# astrodynamics

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## astrodynamics

(ass-troh-dÿ-**nam**-iks) The application of celestial mechanics, the ballistics of high-speed solids through gases, propulsion theory, and other allied fields to the planning and production of the trajectories of spacecraft.

## Astrodynamics

the most common designation of that branch of celestial mechanics which deals with the study of the motion of artificial celestial bodies such as artificial earth satellites, artificial lunar satellites, automatic interplanetary stations, and the like. Astrodynamics underwent intensive development following the launching of the first artificial satellite from the Soviet Union (1957). The terms “cos-modynamics,” “space dynamics,” “celestial ballistics,” and “spaceflight mechanics” are also encountered in the writings on this subject.

Astrodynamics first developed as a branch of classical celestial mechanics studying the motion of natural celestial bodies or hypothetical bodies considered within the framework of various astronomical hypotheses. Its specific features center on the fact that—in contrast to classical celestial mechanics, which with rare exceptions is limited to dealing with the mutual attraction between celestial bodies obeying the Newtonian gravitational law—additional forces, other than forces of gravitational attraction, must, as a rule, be taken into account in problems of astrodynamics. Some of these forces are the resistance (drag) presented by the earth’s atmosphere, solar radiation pressure, and the geomagnetic field. Space vehicles can be controlled and maneuvered with the aid of jet propulsion engines mounted on board the spacecraft and actuated automatically or by command from an earth-based control station. Astrodynamics is based on mathematical investigation of the equations (ordinary differential equations) of motion of artificial celestial bodies, and relies partially on methods developed earlier in classical celestial mechanics. Nevertheless, since the “set” of forces to be taken into account in astrodynamical problems is broader, the equations of motion are often far more intricate than those encountered in classical celestial mechanics; compilation of the equations of astrodynamical motion rests on the achievements of analytical mechanics, aerodynamics, automatic control theory, and so on. New techniques are also being developed for the solution and analysis of problems in astrodynamics. Widespread use is made of numerical methods of orbital calculation involving the use of electronic digital computers. In addition, astrodynamics deals with a number of specific problems that are not encountered in classical celestial mechanics. Among these is the projecting of orbits, which involves determining the conditions governing launches and control programs needed to ensure that the actual motion of an artificial celestial body exhibits prespecified properties. Fuel economy requirements in launch and control maneuvers (rocket-propellant consumption rates) and power expended must also be taken into account in problems of astrodynamics.

An artificial celestial body is usually launched with the aid of a multistage rocket. The rocket moves for some distance from its launch point under the thrust of its jet propulsion engines. This is the active segment of its flight trajectory, in which the future artificial celestial body is still a part of the automatically controlled rocket craft. As soon as the engines of the rocket’s last stage burn out, the space vehicle being launched usually separates from its carrier rocket and becomes an artificial celestial body that moves passively in its initial orbit around the earth on the energy it acquired during the boost phase trajectory of its carrier rocket. The moment of burnout is taken as the instant when the artificial celestial body is launched into orbit. The properties of its subsequent motion are determined completely by the position and velocity of the body at that instant (initial conditions) and by the set of passive and active (control) forces acting on the body. That motion can be analyzed and calculated on the basis of the equations of motion. The calculation of the initial position and velocity of an artificial celestial body to correspond with a preassigned initial orbit is one of the problems in orbit projecting. Moreover, since it is practically impossible to achieve absolute precision in automatic control of motion on the calculated active phase of the trajectory, the problem arises of estimating the allowable errors in position and velocity at the end of the active phase without leading to undesirable deviations from the specified initial orbit.

A very important aspect of orbit projecting is the transfer of an artificial celestial body from one orbit to another, since it is often either impossible or energetically unprofitable to effect a launch directly into the orbit corresponding to the proposed purpose of the investigation. Problems involving either comparatively small orbit corrections or transfers to a completely different orbit can be posed. Such problems are encountered, for example, when carrying out interplanetary flights, when launching artificial lunar satellites, or when launching artificial earth satellites into a stationary orbit around the earth. These problems pertain to controlled artificial celestial bodies, and control can be achieved with the aid of jet engines switched on either temporarily at specified instants (whereupon the space vehicle experiences the effect of an almost instantaneous jolt or impulse imparting an additional velocity) or over a sufficiently protracted time span (whereupon a constantly acting additional thrust is generated).

From a mathematical standpoint, these problems involve calculating momenta or additional thrust (their dimensions and directions and the moment and duration of the action) needed to effect a desirable change in the orbit. The complexity of these problems is due primarily to the fact that transfer from one orbit to another should be optimized (that is, it must be the best possible from some specific standpoint). The most common requirement is that the momenta or additional thrust be accompanied by minimal power expenditure, or that the transfer to the new orbit be completed in the shortest possible time. Problems pertaining to optimum motion of artificial celestial bodies with additional thrust are also undergoing intensive investigation. Prominent among these problems are the selection of the optimum control program for delivery of a maximum payload to a circular orbit at a great height above the earth’s surface within a specified time; calculation of the minimum time required for a low-thrust spacecraft to effect an earth-Mars-earth orbital transfer; optimum multimomentum transfer of artificial earth satellites between arbitrary elliptical orbits; and interplanetary orbit transfer within the shortest possible time from an earth orbit to more remote planets with the aid of a solar-wind sail (equipment utilizing solar radiation pressure). Other problems of this nature are the return of a spacecraft to the earth with deceleration in the earth’s atmosphere taken into account, or landing a craft on the moon or planets.

