**2.2 Dynamics of the Moon's orbit in the initial epoch**

In the initial epoch *T* = 0, on the date of December 30, 1949 with the Julian-day number *JD0* = 2433280.5, consider the variation of the Moon's orbital parameters on a doubled interval of 736 revolutions, or 56.7 years. In order to distinguish between fluctuations, the results in **Figure 2** are shown for an interval of 10 years. The perigee radius *Rp* oscillates with a period *TRp* =0.5637 years around its average value *Rpm* = 3.622069 <sup>10</sup><sup>5</sup> km. The eccentricity of the orbit *<sup>e</sup>* oscillates with the same period around the mean value *em* = 0.0563331. In addition, there is a longer period of 3.719 years, yet exhibiting smaller oscillation amplitude.

The period of revolution of the Moon around the Earth *P* with respect to fixed stars, i.e. the sidereal period, oscillates around the average value *Pm* = 7.47928 <sup>10</sup><sup>2</sup> years. There are two oscillation periods lasting 0.664039 and 3.719 years.

Over the entire interval, the perigee angle *φ<sup>p</sup>* almost linearly increases into the future i.e. the perigee of the Moon's orbit rotates counterclockwise. The sidereal period of this rotation is *Tφ<sup>p</sup>* = 8.8528 years. In addition, the perihelion angle

#### **Figure 2.**

*Dynamics of the Moon's orbital elements in the geocentric equatorial coordinate system: perigee radius* Rp *– in km, period* P *and time* T *– in sidereal years with a duration of 365.25636042 days, angles* φp, iMo *and* φMo *– in radians; the white centerlines 1 and 2 are the approximating dependences (13) and (16), respectively. For other designations, see Figure 1.*

oscillates with a short period *Tφp1* = 0.5637 years and with a long period *Tφp2* =18.6006 years.

The inclination angle *iMo* oscillates around its mean value *iMom* = 0.41526. The oscillations occur with two periods: a short one, of 0.4745 year, and a long one, of 18.6006 years. The angle of the ascending node *φΩMo* oscillates around the mean value of *φΩMom* = 6.5472 � <sup>10</sup>�<sup>4</sup> with the same periods.

#### **2.3 Precession of the Moon's orbital axis**

When studying the orbits of the planets, we introduced the orbital axis *S* ! in the form of a unit-length vector normal to the orbital plane [8]. Using the inclination angle *iMo* and the ascending-node angle *φΩMo*, we can write the projections of the orbital axis onto the axes of the *xyz*-coordinate system:

$$\mathcal{S}\_{\text{Max}} = \cos i\_{\text{Mo}}; \quad \mathcal{S}\_{\text{Moy}} = \sqrt{\mathbf{1} - \mathcal{S}\_{\text{Max}}^{-2}} \cos \rho\_{\Omega \text{Mo}}; \quad \mathcal{S}\_{\text{Max}} = -\mathcal{S}\_{\text{Moy}} \text{tg} \rho\_{\Omega \text{Mo}} \tag{1}$$

The orbital axes of all planets precess about the angular momentum of all Solarsystem bodies. As a result of the study, it was found that the axis *S* ! *Mo* of the Moon's orbit precesses about the moving axis *S* ! *<sup>E</sup>* of the Earth's orbit (**Figure 1**). The same will also be shown below. We introduce a coordinate system *xMyMzM*. Along the *zM*axis of this system, the axis *S* ! *<sup>E</sup>* is directed, and the axis *xM* passes through the ascending node *γ<sup>2</sup>* of the Earth's orbit. Then, using the angles *iE* and *φΩ<sup>E</sup>* specifying the position of the Earth's orbital plane and the projections of the Moon's orbital axis according to formulas (29) in Melnikov and Smulsky [7], one can find the projections *SMoxM*, *SMoyM,* and *SMozM* of the axis *S* ! *Mo* onto the axes of the *xMyMzM*coordinate system. **Figure 3a** shows the motion of the endpoint of the orbital axis *S* ! *Mo*as projected onto the *yMxM*-plane over the examined time interval of 113.4 years. It is evident from the graph that the endpoint of the vector *S* ! *Mo* moves in a circle with slight fluctuations. The rotation period is *TS* = �18.6006 years, and the oscillation period is *Tψ<sup>1</sup>* = 0.4745 years. During the time interval under consider-!

ation, the axis *S Mo* makes six revolutions in the clockwise direction.

