Newton's law of universal gravitation

[1][2][3] This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning.

Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies.

René Descartes started over with a more fundamental view, developing ideas of matter and action independent of theology.

Galileo Galilei wrote about experimental measurements of falling and rolling objects.

Johannes Kepler's laws of planetary motion summarized Tycho Brahe's astronomical observations.

The analysis required assuming that the gravitation force acted as if all of the mass of the Earth were concentrated at its center, an unproven conjecture at that time.

By 1680, new values for the diameter of the Earth improved his orbit time to within 1.6%, but more importantly Newton had found a proof of his earlier conjecture.

[8]: 201 In 1687 Newton published his Principia which combined his laws of motion with new mathematical analysis to explain Kepler's empirical results.

Newton would need an accurate measure of this constant to prove his inverse-square law.

When Newton presented Book 1 of the unpublished text in April 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him, ultimately a frivolous accusation.

[8]: 204 While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" that his equations implied.

In 1692, in his third letter to Bentley, he wrote: "That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it."

In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity (although he invented two mechanical hypotheses in 1675 and 1717).

Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science.

And in Newton's 1713 General Scholium in the second edition of Principia: "I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses. ...

It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies.

where Assuming SI units, F is measured in newtons (N), m1 and m2 in kilograms (kg), r in meters (m), and the constant G is 6.67430(15)×10−11 m3⋅kg−1⋅s−2.

In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies.

For points inside a spherically symmetric distribution of matter, Newton's shell theorem can be used to find the gravitational force.

The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance r0 from the center of the mass distribution:[13] As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration anywhere within the hollow sphere.

It is a generalisation of the vector form, which becomes particularly useful if more than two objects are involved (such as a rocket between the Earth and the Moon).

As per Gauss's law, field in a symmetric body can be found by the mathematical equation: where

Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used.

In situations where either dimensionless parameter is large, then general relativity must be used to describe the system.

The first two conflicts with observations above were explained by Einstein's theory of general relativity, in which gravitation is a manifestation of curved spacetime instead of being due to a force propagated between bodies.

In general relativity, the gravitational force is a fictitious force resulting from the curvature of spacetime, because the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime.

In recent years, quests for non-inverse square terms in the law of gravity have been carried out by neutron interferometry.

Solving this problem – from the time of the Greeks and on – has been motivated by the desire to understand the motions of the Sun, planets and the visible stars.

The classical problem can be informally stated as: given the quasi-steady orbital properties (instantaneous position, velocity and time)[20] of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.

[21] In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.