You ask a very good fundemental question. For things like planets in orbit (under constant acceleration) there really is radiating gravitational waves which car

You ask a very good fundemental question. For things like planets in orbit (under constant acceleration) there really is radiating gravitational waves which carry energy and in principle the orbits must decay over time. Even in particle accelerators the electrons moving in a circle emit radiation which slows them down and must be compensated for. So why don't atoms give off a ton of radiation quickly while the electron decays and crashes into the nucleus? The answer is on that scale it no longer makes sense to think of an electron as a tiny planet orbiting another tiny planet. The rules of quantum mechanics apply, and quantum mechanics says the orbits are quantized and there are no tinier orbits to decay into. So the full answer is that there only seems to be a paradox because your notion of when things should and shouldn't radiate is an approximate truth which only holds for "normal" cases (neither microscopic nor cosmological).