        SOLUTIONS TO EXERCISE 17

 back to exercise frontiers: TOC for Knowledge Concepts, Exercises, and Solutions

```
(a) If dark matter is, as current theories suggest, not made of protons and
neutrons, then we can approximate the total mass of protons and neutrons in
the observable universe by the total mass of stars.  There are about 100
billion galaxies in the observable universe, and about 100 billion stars
per galaxy.  A typical star (such as our Sun) has a mass of 2 * 10^30 kg.
Thus, the mass of all the stars in the observable universe is

M = 10^11 * 10^11 * 2 * 10^30 = 2 * 10^52 kg.

Protons and neutrons have nearly identical mass, 1.67 * 10^(-27) kg.  Electrons
have a much smaller mass (1836 times smaller than a proton, in fact), so we can
leave them out of the calculation without adding much error.  Thus, after all
neutrons decay into protons, there will be about

(2 * 10^52 kg)
---------------------------  =  1.2 * 10^79 protons.
(1.67 * 10^(-27) kg/proton)

(b) After every half-life, half of all the protons will be gone.  Doing some
simple calculations, we see a pattern:

After 10 half-lives, there will be about 1/(10^3) as many protons as before.
After 20 half-lives, there will be about 1/(10^6) as many protons as before.
After 30 half-lives, there will be about 1/(10^9) as many protons as before.

So let's keep counting:

After  40 half-lives, about 1/(10^12) the number of protons will remain.
After  50    "      , about 1/(10^15)     "
After  60    "      , about 1/(10^18)     "
After 100    "      , about 1/(10^30)     "
After 200    "      , about 1/(10^60)     "
After 260 half-lives, about 1/(10^78) the number of protons will remain.

We are assuming that the half-life of a proton is 10^33 years.
So, after (260 * 10^33) years, there will be about (1.2 * 10^79)/(10^78)

= 12 protons left.  They will decay in about 4 more half-lives.

So all the protons in our current observable universe will have decayed away
after (260 + 4) * (10^33 yr)

= 264,000,000,000,000,000,000,000,000,000,000,000 years.

``` 