**4.3 Imbalance of the TSI depositions in the two millennia**

Since there is a shift of the minimum point of the TSI annual variations (**Figure 13**) from 21 June (M1) to 15 July (end of M2), this indicates a possible imbalance between the annual TSI input and output in M2 (1600–2600). From the daily magnitudes of TSI shown in **Figures 11** and **12**, it is possible to count the total annual amount of TSI emitted by the Sun towards the Earth in each year of the both millennia. If this amount does not change from year to year, then TSI is, indeed, the same for each year for both centuries, as currently assumed.

However, the real annual magnitudes of TSI deposited to the Earth during the two millennia are shown in **Figure 14** calculated for the two cases: (a) added together the averaged monthly TSI magnitudes (left plot) calculated for the S-E distances shown in **Figure 13** when only 12 magnitudes per year (for 12 months);

#### **Figure 14.**

*The total annual TSI variations (W=m*<sup>2</sup>*) in the millennia M1 and M2 derived by summation of the mean monthly (top) and daily TSI magnitudes (bottom). Axis X shows the years of the millennia.*

*Millennial Oscillations of Solar Irradiance and Magnetic Field in 600–2600 DOI: http://dx.doi.org/10.5772/intechopen.96450*

(b) added together the daily TSI magnitudes (right plot) taken from **Figures 11** and **12** associated with the daily magnitudes of TSI (for 366 days for the leap years used).

These two plots clearly demonstrate that the monthly TSI variations (case a) show the increase of TSI by about 1–1.3 *W=m*<sup>2</sup> in 2020 compared to 1700 (**Figure 14**, left plot). This TSI increase found from the S-E distance ephemeris is close to the magnitude of 1–1.5 *W=m*<sup>2</sup> reported from the current TSI observations [34]. However, the annual TSI magnitudes, calculated from the daily S-E distances (case b) reveal a much larger annual increase of the total solar irradiance by about 20–25 *W=m*<sup>2</sup> (> 1.8%) in M2 (by 2500) than in millennium M1 (**Figure 14**, right plot). This analysis gives the indication of the averaged TSI increase in M2 could be 2.5–2.8 *W=m*<sup>2</sup> per century, or (0.18–0.20)%, comparing to the TSI in 1700. This is the very important hidden solar irradiance input in millennium M2 (1600–2600) caused by the SIM effects, which was significantly underestimated if only the averaged monthly TSI magnitudes are used (**Figure 14**, compare left and right plots). The essential issue is how much of this extra solar radiation is distributed between the hemispheres owing to the Earth tilt, its position on the orbit or the level of exposure to solar radiation [42, 49].

Our study of the S-E distance variations shows that at the start of any year, in January, the Earth is turned to the Sun with its southern hemisphere, meaning that any decrease and increase of solar radiation during this time is mostly absorbed by the parts in Southern hemisphere. When the Earth's orbiting approaching March, the distribution of solar irradiance between the hemispheres becomes nearly even, while in April–June the main part of the solar radiation input is shifted towards the Northern hemisphere, having its maximum theoretically (by Kepler's law) on 21 June, while in reality, shifted to 5 July in 2020 and to 16 July in 2500. Hence, in M2 the Northern hemisphere should get the extra solar radiation not only in the first six months of a year but also in the 25 days from 21 June by approaching the local aphelion on 16 July, which is not compensated later by its expected cooling because of a shift of the local perihelion to 16 January.

By comparing the mean-by-time and mean-by-arc S-E distances for an elliptic orbit (see Appendices B and C) based on the calculated shifts of aphelion and perihelion [49] with the real S-E distances derived from the ephemeris one can conclude that the ephemeris of the S-E distances have to reflect the Sun shifts in SIM, in addition to the Earth revolution about the ellipse focus. Therefore, the solar radiation deposition in the millennium M2 is expected to be essentially higher than in in millennium M1 and different from the standard seasonal changes because of the uneven shifts of Sun-Earth distances on the orbit owing to SIM. This extra TSI amount caused by SIM (from the variations of a distance *d* in formula (2) will undoubtedly add to the magnitude of solar irradiance coming from the solar activity itself (or the parameter *I*<sup>⊙</sup> in formula (2)) shown in **Figure 4** (bottom plot, blue lines) leading to the overall solar irradiance increase that, in turn, can account for a large amount of the terrestrial temperature increase shown by the red curves in **Figure 4** (bottom plot). This extra solar forcing caused by SIM needs to be taken into account in any climate models.
