Coakley and Chylek (1975) provide a formulation of the plane-parallel equation of radiative transfer: $$ \mu \frac{dI^{+}(\tau,\mu)}{d\tau} = I^{+}(\tau,\mu) - \frac{1}{2} \int_{0}^{1} d\mu'\, p(\mu,\mu') I^{+}(\tau,\mu') - \frac{1}{2} \int_{0}^{1} d\mu'\, p(\mu,-\mu') I^{-}(\tau,\mu'), \tag{2} $$
$$ -\mu \frac{dI^{-}(\tau,\mu)}{d\tau} = I^{-}(\tau,\mu) - \frac{1}{2} \int_{0}^{1} d\mu'\, p(\mu,\mu') I^{-}(\tau,\mu') - \frac{1}{2} \int_{0}^{1} d\mu'\, p(\mu,-\mu') I^{+}(\tau,\mu'). \tag{3} $$
I wrote a python program to solve this equation numerically assuming only two directions for scattering ('forwards' and 'backwards'), for a homogenous medium that features an attenuation coefficient of 1 (fully scattering without absorption), with an isotropic phase function (50% of light moving right, 50% moving left at each scatter). Each bounce, I gathered the amount of light (both forward and backward traveling) present at each optical depth and summed them in bins. I noticed a peculiar function emerges:
$$ ae^{-b\tau}-e^{-\tau} $$
When fitted with the values $a = 1.850522$ and $b = 0.070484$, $R^2 = 0.999999$ up to an optical depth of 100. Given more bins and more bounces, from what I have been able to discern this function closely matches what the radiative transfer equation seems to converge towards. I have assumed that the physical interpretation is that when light initially enters the medium, it has not yet attenuated strongly, as it has only passed through a very thin section of the volume, and thus the amount of scattered energy is low. But, as the light moves further in the medium, scattering increases, up until a peak of both 1st-order and nth-order scattering occurs, until scattering weakens at further depths.
Surely this behavior is known in literature, but I have not been able to find any information about it online - perhaps it's buried in a textbook. Does it have a name? Is my interpretation correct? Can it be exactly derived from the RTE? I'd love to know why such peculiar constants appear in my fit. Or, alternatively, is this behavior not expected at all (indicating I'm doing something wrong)?
Coakley , J. A., and P. Chylek, 1975: The Two-Stream Approximation in Radiative Transfer: Including the Angle of the Incident Radiation. J. Atmos. Sci., 32, 409–418, https://doi.org/10.1175/1520-0469(1975)032<0409:TTSAIR>2.0.CO;2.
