Open access peer-reviewed chapter

Astrodynamics in Photogravitational Field of the Sun: Space Flights with a Solar Sail

Written By

Elena Polyakhova and Vladimir Korolev

Submitted: 24 June 2021 Reviewed: 13 December 2021 Published: 13 October 2022

DOI: 10.5772/intechopen.102005

From the Edited Volume

Gravitational Field - Concepts and Applications

Edited by Khalid S. Essa

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Abstract

Mathematical models of the controlled motion of a spacecraft with a solar sail and the possibilities are considered, taking into account the translational motion in the gravitational field, the forces of light pressure, and rotational motion relative to the center of mass. The structure of possible problems of photogravitational celestial mechanics is proposed. To control movement, it is possible to change the size, shape, surface properties, and orientation of the elements of the sail system in relation to the flow of sunlight. The equations of motion can be presented on the basis of the problem of motion in a photogravitational field, taking into account the action of other disturbing forces. When studying the orbits of motion in the vicinity of the Earth, one should use a more general model of the photogravitational field of the restricted three-body problem. In this case, the gravitational action of the Sun and the Earth is supplemented by the field of forces of light pressure, which makes it possible to simulate real problems of dynamics.

Keywords

  • light pressure
  • photogravitational field
  • space flight
  • solar sail
  • spacecraft

1. Introduction

Even in ancient times, scientists, studying the relative movements of various bodies, have always tried to determine the reasons, principles and patterns that determine these movements. The combination of modern methods of mathematics, physics and mechanics makes it possible to form mathematical models of the interaction of material objects and predict possible new states based on the observed initial values of parameters and quality characteristics for the selected system of bodies.

The gravitational field for determining the forces of mutual attraction was originally considered as the field of gravity on the surface of the Earth. The famous experiments of Galileo Galilei (1564–1642) made it possible to determine the magnitude of acceleration and the direction of free fall, which is considered the same for any falling body. This defines a uniform gravitational field, which gives the simplest version of a mathematical model of gravity. The appearance of the basic laws of Johannes Kepler (1571–1630) after processing the results of many years of astronomical observations by the Danish astronomer Tycho Brahe (1546–1601) makes it possible to describe the motions of the planets of the solar system and other bodies. As a result of the publication of “Mathematical Foundations of Natural Philosophy” by Isaac Newton (1642–1727), a substantiation of the law of universal gravitation appeared, which led to the creation of new models of gravitational forces and new possible solutions to the problems of celestial mechanics [1].

Theoretical astronomy studies various variants of the motion of a selected system of bodies: stars and planets, asteroids or comets. A special class of problems appears when the forces of attraction are supplemented by the forces of light pressure [1, 2, 3, 4, 5, 6, 7], acting on the surface of bodies. The history of the emergence of new directions of science in the form of photogravitational celestial mechanics (PhCM) is determined by remarkable scientists: Kepler, Bessel, Maxwell, Bredikhin, Lebedev and some authors.

The principle of movement in space under a solar sail is based on the effect of light pressure, which Johannes Kepler guessed about when observing the movement of comets. Kepler was the first to suggest that cometary tails are a stream of particles thrown by the action of light away from the Sun as the comet approaches the Sun (as was noticed by ancient Chinese astronomers, who did not try to explain this, however). Back in 1619, he wrote: “Dirty matter clumps together, forming the head of a comet. The sun’s rays, falling on it and penetrating through its thickness, again transform it into the thinnest substance of the ether and, leaving it, form a strip of light on the other side, which we call a comet’s tail. Thus, a comet, throwing out a tail from itself, thereby destroys itself and is destroyed “[2]. Kepler was the first to formulate the basic laws of planetary motion, and also realized and pointed out the essential role of solar radiation in the evolution of bodies in the solar system, in particular for comets.

