**4. Mathematical models and equations of motion**

For many years, observing the world around us, scientists have tried to understand, describe, or predict the dynamics of various objects: the movement and flight of a stone or spear after a throw, and the movement of stars or comets in the sky. The properties, principles, and patterns of processes were gradually formed.

**Aristotle** formed the general principles of movement, created a theory of the movement of the celestial spheres, and considered the source of movements to be forces caused by external influences.

**Kepler**, based on the results of processing tables of observations of planetary motion, taking into account the **Copernican** hypothesis, discovered the laws of planetary motion, proposed convenient parameters for describing possible orbits, and laid the foundation for celestial mechanics.

**Laplace**, in his multivolume monograph, completed the creation of celestial mechanics based on the law of universal gravitation.

The use of mathematical modeling methods makes it possible to form various equations of dynamics, to study solutions or properties for a selected set of bodies, taking into account the main forces of interaction, when all other conditions are considered insignificant. This allows more complex variants, conditions, and models to be studied later, as well as to obtain new results.

For the forces of attraction, **Galileo's** experiments made it possible to introduce a uniform gravitational field and motion with constant acceleration *g* ¼ const:

$$mw = \mathbf{F} = m\mathbf{g} \tag{1}$$

The law of universal gravitation, when describing the motion of bodies in absolute space, allowed **Newton** to obtain mathematical models of the gravitational field and exact solutions that confirm **Kepler's** laws for the motion of the planets of the solar system.

$$m\_1 w\_1 = F\_1 = -f \frac{m\_1 m\_2}{r^3} \text{ } r\_{12} \tag{2}$$

$$
\sigma\_2 \mathbf{w}\_2 = -f \frac{m\_1 m\_2}{r^3} \ \mathbf{r}\_{21} = \mathbf{F}\_2 = -\mathbf{F}\_1 \tag{3}
$$

Considering a system of only two interacting bodies, we can write down the equations of motion, which, taking into account Newton's laws, determine the properties for the center of mass of the system.

$$m\_1\mathfrak{w}\_1 + m\_2\mathfrak{w}\_2 = 0, \ \ m\_1\mathfrak{v}\_1 + m\_2\mathfrak{v}\_2 = \text{const} \tag{4}$$

Galileo's principle of relativity asserts the existence of a frame of reference in which the center of mass retains its position, and two bodies move in their orbits around it. The motion in the two-body problem determines the gravitational parameter *μ* of the central body. Equations for any selected system of bodies can be written in a similar way:

$$
\omega v\_i = -\frac{\mu}{r^3} \text{ } r\_i, \quad \mu = f(m\_0 + m\_i). \tag{5}
$$

The motion of a body in the Cartesian coordinate system relative to the central body can be written by equations

$$\frac{d^2\mathbf{x}\_i}{dt^2} = -\frac{\mu}{r^3}\mathbf{x}\_i, \quad i = 1, 2, 3\tag{6}$$

The parameters that determine the action of the central force are formed on the basis of indirect observations for the coefficient of the gravitational constant and the masses of the bodies involved. And if you need to write down equations for a planetary problem, where formally there are ten planets and hundreds or thousands of small bodies of the solar system, then with what accuracy are all parameters represented?

*Astrodynamics in Photogravitational Field of the Sun: Space Flights with a Solar Sail DOI: http://dx.doi.org/10.5772/intechopen.102005*

The principle of determinism makes it possible to obtain coordinates at any moment in time for such a model with constant parameters and orbital elements.

The motion in the central gravitational field has a solution, which in the absence of disturbing forces is determined by the initial values of the radius vector, the velocity vector, and the gravitational parameter of the central body. This determines the constant Keplerian elements ¼ ð Þ *a*, *e*, *i*, *Ω*, *ω*, *M*<sup>0</sup> , which allow calculating the Cartesian coordinates *xi*ð Þ*t* and velocities *vi*ð Þ*t* for unperturbed motion at an arbitrary moment in time *t* by formulas [19]:

$$\begin{cases} \begin{aligned} \mathbf{x}\_1 &= r(\cos u \cos \Omega - \sin u \sin \Omega \cos i) \\ \mathbf{x}\_2 &= r(\cos u \sin \Omega + \sin u \cos \Omega \cos i), \\ \mathbf{x}\_3 &= r \sin u \sin i, \end{aligned} \end{cases} \tag{7}$$

$$\begin{cases} \begin{aligned} v\_1 &= \alpha (\cos u \cos \Omega - \sin u \sin \Omega \cos i) \\ &- \beta (\sin u \cos \Omega + \cos u \sin \Omega \cos i), \\ v\_2 &= \alpha (\cos u \sin \Omega + \sin u \cos \Omega \cos i) \\ &- \beta (\sin u \sin \Omega + \cos u \cos \Omega \cos i) \\ &v\_3 &= \arcsin u \sin i + \beta \cos u \sin i. \end{aligned} \tag{8}$$

