@George (Murphy)
and that would be a lower limit, of course, since on average, photons are not going to all execute their random walks in the outward radial direction.   I've seen calculations that range from 10,000 to 170,000 years in the literature; the Open Source Physics site has a Java executable that does the calculation, and allows the user to input values for various parameters here; a short discussion at a NASA site,here, based on a shell model of the solar interior (see here ).

My question is why Tyson didn't correct the writers on the 10 million year escape time?  Surely he would know how the calculation works and what realistic models produce.  It also would have been a good talking point on how physicists build models and evaluate them.

They could have even used the OSP computation model to illustrate the process.  This is why I always tell my non-scientist friends to not get their science education from TV documentaries :)

Victoria

The equations look correct, thanks to my Google sources.  The opacity and density, however, are not constants but variables that, apparently, are not agreed upon.  The results from variations used seems to produce a walk time ranging from 100,000 years to 1 million years.   The Random Walk time in the convective zone is very fast as photons ride the up elevator.  

Harwit, who I quoted, didn't sat that κρ was constant but that it's "of order unity" for a star like the sun. & as Messrs. Dassen & Cooper note, there are other complications. I would add also that the photons were dealing with don't start at the very center of the sun so using the solar radius for R overestimates the time. (& of course to be pedantic, the "photon" that is pictured as random walking out to the surface is actually a many times removed descendant of what started out as an X-ray or gamma ray photon.) Not all the photons that start out at a given time & depth from the surface get out at the same time.

Here is a 1992 paper cutting the MFP by ~ 1/10th the common figure of 1 cm.  Their result is 170,000 years, but it too does not address your point about delay times during absorption, which I assume would greatly increase the time.