In 1836, Bessel published a work on the motion of cometary particles under the influence of the Sun’s gravitational force and its repulsive force, which (as assumed) changes like the gravity force inversely proportional to the square of the distance from the center of the Sun to the point of observation. Later, it was noted that the magnitude of the force of light pressure depends on the shape and surface area of the body, as well as on the reflection coefficient and location relative to the flow of sunlight, all other things being equal.

The mechanical theory of cometary forms was further developed in the works of the Moscow astronomer academician F.A. Bredikhin. Improving Bessel’s theory, he corrected a number of inaccuracies in it and found the law of motion of a particle under various physical conditions.

New possibilities in explaining the nature of these forces appeared after it was predicted and then experimentally confirmed the effect of light (as a particular manifestation of an electromagnetic field) on material bodies. D.K. Maxwell in the middle of the nineteenth century, developing the theory and equations of the action of the electromagnetic field of forces, showed that light must produce pressure on the surface placed in the path of the light flux. Maxwell in 1873 succeeded in theoretically predicting the magnitude of the light repulsion force and substantiating the dynamic essence of light pressure as a physical effect. Experiments that confirmed Maxwell’s prediction were carried out by the Russian physicist Pyotr Nikolaevich Lebedev (1866–1912).

Prominent Russian experimental physicist P.N. Lebedev, the founder of the first Russian scientific physics school, who made a significant contribution to the development of astrophysics, primarily by his virtuoso experimental proof and measurement of light pressure on solids and gases (1899; 1910). He worked in close contact with the largest Russian astrophysicist F.A. Bredikhin in the study of comets. It has been proven that comet tails are a real formation from matter flowing from the comet’s nucleus. He began to experimentally prove and measure the pressure of light on solids in 1897. In 1899, for the first time in the history of science, he experimentally discovered and measured the pressure of light on solids and reported this in Lausanne (1899), and later more fully at the International Congress of Physicists in Paris (1900). The results were published in 1901. in Leipzig in “Annalen der Physik” in the article “Experimental investigation of light pressure.”

It is important to note that the most consistent and logically grounded explanation of the phenomenon of light pressure was given only within the framework of quantum theory, on the basis of corpuscular concepts of electromagnetic radiation. It is here that it is possible to obtain an explicit analytical expression for the force of light pressure acting on a particle of a comet’s tail, relying on a number of basic principles and relations of quantum theory.

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2. Flying in space under a solar sail

The principle of movement in space under a solar sail is based on the effect of light pressure. Friedrich Arturovich Tsander (or F.A. Zander) (1887–1933), a Soviet scientist and inventor, one of the pioneers of rocketry, determines the emergence and development of ideas for space navigation under a solar sail. F.A. Tsander was one of the creators of the first Soviet liquid-propellant rocket and the author of the first technical design for a solar-sail spacecraft. The development of his is an engineering project for a space flight with a low-thrust engine in the form of a metal mirror. The idea of such spacefaring was first expressed by him in 1910–1912 as having a scientific and engineering sense, whereas previously it appeared in only a few works of science fiction.

In 1920, F.A. Tsander and K.E. Tsiolkovsky (1857–1935) discussed the possibility that a very thin flat sheet, illuminated by sunlight, could reach high speeds in space. As for the question of whether it is possible to use the property of photons for cosmic motion, they answered in the affirmative. Tsander was the first who not only expressed the idea, substantiating its scientific reliability and technical feasibility of its implementation, but also embodied this in 1924 into a calculated engineering design of a spacecraft with a reflecting mirror.

In 1921, a report on this project was presented by Tsander at a conference of inventors, and in 1924, it was revised and published in the journal “Technics and Life” under the title “Flights to other planets.” In the same article, Tsander expressed ideas about the benefits of using ramjet engines, and about the possibility of using and constructing a solar sail and transferring energy to a moving rocket.

