Points to Ponder

Both Thomson's and Rutherford's atomic models turn out to be unstable, just for different reasons. Thomson's positively-charged "cloud" model is electrostatically unstable; Rutherford's nuclear model is unstable because an orbiting (and therefore constantly accelerating) electron must, by classical electromagnetic theory, radiate energy continuously and spiral into the nucleus.

Why did Bohr choose to quantise angular momentum specifically, rather than some other quantity? Planck's constant $h$ has the same dimensions as angular momentum, and angular momentum is the natural conserved quantity for a particle moving in a circular orbit — so once you accept that something about the orbit must be quantised, angular momentum is the very natural choice, not an arbitrary one.

Bohr's picture of an electron following one definite orbital path is not really consistent with the Heisenberg uncertainty principle. Modern quantum mechanics replaces these sharp, well-defined orbits with regions of space where the electron has a large probability of being found — Bohr's orbits survive only as a useful approximate picture, not the literal truth.

In the solar system, the gravitational pull between any two planets is negligible compared to the Sun's pull on each planet individually, because the Sun is so much more massive than any planet. There is no such convenient simplification inside a multi-electron atom: the electron-electron repulsion is comparable in size to the electron-nucleus attraction, since both act over similar distances with similarly-sized charges. This is exactly why the simple, one-electron-at-a-time Bohr picture cannot be extended to atoms with more than one electron.

Bohr's original theory used a single quantum number, $n$, to label a state. Full quantum mechanics — which keeps Bohr's core insight that certain states don't radiate — generally needs four quantum numbers ($n$, $l$, $m$, $s$) to describe a state completely. It happens that for a purely Coulombic potential, as in hydrogen, the energy depends on $n$ alone, which is precisely why Bohr's simpler one-number picture works so well for hydrogen specifically, even though it isn't the whole story.

It's tempting to assume the electron's frequency of revolution around the nucleus equals the frequency of light it emits — but in the Bohr model these are two genuinely different quantities. The frequency of the EMITTED photon comes from the ENERGY DIFFERENCE between two orbits, divided by $h$ ($h\nu = E_i - E_f$), not from how fast the electron happens to be orbiting in either state. The two frequencies do converge for transitions between very large, closely-spaced quantum numbers (large $n$ to $n-1$), which is a nice check that the quantum picture smoothly reduces to the classical one at large scales. …