From the projection onto the *zMxM*-plane (**Figure 3b**), it can be seen that the endpoint of the vector *S* ! *Mo* executes small-amplitude oscillations along the *zM*-axis with a swing of *<sup>Δ</sup>zM* = 4.43 � <sup>10</sup>�<sup>4</sup> . Those oscillations are symmetrical about the *xM*-axis.

Thus, the Moon's orbital axis precesses in a clockwise direction relative to the Earth's orbital axis. The precession period *TS* is 18.6006 years. The precession proceeds with oscillations, which are called nutational. The period of the latter oscillations is *Tψ<sup>1</sup>* = 0.4745 years.

Such studies were carried out for each epoch following 10 thousand years over time intervals of 0 ÷ �2 million years and �98 ÷ �100 million years. As a result, it was found that in all these cases the Moon's orbital axis *S* ! *Mo* precesses relative to the moving axis *S* ! *<sup>E</sup>* of the Earth's orbit (**Figure 1**).

In the *xMyMzM*-coordinate system, the Moon's orbital axis *S* ! *Mo* (**Figure 1**) is specified by the inclination and precession angles (respectively, *θMo* and *ψMo*):

$$\theta\_{Mo} = \arccos S\_{MaxM}; \dots \\ \mu\_{Mo} = \text{arctg S}\_{MoyM}/S\_{MaxM} + \text{0.5} \pi \tag{2}$$

*The Evolution of the Moon's Orbit Over 100 Million Years and Prospects for the Research… DOI: http://dx.doi.org/10.5772/intechopen.102392*

#### **Figure 3.**

*Projections of the Moon's precession axis S*! *Mo onto the axes of the* xMyMzM *coordinate system over a period of 113.4 years (*a*,* b*) and from* T *= 0 to* T *= 18.6 years (*c, d*):* d *– in three-dimensional form; points: (1) projection of the axis S*! *Mo onto the* yMxM*-plane; (2) hypocycloid equation* yMhp*(*xMhp*) (4)–(5); and (3) position of the axis S*! *Mo at* T *= 0.*

Since the precession angle *ψMo* varies over ranges wider than 2π, for calculating continuous values of this angle, its values at adjacent time intervals were calculated, and then the values obtained were summed using certain rules. As a result of studying the variation of the angle *ψMo*, it was found that this angle decreases into

the future, i.e., the axis *S* ! *Mo* rotates in the clockwise direction with the rotation period of *TS* = �18.6006 years. In addition, the angle *ψMo* oscillates with a period of *Tψ<sup>1</sup>* = 0.4745 years and an amplitude of *ΔψMoA* = 0.023662. The inclination angle also oscillates with the latter period and with an amplitude *θMoA* = 0.002464 = 8.4692*'* about the mean value *θMom* = 0.09006 = 5.1544°.

**Figure 3c** shows, on a larger scale, the projection of the precessing orbital axis *S* ! *Mo* onto the *yMxM*-plane for one precession period *TS* = �18.6006 years. Starting from the moment *T* = 0, marked by point 3, the end of the axis moves clockwise. It is evident from the graph that the endpoint of the vector *S* ! *Mo* moves exactly around a circle, and the nutational oscillations here are regular. In **Figure 3d** dots and a line show the precession of the orbital axis in three dimensions. On the graph, the scale along the vertical *zM* axis is increased 40 times.