He applied to the Committee on Inventions for a space plane that can use huge and very thin mirrors to travel in interplanetary space. The project was presented in the form of two manuscripts, which, however, remained unpublished at that time. They will see the light only in 1961. In the middle of the twentieth century, science fiction writers again returned to solar sails (e.g., in A. Clark’s story “The Solar Wind”), and then engineers and scientists. Around the same time, the short term “solar sail” appeared and took root as a successful borrowing from foreign science fiction. The idea of using a solar sail as a low-thrust engine corresponds to the historically important ideas of F.A. Tsander on space flight under the influence of light pressure and makes a feasible contribution to the development of the scientist’s scientific heritage. Spacecraft flights using the energy of light pressure are no longer fiction, but the reality of projects of the present time and the near future [1, 8, 9, 10, 11, 12, 13].

In 1924, Tsander, based on the formulas and results of Lebedev’s experiment, proposed the first engineering design of a space sailing ship with a mirror-screen sail made of the finest metal foil. However, the problem of space navigation has remained outside the field of vision of scientists for a long time and gains popularity only in the space age. Theoretical developments and design projects are resumed, and variants of spacecraft with solar sails, different in design and shape, appear: with flat (solid, round or rectangular), such as parachutes or inflatable balloons, multi-bladed, that is, split like a helicopter propeller, honeycomb type, such as complex mirror systems and so on.

The main guarantee of the sail’s efficiency is a high, close to unity, reflectivity of a light film mirror, which creates a high windage of the entire structure. At the level of modern technical capabilities, the value of windage, that is, the surface-to-mass ratio, is of the order of 1000 cm2/g, which makes it possible to achieve an acceleration of the order of 0.1 cm/s2, which is only six times less than the acceleration in the Earth’s orbit from the gravitational actions of the Sun.

Since the second half of the twentieth century, the rapid development of the sailing theme was accompanied by an intensive increase in the number of publications in Russia and abroad. Numerous articles began to be published in magazines, and monographs appeared, in whole or in part, related to solar sails. Bold projects of sailing flights to the Moon, Mars, or Halley’s comet, projects of illuminating the Earth from space with the help of a mirror sail-illuminator on the Artificial Earth Satellite, and many other fascinating technical ideas were described. In the 90s. the domestic device “Znamya” has been successfully developed and implemented, but it was not yet an interplanetary flight, but a launch into a satellite orbit in the vicinity of the Earth.

The first attempt to implement the “Znamya” project and deploy a solar sail in space was successfully carried out in 1993 [2, 4, 13]. Such space sailing ships can be used for flights to large and small planets, to meet with asteroids or comets, and to form special orbits of motion in the vicinity of the Sun or the Earth. New technologies should bring visible results in the creation of space engines based on the direct use of an unlimited source of solar energy. Over the years, numerous variants of motion patterns and new possible forms of solar sails have appeared. The technology for large-scale solar reflector designs is in its infancy.

After constructing and orbiting such mirrors of certain proportions, we obtain a self-adjusting systemic orientation with respect to the Sun for coplanar or spatial trajectories. More sophisticated options and models make it possible to control the orbital and rotational movements of the spacecraft while in motion using a solar sail.

One of the most difficult problems of mankind at the turn of the twentieth/twenty-first centuries was the problem of providing energy. One of the most sensible ways to save energy resources in space is to develop renewable or “perpetual” energy sources. Such promising energy resources primarily include the energy of the sun’s rays, both thermal for filling special devices and mechanical for the formation of additional pressure forces on the surface of spacecraft or special installations of “solar sails.”

The latest version involves the use of the pressure of the sun’s rays on all the bodies encountered in the luminous flux. This “eternal”, and at the same time environmentally friendly type of energy resources is also beneficial because it does not require any expenses for its transportation to the place of consumption.

The forms of using the thermal energy of the Sun on Earth are widely known, but the problem of using a very small mechanical pressure of the light flux for the purposes of the so-called “space navigation” turns out to be much more complicated. Flying in space under a solar sail is just a real embodiment of the idea of full or partial replacement of the energy of jet engines with the “donated” energy of the sun’s rays, the pressure of which on a mirror reflective sail is able to create, albeit a small, but quite tangible thrust force in space for a spacecraft… With the help of a solar sail, you can determine or change the direction of movement in orbit or perform complex gravity assist maneuvers around large planets.

