Geometric formalism to account for sunspot effects on granulation size distribution
Authors: Sharov Alexey , Hanslmeier Arnold 
Published: March 2024

Analytical formulations remain an important and most accessible tool for modelling heliophysical processes and analysing solar activity. In this three-year study, we formulated and proved a simple surface equation to explain the effects of magnetic activity on the size distribution of solar granulation, assuming that the total surface area of the Sun is the sum of the areas of granular cells, pores and sunspots. The combination of this global geometric expression with the mass conservation equation for a single cell provided stimulating insights into the physical geometry of solar granulation and helped describe the interdependence and covariation of the size and total number of granular cells with the total sunspot area. This is a significant difference from other studies and scaling relationships known from the literature that describe star granulations without direct analytical reference to the spot index.
We wrote two versions of the solar surface equation that treat the granule tops as planar polygons and spherical caps, thereby reducing the equation degree and eliminating the average cell size in the second version. The most interesting consequence of these equations is the inverse relationship between the total sunspot area and the total number of granular cells, which contradicts the estimates of C.Macris but does not seem unlikely given the smaller extent of surface convection due to the area covered by sunspots and pores. Substituting standard values for the global solar parameters into our formulations gave a total number of granular cells on the Sun of 3,104 &#177; 0,012 million and showed that this number decreases by 0.4% as the sunspot area increases from 0.001 to 0.005 mhs.
To test this finding we used the total differential of the resulting equations and estimated the relative change in horizontal size and number of granular cells due to variations in the sun's radius and sunspot area. The total number of granular cells and their mean size were defined to decrease while the solar radius, effective temperature, luminosity and granulation timescale increase with increasing sunspot area. The maximum magnitude of change was given as 5x10-3 for the total sunspot area variation, 1x10-3 for the luminosity variation, 1x10-4 and 2x10-4 for the change in effective temperature and solar radius, respectively, and 2x10-5 for changes in surface gravity. We found that changes in sunspot area have the dominant influence on the mean equivalent radius of granular cells, with the maximum relative variation of the latter being 0.55%.
Further empirical evidence for the anti-correlated changes in granulation size and number relative to sunspot activity in cycle 24 was derived from the high-resolution image series acquired by the Hinode SOT in the blue continuum and a concurrent series of daily sunspot areas using an original colour-coding technique, controlled watershed segmentation and Mathlab statistical tools. It was concluded that the size distribution of solar granulation is related to sunspot activity. The differences between expected and observed variations were attributed to a lack of knowledge of the polar granulation size rather than the variable quality of the image data.