Can Gravity alone produce a simple Hydrogen or Helium atom?

Question

Imagine there is no EM or Nuclear Force, and Gravity is the only force.

Can gravity alone produce an atomic structure similar to a basic atom?

In this scenario, the nucleus is just a single particle. The electron is another particle with only mass and angular momentum, like a planet orbiting a star.

How feasible is it for Gravity alone to produce such a structure and hold it together, with energy and mass equivelent to those of simple hydrogen nuclei, electrons and matching speed/energy of orbits and angular momentum

Answer 1

In a universe where only gravity exists, every form of mass would eventually collapse to a region that is within its own event horizon. So naively, one may assume the particles in question to be tiny black holes, although, at these tiny scales, general relativity would likely need to be modified.

However, if naively one thinks of the atom using the simple Bohr model. Analogous to electromagnetic radiation, here we will have gravitational radiation that will remove energy and angular momentum from the orbit of the two-particle system. However, we might postulate certain configurations to be stable, as is done in the Bohr model.

Unfortunately, these configurations would have a very large radius as $$ r_n = \frac{n \hbar}{mv} \,, $$

$n$ being the quantum number, and $v$ is the orbital velocity for a circular orbit. This can be further shown to be (check the link above and calculate)

$$ r_n = \frac{n^2\hbar^2}{G m_\mathrm{e}^3} $$

which will give very large radius.