Opening the sail in orbit will cause the light pressure to partially compensate for the sun’s gravity. If the gravitational field of attraction is supplemented by the field of forces of light repulsion, then for problems of celestial mechanics one can speak of a mathematical model of the photogravitational field [3, 4, 5]. The acceleration imparted to the spacecraft by the flow of solar rays depends on the ratio of the sail area to the mass of the entire structure. With such modeling, it is sufficient to restrict oneself to taking into account two main forces: the gravitational interaction of bodies Fg and the light pressure Fp on the body from the flow of solar radiation. But for problems of motion in the vicinity of the Earth, it is necessary to take into account the features of the Geopotential and other forces (atmospheric resistance, the influence of other bodies, etc.).

The use of a solar sail will provide the spacecraft with a low-thrust engine, which has an almost unlimited supply of fuel. However, it has a disadvantage: Unlike jet engines, we cannot use its thrust in an arbitrary direction with the same efficiency. The resulting force is determined by the position of the spacecraft in space, as well as by the orientation of all elements of the solar sail relative to the attracting centers and centers of radiation. It is necessary to specifically orient the sail to obtain the desired change in orbital parameters.

Problems of motion control with a solar sail lead to the study of mathematical models of dynamics in photogravitational fields of orbital motion and problems of control over the rotation of the entire spacecraft complex relative to the center of mass [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

The influence of the main forces determines not only the orbital and rotational motion, but can also be used to implement control during interplanetary flights and maneuvering in the sphere of action of the next planet, or to stabilize the spacecraft orientation during its orbital motion. This will make it possible to form the direction and thrust of such a solar-powered engine with an unlimited margin (solar sails with a good mirror surface for reflecting the light flux), which depends on the distance to the source, and the area and shape of the surface of the spacecraft sail elements [3, 9].

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3. The main problems of photogravitational celestial mechanics

3.1 Photogravitational celestial mechanics (PhCM) with one radiation source

(simulation of motion in the framework of the two-body problem or the restricted. three-body problem for the solar system)

  1. Heliocentric motions: The sun as a point source of radiation. The sun as a non-rotating extended source. The sun as a rotating extended source. Darkening effect toward the edge. Energy luminosity of the Sun (numbers), “solar constant” in the Earth’s orbit.

  2. Heliocentric photogravitational problem of two bodies (beta-meteoroids)

  3. Photogravitational limited three-body problem/evolution of orbits taking into account light pressure in special cases:

    1. Sun-planet-meter asteroid,

    2. Sun-comet-comet tail particle

  4. Libration points of the photogravitational three-body problem: position, behavior, and stability of libration points in a plane-bounded problem in the presence of resonances.

  5. Nonstationary modifications of photogravitational problems with variable physical parameters in the framework of the dynamics of bodies with variable mass.

  6. Heliocentric flights of the solar sail to the Sun, large planets, asteroids, comets, planetary satellites. Interplanetary flights over ellipses of the Hohmann-Tsander type in a photogravitational field.

  7. A chain of resonant encounters and repeated hyperbolic gravity assist maneuvers on the Earth-Venus or Venus-Sun flight.

  8. The exit of a spacecraft with a solar sail from the Solar System, a gravity assist of a spacecraft with a sail near the rotating Sun, taking into account the relativistic effects (Kerr and Schwarzschild metrics, Lense-Thirring effect). Flight to the focus of the Sun’s gravitational lens—550 AU from the Earth.