As a result of an analysis, it was found that the endpoint of the vector *S* ! *Mo* moves exactly along a hypocycloid. The hypocycloid is formed by some point of a circle of radius *r* rolling without slippage on the inner side of another circle of radius *R*.

In the *yMxM*-plane (**Figure 3c**), the radius of the great circle *R* = sin*θMom* is the

mean value of the projection of the orbital axis *S* ! *Mo* onto this plane, and the radius of the small circle *r*= sin*θMoA* is the oscillation amplitude of this axis. The center of the small circle moves in the clockwise direction with the angular velocity 2π/*TS*. In

this translational motion, the oscillations of the vector *S* ! *Mo* occur with the period *Tψ1*, or with the angular velocity 2π/*Tψ1*. Then, the absolute angular velocity of rotation of the small circle, 2π/*Tn*, will be equal to the sum of these velocities: 2π/*Tn* = 2π/*Tψ<sup>1</sup>* + 2π/*TS*. That is why the period of the nutational rotation will be

$$T\_n = \frac{T\_\ $ \cdot T\_{\psi1}}{T\_\$  + T\_{\psi1}} = 0.48692 \text{ years} \tag{3}$$

Then, in the *yMxM*-plane the equation of the hypocycloid can be written as follows:

$$\mathcal{X}\_{\text{Mhp}} = R \cos \left( \varphi\_{10} + 2\pi \frac{T}{T\_S} \right) + r \cos \left( \varphi\_{20} + 2\pi \frac{T}{T\_n} \right) \tag{4}$$

$$\varphi\_{Mhp} = R\sin\left(\varphi\_{10} + 2\pi\frac{T}{T\_S}\right) + r\sin\left(\varphi\_{20} + 2\pi\frac{T}{T\_n}\right) \tag{5}$$

where *φ*<sup>10</sup> = 4.92766 and *φ*<sup>20</sup> = 2.19315 are the initial phases that specify the position of the vector *S* ! *Mo* on the circles at the initial time *T* = 0.

The line in **Figure 3c** shows the trajectory of the motion along the hypocycloid, given by Eqs. (4) and (5), and the points are the projection of the motion of the Moon's orbital axis. Both are perfectly coincident. Thus, the Moon's orbital axis *S* ! *Mo* executes an averaged clockwise motion around the Earth's orbital axis *S* ! *<sup>E</sup>* with a period *TS* = �18.6006 years. Here, the average angle between the axes *S* ! *Mo* and *S* ! *<sup>E</sup>* is *θMom* = 5.1544°.

The orbital axis *S* ! *Mo* executes a second counterclockwise rotational motion about the averaged motion with a period *Tn* and an angular deviation from the median axis *θMoA* = 8.4692*'*. During the complete revolution of the averaged axis of the Moon's orbit, �*TS*/*Tψ<sup>1</sup>* = 39.2 nutational revolutions of the instantaneous axis *S* ! *Mo* occur.

The dynamics of the inclination and precession angles, *θMo* and *ψMo*, of the Moon's orbit over an interval of 20 years is shown in **Figure 4**. The oscillations of the angle *θMo* are more regular than those of the angle *iMo* of inclination of the Moon's orbit to the equatorial plane. They are harmonic with one and the same oscillation period. The precession angle *ψMo* executes similar oscillations. At the same time, it monotonically decreases, this decrease being indicative of a clockwise precession of the orbital axis *S* ! *Mo*. The light line shows the approximating time dependence of the precession angle

$$
\psi\_{Mo}(T) = \psi\_{Mo0} + 2\pi \cdot T/T\_S + \Delta\psi\_0 \tag{6}
$$

where *ψMo0* = 0.202798, and *TS* = �0.186006 is the period of precession of the Moon's orbital axis *S* ! *Mo*.

