here, $T_0$ is the age of the universe and $H_0$ is the Hubble constant.

Explanation:

Recall the relation between the age of the universe and the Hubble constant.

$T_0=\frac{1}{H_0}$

Substitute $100\text{km/s/Mpc}$ for $H_0$,

$\begin{array}{c}{T}_0=\frac{1}{100\text{ km/s/Mpc}}\\ =\frac{1\text{ Mpc}}{100\text{ km/s}}\left(\frac{3.09\times {10}^{19}\text{ km}}{1\text{ Mpc}}\right)\\ =6.18\times {10}^{17}\text{ s}\left(\frac{1\text{ ly}}{3.156\times {10}^7\text{ s}}\right)\\ =9.79\times {10}^9\text{ yr}\end{array}$

Upon further solving,

$T_0=9.79\text{ billion years}$.

From the calculation in all the three parts, the value of $H_0$ gives a corresponding value of $T_0$, which is less than the ages of oldest stars present in a globular cluster.

Conclusion:

Therefore, the universe must be older than the oldest stars of the globular cluster. So, the Hubble constant should be such that the age of the universe is older than that of the oldest star in the globular cluster.