# Newton's law of gravitation

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## Newton's law of gravitation:

see gravitation**gravitation,**

the attractive force existing between any two particles of matter.

**The Law of Universal Gravitation**

Since the gravitational force is experienced by all matter in the universe, from the largest galaxies down to the smallest particles, it is often called

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## Newton's law of gravitation

See gravitation.## Newton’s Law of Gravitation

one of the universal laws of nature. According to Newton’s law of gravitation, all physical bodies attract one another, and the magnitude of the force of attraction is independent of the physical and chemical properties of the bodies, the state of motion of the bodies, and the properties of the medium in which the bodies are located. On the earth, gravitation is manifested primarily in the existence of a gravitational force, which is a result of the attraction of any physical body by the earth.

Newton’s law of gravitation, which was discovered in the 17th century by I. Newton, may be stated in the following way. Any two mass points attract each other with a force *F* that is directly proportional to their masses *m*_{1} and *m*_{2} and inversely proportional to the square of the distance *r* between them:

Here, the force *F* is directed along the line that connects these points. The proportionality factor *G*, which is a constant, is called the constant of gravitation. In the cgs system, *G* ≈ 6.7 × 10^{−8} dyne · cm^{2} · g^{−2}. Here, the term “mass points” is understood to mean bodies whose dimensions are negligibly small in comparison with the distances between the bodies. Newton’s law of gravitation may be interpreted differently, if we assume that any mass point with a mass *m*_{1} creates around itself a field of attraction (a gravitational field) in which any other free mass point located at a distance *r* from the center of the field receives an acceleration that is independent of the mass of this second particle and that is equal to

and is directed toward the center of the field.

The gravitational forces and gravitational fields of separate particles possess the property of additivity; that is, the force acting on a certain particle produced by several other particles is equal to the geometric sum of the forces produced by each particle. It follows that the attraction between real physical bodies, taking into account the size, shape, and density distribution of the bodies, can be determined by calculating the sum of the attractive forces of separate small particles into which the bodies may be mentally divided; in such a calculation, the direction of the components of the forces is taken into account. It has been established in this manner that a spherical body, homogeneous or having a spherical distribution of mass density, has precisely the same force of attraction as that of a mass point if the distance *r* is measured from the center of the sphere.

The nature of the motion of celestial bodies in space is determined primarily by gravitational forces. Indeed, Newton’s law of gravitation was discovered and subsequently rigorously substantiated in the study of the motion of the planets and planetary satellites. In the early 17th century, J. Kepler empirically established the fundamental laws governing planetary motion, which are called Kepler’s laws. Proceeding from these laws, Newton’s contemporaries, such as the French astronomer I. Boullian, the Italian physicist G. A. Borelli, and the British physicist R. Hooke, reasoned that planetary motions may be attributed to the action of a force that attracts every planet to the sun and that decreases in inverse proportion to the square of the distance from the sun. However, this was not rigorously proved until Newton did so in 1687 in the *Philosophiae naturalis principia mathematical*, his proof was based on his first two laws of motion and his newly devised mathematical methods that constituted the foundation of the differential and integral calculi. Newton proved that the motion of every planet must obey Kepler’s first two laws if it moves under the gravitational force of the sun in accordance with formula (1). Newton further showed that the motion of the moon can be approximately explained by using an analogous force field for the earth and that the gravitational force of the earth results from the action of this force field on physical bodies near the surface of the earth. Newton concluded, on the basis of his third law of motion, that attraction is a reciprocal property, and he formulated his law of gravitation for all physical bodies. Derived from empirical data based on necessarily approximate observational results, Newton’s law of gravitation was originally a working hypothesis. An enormous amount of work over a period of 200 years was subsequently required in order to rigorously substantiate this law.

Newton’s law of gravitation was the foundation for celestial mechanics. In the 17th to 19th centuries, one of the fundamental tasks of celestial mechanics was to prove that gravitational interaction according to Newton’s law explains precisely the observed motions of celestial bodies in the solar system. Newton himself showed that the mutual attraction among the earth, moon, and sun explains quite accurately a number of peculiarities in the motion of the moon that had been observed much earlier, such as the lunar variations, the regression of the nodes, the motion of the perigee, and fluctuations in the inclination of the lunar orbit; he also showed that the earth, because of its rotation and the action of gravitational forces between the particles that make up the earth, should be flattened at the poles. Newton also attributed to gravitational forces such phenomena as the tides and the precession of the earth’s axis. One of the most brilliant confirmations of the validity of the law of universal gravitation in the history of astronomy was the discovery in 1845–46 of the planet Neptune as a result of preliminary theoretical calculations that predicted the planet’s position. Modern theories of the motion of the earth, moon, and planets that are based on Newton’s law of gravitation account for the observed motions of these bodies in all details, except for certain effects, such as the motions of the perihelia of Mercury, Venus, and Mars; these effects are explained in relativistic celestial mechanics, which is based on Einstein’s theory of gravitation.

According to Newton’s law of gravitation, gravitational interaction plays the primary role in the motion of such stellar systems as binary and multiple stars and within star clusters and galaxies. However, the gravitational fields within star clusters and galaxies are quite complex and have not yet been adequately studied. Consequently, the motions within these clusters are studied by methods that differ from those of celestial mechanics (*see*STELLAR ASTRONOMY). Gravitational interaction also plays a significant role in all cosmic processes in which concentration of large masses takes part. Newton’s law of gravitation is the basis for the study of the motion of artificial celestial bodies, in particular, space probes and artificial earth and lunar satellites. Gravimetry is based on Newton’s law of gravitation. The attractive forces between ordinary macroscopic bodies on the earth can be detected and measured but do not have any significant practical role. In a microcosm, gravitational forces are negligibly small in comparison with intramolecular and intranuclear forces.

Newton left unanswered the question of the nature of gravitation and did not explain the hypothesis of instantaneous propagation of gravitation in space, that is, the hypothesis that as the positions of bodies change there is an instantaneous change in the gravitational force between the bodies. This hypothesis is closely related to the nature of gravitation. The associated difficulties were not eliminated until Einstein’s theory of gravitation, which represents a new stage in the understanding of objective natural laws.

### REFERENCES

*Isaak N’iuton, 1643–1727*(a collection of articles commemorating Newton’s 300th birthday). Edited by Academician S. I. Vavilov. Moscow-Leningrad, 1943.

Berry, A.

*Kratkaia istoriia astronomii*. Moscow-Leningrad, 1946. (Translated from English.)

Subbotin, M. F.

*Vvedenie*ν

*teoreticheskuiu astronomiiu*. Moscow, 1968.

IU. A. RIABOV