*The Evolution of the Moon's Orbit Over 100 Million Years and Prospects for the Research… DOI: http://dx.doi.org/10.5772/intechopen.102392*

#### **Figure 4.**

*Dynamics of the inclination and precession angles,* θMo *and* ψMo *of the Moon's orbit relative to the Earth's moving orbit. In the graph of* ψMo*, the line shows the approximating dependence (7).*

Changes in the Moon's orbit occur in the form of two groups of motions. In the first group, changes occur in the orbital plane with variation of the following parameters: perigee radius *Rp*(*T*), perigee angle *φр*(*T*), orbital eccentricity *e*(*T*), and the orbital period *P*(*T*). Here, the character *T* denotes the time dependence of the elements. In the second group, changes occur of the Moon's orbital plane *γMoA1B* (**Figure 1**), specified by the angles *φΩ<sup>E</sup>* and *iMo* relative to the equatorial plane *A0A0'* or by the angles *ψMo* and *θMo* relative to the moving plane of the Earth's orbit *EE'*. Since the latter angles change more regularly, using them is more preferable in describing the motion of the Moon's orbital plane.

#### **2.4 Approximation of the orbital-plane elements**

As it was noted above, the behavior exhibited by the Moon's orbital elements was studied for the period of 736 its continuous revolutions in different epochs over the interval from 0 to 100 million years. In addition, the elements of the Moon's orbit were investigated following the adoption of different initial conditions in the integration of the equations of motion using the Galactica program [7]. As a result of these studies, regularities of the dynamics of the elements were established, and the approximating dependences for them were chosen. The final form of the approximations was refined on a doubled interval from �736 to +736 revolutions, in which the meantime falls onto the epoch of December 30.0, 1949 with the Julianday number *JD0* = 2433280.5. The perigee radius is defined by the expression

$$R\_p(T) = R\_{pm} + R\_{pA} \cdot \sin\left(\rho\_{Rp0} + 2\pi \cdot T/T\_{Rp}\right) \tag{7}$$

where *Rpm* = 3.622069 � <sup>10</sup><sup>5</sup> km is the average value of the perigee radius, *RpA* <sup>=</sup> 6.2754 � <sup>10</sup><sup>5</sup> km is the amplitude of oscillations, *<sup>φ</sup>Rp0* = 0.942478 is the initial phase of oscillations, and *TRp* = 0.005637 is the period of oscillations of the perigee radius. The time *T* and the periods of oscillations are expressed here in sidereal centuries of 36525.636042 days per century, and they are counted from the *JD0* epoch, December 30, 1949.

As it is evident from the graph *Rp*(*T*) in **Figure 2**, there are oscillation beats of the perigee radius, which can be described by invoking a second harmonic with a large period. However, due to the irregularity of these beats over large time intervals, the second harmonic did not significantly improve the approximation of the perigee radius.

The eccentricity is approximated with two harmonics:

$$e(T) = e\_0 + e\_{A1} \cdot \sin\left(\wp\_{e01} + 2\pi \cdot T/T\_{\epsilon1}\right) + e\_{A2} \cdot \sin\left(\wp\_{e02} + 2\pi \cdot T/T\_{\epsilon2}\right) \tag{8}$$

whose characteristics are given in **Table 1**.

The perigee of the Moon's orbit rotates counterclockwise and, in addition, it executes oscillatory movements, which were also approximated with two harmonics:

$$\begin{aligned} \rho\_p(T) &= \rho\_{p0} + 2\pi \cdot T/T\_{qp} + \Delta\rho\_{p01} + \rho\_{pA1} \cdot \sin\left(\rho\_{p01} + 2\pi \cdot T/T\_{qp1}\right) + \Delta\rho\_{p02} + \rho\_{pA2} \\ &\cdot \sin\left(\rho\_{p02} + 2\pi \cdot T/T\_{qp2}\right) \end{aligned} \tag{9}$$

where *Tφ<sup>p</sup>* is the period of revolution of the Moon's perigee, and *Tφp1* and *Tφp2* are the first and second periods of oscillations of the perigee angle. The coefficients entering Eq. (9) are given in **Table 2**.

