Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to their masses and inversely proportional to the square of the distance between their centers of mass. Separated, spherically symmetrical objects attract and are attracted as if all their mass were concentrated at their centers. The publication of the law has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.

This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica (Latin for 'Mathematical Principles of Natural Philosophy' (the Principia)), first published on 5 July 1687.

The equation for universal gravitation thus takes the form:

<math display="block">F=G\frac{m_1m_2}{r^2},</math>

where F is the gravitational force acting between two objects, m<sub>1</sub> and m<sub>2</sub> are the masses of the objects, r is the distance between the centers of mass, and G is the gravitational constant, ().

The first test of Newton's law of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. It took place 111 years after the publication of Newton's Principia and approximately 71 years after his death.

Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has charge in place of mass and a different constant.

Newton's law was later superseded by Albert Einstein's theory of general relativity, but the universality of the gravitational constant is intact and the law still continues to be used as an excellent approximation of the effects of gravity in most applications. Relativity is required only when there is a need for extreme accuracy, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at small distances (such as Mercury's orbit around the Sun).

History

Before Newton's law of gravity, there were many theories explaining gravity. Philosophers made observations about things falling down − and developed theories why they do – as early as Aristotle who thought that rocks fall to the ground because seeking the ground was an essential part of their nature.

Around 1600, the scientific method began to take root. 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.

In 1687, Newton published his Principia which combined his laws of motion with new mathematical analysis to explain Kepler's empirical results. Newton's formulation was later condensed into the inverse-square law:<math display="block">F = G \frac{m_1 m_2}{r^2}, </math>where is the force, and are the masses of the objects interacting, is the distance between the centers of the masses and is the gravitational constant While is also called Newton's constant, Newton did not use this constant or formula, he only discussed proportionality.

Newton made quantitative analysis based on this formula around 1665, considering the period and distance of the Moon's orbit and considering the timing of objects falling on Earth. Newton did not publish these results at the time because he could not prove that the Earth's gravity acts as if all its mass were concentrated at its center. That proof took him twenty years. 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.

Newton's 1713 General Scholium in the second edition of Principia explains his model of gravity, translated in this case by Samuel Clarke:

{=}\ \frac{\mathbf{r_2 - r_1{|\mathbf {r_2 - r_1}|} </math> is the unit vector from body 1 to body 2.

It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that F<sub>12</sub> = −F<sub>21</sub>.

Gravity field

<!-- This section is linked from Center of mass -->

The gravitational field is a vector field that describes the gravitational force that would be applied on an object in any given point in space, per unit mass. It is actually equal to the gravitational acceleration at that point.

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). For two objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write r instead of r<sub>12</sub> and m instead of m<sub>2</sub> and define the gravitational field g(r) as:

<math display="block">\mathbf g(\mathbf r) =

- G {m_1 \over

\, \mathbf{\hat{r

</math>

so that we can write:

<math display="block">\mathbf{F}( \mathbf r) = m \mathbf g(\mathbf r). </math>

This formulation is dependent on the objects causing the field. The field has the dimension of acceleration; in the SI, its unit is m/s<sup>2</sup>.

Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent. This has the consequence that there exists a gravitational potential field V(r) such that

<math display="block"> \mathbf{g}(\mathbf{r}) = - \nabla V( \mathbf r).</math>

If m<sub>1</sub> is a point mass or the mass of a sphere with homogeneous mass distribution, the force field g(r) outside the sphere is isotropic, i.e., depends only on the distance r from the center of the sphere. In that case

<math display="block"> V(r) = -G\frac{m_1}{r}. </math>

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

where <math>\partial V</math> is a closed surface and <math>M_\text{enc}</math> is the mass enclosed by the surface.

Hence, for a hollow sphere of radius <math>R</math> and total mass <math>M</math>,

<math display="block">|\mathbf{g(r)}| = \begin{cases}

0, & \text{if } r < R \\

\\

\dfrac{GM}{r^2}, & \text{if } r \ge R

\end{cases}

</math>

For a uniform solid sphere of radius <math>R</math> and total mass <math>M</math>,

<math display="block">|\mathbf{g(r)}| = \begin{cases}

\dfrac{GM r}{R^3}, & \text{if } r < R \\

\\

\dfrac{GM}{r^2}, & \text{if } r \ge R

\end{cases}

</math>

Limitations

Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities <math>\phi / c^{2}</math> and <math>(v/c)^2</math> are both much less than one, where <math>\phi</math> is the gravitational potential, <math>v</math> is the velocity of the objects being studied, and <math>c</math> is the speed of light in vacuum. For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since

<math display="block">\frac{\phi}{c^2}=\frac{GM_\mathrm{sun{r_\mathrm{orbit}c^2} \sim 10^{-8},

\quad \left(\frac{v_\mathrm{Earth{c}\right)^2=\left(\frac{2\pi r_\mathrm{orbit{(1\ \mathrm{yr})c}\right)^2 \sim 10^{-8} ,</math>

where <math>r_\text{orbit} </math> is the radius of the Earth's orbit around the Sun.

In situations where either dimensionless parameter is large, then general relativity must be used to describe the system. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity.

Observations conflicting with Newton's formula

  • Newton's theory does not fully explain the precession of the perihelion of the orbits of the planets, especially that of Mercury, which was detected long after the life of Newton. There is a 43 arcsecond per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th century.
  • The predicted angular deflection of light rays by gravity (treated as particles travelling at the expected speed) that is calculated by using Newton's theory is only one-half of the deflection that is observed by astronomers. Calculations using general relativity are in much closer agreement with the astronomical observations.
  • In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey both Newton's law of universal gravitation and general relativity. Astrophysicists, however, explain this marked phenomenon by assuming the presence of large amounts of dark matter.

Einstein's solution

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 Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. This allowed a description of the motions of light and mass that was consistent with all available observations. 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.

Extensions

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

Solutions

The problem of predicting the motion of n objects subject to gravity is known as the n-body problem. The two-body problem has been completely solved, but for more bodies the solution is in general chaotic and can only be obtained numerically. The most-studied case is the three-body problem, for which several solutions for particular cases are known, for example those giving rise to the Lagrange points.

See also

Notes

References

  • Newton's Law of Universal Gravitation Javascript calculator