The Total Solar Irradiance variability in the Evolutionary Timescale and its Impact on the Mean Earth's Surface Temperature

The Sun is the primary source of energy for the Earth. The small changes in total solar irradiance (TSI) can affect our climate in the longer timescale. In the evolutionary timescale, the TSI varies by a large amount and hence its influence on the Earth's mean surface temperature (T$_{s}$) also increases significantly. We develop a mass-loss dependent analytical model of TSI in the evolutionary timescale and evaluated its influence on the T$_{s}$. We determined the numerical solution of TSI for the next 8.23 Gyrs to be used as an input to evaluate the T$_{s}$ which formulated based on a zero-dimensional energy balance model. We used the present-day albedo and bulk atmospheric emissivity of the Earth and Mars as initial and final boundary conditions, respectively. We found that the TSI increases by 10\% in 1.42 Gyr, by 40\% in about 3.4 Gyrs, and by 120\% in about 5.229 Gyrs from now, while the T$_{s}$ shows an insignificant change in 1.644 Gyrs and increases to 298.86 K in about 3.4 Gyrs. The T$_{s}$ attains the peak value of 2319.2 K as the Sun evolves to the red giant and emits the enormous TSI of 7.93$\times10^{6} Wm^{-2}$ in 7.676 Gys. At this temperature Earth likely evolves to be a liquid planet. In our finding, the absorbed and emitted flux equally increases and approaches the surface flux in the main sequence, and they are nearly equal beyond the main sequence, while the flux absorbed by the cloud shows opposite trend.