Here

$$r = a(1 - 
ewsE), \quad p = a\left(1 - e^2\right),$$

$$a = \sqrt{\frac{\mu}{p}}er^{-1}\sin\theta, \quad \beta = \sqrt{\mu p}r^{-1},\tag{9}$$

The time of movement between two points of the orbit can be determined from the equation, which is called the Kepler equation

$$E - e \sin E = M\_0 + n(t - t\_0) = M.\tag{10}$$

Osculating elements ðÞ¼ *t* ð Þ *a*, *e*, *i*, *Ω*, *ω*, *M*<sup>0</sup> can be used to describe the motion with allowance for perturbations, when the spacecraft orbital elements are functions of time. Differential equations can be used, where the right-hand sides are determined by the current values of the elements and the projections of the disturbing accelerations on the axis of the orbital coordinate system.

$$\begin{cases} \begin{aligned} \frac{da}{dt} &= 2a^2(\sin\theta P\_1 + pr^{-1}P\_2), \\ \frac{de}{dt} &= p(\sin\theta P\_1 + \cos\theta P\_2 + \cos E P\_2), \\ \frac{di}{dt} &= r\cos\theta P\_3, \quad \frac{d\Omega}{dt} = r\sin u \sin^{-1} i \ P\_3, \\ \frac{d\alpha}{dt} &= e^{-1}[(r+p)\sin\theta P\_2 - p\cos\theta P\_1] - \cos i \frac{d\Omega}{dt}, \\ \frac{dM\_0}{dt} &= \sqrt{e^{-2}-1}[(p\cos\theta - 2er)P\_1 - (r+p)\sin\theta P\_2]. \end{aligned} \end{cases} \tag{11}$$

In the general case, the vector projections on the radial and transverse directions will affect the change in the parameters of the motion orbit. Projection to the normal to the orbital plane will allow you to change its inclination relative to its original position.

The third and fourth equations of system (6) show that the initial position of the orbit is preserved in the absence of a projection of the disturbing forces on the normal to the plane.

If the mathematical model defines the equations, and the properties of the solutions suit the researchers, then great. Analytical or numerical methods make it possible to refine objects and predict movement.

The discovery and experimental confirmation of the effect of light pressure by **Lebedev** makes it possible to use the photogravitational field, which introduces a correction (reducing the force of attraction to the central body by the amount of light pressure on the surface of the body) *μ<sup>f</sup>* ¼ *μ<sup>g</sup>* � *μ<sup>p</sup>* ¼ *μ<sup>g</sup>* 1 � *δ<sup>p</sup>* . For asteroids with similar parameters, the coefficients of light pressure will be close.

A new level of influence of light pressure appeared after Tsander **'s** idea of spacecraft flights under a solar sail, where the parameters of the ratio of body surface area and mass are significantly different. Additionally, it becomes possible to change the direction of the thrust vector, which takes into account the position of the sail and the properties of the mirror surface that reflects the streams of sunlight. In this case, the central gravitational field is supplemented by a special governing force. In the projection on the axis of the Cartesian coordinate system, you can write (2).

The equations can be written, as suggested by **Euler**, using **Kepler's** osculating elements to use the projections of forces on the axis of the orbital coordinate system to change the parameters of the orbit of motion over time as a feedback control.

The problems of motion of the CASP in the vicinity of the Earth or another planet can be investigated on the basis of the limited problem of three bodies, supplemented by a special control force, depending on the relative position of the two main bodies: the Sun and the planet.

The ability to control the orientation of the CASP and individual elements of the sails system has a special character to form the best control of the main thrust vector and the moment of forces for turning the CASP relative to the center of mass or maintaining the desired position.

The main problem of the relative motion of two bodies under the action of gravitational forces of interaction (without taking into account other perturbing forces) is reduced to the equations of the central force field, describing the movement of a material point along an elliptical trajectory, in the focus of which is the attracting center of the main body. The magnitude of the force *Fg* depends on the square of the distance and the gravitational parameter. In the projection on the axis of the Cartesian coordinate system, you can write Eqs. (6).

