Modeling brightness temperature of sunspots using ALMA single-dish observations

We model sunspot brightness temperature in the wavelength range 0.3-10mm using the Atacama Large Millimeter/submillimeter Array (ALMA) single-dish measurements together with other instruments. Our modeling idea starts from an existing model, the 1D semi-empirical Avrett-Tian-Landi-Curdt-W\"ulser (ATLCW) model for sunspots. The original model is then modified in density and temperature with a constant factor over all heights. A least-square fit in both density and temperature to the observed values yields an improved model for sunspot brightness temperature. Observations show that sunspots are darker than QS at wavelengths below 2 mm and brighter beyond, while fitting of the ATLCW sunspot model imply that sunspots should be darker than QS over the entireALMArange. The best-fitted sunspot brightness temperature profiles resulted in 3.6-4.0 times lower sunspot density compared to the original sunspot ATLCW model, or 1.4-1.8 times higher density compared to QS. The temperature came out around 1.2 times higher than the original ATLCW predictions for sunspots, or 1.2 times lower than QS temperature. Even with improved density and temperature parameters, the remaining discrepancy between the bestfitted models and observations indicated that additional improvements are needed in the ATLCW model and modeling assumptions.