Effect of particulates on the visibility of stars?

Question

I am looking for a (mathematical) relationship - either empirical or theoretical - which quantifies how the visibility of celestrial objects decreases with increasing amount of particulates in the air. I would be already happy if somebody knows such a relationship for the atmospheric light attenuation effects of black carbon or soot. Those two are actually not the same as Alfred Wiedensohler from the Leibniz Institute for Tropospheric Research points out in a presentation on the topic:

Soot is carbon particles resulting from the incomplete combustion of hydrocarbons. Soot contains polycyclic aromatic hydrocarbons (PAHs) and heavy metals. Black carbon (BC) itself is neither a toxic nor a carcinogen.

What I expect (but I am happy to be proven wrong) is that I am actually after the three constants in the following expression:

$$ f_{\rm attenuation}(d) = \frac{1}{k_c + k_l \cdot d + k_q \cdot d^2}$$

In this formula, $d$ is the distance between the light source and the observers, so in our case it would be the height of Earth's atmosphere (if we assume homogenous distribution of soot/ BC over the air column). $k_c$ is called the constant attenuation factor, $k_l$ is the linear attenuation factor, and $k_q$ stands for quadratic attenuation factor.

Such an expression would already be very helpful, but if there is science on the overall effect of particulates, that would be much cooler. I am dreaming of having a formula for visible magnitude of stars with naked eye in dependence of particulate concentration (for ideal, pitch-black nights without clouds).

Background

My question is inspired by What effect do aircraft have on night-time visibility? which does not have an answer yet.

I am aware that in the physical oceanography and in marine biology, there are extensive studies on light attenuation due to BC or other dissolved substances in water, and most search results are on that.

References and related questions

• C.Liousse, H.Cachier, S.G.Jennings: Optical and thermal measurements of black carbon aerosol content in different environments: Variation of the specific attenuation cross-section (1993)
• H. Horovath: Atmospheric light absorption — A review (1993)
• ESA on Desert dust plume over the Atlantic observed by Aeolus and Sentinel-5P

Answer 1

Aerosol physics is actually an old topic - yet still a very vibrant one.

One main factor on the extinction (which sums up scattering and absorption) other than the distance (which is about constant when looking at a a particular zenith distance) is the wavelength $\lambda$ you look at compared to the typical aerosol particle radius $r$ as its ratio affects the scattering regime we look at ($\lambda \gg r, \lambda \approx r, \lambda \ll r$). There's multiple papers on the aerosol influence on observations, like this (Stubbs et al) for PanSTARRS or Patat et al (2018) on extinction over Paranal which also looks at the temporal variability.

Looking at the typical aerosol concentrations (e.g. here by DWD), it seems that the smaller particles are more abundant. Thus we are in the Rayleigh or Mie scattering regimes where the extinction is due to scattering and does not depend much on the actual particle properties other than their size. The actual absorption only becomes important for the larger particles which are the least abundant.

All in all this might only be half an answer, but too long for a comment and maybe gives you leads to follow-up on. If you get a hand on Friedlander's "smoke, dust and haze", it might be a good read, too (especially chapters 5 and 13).