Why would an electron in an orbit be accelerating continuously and would thus radiate away its energy and fall into the nucleus in a classical model? [duplicate]

Question

I was reading this answer by madame anna v:

You are right, the planetary model of the atom does not make sense when one considers the electromagnetic forces involved. The electron in an orbit is accelerating continuously and would thus radiate away its energy and fall into the nucleus.

One of the reasons for "inventing" quantum mechanics was exactly this conundrum.

The Bohr model was proposed to solve this, by stipulating that the orbits were closed and quantized and no energy could be lost while the electron was in orbit, thus creating the stability of the atom necessary to form solids and liquids. It also explained the lines observed in the spectra from excited atoms as transitions between orbits.

But I don't understand one assumption: Why would an electron in an orbit be accelerating continuously, and would thus radiate away its energy and fall into the nucleus in a classical model?

Also in a planetary model, would planets also be accelerating and thus fall into the sun?

I mean for example: a planet would accelerate towards the sun, but because it will then have bigger velocity, it will escape a little bit the sun, and then it will accelerate towards the sun... and the whole process repeats again. But, why wouldn't that be true for an electron and proton?

Thank you.

Answer 1

I suspect you have misunderstood what is meant by acceleration in the context of circular motion. Acceleration is the rate of change of velocity, but remember that velocity is a vector so it has both direction and magnitude. You are probably thinking that acceleration means a change in the magnitude of the velocity, e.g. speeding up from 1 m/s to 2 m/s, but it can also mean a change in the direction. For a planet orbiting a sun, or a classical electron orbiting the nucleus, the direction of its motion is continually changing so it is accelerating even when the magnitude of its velocity stays constant. So when you say ...

I mean for example a planet would accelerate towards the sun, but because it will then have bigger velocity, it will escape a little bit the sun, and then it will accelerate towards the sun

... this isn't true because in a circular orbit only the acceleration is only changing the direction of motion. The planet stays at the same radial distance from the sun.

Incidentally, a planet orbiting a star, and therefore accelerating towards it, does radiate energy in the form of gravitational waves. However because gravity is a vastly weaker force than electromagnetism the rate of energy loss is vastly slower.

There is more about this in the answers to the question Uniform Circular Motion

Answer 2

I think the answer to this is fairly simple, an electron undergoing acceleration will emit a photon, thus losing energy. So if we assume a planetary model involving an orbit the electron would have to emit photons continuously, lowering its energy state and collapsing the atom.