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.


INTRODUCTION
The Sun is the largest energy source to the Earth's atmosphere (Kren 2015;Kren et al., 2017). The small changes in total solar irradiance (TSI), which is also called solar constant (S 0 ), is the energy radiated by the Sun that is incident on the Earth's atmosphere. TSI varies by about 0.1% over the solar-cycle timescale (Fröhlich 2006;Kopp 2016). It is varying with time and its tiny changes in TSI affect the Earth's climate in the longer timescale (Eddy 1976). This idea is strengthened by many other studies (Haigh 2007;Lean & Rind 2008;Gray et al., 2010;Ineson 2011;Ermolli et al., 2013;Solanki et al., 2013;Kopp 2016). The space-borne instruments have been measuring the TSI since 1978. Its average value is 1361 W m −2 at 1 au (Kopp & Lean 2011).
In the evolutionary timescale, the Sun's luminosity changes by a large amount.The term "evolution" refers a star's change over the course of time. Currently, the solar luminosity is increasing by 0.009% per million years ( Hecht 1994). At this rate, the TSI increases by 0.1% in 10 million years. Following this rate, TSI is expected to increase by 10% in 1 Gyr at which time the Earth will become uninhabitable. In the same way, the luminosity will increase by 40% in about 3.5 Gyrs ( Hecht 1994;Kopp 2016). As Kopp (2016) indicated, the TSI increases as the solar luminosity increases in the evolutionary timescale. However, there is a limitation of estimation of the TSI in the evolutionary timescale. Kaplan (1981) estimated the TSI to be varying from 1.1S 0 to 1.4S 0 . This variation is in about 1.1 to 3.5 Gyrs as indicated by Sackmann (1993); Hecht (1994). However, to study the influence of the TSI on the Earth's mean surface temperature (T s ) during Sun's lifetime, modeling of the TSI in relation to the evolutionary timescale is needed.
O'Malley-James (2013) used time-dependent luminosity to study the TSI variability and found its influence on T s in the evolutionary timescale. However, the solar mass-loss has an influence on the solar luminosity on this timescale. Moreover, their work is limited to the next 3 Gyrs, although further study is needed to understand the variability of an extended time interval. Schröde et al., (2001) presented a model of T s in which they include the effect of solar mass variation, and predict the T s of different planets including Earth over the lifetime of the Sun. They conclude that we can survive although the maximum ratio of the future to the present temperature is about 6.6. However, in their modeling, they assume that albedo (α) is constant, and neglect the effect of bulk emissivity (ε). But, α is expected to be changing, perhaps decreases in the future. Therefore, in our model, we incorporate the climate parameters: non-constant α and ε. Moreover, in our modeling, we include the solar mass-loss in both TSI and T s models.
The aim of this work is to model the TSI variability and T s in the evolutionary timescale and to study the link between the two parameters.
In this study, we developed an analytical model of the TSI variability and T s in the evolutionary timescale. The base of our formulation is the models developed by Shukure et al., (2019) andN. T. Shukure et al. (2021, submitted to APJ). They developed a mass-loss dependent solar luminosity in the evolutionary timescale. To calculate the solar mass-loss, they first calculate the solar mass reduction every 1 million years for the next 8.23 Gyrs. Then, they calculate the mass-loss at each interval of 1 million years by subtracting every newly reduced solar mass from the present-day solar mass. Following this method, we calculate the mass-loss to model TSI in the evolutionary timescale. Using the newly formulated TSI model and zero-dimensional energy balance model (EBM), we formulate T s . We used the numerical method to solve the newly formulated models by setting some boundary conditions to ε and α that will be explained in detail in the method part.
where M is the present-day solar mass, δ determines the rate of decay ( Eq. 2 estimates the solar ∆M at any time t.

Luminosity-mass Loss Relation
The ∆M dependent luminosity that is modeled by N. T. Shukure et al. (2021, submitted to APJ) is: where, L = 3.85×10 26 W is the present-day solar luminosity, and t = 4.57 Gyr is the present-day solar age (?Feulner 2012). Eq. 3 used as an input to model TSI in the evolutionary timescale.