If we restrict ourselves to motion while maintaining the initial plane of the orbit, then we can use polar coordinates *r t*ð Þ,*ϕ*ð Þ*t* and projections of forces on the radial and transverse directions

$$\frac{d^2r}{dt^2} + \frac{\mu}{r^2} - r(\dot{\rho})^2 = P\_1, \quad \frac{d}{dt} \left(r^2 \dot{\rho}\right) = P\_2,\tag{12}$$

The motion in the central gravitational field has a solution, which in the absence of disturbing forces is determined by the initial values of the radius vector, the velocity vector and the gravitational parameter of the central body.

The discovery and experimental confirmation of the effect of light pressure by **Lebedev** makes it possible to use the photogravitational field, which introduces a correction (reducing the force of attraction to the central body by the amount of light pressure on the surface of the body)

$$
\mu\_f = \mu\_\mathfrak{g} - \mu\_\mathfrak{p} = \mu\_\mathfrak{g} \left( \mathbf{1} - \delta\_\mathfrak{p} \right).
$$

*Astrodynamics in Photogravitational Field of the Sun: Space Flights with a Solar Sail DOI: http://dx.doi.org/10.5772/intechopen.102005*

For asteroids with similar parameters, the coefficients of light pressure will be close.

A new level of influence of light pressure appeared after Tsander's idea of spacecraft flights under a solar sail, where the parameters of the ratio of body surface area and mass are significantly different. Additionally, it becomes possible to change the direction of the thrust vector, which takes into account the position of the sail and the properties of the mirror surface that reflects the streams of sunlight. In this case, the central gravitational field is supplemented by a special governing force.

In the projection on the axis of the Cartesian coordinate system, you can write (3).

The equations can be written, as suggested by Euler, using Kepler's osculating elements to use the projections of forces on the axis of the orbital coordinate system to change the parameters of the orbit of motion over time as a feedback control.

The problems of motion of the CASP in the vicinity of the Earth or another planet can be investigated on the basis of the limited problem of three bodies, supplemented by a special control force, depending on the relative position of the two main bodies: the Sun and the planet.

The photogravitational field in the three-body problem for motion in the vicinity of the Earth can be considered a combination of the central field or geopotential and the disturbing action of the Sun's gravitational force and the forces of light pressure. A modification of the restricted three-body problem is obtained, taking into account the additions that can form control forces for the implementation of the CASP flight program along a given trajectory in the vicinity of the Earth or for spinning the initial orbit in the case of the problem of getting out of the sphere of the Earth's gravity for flights into the distant expanses of the Solar system.

If motion in the vicinity of the Earth is considered, then the directions of the two main forces do not coincide, but it can be assumed in a first approximation that the luminous flux determines an almost constant pressure force, collinear with a straight line that passes through the two main bodies of the system. The position of the sail plane allows you to form the direction of the control force **P***(t,u)* to change the trajectory or stabilize in the vicinity of the singular libration points. Then, you can use the equations of motion in the framework of the limited circular problem of three bodies

$$\ddot{\mathbf{x}} - 2\eta \ \dot{\mathbf{y}} = \frac{\partial U}{\partial \mathbf{x}} + P\_1, \quad \ddot{\mathbf{y}} + 2\eta \ \dot{\mathbf{x}} = \frac{\partial U}{\partial \mathbf{y}} + P\_2, \quad \ddot{\mathbf{z}} = \frac{\partial U}{\partial \mathbf{z}} + P\_3. \tag{13}$$

The position of the center of a spacecraft of infinitely small mass relative to the main bodies (Earth and Sun) of mass *μ*< <1 and 1ð Þ � *μ* in a rotating barycentric Cartesian coordinate system is determined by the radius vectors

$$r = (\mathbf{x}, \mathbf{y}, \mathbf{z}), \ r\_1 = (\mathbf{x} + \boldsymbol{\mu}, \ \mathbf{y}, \ \mathbf{z}), \ r\_2 = (\mathbf{x} - \mathbf{1} + \boldsymbol{\mu}, \ \mathbf{y}, \ \mathbf{z}).\tag{14}$$

The force function of the gravitational interaction has the form

$$U = \frac{1}{2}\eta^2(\varkappa^2 + \jmath^2) + \kappa^2 \left(\frac{\mathbf{1} - \mu}{r\_1} + \frac{\mu}{r\_2}\right) \ . \tag{15}$$

Here, *η* is the constant angular velocity of rotation of the coordinate system relative to the center of mass of the system together with the main bodies. Thus, there is an additional simplification at *P*<sup>1</sup> ¼ *const*, *P*<sup>2</sup> ¼ *P*<sup>3</sup> ¼ 0. When moving in the vicinity of the Earth, for an approximate solution, one can use the intermediate orbits of the Hill problem [17], including when studying the stability of motion in the vicinity of libration points.
