**C. Average distances versus aphelion/perihelion variations.**

The first step in investigation of millennial variations of solar irradiance and baseline magnetic field came from a suggestion of changing Sun-Earth distances because of a change of the Earth orbit shape. In fact, the ephemeris show [48] that since 1600 the Earth orbit's aphelion is found steadily decreasing while its perihelion is increasing (see **Figure 16**). Could this change of the Earth orbit cause the millennial changes of solar irradiance and magnetic field baseline? Let us explore this option. Sun is located in the ellipse focus, the S–E distance variations are affected by the variations of the Earth orbit parameters: aphelion and perihelion distances which are calculated by JPL. The link between the semi-minor axis *b* and the semi semi-major axis *a* can be written as follows:

$$2b = \sqrt{\left(d\_1 + d\_2\right)^2 - f^2};\tag{24}$$

where *d*<sup>1</sup> and *d*<sup>2</sup> are the distances from the two ellipse foci to any point on the orbit, f is the distance between the foci of the ellipse, e.g. *f* ¼ *Ra* � *Rp*. For

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

**Figure 16.**

*dt* ¼ *a* 1 þ

The integral for the arc length of an ellipse cannot be evaluated in finite terms, we need to proceed indirectly utilising the defining property of an ellipse that the sum of the distances from any point of the ellipse for the two foci is constant as

> ð*L* 0

which is larger than the semi-major axis a by a factor of (1 <sup>þ</sup> *<sup>e</sup>*2*=*2).

described by Eq. (9). Let *L* to be the whole length of the orbit.

ð*L* 0

*ds* <sup>¼</sup> <sup>1</sup> *L* ð*L* 0

of the star if there are no other gravitational effects are considered.

**C. Average distances versus aphelion/perihelion variations.**

the semi semi-major axis *a* can be written as follows:

**54**

2*b* ¼

q

orbit, f is the distance between the foci of the ellipse, e.g. *f* ¼ *Ra* � *Rp*. For

ð*L* 0 *d*1*ds* ¼

Since by the definition of the ellipse *d*1 þ *d*2 ¼ 2*a* (see Eq. (9), hence:

ð*L* 0

Then the distance *rs* averaged by the arc length is given by the expression:

Therefore, for the ideal revolution of a planet in ellipse about the star located in the ellipse focus, the average distance of the planet from the focus is defined by the parameters of the ellipse, along which the planet moves about the assumed location

The first step in investigation of millennial variations of solar irradiance and baseline magnetic field came from a suggestion of changing Sun-Earth distances because of a change of the Earth orbit shape. In fact, the ephemeris show [48] that since 1600 the Earth orbit's aphelion is found steadily decreasing while its perihelion is increasing (see **Figure 16**). Could this change of the Earth orbit cause the millennial changes of solar irradiance and magnetic field baseline? Let us explore this option. Sun is located in the ellipse focus, the S–E distance variations are affected by the variations of the Earth orbit parameters: aphelion and perihelion distances which are calculated by JPL. The link between the semi-minor axis *b* and

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *d*<sup>1</sup> þ *d*<sup>2</sup>

where *d*<sup>1</sup> and *d*<sup>2</sup> are the distances from the two ellipse foci to any point on the

<sup>2</sup> � *<sup>f</sup>* 2

; (24)

**B.3 Average distance in arc length**

*Solar System Planets and Exoplanets*

From the point of symmetry:

*e*2 2 � �

, (19)

*d*2*ds:* (20)

ð Þ *d*<sup>1</sup> þ *d*<sup>2</sup> *ds* ¼ 2*aL*, *or* (21)

*d* � *ds* ¼ *aL:* (22)

*d* � *ds* ¼ *aL=L* ¼ *a:* (23)

*Variations of the aphelion (left) and perihelion (right) distances of the Earth orbit in the millennium M2.*

calculation of *d*<sup>1</sup> þ *d*<sup>2</sup> we use the Eq. (9) that provides the relations for the semimajor *a* and the semi-major *b* distances via the aphelion *Ra* and perihelion *Rp* as follows:

$$2b = \sqrt{\left(d\_1 + d\_2\right)^2 - f^2} = 2\sqrt{R\_a R\_p} \tag{25}$$

$$a = \frac{R\_a + R\_p}{2}.\tag{26}$$

Hence, if we fix for some time *t*<sup>0</sup> the aphelion *Ra*<sup>0</sup> and perihelion *Rp*<sup>0</sup> distances, and assume that they are proportionally changed after some time by a magnitude Δ, so that *Ra* ¼ *Ra*<sup>0</sup> � Δ and *Rp* ¼ *Rp*<sup>0</sup> þ Δ, so that *Ra* þ *Rp* ¼ *Ra*<sup>0</sup> þ *Rp*<sup>0</sup> and *f* ¼ *Ra*<sup>0</sup> � *Rp*<sup>0</sup> � 2Δ. Then from Eq. (24) the semi-minor axis can be calculated as follows:

$$\begin{split} \mathbf{2b} &= \sqrt{\left(d\_1 + d\_2\right)^2 - f^2} = \sqrt{\left(R\_{a0} + R\_{p0}\right)^2 - \left(R\_{a0} - R\_{p0} - \mathbf{2}\Delta\right)^2} = \\ &= \sqrt{4\mathbf{R}\_{a0}\mathbf{R}\_{p0} + 4\boldsymbol{\Delta}\left[\mathbf{R}\_{a0} - \mathbf{R}\_{p0} - \boldsymbol{\Delta}\right]}. \end{split} \tag{27}$$

Giving the relationships between the the ellipse axises *a* and *b* and aphelion and perihelion distances.

$$\mathcal{a} = \mathcal{R}\_a + \mathcal{R}\_b = \mathcal{R}\_{a0} + \mathcal{R}\_{p0};\tag{28}$$

$$b = \sqrt{R\_{a0}R\_{p0} + \Delta \left[R\_{a0} - R\_{p0} - \Delta\right]}.\tag{29}$$

It is evident from the equations above that variations of the aphelion and perihelion distances will affect only the Earth semi-minor axis *b* and, thus, eccentricity *e*. Let us use them for estimation of the Earth orbit parameters from the ephemeris of aphelion and perihelion presented in **Figure 16**.

In the case of decreasing aphelion and increasing perihelion distances for a elliptic orbit with the star in its focus shown in **Figure 16**, it occurs from Eq. (29) that *b* will increase, while the orbit eccentricity *e* would decrease. By comparing the variations of the Earth aphelion and perihelion distances in 1500–2500 we evaluated the following changes. The variations of the aphelion and perihelion distances produce the reduction of eccentricity from 0.0170 in 1500 to 0.0163 in 2500. This would lead to a change of the average-by-time Sun-Earth distance from 1.0001462 au in 1600 to 1.0001328 au in 2500, e.g. the difference is virtually negligible and cannot be reflected in a noticeable change of solar irradiance. This indicates that over the whole millennium 1600–2600 the Earth orbit remains, in fact, a pretty stable elliptic orbit. However, the average S-E distances in this elliptic orbit do not change to such the extent to produce noticeable variations of solar irradiance or magnetic field baseline.

*Solar System Planets and Exoplanets*
