Newton's law of universal gravitation is usually stated as that every
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
attracts every other particle in the universe with a
force that is
proportional to the product of their masses and
inversely proportional
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
to the square of the distance between their centers.
[It was shown separately that separated spherically symmetrical masses 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
Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
derived from
empirical observation
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
s by what
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
called
inductive reasoning. It is a part of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
and was formulated in Newton's work ''
Philosophiæ Naturalis Principia Mathematica'' ("the ''Principia''"), first published on 5 July 1687. When Newton presented Book 1 of the unpublished text in April 1686 to the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
,
Robert Hooke made a claim that Newton had obtained the inverse square law from him.
In today's language, the law states that every
point
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Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
attracts every other point mass by a
force acting along the
line intersecting the two points. The force is
proportional to the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of the two masses, and inversely proportional to the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the distance between them.
The equation for universal gravitation thus takes the form:
:
where ''F'' is the gravitational force acting between two objects, ''m
1'' and ''m
2'' are the masses of the objects, ''r'' is the distance between the
centers of their masses, and ''G'' is the
gravitational constant.
The first test of Newton's law of gravitation between masses in the laboratory was the
Cavendish experiment
The Cavendish experiment, performed in 1797–1798 by English scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational con ...
conducted by the
British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity and common culture
* British English, ...
scientist
Henry Cavendish
Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English natural philosopher and scientist who was an important experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "infl ...
in 1798.
[The Michell–Cavendish Experiment](_blank)
Laurent Hodges 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
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
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 the product of two charges in place of the product of the masses, and the
Coulomb constant in place of the gravitational constant.
Newton's law has later been superseded by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, but the universality of 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
Early history
In 1604,
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He wa ...
correctly hypothesized that the distance of a falling object is proportional to the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the time elapsed. The relation of the distance of objects in
free fall
In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on ...
to the square of the time taken was confirmed by Italian
Jesuits
, image = Ihs-logo.svg
, image_size = 175px
, caption = ChristogramOfficial seal of the Jesuits
, abbreviation = SJ
, nickname = Jesuits
, formation =
, founders = ...
Grimaldi and
Riccioli
Giovanni Battista Riccioli, SJ (17 April 1598 – 25 June 1671) was an Italian astronomer and a Catholic priest in the Jesuit order. He is known, among other things, for his experiments with pendulums and with falling bodies, for his discussion ...
between 1640 and 1650. They also made a calculation of the
gravity of Earth by recording the oscillations of a pendulum.
A modern assessment about the early history of the inverse square law is that "by the late 1670s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons".
[Discussion points can be seen for example in the following papers:
*
*
*
] The same author credits
Robert Hooke with a significant and seminal contribution, but treats Hooke's claim of priority on the inverse square point as irrelevant, as several individuals besides Newton and Hooke had suggested it. He points instead to the idea of "compounding the
celestial motions" and the conversion of Newton's thinking away from "
centrifugal" and towards "
centripetal
A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved trajectory, path. Its direction is always orthogonality, orthogonal to the motion of the body and towards the fixed po ...
" force as Hooke's significant contributions.
Newton gave credit in his ''Principia'' to two people:
Bullialdus (who wrote without proof that there was a force on the Earth towards the Sun), and
Borelli (who wrote that all planets were attracted towards the Sun).
[Bullialdus (Ismael Bouillau) (1645), "Astronomia philolaica", Paris, 1645.][Borelli, G. A., "Theoricae Mediceorum Planetarum ex causis physicis deductae", Florence, 1666.] The main influence may have been Borelli, whose book Newton had a copy of.
[See especially p. 13 in ]
Plagiarism dispute
In 1686, when the first book of
Newton's ''
Principia'' was presented to the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
,
Robert Hooke accused Newton of
plagiarism by claiming that he had taken from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". At the same time (according to
Edmond Halley
Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720.
From an observatory he constructed on Saint Helena in 1676–77, H ...
's contemporary report) Hooke agreed that "the Demonstration of the Curves generated thereby" was wholly Newton's.
[H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676–1687), (Cambridge University Press, 1960), giving the Halley–Newton correspondence of May to July 1686 about Hooke's claims at pp. 431–448, see particularly page 431.]
Hooke's work and claims
Robert Hooke published his ideas about the "System of the World" in the 1660s, when he read to the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
on March 21, 1666, a paper "concerning the inflection of a direct motion into a curve by a supervening attractive principle", and he published them again in somewhat developed form in 1674, as an addition to "An Attempt to Prove the Motion of the Earth from Observations".
[Hooke's 1674 statement in "An Attempt to Prove the Motion of the Earth from Observations" is available i]
online facsimile here
Hooke announced in 1674 that he planned to "explain a System of the World differing in many particulars from any yet known", based on three suppositions: that "all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers" and "also attract all the other Celestial Bodies that are within the sphere of their activity"; that "all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a straight line, till they are by some other effectual powers deflected and bent..." and that "these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers". Thus Hooke postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, together with a principle of linear inertia.
Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses. He also did not provide accompanying evidence or mathematical demonstration. On the latter two aspects, Hooke himself stated in 1674: "Now what these several degrees
f attractionare I have not yet experimentally verified"; and as to his whole proposal: "This I only hint at present", "having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e. "prosecuting this Inquiry").
It was later on, in writing on 6 January 1679, 80 to Newton, that Hooke communicated his "supposition ... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as
Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
Supposes Reciprocall to the Distance." (The inference about the velocity was incorrect.)
