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

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

$$\overline{d}\_{\theta} = \frac{1}{2\pi} \int\_{0}^{2\pi} d\cdot d\theta = \frac{1}{2\pi} \int\_{0}^{2\pi} \frac{P\_{\epsilon}}{1 - e \cos \theta} d\theta = \frac{2\pi P\_{\epsilon}}{2\pi \sqrt{\left(1 - \epsilon^{2} = \frac{a(1 - \epsilon^{2})}{\sqrt{1 - \epsilon^{2}}}\right)}},\tag{11}$$

so that

*<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> <sup>1</sup> � *<sup>e</sup>*

*Elliptic orbit of the earth revolution about the sun located in the left focus of the ellipse with a semi-major axis a,*

The planet is located in the point (x,y) on the orbit on a distance *d* from the

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

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*

The sum of the distances *d*<sup>1</sup> and *d*<sup>2</sup> from a planet location on the orbit to the both

where bf d1 and *d*<sup>2</sup> are distances from the two foci to the current position of a

There are three average distances of a planet from the star can be calculated averaged in: (a) the angle *θ*; (b) time; and (c) arc length *s* [62]. The formula (6) for the planet distance from the ellipse focus, according to Kepler's third law, can be re-

> *<sup>d</sup>* <sup>¼</sup> *Pe* 1 � *ecosθ*

**B. Average distances of a planet from the focus of ellipse**

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:

equal to zero for aphelion and 180<sup>∘</sup> for perihelion, e.g.

foci of the ellipse is constant and equal 2*a*, e.g.

planet (see **Figure 15**, left plot).

written as follows:

**52**

where *Pe* <sup>¼</sup> *<sup>a</sup>* <sup>1</sup> � *<sup>e</sup>*<sup>2</sup> ð Þ.

where 0 ≤*e*≤1.

*a semi-minor axis b and the eccentricity e.*

*Solar System Planets and Exoplanets*

**Figure 15.**

<sup>2</sup> *:* (5)

, (6)

*:* (10)

*Ra* ¼ *a*ð Þ 1 þ *e* , (7) *Rp* ¼ *a*ð Þ 1 � *e :* (8)

*d*<sup>1</sup> þ *d*<sup>2</sup> ¼ 2*a* ¼ *Ra* þ *Rp:* (9)

$$
\overline{d}\_{\theta} = a\sqrt{1 - e^2} = b,\tag{12}
$$

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