It's a complex question. CSR is actually not a function of the sun's diameter (which is itself only a function of date). It is a function of air mass (thus solar altitude), but it is mostly a function of the optical depth of the main scatterers, i.e. cirrus clouds and aerosols.

See http://www.researchgate.net/profile/Chris_Gueymard/publication/234909926_Spectral_Circumsolar_Radiation_Contribution_To_CPV/file/72e7e524c9449a6338.pdf

http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.3509220

Article Spectral Circumsolar Radiation Contribution To CPV

Additionally, regarding aerosols, their size (hence type) is important too since it determines their phase function. Typically, large particles (e.g., desert dust) will result in steeper phase functions, hence more circumsolar ,than small particles (e.g. pollution).

Your article "Spectral Circumsolar Radiation Contribution To CPV" is very interesting. Obviously it should be a relationship between the broadband circumsolar contribution (in your work) from one side and CSR and the equation (13) about the amount of the energy from the circumsolar region in the paper of D. Buie et al. / Solar Energy 74 (2003) 113–122: https://www.researchgate.net/publication/256854372_Sunshape_distributions_for_terrestrial_solar_simulations?ev=srch_pub&_sg=uBHi2w3S9zV4PDFOqG7dZTpoU5%2BSDbDRp%2FHJYQgcbFsafezgOQbxLWKWdOG4%2BBj2_20dQdeQWvbiJhBh9A8BVYWGIdnCQ1TnkFHWxy%2FhrcH6XeC2OVZsozyESUCOi5vH3

Article Sunshape distributions for terrestrial solar simulations

It seems from your graphics that there has to be a relationship between Ångström’s turbidity coefficient β, the air mass m, circumsolar contribution and CSR. From other side I'm not very sure that I understand what exactly do you mean under "circumsolar contribution" in your paper (Fig. 5). It should be the ratio between the summary circumsolar flux for different angular distances (opening half-angle) and DNI. On your fig 5 for angular distance 2.9 degrees from the center of solar disk the vertical value has to be CSR...

Yes, there is such a relationship, which was explored in a 1998 paper: http://cl.ly/0B063R1Z3c2z0g0Q3e2N

Note that cirrus clouds have a much stronger effect on CSR than aerosols.

In Fig.5 I defined "circumsolar contribution" as CC=CS/DNIth where DNith is the theoretical (sun only) DNI, whereas the classic definition of CSR is (CS+DNIth)/DNIth. Therefore CSR=1+CC.

Thank you for the definition of CC!

I was thinking (according Buie) that CSR is defined as the radiant flux contained within the circumsolar region of the sky, divided by the incident radiant flux from the direct beam and aureole. In "MEASUREMENT OF SOLAR RADIANCE PROFILES WITH THE SUN AND AUREOLE MEASUREMENT SYSTEM (SAM)" (by S. Wilbert, B. Reinhardt, J. DeVore, M. Röger, R. Pitz-Paal and C. Gueymard) CSR is defined as the ratio of the circumsolar irradiance and the sum of circumsolar and disk irradiances.

In this case I think it should be:

CSR=CS / (CS+DNIth)

1-CSR=DNIth / (CS+DNIth)

CC=CS / DNIth

1+CC=(CS+DNIth) / DNIth

1 / (1+CC)=1-CSR

CSR=1 - 1 / (1+CC)

CSR= CC / (1+CC)

and

CC = CSR / (1 - CSR)

In that your 1998 paper the circumsolar contribution is under name "Circumsolar magnification factor" or percent circumsolar radiation added to the true direct irradiance:

CMF = Edc / Ebn * 100

Well, there might be different definitions of CSR floating around. The one you describe is indeed that of Buie.

Incidentally, the document you mentioned has been superseded by http://cl.ly/2y250C3k341z

In a clear day the circumsolar radiance at angular distance 90 degrees from the center of the solar disk is approximately 0 (as a difference between the diffuse radiance and the background diffuse radiance). This means that the curve of the circumsolar contribution (as on your Fig. 5) at this distance has to be close to a horizontal line...

Attached image of diffuse luminance (radiance) as a sum of background and circumsolar components:

The distinction between a circumsolar component and a "background" radiance component is rather arbitrary. What creates both is actually the aerosol phase function. It is highly peaked forward, which creates the circumsolar aureole in the first place. The same phase function reaches its lowest point for a scattering angle of about 90°, but then rises again, which is why the radiance around the horizon is larger than at 90° from the sun.

In other words, a physical radiance model (based on an realistic or actual aerosol phase function) would just have a single continuous function rather than a superposition of many empirical ones...

The decomposition of the diffuse radiance (irradiance) to background and circumsolar components could be useful for the estimation of the diffuse irradiance under partially obstructed sky (this is an important problem for the solar architecture in an urban environment). For this we have to be able to estimate the emitted diffuse radiance in every point of the sky. If it is described by a single (and very complicate) continuous function, this will make impossible hard to find an analytic solution of the integrated diffuse radiance from the visible sky areas. The only solution will be the numeric integration of the radiance from many hundreds sky patches... which is a very slow process... From other side the equation by Moon and Spencer Lt=Lz*(1+b*cos t)/(1+b), which can be used to describe the background diffuse radiance, is relatively easy to be analytically integrated, even under partially obstructed sky. The problem of the analytic integration of the circumsolar radiance under partially obstructed sky is not decided yet, as far as I know... and I have plans to include it in the scope of my research...

https://www.researchgate.net/publication/236031907_Estimation_of_background_diffuse_irradiance_on_orthogonal_surfaces_under_partially_obstructed_anisotropic_sky._Part_I__Vertical_surfaces?ev=prf_pub

Article Estimation of background diffuse irradiance on orthogonal su...

The circumsolar radiance (and the circumsolar contribution) is not rotational symmetric for scattering angles more than 10 degrees. This is very easy to be noticed in the results from the sky scans, especially for low sun. In the best case the circumsolar radiance has a reflectional (mirror) symmetry regarding the solar meridian. I believe, the creation of a single continuous function, which reflects the angular variations of the diffuse radiance, could be very difficult... I know some anysotropic sky radiance models (for instance the models of Brunger and Norio Igawa), where such continuous functions exist. But they use also a decomposition of the diffuse radiance to circumsolar radiance, uniform background and horizon brightening...

From other hand I do understand and very high appreciate your idea and work to create a realistic physical radiance model, based on real existing physical processes (Rayleigh scattering, aerosol scattering, backscattering, etc.), not only on statistical processed measurements...

I agree with you that a single continuous radiance function would be very difficult to develop. That's why all empirical models (including the one I developed in the early '80s) use the superposition principle to obtain something manageable. I was just pointing out that this is an extreme simplification of the physical reality.

I did a comparison of various radiance models in chapter 19 of http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470446331.html

The plots I did then are attached.

Since I'm not satisfied with my old radiance model (or any more recent one for that matter), I'd like of develop a better one, based on more physics. Maybe we could collaborate on this. I know what to do, but I don't have the time...

Yes, may be we could collaborate on this. This will be a big challenge... I'll have more free time after September, but for now I can use next 4-5 months to understand better your last work...

OK, let me know what you need...