3.2 Geocentric movements within the photogravitational celestial mechanics

  1. Theory of perturbed satellite motion under solar radiation pressure.

  2. Controlled low-thrust solar sail flights.

    1. Sailing geocentric separation. Spinning to the Moon inside the Earth’s sphere of action.

    2. Environmental aspects of light pressure: a solar sail or a bundle of sails as a solar shield (shield) against global warming, a passive relay, an orbital illuminator (reflector) of near-polar regions, an element of a cable system for changing the orbit of a dangerous asteroid to deflect it from the Earth. Deviation of an asteroid from the orbit of a dangerous approach to the Earth due to an artificial increase in its albedo (black-and-white control coatings), the use of axial spinning of the asteroid due to moments of light pressure forces (the Yarkovsky-Radzievsky effect, i.e., the transverse rotational thermodynamic effect).

    3. Solar sail in a perturbing Coulomb field—the effect of an electric static charge induced on the sail surface on its strength characteristics and on the dynamics of movement of a vehicle with a sail along a Zander trajectory, charging a thin sailing film in space plasma.

3.3 Photogravitational celestial mechanics with two sources of radiation

(movement of interstellar dust within the limited three-body problem)

  1. Points of libration: positions, behavior, and stability

  2. Instability of circumstellar dust complexes and cloud clusters of microparticles in binary and multiple stellar systems.

3.4 Photogravitational celestial mechanics in the solar system

(objects of research—objects of high windage)

  1. Natural small bodies in the solar photogravitational field.

    1. Orbital motions of small bodies taking into account the pressure of solar radiation: dust particles (beta-meteoroids), dust particles of a comet tail, small (meter) asteroids with high albedo, dust complexes around major planets.

    2. Rotational movements under the influence of solar radiation pressure (the thermodynamic effect of Yarkovsky-Radzievsky, thermal inertia of a rotating body).

    3. Orbital relativistic aberration Poynting-Robertson effect: a spiral orbital twist of particles toward the Sun, followed by a catastrophic fallout on it.

    4. Dynamics of emission of dust particles from the head of a comet under the action of light pressure (the parachute effect according to Radzievsky, the movement of particles from the nucleus to the tail in the framework of the limited three-body problem).

  2. Artificial celestial bodies with high windage: spherical and ellipsoidal inflatable satellites, spacecraft with solar sails, large transformable space structures, etc. Calculation of light pressure on bodies of various shapes—regular and irregular. Spherical AES—the theorem about the light pressure on an ideal sphere.

    1. Controlled flights of spacecraft with low thrust of the solar sail. Closed interplanetary flights with a sail, near-solar maneuvers.

    2. Near-planetary and circumsolar gravity-assist maneuvers with a solar sail.

    3. Orbit correction (satellite retention in Geo Stationary Orbit).

    4. Rotational movements: 3-axis stabilization, orientation, and control in space under the influence of the moment of radiation pressure forces.

    5. A solar sail as an element of a cable system for transporting a dangerous asteroid from orbit.

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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=const:

mw=F=mgE1

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.

m1w1=F1=fm1m2r3r12E2
m2w2=fm1m2r3r21=F2=F1E3

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.

m1w1+m2w2=0,m1v1+m2v2=constE4

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:

wi=μr3ri,μ=fm0+mi.E5

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

d2xidt2=μr3xi,i=1,2,3E6

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?

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 k=aeiΩωM0, which allow calculating the Cartesian coordinates xit and velocities vit for unperturbed motion at an arbitrary moment in time t by formulas [19]:

x1=rcosucosΩsinusinΩcosix2=rcosusinΩ+sinucosΩcosi,x3=rsinusini,E7
v1=αcosucosΩsinusinΩcosiβsinucosΩ+cosusinΩcosi,v2=αcosusinΩ+sinucosΩcosiβsinusinΩ+cosucosΩcosiv3=αsinusini+βcosusini.E8

Here

r=a1ecosE,p=a1e2,α=μper1sinϑ,β=μpr1,E9

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

EesinE=M0+ntt0=M.E10

Osculating elements kt=aeiΩωM0 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.

dadt=2a2esinϑP1+pr1P2,dedt=psinϑP1+cosϑP2+cosEP2,didt=rcosϑP3,dt=rsinusin1iP3,dt=e1r+psinϑP2pcosϑP1cosidt,dM0dt=e21pcosϑ2erP1r+psinϑP2.,E11

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) μf=μgμp=μg1δp. 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 rt,ϕt and projections of forces on the radial and transverse directions

d2rdt2+μr2rφ̇2=P1,ddtr2φ̇=P2,E12

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)

μf=μgμp=μg1δp.

