**4. Millennial oscillations of solar irradiance with the Sun-Earth distances**

#### **4.1 Method of inverse squares**

Following the variations of the S–E distances discussed in section 3, let us evaluate the variations of total solar irradiance (TSI) imposed by a change of these S-E distances in the millennia M1 and M2 using the method of inverse squares. A magnitude of the total solar irradiance *S* variations at the solar-Earth distance *d* by considering the Sun as a point body emitting radiation with an intensity *I*<sup>⊙</sup> [53]:

$$\mathcal{S} = \frac{I\_{\odot}}{d^2}.\tag{2}$$

help of Appendices A and B. However, the S-E distance reductions and growths reported here deviate from the Kepler's third law (Eq. 6 in Appendix A). By comparing the mean-by-time and mean-by-arc S-E distances for an elliptic orbit (see Appendices B) with the expected changes imposed by the calculated shifts of aphelion and perihelion [49] shown in Appendix C, it is evident that the real S-E distances derived from the ephemeris are different from Kepler's 3rd law (see Eq. 6

This can can only happen if these S-E ephemeris reflect the additional motion: the revolution of the Sun about the barycentre, which is induced by the action of large planets of the solar system. The similar effect is observed in the stars, which have planetary systems, leading to a wobbling star effect that is used to trace possible exoplanets [50, 51]. The shift of S-E distances reported above should be caused by the increasing shift of the Sun's location from the focus of the ellipse, where it is supposed to reside, according to Kepler's laws, towards the spring equinox of the Earth orbit. This shift of the position of the Sun with respect to the barycentre has been recognised as the solar inertial motion - SIM [25, 27, 30]. The resulting S-E distances are defined by the superposition of these two motion: Earth

In fact, the variations of the S-E distances during the two millennia are likely to be affected by the gravitational effects of Jupiter, Saturn, Uranus and Neptune on the Sun's inertial motion [30] revealing the oscillations of the planet orbits with a period of 8.5 thousand years (see Fig. 1 in [30], affecting SIM. From the whole period of 8.5 thousand years reported in the paper the semi-period with maximum of 4.2–4.3 thousand years with the ascending part of 2.1 thousand years are similar to the period of decreasing S-E distances reported in section 3.1 for 600–2600. Also the reported S-E distances reveal the noticeable shifts of the aphelion and perihelion from the major axis of the ellipse that coincides also with the oscillations of magnetic field baseline [19, 52] and solar irradiance [22]. It seems that in the two millennia 600–2600 the large planets continuously shifted the Sun from its focus towards the spring equinox as detected from the S-E ephemeris in **Figures 5** and **6**. Therefore, it can be noted that owing to SIM, the shortest and longest Sun-Earth distances (perihelion and aphelion) in the elliptic orbit of the Earth are shifted to the local aphelion and perihelion, which are located on the shorter axis of the ellipse than the major axis. This line has an angle *ϕ* to the semi-major axis roughly defined

tan *<sup>ϕ</sup>* <sup>¼</sup> <sup>2</sup>*ds*

semi-minor axis *b* of the Sun from the focus of the ellipse. Naturally, by the definition of an ellipse, this line is shorter than the semi-major axis *a* of the Earth

elliptic orbit, which is the longest axis in the ellipse.

where *f* is a distance between the foci of the ellipse and *ds* is the shift along the

Furthermore, the calculations of the double differences between the maximal distance shifts occurred in millennia M1 and M2 (M1-M2) for daily data shown in **Figure 8** and their annual variations shown in **Figure 10** reveal that the double differences become negative in April and remain such until the end of October. This means that in M2 (1600–2600) the S-E distance decreases in April–July and its increases in July–December are much larger that in M1 (600–1600). This also indicates that in M2 the Sun becomes closer and closer to the Earth during April– October before the Earth revolution will make the S-E distance increases in November–February, since these increases are larger than expected from Kepler's third law. This, in turn, can lead to a significant solar radiation input to the Earth in

*<sup>f</sup>* , (1)

in Appendix A).

*Solar System Planets and Exoplanets*

revolution and SIM.

by the formula for tan *ϕ*:

**40**

Hence, the solar irradiance *S* can vary either because of the variations of intensity *I* of solar radiation at the Sun itself or because of the variations of a distance *d* between the Sun and Earth. The variations of the solar intensity *I* is caused by the variations of solar activity induced by the electro-magnetic dynamo action in the solar interior.

If the intensity *I*<sup>⊙</sup> of radiation on the Sun is considered to be constant at a given time (*I*⊙=const), then the solar irradiance *S* can also change because of a variation of the Sun-Earth distance caused by the Earth orbital motion itself leading to the terrestrial seasons and by solar inertial motion whose effects are not yet fully investigated. In any case, by knowing the ephemeris of the S-E distances and using Eq. (2) above for calculating solar irradiance at two different distances *d*<sup>1</sup> and *d*2, one can find the relationship between the solar irradiance, *S*<sup>1</sup> and *S*<sup>2</sup> at these distances, which follows the inverse square law [53]:

$$\mathbb{S}\_1 \cdot d\_1^2 = \mathbb{S}\_2 \cdot d\_2^2. \tag{3}$$

Therefore, if at a distance *d*<sup>1</sup> the average solar irradiance is 1366 *W=m*<sup>2</sup> [22, 31] then if the distance is changed to *d*2, the solar irradiance *S* should also change following the Eq. (3). For example, if the distance *d*<sup>2</sup> between the Earth and Sun was to be decreased by 0.016 au (as shown in section 3 for two millennia 600–2600) so that the initial irradiance of 1366 *W=m*<sup>2</sup> divided by the square of the new distance results in the irradiance of 1411 *W=m*2. The difference in the irradiance is 1411–1366 = 45 *W=m*2, that is 3.3% that is exactly the magnitude mentioned in the first paragraph of the last section of paper [19].

In section below the solar irradiance is explored in more details for the two millennia from 600 to 2600 AD for the S-E distances presented in section 3.