The Moon's orbital period *P* oscillates around its mean value *Pm*. The analysis of these oscillations was carried out considering the relative difference *δP* = (*P*�*Pm*)/ *Pm*. Since the period *P* and the semi-major axis *a* of the Moon's orbit vary consistently, the analysis of those oscillations in relative differences allows their consistent approximation. The period *P* is also approximated with two harmonics:

$$P(T) = P\_m \left[ \left( 1 + \Delta P\_0 + \Delta P\_{A1} \cdot \sin \left( \wp\_{pro2} + 2\pi \cdot T/T\_{p1} \right) + \Delta P\_{A2} \cdot \sin \left( \wp\_{pro2} + 2\pi \cdot T/T\_{p2} \right) \right) \right] \tag{10}$$

where *Tp1* and *Tp2* are the first and second oscillation periods of the orbital period, and the values of the coefficients are given in **Table 3**.


#### **Table 1.**

*Coefficients in Eq. (8).*


#### **Table 2.**

*Coefficients in Eq. (9).*


**Table 3.** *Coefficients in Eq. (10).* *The Evolution of the Moon's Orbit Over 100 Million Years and Prospects for the Research… DOI: http://dx.doi.org/10.5772/intechopen.102392*

Evidently, some parameters have identical oscillation periods. Perigee radius *Rp*(*T*), eccentricity *e*(*T*), and perigee angle *φp*(*T*) have identical first periods of 0.005637 century, whereas the eccentricity and the orbital period *P*(*T*) have identical second oscillation period of 0.03719 century.

### **2.5 Approximation of the orbital angles**

As a result of studies, it was found that the precession angle *ψMo* oscillates with two periods, a shorter (0.4745 years) and a longer (2.995 years) one. Since the amplitude of the large-period oscillations is small, we neglect those oscillations. As a result, the precession angle can be approximated with the following expression:

$$\Psi\_{\rm Mo}(T) = \psi\_{\rm Mo0} + 2\pi \cdot T/T\_S + \Delta\psi\_{\rm Mo0} + \Delta\psi\_{\rm MaA} \cdot \sin\left(\wp\_{\rm y} + 2\pi \cdot T/T\_{\rm y1}\right) \tag{11}$$

where *ψMo0* = 0.202798,*TS* = �0.186006 is the precession period of the Moon's orbital axis *S* ! *Mo*, *ΔψMo0* = 2.3024710�<sup>4</sup> , *ΔψMoA* = 0.023662, *φψ* = 2.82743, and *Tψ<sup>1</sup>* = 0.004745 is the period of oscillations of the precession angle *ψMo*.

The inclination angle *θMo* also oscillates with two periods. The longer period, equal to 2.995 years, has an amplitude of 5.978�10�<sup>5</sup> radians, which value is almost two orders of magnitude smaller than the amplitude of the first period. Therefore, the second harmonic, i.e. the one with the period of 2.995 years, can also be neglected, and the approximation for the nutation angle, therefore, has the form:

$$
\theta\_{\rm Mo}(T) = \theta\_{\rm Mo0} + \theta\_{\rm MoA} \cdot \sin\left(\wp\_{\theta} + 2\pi \cdot T/T\_{\rm \varphi 1}\right) \tag{12}
$$

where *θMo0* = 0.09006, *θMoA* = 0.002464, and *φθ* = �2.19911

The angles *ψMo* and *θMo* are tied to the moving plane of the Earth's orbit *EE'* (see **Figure 1**), so they are inconvenient to use. We, therefore, pass to the angles *φΩMo* and *iMo* , which specify the position of the Moon's orbital plane relative to the fixed plane of the equator *A0A0'* (**Figure 1**), with which the main coordinate system *xyz* is associated. In the spherical triangle *γ*2*γMoA1*, the side *γ*2*A1* = *ψMo* and the two angles *γ*<sup>2</sup> = *iE* and *A1* = *θMo* are known. The cosine theorem can be used to determine the obtuse angle *γ*2*γMoA1*, from which the acute angle *iMo* can be subsequently found: *iMo = π – γ*2*γMoA1*. As a result, for the angle of inclination of the Moon's orbital plane to the plane of the stationary equator, we obtain the following expression:

$$i\_{Mo} = \pi \text{-arccos}\left(-\cos\ i\_E \cdot \cos\theta\_{Mo} + \sin\ i\_E \cdot \sin\theta\_{Mo} \cdot \cos\varphi\_{Mo}\right) \tag{13}$$