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

x¨2ηẏ=Ux+P1,y¨+2ηẋ=Uy+P2,z¨=Uz+P3.E13

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 1μ in a rotating barycentric Cartesian coordinate system is determined by the radius vectors

r=xyz,r1=x+μyz,r2=x1+μyz.E14

The force function of the gravitational interaction has the form

U=12η2x2+y2+κ21μr1+μr2.E15

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 P1=const,P2=P3=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.

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5. Control and stabilization capabilities

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.

Heliocentric motion of a spacecraft with a solar sail during flights to the Sun, planets, asteroids, or comets; and also to create special orbits of motion in the vicinity of the Sun, taking into account the force of light pressure, leads to the equations of motion in the form of a system

d2xidt2=μr3xi+Uxi+fix+uitx,i=1,2,3,E16

where the notation is used: xi are the Cartesian coordinates of the spacecraft, r is the modulus of the radius vector, μ is the gravitational parameter of the central body, the function U is determined by the influence of disturbances from potential forces, and the functions fix,uitx are the components of the acceleration vectors non-potential forces and the vector control contribution utx, including the action of light pressure forces or jet engines on active sections of motion, when expanded on the axes of the orbital coordinate system [20]. In this case, the forces of light pressure on the elements of the sail system and the moments relative to the center of mass of the system determine the vector quantities:

F=Fi=kiSibθir2niθi,M=iρi×Fiθi.E17

The contribution of the light pressure is determined by the angle of deviation of the normal vector n from the direction of the flux of the sun’s rays e. If the flat mirror sail is angled θi to the beams, then the transmitted pulse will be directed almost perpendicular to the reflective surface. The photons will retain a part of the impulse directed parallel to the sail, so that the sail will get less than with full opening to the rays. The magnitude of the light pressure decreases, and the direction will almost coincide with the normal to the sail, laid down from its shadow side. By turning the sail, we get the opportunity to change the direction of the thrust vector and control the spacecraft. However, this changes the value. If the normal of a flat sail is perpendicular to the stream of rays, then the sail exerts no thrust at all. In the general case, the vector projections onto 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. Acceleration also depends on the ratio of the sail area S to the mass of the entire structure and on the surface properties or surface reflection coefficient. Here, the designations are additionally used when summing over all structural elements.

The design features of the spacecraft in the problems under consideration make it possible to use a solar sail to control motion. By turning the sail, taking into account the pressure of sunlight, it is possible to control the orbital dynamics and the relative position of the spacecraft in space, as well as perform stabilization.

For rotational motion relative to the center of mass, the kinetic energy of the body is determined by the moments of inertia and angular velocity T=12Ixωx2+Iyωy2+Izωz2.

Differential equations can be written under the action of the moment of forces in the form

Ixω̇x=IyIzωyωz+Mx,Iyω̇y=IzIxωxωz+My,Izω̇z=IxIyωyωx+MzE18

Taking into account the effect of light pressure on the spacecraft sail leads to the appearance of special stability conditions that can be used to control the movement. Correctly chosen shape of the sail allows you to keep the spacecraft in the desired position. The effect of disturbances can be compensated for by changing the size or reflective properties of the elements of the sail of the spacecraft, as well as their relative position. This creates the additional moments of force that can be used as controls.

