Set in the not so distant future nearly all commercial, private and military spaceships capable of fractional speed of light travel would always have a time keeping device powered by radioactive isotope. This is contradicting a doctrine which would have eliminated every traces of nuclear materials on the surface of Earth, actually most of it were buried deep beneath the dirt and sea floor. What is the reason for it not being banned and replaced by advanced technology with unparalleled accuracy and precision?
Atomic clocks are not radioactive. They do not rely on atomic decay. Rather, they have an oscillating mass and a spring, just like ordinary clocks.
The big difference between a standard clock in your home and an atomic clock is that the oscillation in an atomic clock is between the nucleus of an atom and the surrounding electrons. This oscillation is not exactly a parallel to the balance wheel and hairspring of a clockwork watch, but the fact is that both use oscillations to keep track of passing time. The oscillation frequencies within the atom are determined by the mass of the nucleus and the gravity and electrostatic "spring" between the positive charge on the nucleus and the electron cloud surrounding it.
a doctrine which would have eliminated every traces of nuclear materials on the surface of Earth
The doctrine is plainly dumb and ill conceived.
There are no materials which are radiation free, and we live in a radioactive background. Just to give you a few examples:
- ever heard of C14 dating? we are all radioactive, since we have unstable carbon-14 nuclei in our bodies. Same does our food, wooden furniture, anything organic. And that C14 is constantly produced in our atmosphere.
- water has traces of tritium in it, making it radioactive, again.
- bananas are notoriously radioactive, due to having potassium isotopes
Following that doctrine to the letter would mean burying everything under a layer of still radioactive material.
The most obvious solution seems to be that the phrase, "the law applies to the surface of the Earth" is too limited to prevent nuclear technology use among spacecraft. First, the law doesn't specify that nuclear materials are illegal in space. Second, even if it did, Earth governments have no jurisdiction off the surface of the Earth, or at least no real way to enforce their laws. Third, even if they could enforce them, how would they practically find every single atomic clock?
We can take this in several different directions. Perhaps radioactive isotopes are mined in the asteroid belt, on the moon, or in Martian colonies; that way, no part of production ever violates the law on Earth.
Importantly, space colonization as we know it can't exist without nuclear power. Today, many probes are powered by onboard reactors, which is one of the most efficient, dependable ways to keep them running. In the future, the advent of relativistic spacecraft implies interstellar travel - resulting in distances at which solar panels are effectively useless. Nuclear power may be the only option for spaceships, so why would something as trivial as an atomic clock matter next to a reactor - unless the law changes?