Modeling Total solar irradiance in the evolutionary timescale
The TSI for the present-day solar constant is (O'Malley-James 2013): where k is a constant (k ≈ 1) and d E is the Earth-Sun distance. Eq. 4 estimates the amount of energy which cross an area A = 1m 2 (one square meter). For our new model of luminosity in the evolutionary timescale, Eq. 4 can be re written as: Re writing Eq. 5 in terms of Eq. 3 become Eq. 6 estimates the TSI variability in the evolutionary timescale as solar mass-loss significantly increases.

Earth's Mean Surface Temperature Variation
The temperature of any planet varies as the Sun evolves. Our main interest in this section is to model T s . In the modeling, we excluded the greenhouse effect on the T s variation. The base of our modeling is the EBM. The energy flux that crosses a unit area is estimated by Eq. 4. If we consider the region of our Earth that faces the Sun as a circular area, the total amount of energy flux (φ) that intercept the Earth's circular area is the product of the flux through the unit area (S 0 ) and the area of the circle: where R E is the radius of Earth. However, the total surface area of Earth (A E ) is The average energy absorbed by the Earth is the ratio of the total energy absorbed by the Earth to the surface area of the Earth and it is written as: Eq. 9 can be the flux absorbed by a completely dark planet whose albedo is zero. However, planet Earth has an albedo different from zero. Hence, the absorbed φ by Earth is defined as Mcguffie & Hendersonsellers (1987): where α is the temperature-dependent Earth's albedo taken to be varying in the interval 0 ≤ α ≤ 1. Energy emitted by Earth's surface is estimated by the Stefan-Boltzmann law: where φ s is the flux that escapes from the surface of the Earth, σ is Stefan-Boltzmann constant and T s is the Earth's mean surface temperature. However, Eq. 11 needs a correction for the cloud absorption.The flux absorbed by clouds is: where φ c flux absorbed by cloud, ε is dimensionless bulk emissivity that can be in the interval 0 ≤ ε < 1 (Swift 2013). The flux radiated out of the atmosphere is : When the energy of a planet is at thermal equilibrium, the absorbed and emitted φ by Earth become equals.
Substituting Eq. 10 and 13 in to Eq. 14 one can calculate the T s as: Eq. 15 estimates the expected present-day Earth's mean temperature. In the evolutionary timescale, the TSI is expected to be changing (i.e., S 0 → S evolution ) and hence, it influences T s . Therefore, the expected T s in the evolutionary timescale can be estimated by rewriting Eq. 15 in terms of S evolution .
where T s changes as the Sun evolves. Since the Earth's surface covered by ice will reduce in the future, which will result in the decrease of Earth's albedo. Substituting Eq. 6 in to 16, T evolution is estimated as: Eq. 17 predicts the T s of our planet in the evolutionary timescale.

Numerical Computations of the Model
In the numerical analysis, we assumed that the present-day position of the Earth's orbit is not changing as the Sun evolves. We numerically solved Eq. 6 by setting δ to be varying as 10 −13 yr −1 ≤ δ ≤ 6×10 −11 yr −1 based on the presentday value ofṀ = 10 −13 M yr −1 reported by Feulner (2012). We also assumed value ofṀ = 6 × 10 −11 M yr −1 during red giant phase and we used the solar observables to be S 0 = 1361W m −2 (Schmutz et al., 2013), M = 1.99 × 10 30 (Kaplan & Union 1981) and the present-day solar age as initial time t = 4.57 × 10 9 (Bahcall et al., 1995;Feulner 2012). The solar mass reduction is calculated for about 8.23 Gyrs from now. The new mass of the Sun, M(t) is recorded at the intervals of every 1M yrs based on Eq. 1 to have a total of 8231 numerical values. The total numerical values are used for further analysis.
similarly, the solar ∆M in the same time interval is calculated using Eq. 2. Each of the new numerical values is subtracted from the present-day mass of the Sun in each time step (1M yrs). The new values of ∆M are used as an input to calculate the TSI in the same time intervals.
To solve Eq. 17 we set some boundary conditions of α and ε. We set the present-day values of the Earth's α and ε to be 0.3 and 0.407, respectively, as initial boundary conditions. By assuming the future Earth's atmosphere to be the present-day atmosphere of Mars, we set the final boundary condition to be 0.250 and 0.0038, respectively, for α and ε (Swift 2013).
We numerically solved Eq. 17 for the next 8.23 Gyrs by using the numerical solution of TSI as an input. Our program determines the T s every 1 million years. We then calculate φ absorbed , φ s , φ c , and φ emitted based on Eq. 10, 11, 12 , and 13, respectively, by using the T s solution as an input.

