Abstract
NASA’s Solar Radiation and Climate Experiment (SORCE) Spectral Irradiance Monitor (SIM) instrument produced about 17 years of daily average Spectral Solar Irradiance (SSI) data for wavelengths 240 – 2416 nm. We choose a day of minimal solar activity, August 24, 2008 (2008-08-24), during the 2008 – 2009 minimum between Cycles 23 and 24, and compute the brightness temperature (T_{o}) from that day’s solar spectral irradiance (mathit{SSI}_{o}). We consider small variations of T and SSI about these reference values, and derive linear and quadratic analytic approximations by Taylor expansion about the reference-day values. To determine the approximation accuracy, we compare to the exact brightness temperatures T computed from the Planck spectrum, by solving analytically for T, or equivalent root finding in Wolfram Mathematica. We find that the linear analytic approximation overestimates, while the quadratic underestimates, the exact result. This motivates the search for statistical “fit” models “in between” the two analytic models, with minimum root-mean-square-error, RMSE. We make this search using open-source statistical R software, determine coefficients for linear and quadratic fit models, and compare statistical with analytic RMSEs. When only linear analytic and fit models are compared, the fit model is superior at ultraviolet, visible, and near-infrared wavelengths. This again holds true when comparing only quadratic models. Quadratic is superior to linear for both analytic and statistical models, and statistical fits give the smallest RMSEs. Lastly, we use linear analytic and fit models to find an interpolating function in wavelength, useful when the SIM results need adjustment to another choice of wavelengths, to compare or extend to any other instrument. Advantages of the quadratic T over the exact T include ease of interpretation, and computational speed.
Highlights
The Sun’s temperature and its variations over timescales from hours to decades have been determined since 1978 from satellite measurements of associated variations in Total Solar Irradiance (TSI)
The average radiative temperature of the Earth is determined by an approximate balance between the amount of energy it receives from the Sun, which can be calculated from the TSI and the Earth’s albedo, and the amount of energy that Earth emits into space that depends on Earth’s emissivity (Stephens et al, 2015)
We employ a root-finding algorithm developed in Wolfram Mathematica, using the following initial condition T = 5770 K, where T is chosen near the effective radiative temperature computed using TSI = 1360.8 W/m2 as provided by the Solar Radiation and Climate Experiment (SORCE) Total Irradiance Monitor (TIM) (Kopp and Lean, 2011)
Summary
The Sun’s temperature and its variations over timescales from hours to decades have been determined since 1978 from satellite measurements of associated variations in Total Solar Irradiance (TSI). Since the deployment of the SORCE satellite in 2003, the Sun’s temperature has been determined for a continuous range of wavelengths that span ultraviolet, visible and near-infrared wavelengths, from solar spectral irradiance (SSI) measurements across the peak of the SSI distribution. These are of great interest due to the fundamental role that solar variations play in understanding the variations of the Earth’s climate (Harder et al, 2005; Eddy, 2009). To determine how solar variations impact Earth’s atmosphere–ocean system at various heights, SSI must be monitored in addition to TSI
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