Radiative Transfer Calculations

Energy incident normally on an area dA in primed direction:

Energy scattered, attenuated by emerging in non-primed direction:

Emission coefficient is then:

Source function is then:

Including thermal emission:

Single scattering albedo (av)

Separate intensity into direct (no scattering) & diffuse components, Is and Id

Then Is is simply

Consider an atmosphere with no emission or reflection from the surface, as might be applicable for the ultraviolet illumination of Titan’s atmosphere.

What are the boundary conditions?

Guess: How much direct (as opposed to diffuse) light hits the surfaces of Earth and Titan?

Boundary Conditions: 1) Incident solar beam 2) No reflection or emission from the surface

2-stream: +=upward -=downward

2-stream

I_s not effected by scattering/ emission -> obeys no source function eq.

Equation for diffuse intensity:

Where:

Since:

Substitute this term

DOWNWARD INTENSITY

incoming scattered

Similar to I- RT equation:

We have:

UPWARD INTENSITY

That is Id = I+d + I-d can be written as:

Include new effects as long as direct intensity is represented as an extra source function:

Looks like expression for I with extra source function

Include new effects as long as direct intensity is represented as an extra source function:

Modify source function to include direct intensity from from surface, scattered once:

Both downward and direct components are reflected from the surface, thus in top equation:

Surface reflection Surface emission

RT equation in form:

Solution:

S includes all of the source functions

Why is that?

Hint: Specular reflection must derive from scattering of direct rather that diffuse radiation

Observations

My RT calculations

From Stephan et al 2007