How much do anthropogenic carbon aerosols (soot) affect the atmospheric radiation budget?
For the purposes of the answer, anthropogenic carbon sources, will be referred to as BC (black carbon).

The effects of BC on atmospheric radiative forcing has come under considerable scrutiny in the past few years, several studies have sought to determine exactly the effect of this aerosol has on the Earth's radiation budget.  A major study was performed by Bond et al. (2013), in the article Bounding the role of black carbon in the climate system: A scientific assessment.

A summary of the emissions and effects are shown in the image below:



Caption: Schematic overview of the primary black-carbon emission sources and the processes that control the distribution of black carbon in the atmosphere and determine its role in the climate system. (Bond et al. 2013)

Their findings was that the total direct forcing of BC, independent of co-emitted aerosol species and including all sources (direct emissions, pre-industrial background, cryosphere and clouds) is +1.1 Wm-2 with an uncertainty range of +0.08 Wm-2 to +2.1 Wm-2. 

This places BC itself as the 2nd most important anthropogenic emission, behind $\ce{CO2}$. 

However, when the total radiative effects of BC with co-emitted aerosols, including organic carbon sources are taken into consideration, there is a slight cooling effect, with the total direct radiative forcing of -0.06 Wm-2, with a greater uncertainty range of -1.45 Wm-2 to +1.29 Wm-2.



Bond et al. 2013.

A major source for the uncertainties is described by Bond et al. (2013) as being due to a lack of knowledge of how black carbon and co-emitted species interact with clouds.

In summary, from the research that Bond et al. (2013) performed, the presence of black carbon itself causes a significant positive radiative forcing, but when considered with its co-emitted aerosols, there is a slight negative radiative forcing.  However, the uncertainties calculated demonstrate that there are still mechanisms that are not completely understood.

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