Problems pertaining to the development of programs of optimum control of vehicle motion in orbital transfer are entirely new and outside the framework of the problems of classical celestial mechanics, and their solution requires, as a rule, the application of methods developed in mathematical automatic control theory (the dynamic programming method, the Pontriagin maximum method, and so on). The practical utilization of the mathematical results of astrodynamics in orbital transfer problems is closely related to engineering problems in the design and automatic control of spacecraft. Examples of such orbital transfers executed for the first time in the USSR are the return to the earth of the second spaceship-satellite (Aug. 20,1960), the soft landing of the spacecraft Luna 9 on the moon (Feb. 3, 1966), the mission of the deep-space probe Venera 4 to the planet Venus (Oct. 18, 1967), the orbiting of the artificial lunar satellite Luna 10 (Apr. 1, 1966), and the return to earth of the spacecraft Zond 5 (Sept. 21,1968). The United States achieved the first landing of astronauts on the moon (July 20, 1969), accompanied by several orbital transfers, including a takeoff from the lunar surface into a selenocentric orbit and a subsequent transfer to an earthward flight orbit.

The construction of analytical, semianalytical, or numerical theories of the motion of artificial celestial bodies, making it possible to compute the position of such bodies in space at any instant of time depending on the initial position and initial velocity and the parameters of gravitational and other passive and active forces at work, occupies as significant a place in astrodynamics as in classical celestial mechanics. The development of these theories encounters various specific difficulties of a mathematical nature owing to the complexity of the equations of motion and the impossibility of restricting the discussion to the methods developed in classical celestial mechanics.

Topics pertaining to analysis and projection of the rotational motion of artificial celestial bodies about their center of inertia also occupy a prominent place in astrodynamics. In many instances fulfillment of a proposed space research program calls for knowledge of how the orientation of a spacecraft in space varies as the craft experiences translational displacements in orbit; it is often required that the spacecraft remain oriented in some specified manner—for example, relative to the earth and the sun—over a protracted time. The resulting problem of how to study the rotational motion of the craft is much more complicated than the analogous problem of studying the rotation of natural celestial bodies in classical celestial mechanics because the rotation of artificial celestial bodies is seriously influenced by the rotational moments generated as a result of the resistance presented by the atmosphere (aerodynamic effects), the effect of magnetic forces, and radiation pressure. In addition, space vehicles generally exhibit a complex dynamic shape, which adds to the mathematical difficulties in treating rotational moments of gravitational forces.

The projection of rotational motion reduces mainly to a problem of stabilizing the spacecraft orientation relative to a selected frame of reference. Stabilization techniques involving the use of rotating flywheels on board the spacecraft (gyroscopic stabilizers) and reaction propulsion engines, as well as additional design structures (known as passive stabilizing systems) employed to stabilize the effect of natural forces (gravitational forces, magnetic forces, and so on), are being developed. The problems solved in this division of astrodynamics involve the optimal stabilization of an axisymmetric artificial earth satellite with the aid of reaction engines, design of a system of gravitational stabilization of an artificial earth satellite moving in circular orbit, and the utilization of the effects of the gravitational and light fields of the sun on the spacecraft in interplanetary space in order to achieve a stable orientation of the craft relative to the sun.

Astrodynamics not only advances new problems and new requirements in the development of new methods but also prompts a revision of many “old” problems in classical celestial mechanics dealing with natural celestial bodies. For example, exact calculations of interplanetary flights would be impossible without the most exact data on the motion of the planets, on planetary masses, and on the distance separating planets. Theories of planetary motion extant prior to the development of astrodynamics have been found insufficiently exact in some applications. Improved theories making it possible to refine data on planetary masses are being developed. Research is continuing on refinements of the astronomical unit—the basic unit of scale in celestial mechanics.

### REFERENCES

Duboshin, G. N., and D. E. Okhotsimskii. “Nekotorye problemy astrodinamiki i nebesnoi mekhaniki.”*Kosmicheskie issledovaniia*, 1963, vol. 1, issue 2.

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Balk, M. B.

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*Vvedenie v teoriiu poleta iskusstvennykh sputnikov Zemli*. Moscow, 1965.

*Problemy orientatsii iskusstvennykh sputnikov Zemli*. Moscow, 1966. (Translated from English.)

King-Hele, D.

*Teoriia orbit iskusstvennykh sputnikov v atmosfere*. Moscow, 1966. (Translated from English.)

Beletskii, V. V.

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Levantovskii, V. I.

*Nebesnaia ballistika*. Moscow, 1965.

Demin, V. G.

*Dvizhenie iskusstvennogo sputnika v netsentral’nom pole tiagoteniia*. Moscow, 1968.

IU. A. RIABOV