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It seems that both Bohr and Rutherford assumed circular orbits of electrons around the atom. But why did they do so, instead of assuming e.g. elliptical orbits?

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It's a common assumption for planetary - ie central Newtonian field - models to consider the orbits to be circular. I don't think Bohr or Rutherford ever believed in circular orbits for electrons, but rather decided to use a circle to describe an orbit they believed was elliptic.

Rutherford-Bohr Model use the planetary system as an analogy, where the movement is in fact elliptic but can be approached by circular paths. Using circles allows you to handle discrete set of radii rather than discrete areas for the movement, hence making the description of orbital jumps easier, but I don't believe Bohr or Rutherford had any fundamental reason to believe the orbits where circular. Check the maths (it's an exercise we did in uni) and everything works fine for fixed areas and elliptic orbits. Actually modern view of the ground state of an electron tend to represent it as orbiting in a path of nil-area, so you would need elliptic orbits if you want it to have a speed. I believe Sommerfeld published a more refined classic model for electrons using elliptic orbits.

Short answer: they wanted to keep it simple

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    $\begingroup$ Sommerfeld studied elliptic orbits to explain the fine structure of the spectrum. I couldn't find a better source, but there's more info here: en.wikipedia.org/wiki/Old_quantum_theory $\endgroup$ – hjhjhj57 Apr 26 '15 at 18:58
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    $\begingroup$ Bohr's original motivation was to explain Rydberg's formula for spectral lines of hydrogen, which suggests that electrons jump between orbits described by a single parameter (energy level). Circles are determined by a single parameter, say radius, but ellipses require two, say major axis and eccentricity. Sommerfeld wanted to explain more elaborate "fine structure" of the spectral lines, so he introduced a second parameter (angular momentum), which enabled ellipses. Bohr tried that himself first, but apparently got stuck with math :) See philsci-archive.pitt.edu/5547 $\endgroup$ – Conifold Apr 27 '15 at 19:26
  • $\begingroup$ @Conifold, Javier feel free to edit my answer, the story about Bohr is funny :) . How such a genius could be stuck by a variable change is a wonder! $\endgroup$ – VicAche Apr 27 '15 at 20:34
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    $\begingroup$ Well, they wanted to incorporate special relativity, so the ellipses precess and it gets hairy. But the best part is this. After Sommerfeld derived the correct formula for the fine structure Einstein said "Bohr's theory must be right" and Planck compared it to Leverier predicting Neptune. As it turns out however, fine structure depends on spin-orbital interaction, which Sommerfeld's theory doesn't account for, so correctness of his formula is a fluke, that happens only for hydrogen and by accident. $\endgroup$ – Conifold Apr 27 '15 at 23:27

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