RESULT AND DISCUSSION
3.1. Total solar irradiance variability in the evolutionary timescale Figure 1 shows the TSI variability as the Sun evolves. Figure 1a indicates that the TSI increases to 1448.1 W m −2 in the first 1 Gyr from now, representing an increase by 6.4% with respect to the present-day value. At about 1.42 Gyr from now, the TSI changes to 1497.1 W m −2 which is about 10% increase compared to the present value. At this value of TSI, the Earth will lose water ( Hecht 1994). However according to our model, the 10% variability achieved at about 1.42 Gyrs that lags by 0.32 Gyrs as compared with Hecht's finding.
In 3.374 Gyrs, the TSI increases to 1905.4 W m −2 , and it is about 40% increment. This is the Kaplan (1981) flux at which an ocean evaporates entirely. This value achieved at about 3.5 Gyrs as indicated by Sackmann (1993); Hecht (1994). In our model, the 40% increment of the TSI occurs before 3.5 Gyrs.
At the end of the time in the main sequence, TSI increases to 2994.2 W m −2 at about 5.23 Gyrs from now. As compared to S 0 it increased by about 120%. Beyond the main sequence, the TSI increases enormously. Figure 1b shows the peak value of 7.93×10 6 W m −2 at about an age of 7.676 Gyrs from now. This age is the age at which the Sun approaches the final stage of RGB (Sackmann 1993). N. T. Shukure et al. (2021, submitted to APJ) indicated that the present-day Sun evolves into the white dwarf of mass 0.46 M in the next 8.23 Gyrs. At this age, the dwarf will have the TSI of -1.13×10 4 W m −2 . The negative sign indicates the TSI is below the present-day value of the Sun.