As it is seen from **Figure 1**, the angle specifying the position of the ascending node of the Moon's orbit is equal to the sum of two arcs,

$$
\rho\_{\Omega \text{Mo}} = \chi\_0 \chi\_{\text{Mo}} = \rho\_{\Omega \text{E}} + \chi\_2 \chi\_{\text{Mo}} \tag{14}
$$

By the sine theorem, in the triangle *γ*2*γMoA1* we have

$$
\sin\ \chi\_2 \chi\_{\rm Mo} / \sin\theta\_{\rm Mo} = \sin\psi\_{\rm Mo} / \sin\left(\pi \text{-} i\_{\rm Mo}\right) \tag{15}
$$

and, therefore, the arc *γ*2*γMo* can be found. Then, according to Eq. (14), the position of the ascending node can be found as

$$
\rho\_{\Omega Mo} = \rho\_{\Omega E} + \arcsin\left[\sin\psi\_{Mo} \cdot \sin\theta\_{Mo}/\sin\left(\pi - i\_{Mo}\right)\right] \tag{16}
$$

In order to check the validity of the obtained approximations of the Moon's orbital elements (13) and (16), we superimposed onto **Figure 2** the calculated

#### **Figure 5.**

*Comparison of the dynamics of the angles* iMo *and* φΩMo*, specifying the position of the Moon's orbital plane relative to the equatorial plane, as obtained in two ways: thick lines – numerical integration; light thin lines 1 and 2 – approximating dependences (13) and (16), respectively.*

elements that were obtained using the Galactica program for the integration of the equations of motion. **Figure 5** shows, over the entire interval of � 56.7 years, the dynamics of the angles *iMo* and *φΩMo* obtained by two methods: using numerical integration (thick lines) and using approximations (13) and (16) (light line). As it is seen from the graphs, the approximations yield data perfectly coincident with the short- and long-period oscillations of the angles *iMo* and *φΩMo*. Thus, this check has fully confirmed the validity of the adopted approximations.

#### **2.6 Evolution of orbital elements over an interval of 100 million years**

So, the dynamics of Moon's orbital elements *Rp, e, φp, P, iMo*, and *φΩMo* relative to the fixed plane of the equator in the geocentric coordinate system *xyz* is described by Eqs. (7)–(10), (13), and (16). This description was obtained over a time interval of 113.4 years. As already mentioned, for establishing the validity of this description over large time intervals, studies were carried out over intervals of 0 ÷ �2 million years and �98 ÷ �100 million years. Following each 10 thousand years, the dynamics of the Moon's orbital elements were investigated during 736 continuous orbital revolutions of the Moon. The dynamics in different epochs did not differ qualitatively from that shown in **Figure 2**. With the purpose of comparison of those dynamics, the average values of individual elements during 736 orbital revolutions were calculated. Then, the evolution of these average values, as well as the periods of rotation, periods of oscillations, and oscillation amplitudes overtime periods of 2 million years with an interval between points of 10 thousand years, was investigated.