If we introduce into consideration the control moments ui (i = 1,2,3), relative to the main axes of inertia, and use the projections of the angular momentum as unknowns x7=Ixωx,x8=Iyωy,x9=Izωz, then the system of equations of motion (1) and (4) for the considered set of generalized coordinates can be represented in the normal form:

ẋi=xi+3,i=1,2,3,ẋi+3=μr3xi+fitxiui,ẋ7=β1x8x9+u4,ẋ8=β2x9x7+u5,ẋ9=β3x8x7+u6,β1+β2+β3=0.E19

In the case of possible oscillations [13], while maintaining the orientation of one of the main axes orthogonal to the plane of motion with angular orbital velocity, the change in the angular momentum taking into account the action of the geopotential can be investigated using the equation

Izω̇z=kω02IxIysinϕ=Mz.E20

Taking into account the additional force leads to new, different from the classical, formulations of optimal control problems, and to new mathematical models. In these models, additional equations appear and other control functions are considered. To solve such problems, the methods of Pontryagin or the Bellman equation are used. There are analytical and numerical methods of research and analysis of the main properties of new equations, which allow obtaining exact or approximate solutions that deliver an extremum to the quality criterion and satisfy the necessary conditions [8, 21, 22, 23].

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6. Inverse problem of photogravitational dynamics

Considering the inverse problem for finding forces for a known or given motion in the case of two interacting bodies, we find the control accelerations that can realize the trajectory in the central field

fi=d2xidt2+μr3xi=uit,i=1,2,3,E21

when the parametric functions xit are predefined for t ∈ [0,T].

The results of solving the inverse problem of dynamics are presented as examples. Simulation of closed trajectories x=r0cos2t,y=r0sintcost,z=h3sint of a spacecraft with a controlled solar sail is presented to reach the heliopolar regions (Figure 1, Viviani curve), and fly over the North and South poles of the Sun and return to near-earth orbit [2]. Assuming that the required control accelerations are created by the position of the solar sail [5], we write them in the form

Figure 1.

The Viviani curve is slightly flattened in the vertical direction at h3 = 0.3r0.

u1=qr2cosγ1cosγ2,u1=qr2sinγ1cosγ2,u1=qr2sinγ2E22

where γ1 and γ2 are the angles of the three-dimensional orientation of the normal vector to the shadow side of the mirror sail on both sides, and the dimensionless factor k depends on the solar radiation pressure on the surface of the spacecraft sail (Figure 2). From here, we obtain the formulas for finding the angles and the construction of the control.

Figure 2.

Graph of γ1 and γ2 for Viviani curve at h3 = 0.3r0, T3 = 1.03 years.

The spiral trajectory (Figure 3) of the possible motion of the CASP is determined by the equations

Figure 3.

The spiral trajectory is defined by expressions in the orbital coordinate system.

x=r0cosω0t,y=r0sinω0t+r1cosω1t,z=r1sinω1t.E23

The helical trajectory can be supplemented by the selected law of control over the position of the sail elements.

The solution of the inverse problem of dynamics when moving along a given trajectory makes it possible to obtain an initial approximation for control and to evaluate the possibility of implementing a system of spacecraft sail elements for the selected model. With this solution, you can also refine the rotation or steering of the entire structure with respect to the center of mass.

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7. Conclusion

The functional features of the motion of a spacecraft with solar sails are considered. The main tasks of photogravitational celestial mechanics are listed, and the possibilities of their mathematical modeling are given. The main differences in this area of research are sophisticated spacecraft models, supplemented by solar sails of various shapes, sizes, and physical properties. Taking into account all the characteristics of the sail complicates the process of mathematical modeling of the spacecraft, but it is possible with the help of classical analytical dynamic tools.

The dynamic aspect of the simulation is to take into account the solar pressure force on the sail. The presence of this force determines the specifics of the formulation of control tasks for such spacecraft, closely related to their design features. Despite the differences between such formulations from the usual optimal control problems, their solution can be obtained by classical optimization methods.

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Written By

Elena Polyakhova and Vladimir Korolev

Submitted: 24 June 2021 Reviewed: 13 December 2021 Published: 13 October 2022