Help understanding why Newtonian mechanics doesn't allow planets to follow Rosetta orbits The argument I am trying to understand is the following:
For an orbit to be closed, the angle between successive perihelions (points where the planet at its closest to the sun) must be a rational number multiplied by pi :
$$ \Delta \phi = \dfrac{m}{n} \pi, \quad m, n \in \mathcal{N} $$
However, since the set of rational numbers is countably infinite, whereas the set of irrational numbers
is uncountably infinite, the probability of a rosetta orbit meeting this requirement is zero and therefore rosetta orbits aren't closed.
What I don't understand is why the angle between successive perihelions must be a rational multiple of pi : why isn't it theoretically possible for a planet to be in a stable orbit around the Sun without ever returning to the same spot ?
In addition, planetary orbits aren't an exact science. Due to the minor effects of other bodies, real planets obviously don't follow the above  mathematically ideal requirement. Why then, does the argument hold?
 A: By Bertrand's theorem, if the strength force is inveresly proportional to the square of the distance then the orbit will be closed, in that will return to the exact same spot each time.
Which means that, in this case $$ \dfrac{m}{n} \pi =2\pi$$
Though however, this is only the case for a two body system. When there's three or more bodies this become much more complicated.
A: In messy systems, like the solar system, orbits aren't closed. When you look at the this page on wikipedia (Tests_of_general_relativity#Perihelion_precession_of_Mercury)  you see that the precession of mercury is only a small part due to general relativity. The majority of the precession is due to interactions with other planets. Here is a table from said page:
$$\begin{array}{ll}
\textbf{Amount (arcsec/Julian century)} & \textbf{Cause}                                       \\
\text{532.3035             }                                       & \text{Gravitational tugs of other solar bodies                                  }  \\
\text{0.0286               }                                       & \text{Oblateness of the Sun (quadrupole moment)                                 }  \\
\text{42.9799              }                                       & \text{Gravitoelectric effects (Schwarzschild-like),}\\&\text{ a General Relativity effect }  \\
\text{−0.0020              }                                       & \text{Lense–Thirring precession                                                 }  \\
\text{575.31               }                                       & \text{Total predicted                                                           }  \\
\text{574.10±0.65   }                                       & \text{Observed                                                                  }
\end{array}$$
From this we can conclude that even without general relativity orbits are not closed. Closed orbits are only an approximation. Classical mechanics only predicts closed orbits when there are two, perfectly spherically symmetric bodies in orbit around each other. In that case the chance of getting a periodic orit is not zero but one, which follows from the equations of motions.