As an example, **Figure 6** shows the evolution of the average orbital period *Pm*, eccentricity *em*, inclination angle *θMom*, and the amplitude *θMoA* of nutational oscillations. The graphs show the relative changes of these quantities. These changes were determined in the same way, for example, for the average period of the Moon's orbital revolution this value was calculated as follows:

$$
\delta P\_m = (P\_m - P\_{m0}) / P\_{m0} \tag{17}
$$

where *Pm0* is the value of the average orbital period over 736 orbital revolutions of the Moon in the modern epoch. In calculating the relative changes of the

*The Evolution of the Moon's Orbit Over 100 Million Years and Prospects for the Research… DOI: http://dx.doi.org/10.5772/intechopen.102392*

**Figure 6.**

*Evolution, over the period of 2 million years, of relative averages for 736 revolutions of the deviations of Moon's orbital parameters: period* δPm*, eccentricity* δem*, inclination angle* δθMom*, and the amplitude of nutational oscillations* δθMoA*;* T *– in million years.*

amplitudes (*δθMoA*), instead of the mean values entering Eq. (17), the amplitude *θMoA* yielded by approximation (12) was used.

As it is seen from **Figure 6**, the oscillation amplitude of the relative mean *δPm*, *<sup>δ</sup>em* and *δθMom* are 2<sup>10</sup><sup>4</sup> , 0.003, and 4.5<sup>10</sup><sup>4</sup> , respectively. At the same time, the similar relative oscillation amplitudes during 736 Moon's orbital revolutions are 3.85<sup>10</sup><sup>3</sup> , 0.2, and 2.7<sup>10</sup><sup>2</sup> , respectively. Thus, the analyzed fluctuations of Moon's parameters *P*, *e*, and *θMo* exceed their changes over the interval of 0 ÷ 2 million years by factors of 19, 67, and 60, respectively. This conclusion is also confirmed by the graph *δθMoA*(*T*) in **Figure 6**: over the interval of 0 ÷ 2 million years, the amplitude of nutational oscillations *θMoA* fluctuates within 2%.

The rest approximation parameters exhibit similar behavior. Similar results were obtained for the interval of 98 ÷ 100 million years. This allows us to conclude that, over the interval of 0 ÷ 100 million years, if there occur oscillations with longer periods than those used in our approximations, then the amplitude of such oscillations does not exceed a few percent of the considered oscillation amplitudes.

#### **2.7 Mathematical model for the Moon's motion**

Thus, Eqs. (7)–(10), (13), (16) describe the evolution of Moon's orbital elements *Rp, e, φp, P, iMo*, and *φΩMo* which can be used over the interval 0 ÷ 100 million years. We have developed a mathematical model of body motion in an elliptical orbit [9], which is based on the listed orbital elements. That is why this model, with Eqs. (7)–(10), (13), and (16), allows one to calculate the Moon's coordinates in the equatorial coordinate system at any time in the interval of 0 ÷ 100 million years.

**Figure 7** compares the Moon's orbits calculated using this model with a time step of 1<sup>10</sup><sup>4</sup> years and numerical integration performed with the help of the Galactica program. The same orbital comparisons were made for the planets [9]. The orbits of the planets calculated by the mathematical model are no visual difference from the orbits obtained by numerical integration. As it is seen from **Figure 7**, such differences are observed for the Moon's orbit. This is due to the shorter Moon's orbital period compared to that of the planets. Nevertheless, this mathematical model of the Moon made it possible to solve the problem of the evolution of the Earth's rotational axis with acceptable accuracy. Comparison of the results of this problem for 200 thousand years, solved with this model of the Moon's orbit and without it, proved differences to be insignificant [1].

#### **2.8 Comparison of calculations with observation data**

The orbital periods of the Moon, the precession of its orbital axis, and the rotation of the perihelion oscillate about the average values of these quantities. Over

#### **Figure 7.**

*Comparison of the projections of the Moon's orbit onto the equatorial plane* xy *calculated in two ways: (1) based on the results of numerical integration, by the Galactica program, of the differential equations of motion of Solar-system bodies; (2) according to the mathematical model of the Moon's motion; and (3) the starting point of the orbit at the moment* T *= 0.*

