# Why did the Bohr model have electrons in circular orbits rather than elliptical orbits, like those of the planets?

The similarities between Coulombic Force and Gravitation Force is that both are inverse square laws and both are central forces.

Can we apply all the three laws proposed by Kepler in Bohr's Atomic Theory?

• First sentence on wikipedia about Bohr model: "In atomic physics, the Rutherford–Bohr model or Bohr model or Bohr diagram, introduced by Niels Bohr and Ernest Rutherford in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar to structure to the Solar System, but with attraction provided by electrostatic forces rather than gravity." The Bohr model is obsolete except as a teaching aid/stepping stone. Aug 5, 2017 at 21:06
• Seems like Bohr has already stated his assumptions. But the point of my question is shouldn't it be elliptical in reality provided Kepler's Law is true? Aug 5, 2017 at 22:02
• Electrons in real life do not actually orbit the nucleus in discrete spatial paths like planets do the sun. Is your question: "Why did the Bohr model have electrons in circular orbits rather than elliptical orbits, like those of the planets?" Aug 5, 2017 at 23:37
• Yeah. In both the models, energy and angular momentum is conserved. How much does the answer for radius and energy deviate from the answer obtained by Bohr? Aug 6, 2017 at 7:39
• I'm voting to close this question as off-topic because it's not clear which question is being asked. The title question is different from the question in the body. In addition, the Bohr model is deprecated. Aug 7, 2017 at 1:28

I asked myself that a while ago and found that Arnold Sommerfeld generalized the quantization condition using Hamiltonian dynamics. He quantized the generalized coordinates $$q_{a}$$ and its conjugate momenta $$p_{a}$$ over a period of motion such that: $$\oint p_{a}\text{d}q_{a}=n_{a}h,$$ where $$n_{a}$$ are integers. This allowed the electron to move with different degrees of freedom, like the radial that allowed elliptical orbits. However that stopped with the development of wave mechanics and all of this is now obsolete.

• I don't quite understand the math, but thanks. :) Aug 12, 2017 at 17:03

Sorry that my answer was a little fuzzy. Here is what I have in my mind.

Bohr just added a picture of "quantum" as an extension to Rutherford's planetary model. The primitive aim was to describe the stability of atom. What Bohr proposed is that the electrons can't occupy any radial distance from the nucleus. It can only have certain discrete spatial orbits, where its angular momentum is an integral multiple of $\hbar$. In short, the orbits in Bohr model represents stationary states.

No such stationary states are available for planets. For example, if the planet makes a little increase in its radius, of course, this will affect its angular momentum, time period of revolution, etc, it can continue its motion in a new orbit. Think about satellites. We can determine the radius at which a geosynchronous satellite should be launched. However, this is not the case with an electron, due to the quantized spatial orbits.

More importantly, there is no such thing as a spatially well-defined trajectory for an electron. Obviously, this is not coming into account in the Bohr model, as the Bohr model is proposed prior to "quantum mechanics". According to Bohr model, the potential energy stored in the electron-nucleus system is quantized, but not with the case of planetary motion. The allowed radius for an electron occupying a quantum state $n$ is given by

$$r_n=\frac{n^2\hbar^2}{Zk_ee^2m_e}$$

Now, still why we can't use elliptical orbits and Kepler's laws in Bohr model?

A simple argument is that the orbits in Kepler's model do cross each other. This is not the case of electrons in Bohr model. In addition, according to Kepler's second law, even though the orbit is symmetric, its motion is not. The planet speeds us near the sun and slows down when far way from the sun. This can be understood in terms of energy in the classical picture. Again, this continuous change in energy is prohibited in Bohr model. Only discrete quantum "jumps" are permitted in Bohr model.

• This answer really doesn't explain that much, as far as I can see. Gravitation also has a spherical potential, but planets move in elliptical orbits. And planets don't change their angular momenta, so why should the electron's angular momentum being quantized have anything to do with elliptical orbits not being allowed? (I don't know the answer to this question, either. The Bohr model happens to work fairly well for reasons unexplained.) Aug 6, 2017 at 19:21
• This answer seems to conflate the Bohr model with the standard textbook treatment of the hydrogen atom. The Bohr model predates quantum mechanics, and justifying it in terms of the latter makes little sense. At best, you could use quantum mechanics to explain why the Bohr model seems to sort of work. Also, as noted above, elliptical classical orbits have constant angular momentum (their orbital speed is not constant).
– AGML
Aug 7, 2017 at 1:03

I haven't dwelled into the math yet so please correct me if I am wrong but according to Bohr's 1922 Nobel Lecture, a circular orbit is just a simplification (which I presume has no effect on the predictions made by his theory):

"Following our picture of atomic structure, a hydrogen atom consists of a positive nucleus and an electron which - So far as ordinary mechanical conceptions are applicable - will with great approximation describe a periodic elliptical orbit with the nucleus at one focus. The major axis of the orbit is inversely proportional to the work necessary completely to remove the electron from the nucleus, and, in accordance with the above, this work in the stationary states is just equal to $$hK/n^2$$. We thus arrive at a manifold of stationary states for which the major axis of the electron orbit takes on a series of discrete values proportional to the squares of the whole numbers. The accompanying Fig. 2 shows these relations diagrammatically. For the sake of simplicity the electron orbits in the stationary states are represented by circles, although in reality the theory places no restriction on the eccentricity of the orbit, but only determines the length of the major axis." The Structure of the Atom, Nobel Lecture 1922, Niels Bohr, p.17

I however wonder if this were the case, why the orbit being circular (or equivalently the angular momentum being quantized) is almost always taught as a postulate of his theory.