A Microscopic Calculator
While a model of an atom had begun to emerge by the start of the 20th century, it was predicated on a structure that was initially thought to be analogous to a miniature planetary system in which electrons 'orbited' a central and much larger nucleus. We might characterize some aspects of this structure in terms of the Bohr model, first published in 1913, and considered by many to have been a major milestone in the development of quantum mechanics. However, within the ongoing development of quantum theory, the idea of any obvious visual model of an atom slowly disappeared into an essentially mathematical model known as Quantum Field Theory (QFT). As such, the idea of any sort of physical substance associated with a sub-atomic particle became subsumed into a description of energy densities that move in an abstracted concept of space-time. possibly as some sort of wave structure; the exact nature of which appears to still be much debated. However, for most practical visualizations, Bohr's original model of an electron 'orbiting' a central nucleus might still provide a comparative measure of the size of the fundamental 'particles' within the basic atomic model. Therefore, in the current context, we might continue to visualise the electron and nucleus as an analogous, but microscopic, planetary system that replaces gravity with electromagnetic forces, but ignores the complexity of an emerging 'quantum reality' that was to follow.
So how might we scale such a model?
On the basis of the planetary analogy, we might wish to define the distance between an 'orbiting' electron and the central nucleus in terms of a microscopic astronomical unit and, in so doing, make a cross-reference to the 'macroscopic calculator'. One of the comparative models within the macroscopic calculator sets the distance between the Earth and the Sun to one inch, which then leads to the scaled distance to the nearest star being over 4 miles away. However, this default within the current model results in the scaled sizes of the electron and proton remaining firmly within the microscopic domain, outside normal human experience. Therefore, it is possibly more informative to use the 'Calculate-2' option, which by default sets the diameter of a proton equal to our Sun.
As indicated above, the default values of the microscopic calculator are predicated on an analogous comparison of the Bohr Radius (BR) to an astronomical unit (AU), which in the macroscopic calculator was set to 1 inch or 2.54 centimetres by default.
The default scaled model is probably less useful in the current context because the scaled values of the proton and electron 'particles' remain in the microscopic domain. In this scaled model, the electron 'orbits' the central proton nucleus at a radius of 2.54 centimetres, i.e. 1 inch, while the proton diameter is less than a thousandth of a millimetre and the electron diameter is reduced to a millionth of millimetre. However, although these sizes seem incredibly small on the human scale, it is worth noting that the scale factor suggests these 'particles' are some 481 million times smaller.
The default value linked to the 'Calculate-2' option sets the size of a proton to that of the Sun, which then allows us to make a direct comparison of the structural separation within the Bohr hydrogen model to its planetary analogy. On this scale, an electron would be some 1580 kilometres in diameter, somewhat smaller than the Earth at 12,742 kilometres, but in the same basic ballpark. However, the scaled Bohr orbit radius would be nearly a thousands times larger than the comparative Earth orbit coming in at some 41 billion kilometres. Of course, the atomic planetary analogy is now governed by the forces of electromagnetism, not gravity, which differ by some 1039 orders of magnitude. However, possibly more surprisingly is that the nearest atom, in interstellar space, would be 835,000 lightyears away, which differs from the 4.3 lightyears to the nearest star. However, it should be recognised that particle density of interstellar space assumed is possibly far too low and could easily be increased by a factor 1000+ in many regions of space.
As stated on several occasions, the actual sizes assumed in this model should not be taken too seriously as they are only intended to provide a very basic comparative model, which is superseded by quantum theory. However, the conclusions of both the microscopic and macroscopic calculators is that it is essentially impossible to draw a scaled model of either an atom or a planetary system on a single sheet of A4 paper without the sizes of the 'proton-electron' or 'Sun-Earth' components disappearing to dots on the page. Again, we may need to reevaluate or redefine our definition of 'empty space' as it might appear that the 'tangible substance' of both the macroscopic and microscopic universes is a negligible component when seen in comparison to its spatial volume.
Matter is mostly ghostly empty space, 99.9999999999999% empty space to be a little more exact. If you could take away the empty space then all the subatomic particles in all the six billion people on planet earth would pack into a volume only a little larger than a grain of rice. Sir Arthur Eddington