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

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

From the point of symmetry:

$$\int\_{0}^{L} d\_{1}ds = \int\_{0}^{L} d\_{2}ds.\tag{20}$$

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

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

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

*<sup>a</sup>* <sup>¼</sup> *Ra* <sup>þ</sup> *Rp*

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

*Ra*<sup>0</sup> þ *Rp*<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4*Ra*0*Rp*<sup>0</sup> þ 4Δ *Ra*<sup>0</sup> � *Rp*<sup>0</sup> � Δ <sup>q</sup> � �*:*

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

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

the ephemeris of aphelion and perihelion presented in **Figure 16**.

Giving the relationships between the the ellipse axises *a* and *b* and aphelion and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Ra*0*Rp*<sup>0</sup> þ Δ *Ra*<sup>0</sup> � *Rp*<sup>0</sup> � Δ

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,

<sup>2</sup> � *<sup>f</sup>* 2

¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*a* ¼ *Ra* þ *Rb* ¼ *Ra*<sup>0</sup> þ *Rp*0; (28)

<sup>q</sup> � �*:* (29)

� �<sup>2</sup> � *Ra*<sup>0</sup> � *Rp*<sup>0</sup> � <sup>2</sup><sup>Δ</sup> � �<sup>2</sup> <sup>q</sup>

ffiffiffiffiffiffiffiffiffiffi *RaRp* q

<sup>2</sup> *:* (26)

¼

(25)

(27)

2*b* ¼

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

¼

<sup>2</sup> � *<sup>f</sup>* 2

*b* ¼

¼

q

*Millennial Oscillations of Solar Irradiance and Magnetic Field in 600–2600*

*DOI: http://dx.doi.org/10.5772/intechopen.96450*

follows:

**Figure 16.**

follows:

2*b* ¼

perihelion distances.

magnetic field baseline.

**55**

q

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

$$\int\_{0}^{L} (d\_1 + d\_2) ds = 2dL, or \tag{21}$$

$$\int\_{0}^{L} d \cdot ds = aL. \tag{22}$$

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

$$\overline{d}\_{\circ} = \frac{1}{L} \int\_{0}^{L} d \cdot ds = aL/L = a. \tag{23}$$

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