Chapter 27, Problem 20Q

(a)

To determine

The average lifetime of a technologically advanced civilization by using the Drake equation, if there are 1000 civilizations present in the galaxy, using the standard values given in the text for the parameters in the equation.

Answer to Problem 20Q

Solution:

$10,000\text{ years}$

Explanation of Solution

Given data:

There are 1000 civilizations in the galaxy.

Formula used:

Write the Drake equation.

$N=R_{*}{f}_p{n}_e{f}_l{f}_i{f}_{c}L$

Here, $N$ is the number of technologically advanced civilizations in the galaxy, $R_{*}$ is the rate at which solar-type stars form in the galaxy, $f_p$ is the fraction of stars that have planets, $n_e$ is the number of planets per solar system that are similar to Earth, $f_l$ is the fraction of those Earth-like planets on which life may actually arise, $f_i$ is the fraction of those life forms that evolve into intelligent species, $f_c$ is the fraction of the species that develop enough technology to be able to send space messages, and $L$ is the longevity of a technologically advanced civilization.

Explanation:

The standard values considered by the astronomers are given in the book. The value of $R_{*}$ is chosen as 1, $f_p$ as 1, $n_e$ as 0.1, $f_l$ as 1, $f_i$ as 1, and $f_c$ as 1 as well.

Recall the Drake equation.

$N=R_{*}{f}_p{n}_e{f}_l{f}_i{f}_{c}L$

Rearrange the expression in terms of $L$.

$L=\frac{N}{R_{*}{f}_p{n}_e{f}_l{f}_i{f}_c}$

Substitute, 1 for $R_{*}$, 1 for $f_p$, 1 for $n_e$, 1 for $f_l$, 1 for $f_i$, 0.1 for $f_c$, and 1000 for $N$ in the above expression,

$\begin{array}{c}L=\frac{1000}{\left(1\right)\left(1\right)\left(1\right)\left(1\right)\left(1\right)\left(0.1\right)}\\ =10,000\end{array}$

Conclusion:

The average lifetime of a technologically advanced civilization, calculated using the drake equation, if there are 1000 civilizations present in the galaxy, is $10,000$ years.

(b)

To determine

The average lifetime of a technologically advanced civilization by using the Drake equation, if there are $1\text{ million}$ civilizations present in the galaxy, using the standard values given in the text for the parameters in the equation.

Answer to Problem 20Q

Solution:

$10,000,000\text{ years}$

Explanation of Solution

Given data:

There are $1\text{ million}$ civilizations in the galaxy.

Formula used:

Write the Drake equation.

$N=R_{*}{f}_p{n}_e{f}_l{f}_i{f}_{c}L$

Here, $N$ is the number of technologically advanced civilizations in the galaxy, $R_{*}$ is the rate at which solar-type stars form in the galaxy, $f_p$ is the fraction of stars that have planets, $n_e$ is the number of planets per solar system that are similar to Earth, $f_l$ is the fraction of those Earth-like planets on which life may actually arise, $f_i$ is the fraction of those life-forms that evolve into intelligent species, $f_c$ is the fraction of the species that develop enough technology to be able to send space messages, and $L$ is the longevity of a technologically advanced civilization.

Explanation:

The standard values considered by the astronomers are given in the book. The value of $R_{*}$ is chosen as 1, $f_p$ as 1, $n_e$ as 0.1, $f_l$ as 1, $f_i$ as 1, and $f_c$ as 1 as well.

Recall the Drake equation.

$N=R_{*}{f}_p{n}_e{f}_l{f}_i{f}_{c}L$

Rearrange the expression in terms of $L$.

$L=\frac{N}{R_{*}{f}_p{n}_e{f}_l{f}_i{f}_c}$

Substitute1 for $R_{*}$, 1 for $f_p$, 1 for $n_e$, 1 for $f_l$, 1 for $f_i$, 0.1 for $f_c$, and $1\text{ million}$ for $N$ in the above expression,

$\begin{array}{c}L=\frac{1\text{ million}\left(\frac{1,000,000}{1\text{ million}}\right)}{\left(1\right)\left(1\right)\left(1\right)\left(1\right)\left(1\right)\left(0.1\right)}\\ =10,000,000\end{array}$

Conclusion:

The average lifetime of a technologically advanced civilization, calculated using the Drake equation, if there are $1\text{ million}$ civilizations present in the galaxy, is $10,000,000\text{ years}$.