In order to explain the maths here it is essential to look at things in a slightly different way. It is a different concept to any that I am aware of before. The way to look at it is to examine the time rate of change of momentum per unit mass. This has the same dimensions as acceleration, and effectively it is acceleration. However there are some cases that we have to deal with where the term "acceleration" would not be applicable, namely vertical (up or down) photons which nevertheless do have a time rate of change of momentum.

It is important to first get out of the way the issues relating to photon mass being zero and the difference between a single photon and a system. In normal convention the mass of a photon is taken as zero. That is the rest mass. However in any system (meaning a conglomeration of matter and radiation in some region of space) the rest mass is the mass as seen in the reference frame where the total momentum is zero or the centre of mass is not moving. So as soon as we have a collection of photons traveling in random directions, they each contribute to the system a mass m as given by e=mc^2 to that system. This is standard physics.

So let us start with a definition of a new variable which I will call "pull" and use the symbol "b" (being right next to "a" in the alphabet). Pull is defined as the time rate of change of momentum per unit mass.

b = 1/m.dp/dt

When we have a conglomerate of matter and radiation (such as the Sun's core or the Sun's outer layers) this variable b is the correct one to use as a measure of the acceleration of any part of that conglomerate. When we add together a bunch of matter and radiation, if we weight each part by its mass, then we will be summing correctly the momentum of that conglomerate. Then dividing again by the total mass we will get acceleration as a result.

When considering photons in a gravitational field (e.g. here on Earth in the laboratory) we can divide the directions into the 6 axes and determine the results for each case. The 4 horizontal axes are all the equivalent and Einstein has already shown that the change in direction is 2x the Newtonian one. So for these 4 photons we have that the acceleration vector and the momentum are changed by 2 times Newtonian gravity (g):

b = 2g

For the vertical photons, there is also a change in momentum predicted by Einstein. This is called "gravitational redshift" and Einstein's formula may be manipulated by using E=mc^2 and E=hf to find that b = g in both the vertically up and down cases.

df/dt = fg/c (as given by Einstein)

so b = 1/m.dp/dt = 1/m/c.dE/dt = h/m/c.df/dt = h/m/c.fg/c = Eg/E = g

That means there is no 2x factor for vertical photons, just the normal Newtonian rate of change of momentum per unit mass.

So taken over the whole sphere of directions for random photons we have 2x in 4 directions and 1x in two directions. That makes an average of 5/3x for random photon directions.