Is there anything special about the Sun's photosphere in terms of density?

Question

The Earth has a definite boundary between rocky/ watery surface and gaseous atmosphere. The same cannot be said of the sun. Even though the photosphere gives an apparent "edge" to the sun via the human eye, the matter of the sun continues on significantly further. I have even heard a claim that from a certain perspective, all of the planets out to the KBOs are orbiting "inside" of the sun.

My question concerns what happens when we graph the density of the sun versus distance from the center. I suspect the result will resemble exponential decay; is this correct? Does the graph have any unusual characteristics (such as a lack of smoothness) at any point along the distance axis, in particular at the photosphere?

Answer 1

The "edge" of the sun that we see (the photosphere) arises not so much from any feature in its density profile, but from the properties of how light travels through the sun as the density drops.

The photosphere is the point where the density drops enough that photons can begin to free stream away without interacting any more with the gas. Formally, this is the point where the optical depth of rays directed inward toward the sun's center equals 1. This can be interpreted to mean that the photosphere provides the average "surface of last interaction" for the photons we see.

The density profile continues to drop smoothly before and after the photosphere. The variation of density with radius can be well-approximated by a power-law near the surface of the sun, and can be roughly derived by finding solutions to the Lane-Emden equation (basically just a combination of the condition for hydrostatic equilibrium and Poisson's equation for the strength of gravity).