*The Evolution of the Moon's Orbit Over 100 Million Years and Prospects for the Research… DOI: http://dx.doi.org/10.5772/intechopen.102392*

the interval of 113.4 years, the average values were designated as *Pm*,*TS*, and *Tφp*, respectively. Their magnitudes in sidereal years are given in **Table 4**. Astronomy considers different months with durations expressed in days. The sidereal month with a period *Pmsid* is specified relative to fixed stars. The synodic month with a period *Pmsyn* is specified in relation to the Earth. The sidereal orbital period of the Earth relative to stars is *PEsid* = 365.25636042 days. Therefore, the angular velocity of the Moon in its orbit around the Earth relative to it is equal to the difference between the angular orbital velocities of the Moon and the Earth in relation to stars. Therefore, the duration of the synodic month is

$$P\_{myn} = \frac{P\_{Eid} \cdot P\_m}{P\_{Eid} - P\_m} \tag{18}$$

where the period *Pm* = 7.479277 � <sup>10</sup>�<sup>2</sup> sidereal years expressed in days is equal 27.318536.

The period *Pmano* of the anomalistic month is specified in relation to the Moon's perigee or its apogee. The period of motion of the Moon's perigee relative to stars is denoted as *Tφp*. Therefore, the period of the anomalous month is

*Pmano* <sup>¼</sup> *<sup>T</sup>φ<sup>p</sup>* � *Pm Tφ<sup>p</sup>* � *Pm* (19)

where the period *Tφ<sup>p</sup>* is expressed in days.

The Draconic month with a period *Pmdra* is specified in relation to the Moon's ascending node. The position of the ascending node *γMo* is specified by the angle *φΩMo* (**Figure 1**), and its motion relative to fixed stars occurs with the precession period *TS* of the Moon's orbital axis *S* ! *Mo*. Therefore, the draconic-month period is

$$P\_{mdrat} = \frac{T\_S \cdot P\_m}{T\_S - P\_m} \tag{20}$$

where the period *TS* is expressed in days.

A tropical month with a period *Pmtro* is defined in relation to the Earth's moving equator *AA'* in **Figure 1**. The moving equator, as well as the Earth's axis of rotation, precess relative to fixed stars with a period of *PprEax* = �25738 sidereal years [1]. Therefore, the period of the tropical month will be:


#### **Table 4.**

*Comparison of calculated and observed average durations of various months: sidereal* Pmsid*, synoptic* Pmsyn*, anomalistic* Pmano*, and draconian* Pmdra*.*

$$P\_{mtro} = \frac{P\_{prExx} \cdot P\_m}{P\_{prExx} - P\_m} \tag{21}$$

where the period *PprEax* is expressed in days.

These periods, as calculated by Eqs. (18)–(21) and as evaluated from the observations of [10] are summarized in **Table 4**. The relative difference between the calculated and observed periods is expressed in terms of a parameter *δ* defined similarly to Eq. (17). As it is seen, the largest value of *<sup>δ</sup>* is 1.24 � <sup>10</sup>�<sup>4</sup> . The main contribution to this difference is made by the sidereal period *Pm* of Moon's orbital revolution. If we use the observed value of *Pm* = 27.321662 days, then the *δ*-values will decrease by two-five orders of magnitude.

As it is seen from **Figure 2**, the Moon's orbital period *P* experiences oscillations with relative amplitudes *ΔPA1* and *ΔPA2*, which in total make up 0.0053 part of the period *P*. In addition, from **Figure 6** it is seen that over time intervals of tens of thousands and more years there exist oscillations of the average period *Pm* with a relative amplitude of the order of 2 � <sup>10</sup>�<sup>4</sup> . For oscillating quantities, their average values depend on the interval over which the averaging is performed. The value of *Pm* given in **Table 4** was obtained by an averaging performed over an interval of 113.4 years, and the value of the sidereal period in astronomy has an averaging interval of about 2 thousand years. This seems to be the main reason for the difference between the calculated and observational data with a relative value of the order of 1 � <sup>10</sup>�<sup>4</sup> .
