**A. Basics of planetary orbits.**

In order to investigate the orbital effects on the distance between the Sun and Earth and the resulting variations of solar irradiance imposed by these variations, let us first remind the basic laws governing the planet revolution about a central star, the Sun. It is suggested that the planets evolution about the central star (Sun) is defined by Kepler's three laws [61]:


The Sun is located in one of the two foci (point -C in **Figure 15**) of the ellipse with a semi-major axis *a* and a semi-minor axis *b* and the eccentricity:

$$e = \sqrt{1 - \frac{b^2}{a^2}},\tag{4}$$

It can be noted that is a link between the semi-major and semi-minor axis:

**Figure 15.**

*Elliptic orbit of the earth revolution about the sun located in the left focus of the ellipse with a semi-major axis a, a semi-minor axis b and the eccentricity e.*

$$b^2 = a^2(1 - e^2). \tag{5}$$

**B.1 Planet distance averaged in angle**

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

*<sup>d</sup>* � *<sup>d</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup>

**B.2 Planet distance averaged in time**

sweeps out an area at a constant rate h:

2*π* ð2*<sup>π</sup>* 0

*<sup>d</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup> 2*π* ð2*<sup>π</sup>* 0

so that

revolution T:

or

*dt* <sup>¼</sup> <sup>1</sup> *T* ð*T* 0

where

*dt* <sup>¼</sup> <sup>1</sup> *T* ð*T* 0

**53**

time can be defined as:

*<sup>d</sup>* � *dt* <sup>¼</sup> *<sup>h</sup>*

*<sup>π</sup>ab* <sup>ð</sup><sup>2</sup>*<sup>π</sup>* 0

1 <sup>2</sup>*πab* <sup>ð</sup><sup>2</sup>*<sup>π</sup>* 0 *d*3

Then the average in time distance is

<sup>2</sup>*πab* <sup>¼</sup> <sup>3</sup>*a=*<sup>2</sup> � *<sup>b</sup>*<sup>2</sup>

*ddt* <sup>¼</sup> *<sup>b</sup>* <sup>3</sup>*a*<sup>2</sup> � *<sup>b</sup>*<sup>2</sup> � �*<sup>π</sup>*

or the averaged by time distance is

Average distance *d<sup>θ</sup>* in angle *θ* is defined by the integral:

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

*Pe* 1 � *ecosθ*

*<sup>d</sup><sup>θ</sup>* <sup>¼</sup> *<sup>a</sup>* ffiffiffiffiffiffiffiffiffiffiffiffi

Average in time distance *dt* is defined by the integral for the period of a planet

*dt* <sup>¼</sup> <sup>1</sup> *T* ð*T* 0

*<sup>h</sup>* <sup>¼</sup> <sup>1</sup> 2 *d*2

*dt=d<sup>θ</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

*<sup>d</sup>* � ð Þ *dt=d<sup>θ</sup> <sup>d</sup><sup>θ</sup>* <sup>¼</sup> *<sup>h</sup>*

We know that the area of the ellipse is *Th* ¼ *πab*. Hence, the average distance in

*<sup>d</sup><sup>θ</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>3</sup>*e*<sup>3</sup> <sup>2</sup> <sup>þ</sup> *<sup>e</sup>*<sup>2</sup> ð Þ*<sup>π</sup>*

*<sup>=</sup>*2*<sup>a</sup>* <sup>¼</sup> <sup>3</sup>*<sup>a</sup>*

*<sup>π</sup>ab* <sup>ð</sup><sup>2</sup>*<sup>π</sup>* 0 *d d*2

Since according to the second Kepler's law, the radial arm of a given planet

Hence, the average distance by *θ* is equal to the semi-minor axis *b*.

*<sup>d</sup><sup>θ</sup>* <sup>¼</sup> <sup>2</sup>*πPe*

s�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>e</sup>*<sup>2</sup> <sup>¼</sup> *<sup>a</sup>* <sup>1</sup>�*e*<sup>2</sup> ð Þ

<sup>1</sup> � *<sup>e</sup>*<sup>2</sup> <sup>p</sup> <sup>¼</sup> *<sup>b</sup>*, (12)

*ddt*, (13)

*dθ=dt*, (14)

*=*2*h:* (15)

<sup>2</sup>*πab* <sup>ð</sup><sup>2</sup>*<sup>π</sup>* 0 *d*3

<sup>2</sup>*<sup>a</sup>* <sup>¼</sup> *<sup>a</sup>* <sup>1</sup> <sup>þ</sup>

*dθ*, (16)

*e*2 2 � �*:* (18)

<sup>2</sup>*<sup>h</sup> <sup>d</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup>

<sup>1</sup> � *<sup>e</sup>*<sup>2</sup>Þð Þ <sup>5</sup>*=*<sup>2</sup> <sup>¼</sup> *<sup>b</sup>* <sup>3</sup>*a*<sup>2</sup> � *<sup>b</sup>*<sup>2</sup> � �*π:* (17)

<sup>2</sup> � *<sup>a</sup>*<sup>2</sup> <sup>1</sup> � *<sup>e</sup>*<sup>2</sup> ð Þ

ffiffiffiffiffiffiffiffi <sup>1</sup>�*e*2<sup>Þ</sup> <sup>p</sup> , (11)

2*π*

The planet is located in the point (x,y) on the orbit on a distance *d* from the focus C under the angle *θ* to the major axis (see **Figure 15**, right plot).

This distance *d* is defined by Kepler's third law as follows:

$$d = \frac{a(1 - e^2)}{1 - e \cos \theta},\tag{6}$$

where 0 ≤*e*≤1.

Let us introduce *Ra* the aphelion distance from the focus where the star is located, to the longest point of the orbit along the major axis and *Rp* - the perihelion distance from the focus to the closest point along the major axis. Using this Eq. (6) one can calculate the aphelion *Ra* and perihelion *Rp* distances by setting the angle *θ* equal to zero for aphelion and 180<sup>∘</sup> for perihelion, e.g.

$$R\_{\mathfrak{a}} = \mathfrak{a}(\mathfrak{1} + \mathfrak{e}),\tag{7}$$

$$R\_p = \mathfrak{a}(\mathfrak{1} - \mathfrak{e}).\tag{8}$$

The sum of the distances *d*<sup>1</sup> and *d*<sup>2</sup> from a planet location on the orbit to the both foci of the ellipse is constant and equal 2*a*, e.g.

$$d\_1 + d\_2 = 2\mathfrak{a} = R\_{\mathfrak{a}} + R\_p. \tag{9}$$

where bf d1 and *d*<sup>2</sup> are distances from the two foci to the current position of a planet (see **Figure 15**, left plot).
