Why are there different time estimates for a solar photon to reach the surface of the Sun?

Question

Some estimates range the photon journey from 10,000 to 170,000 and even millions of years, but only consider the path from the sun's core to its surface. This is called 'Random Walk Problem' and assumes the sun's interior has a constant density and that the 'free path' distance for the photon is about 1cm. 1.

An additional reading pointed that is in the Radiative Zone where most of the 'random walk' occurs. Next, the photon goes to the Convection Zone, which density is less than Radiative Zone, where it has lost most of its energy and shifts to visible light. Photon scattering in the Solar Atmosphere is negligible. [2]

1 https://sunearthday.nasa.gov/2007/locations/ttt_sunlight.php

[2] https://futurism.com/photons-million-year-journey-center-sun

Q. Why such a difference between the estimated travel times? Thank you.

May, 15th There is a neat lecture on the subject

Principles of Astrophysics & Cosmology - Professor Jodi Cooley http://www.physics.smu.edu/cooley/phy3368/lectures/150216_lecture.pdf

1. Calculate how long it takes a photon to travel from the center of the sun and emerge at its surface.

My quote: "In reality, there are regions where electron scattering is more prominent and regions where it is less important. As a result, the typical photon mean free path is 1 mm"

Answer 1

The idea that an identifiable photon "bounces around" inside the Sun and emerges some time later is incorrect.

The photons that are produced in nuclear fusion reactions or indeed by other emission processes (bremsstrahlung etc.) at temperatures of $1.5\times 10^{7}$ K in the solar core are gamma rays and X-rays. Obviously, the radiation we receive from the solar photosphere is mainly in the form of visible and near-infrared light.

What in fact happens is that individual photons are emitted and then absorbed on length scales that can be as short as 1 mm inside the Sun. The exact mean free path of a photon depends on the temperature and density conditions and varies as a function of radius inside the Sun.

The "100,000 year" figure is what you get if you assume indeed that the absorption of a photon is immediately followed by the re-emission of a (different) photon in a random direction - a so-called "random walk process". It is easy enough to show that the time taken for such a random walk to emerge at the solar surface is $$ \tau = \frac{R^2}{lc}\ ,$$ where $R$ is the solar radius and $l$ is the mean free path of a photon. If you reverse-engineer this equation you will see that $l=0.5$ mm corresponds to $\tau= 10^5$ years.

On average, a photon that is emitted from a larger radius within the Sun will have a slightly lower energy, because of the temperature gradient. This effectively transfers energy from the inside to the outside and is called radiative diffusion.

The reason that there is variation in the "100,000 year" number is because of different assumptions about what to use for the average mean free path of a photon; which as I say, varies considerably with depth inside the Sun - anything from around $<0.1$ mm in the core to 2-3 mm nearer the photosphere. It is also because it is not correct to say that each photon travels a set distance $l$ and is then absorbed. In reality there is a distribution of distances, which result in additional numerical factors in this back-of-the-envelope calculation. There is also the issue that convective heat transport rather than radiative diffusion becomes the more dominant heat transfer process in the outer 20% (by radius) of the Sun.