Earth's Mean Surface Temperature Variation in Evolutionary Timescale
It is known that the present-day T s is 288 K (Schröde et al., 2001;Mcguffie & Hendersonsellers 1987). This temperature is expected to change during the Sun's evolution. Figure 2a shows the T s variation as the Sun evolves in the main sequence and Figure 2b for beyond main sequence produced based on Eq.17. The equation predicts the present-day T s to be 290.16 K. This value is close to the experimental mean value.
We assumed that α and ε are decreasing up to the present-day values of Mars. Our basic assumption is that in the future, α and ε of Earth are decreasing. But it is difficult to estimate the final limit. We select the present-day value of Mars's α and ε. This is because the present-day atmosphere of Mars has no right conditions. The right condition for human life is in the temperature range between 270 K and 300 K on the Earth. According to the model of Schröde et al., (2001), the range of temperature ratio and times for which human life might survive on Mars is 1.29 -1.43 and 11.6 -11.7 billion years, respectively. This indicates that about 7 billion years are needed to warm Mars. However, human life might be extinct from the Earth before Mars is warm enough: oceans would evaporate entirely in 3.5 Gyrs from now as estimated by Sackmann (1993) and in 3.374 Gyrs by N. T. Shukure et al. (2021, submitted to APJ). Hence, its α and ε assumed to be decreasing. Therefore, we select its initial value as our final boundary condition by assuming that the parameters decrease in the selected time boundary. The Eq.17 predicts that the T s does not show a significant change in the absence of greenhouse effect for the next 1.644 Gyrs. This shows the influence of TSI in the evolutionary timescale on T s is not significant although the TSI increased to 1528.1 W m −2 . Including the effluence of human activities and the greenhouse effect my change the magnitude of temperature in this interval which results in the Earth losing its water (Kaplan 1981).
Beyond 1.644 Gyrs from now, the temperature of Earth is observed to be rising significantly (see Figure 2a). This is perhaps due to the increment of moisture in the Earth's atmosphere that induces an extra temperature which influences the T s (O'Malley-James 2013). At about 3.374 Gyrs, when TSI increases to 1905.4 W m −2 (40% increment from present-day value ), part of this energy absorbed by the Earth's atmosphere raises the T s to 298.86 K. According Kaplan (1981), the oceans would evaporate entirely when the TSI become 1905.4 W m −2 . T s rises by 3% as predicted by Eq.17. The equation also predicts the T s to be 326.72 K in 5.23 Gyr from now when the Sun completes its main sequence life. This happens when the TSI becomes 2994.2 W m −2 (increases by 120%).
If we consider the assumption of Schröde et al., (2001), that all forms of life will extinct when T s has reached 380 K. Eq.17 predicts that this will happen at the age of 7.13 Gyrs (see Figure 2b). This shows our equation predicts this phenomenon happens 0.1 Gyrs before the Schröder et al., prediction. This is because, in our model, we included ε and α parameters and let them vary. However, humans will have to leave much sooner, similar to the present-day situation, perhaps before the model temperature reaches 291.9 K in 2.2 Gyrs, as our model predicts. Figure 2b also shows that T s attains a peak value of 2319.2 K, where the ratio is 7.7 as the Sun evolves to the red giant phase. This value is much greater than the melting points of rocks and minerals as investigated by Eppelbaum et al., (2014). This finding indicates the bad news that our planet Earth will have the probability of disappearance or have a chance of evolving into the liquid planet due to extreme temperature increment in the next 7.676 Gyrs during Sun's red giant stage. However, beyond this age the good news is since the Sun's energy output observed to be decreasing, the T s also decreases and becomes 311.34 K at the end of 8.23 Gyrs from now. The decrease in the temperature of the planet may have allowed our planet to revert to a solid state. So far we discussed the T s variation due to solar evolution. Since the magnitude of an energy flux entering into Earth's atmosphere is expected to vary with the altitude of our atmosphere, we estimate the energy flux at different levels by assuming that T s is independent of the atmospheric altitude. Figure 3a shows the energy flux variation at different levels of Earth's atmosphere. The φ absorbed and φ emitted that are produced based on Eq. 10 and 13, respectively, are equal in the figure. Their present-day value is 238.41 W m −2 that shows the energy balance of the system. The two values increase as the Sun evolves in the main sequence, and approach the values of φ s whose present-day value is 401.91 W m −2 based on Eq. 11. This value is shown to be greater than φ absorbed since it is dependent only on the T s of the Earth. The φ s does not show significant change for the next 2.83 Gyrs. If the T s varies according to the prediction of our model, the life on the Earth's surface would have a probability of surviving up to 2.83 Gyrs from now. This result is close to that of O'Malley-James (2013), which predicts the maximum lifetime for the life on our planet is 2.8 Gyrs from now. The φ c is found to be decreasing from the present-day values of 163.58 W m −2 . This is because we assumed the cloud emissivity to be decreasing until it becomes equal to the present-day value of Mars. The values of the fluxes at different time intervals are illustrated in Table 1. The variation of various energy fluxes in the Earth's atmosphere beyond the main sequence is shown in Figure 3b.  Table 1.

CONCLUSION
We modeled the TSI variability in the evolutionary timescale. It shows a significant influence on the Earth's mean surface temperature. The following are the conclusions from the above analysis.
1. The TSI varies by 6.4% in 1 Gyr increasing from 1361 W m −2 to 1448.1 W m −2 2. The TSI increases to 1497.1 W m −2 (by 10%) from a present-day value in 1.42 Gyr from now.
3. The TSI increases by 40% at the end of 3.4 Gyrs from now. 4. The model also predicts the TSI variability when the Sun is at the end of the main sequence ton be about 2994.2 W m −2 at about 5.23 Gyrs from now.
5. The Earth's mean surface temperature does not change significantly with a value of 290.16 K at 1.644 Gyrs from now.
6. At about 3.4 Gyrs, when TSI increases to 1905.4 W m −2 , the Earth's surface temperature becomes 298.86 K.
7. The maximum Earth's temperature is predicted when the Sun becomes an RGB to be 2319.2 K turning Earth into a liquid planet.
8. The absorbed flux and emitted flux observed to be increasing equally and approach the surface flux at the end of the main sequence, while the flux absorbed by clouds is found to be decreasing.
9. Beyond the main sequence, the surface flux, absorbed flux and emitted flux become nearly equal attaining a peak value at about 7.676 Gyrs from now. However, the flux absorbed by the clouds decreases relative to others, although it peaks at the same points as other fluxes. code IPPS/AFRO5). N.T.S also thanks Dilla University for covering the living expenses in Ethiopia NG is supported by NASA's Living With a Star Program.