Hooke's correspondence with Newton during 1679–1680 not only mentioned this inverse square supposition for the decline of attraction with increasing distance, but also, in Hooke's opening letter to Newton, of 24 November 1679, an approach of "compounding the celestial motions of the planets of a direct motion by the tangent & an attractive motion towards the central body".
Newton's work and claims
Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hooke's 1679 letter. Newton also pointed out and acknowledged prior work of others,
[Pages 435–440 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676–1687), (Cambridge University Press, 1960), document #288, 20 June 1686.] including
Bullialdus,
(who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and
Borelli (who suggested, also without demonstration, that there was a centrifugal tendency in counterbalance with a gravitational attraction towards the Sun so as to make the planets move in ellipses). D T Whiteside has described the contribution to Newton's thinking that came from Borelli's book, a copy of which was in Newton's library at his death.
Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, he would still have some rights to it in view of his demonstrations of its accuracy. Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center. According to Newton, while the 'Principia' was still at pre-publication stage, there were so many a priori reasons to doubt the accuracy of the inverse-square law (especially close to an attracting sphere) that "without my (Newton's) Demonstrations, to which Mr Hooke is yet a stranger, it cannot believed by a judicious Philosopher to be any where accurate."
This remark refers among other things to Newton's finding, supported by mathematical demonstration, that if the inverse square law applies to tiny particles, then even a large spherically symmetrical mass also attracts masses external to its surface, even close up, exactly as if all its own mass were concentrated at its center. Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles. In addition, Newton had formulated, in Propositions 43–45 of Book 1 and associated sections of Book 3, a sensitive test of the accuracy of the inverse square law, in which he showed that only where the law of force is calculated as the inverse square of the distance will the directions of orientation of the planets' orbital ellipses stay constant as they are observed to do apart from small effects attributable to inter-planetary perturbations.
In regard to evidence that still survives of the earlier history, manuscripts written by Newton in the 1660s show that Newton himself had, by 1669, arrived at proofs that in a circular case of planetary motion, "endeavour to recede" (what was later called centrifugal force) had an inverse-square relation with distance from the center. After his 1679–1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. According to Newton scholar J. Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s. The lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis. This background shows there was basis for Newton to deny deriving the inverse square law from Hooke.
Newton's acknowledgment
On the other hand, Newton did accept and acknowledge, in all editions of the ''Principia'', that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the solar system. Newton acknowledged Wren, Hooke, and Halley in this connection in the Scholium to Proposition 4 in Book 1. Newton also acknowledged to Halley that his correspondence with Hooke in 1679–80 had reawakened his dormant interest in astronomical matters, but that did not mean, according to Newton, that Hooke had told Newton anything new or original: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it ..."
Modern priority controversy
Since the time of Newton and Hooke, scholarly discussion has also touched on the question of whether Hooke's 1679 mention of 'compounding the motions' provided Newton with something new and valuable, even though that was not a claim actually voiced by Hooke at the time. As described above, Newton's manuscripts of the 1660s do show him actually combining tangential motion with the effects of radially directed force or endeavour, for example in his derivation of the inverse square relation for the circular case. They also show Newton clearly expressing the concept of linear inertia—for which he was indebted to Descartes' work, published in 1644 (as Hooke probably was).
[See especially p. 10 in ] These matters do not appear to have been learned by Newton from Hooke.
Nevertheless, a number of authors have had more to say about what Newton gained from Hooke and some aspects remain controversial.
[ The fact that most of Hooke's private papers had been destroyed or have disappeared does not help to establish the truth.
Newton's role in relation to the inverse square law was not as it has sometimes been represented. He did not claim to think it up as a bare idea. What Newton did, was to show how the inverse-square law of attraction had many necessary mathematical connections with observable features of the motions of bodies in the solar system; and that they were related in such a way that the observational evidence and the mathematical demonstrations, taken together, gave reason to believe that the inverse square law was not just approximately true but exactly true (to the accuracy achievable in Newton's time and for about two centuries afterwards – and with some loose ends of points that could not yet be certainly examined, where the implications of the theory had not yet been adequately identified or calculated).
About thirty years after Newton's death in 1727, ]Alexis Clairaut
Alexis Claude Clairaut (; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had ou ...
, a mathematical astronomer eminent in his own right in the field of gravitational studies, wrote after reviewing what Hooke published, that "One must not think that this idea ... of Hooke diminishes Newton's glory"; and that "the example of Hooke" serves "to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated".
Newton's reservations
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."''
He never, in his words, "assigned the cause of this power". 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. He lamented that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer has yet to be found. And in Newton's 1713 ''General Scholium
The "General Scholium" (''Scholium Generale'' in the original Latin) is an essay written by Isaac Newton, appended to his work of ''Philosophiæ Naturalis Principia Mathematica'', known as the ''Principia''. It was first published with the secon ...
'' 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."''
Modern form
In modern language, the law states the following:
Assuming SI units, ''F'' is measured in newtons (N), ''m''1 and ''m''2 in kilograms (kg), ''r'' in meters (m), and the constant ''G'' is
The value of the constant ''G'' was first accurately determined from the results of the Cavendish experiment
The Cavendish experiment, performed in 1797–1798 by English scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational con ...
conducted by the British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity and common culture
* British English, ...
scientist Henry Cavendish
Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English natural philosopher and scientist who was an important experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "infl ...
in 1798, although Cavendish did not himself calculate a numerical value for ''G''. This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's ''Principia'' and 71 years after Newton's death, so none of Newton's calculations could use the value of ''G''; instead he could only calculate a force relative to another force.
Bodies with spatial extent
If the bodies in question have spatial extent (as opposed to being point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses that constitute the bodies. 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
Bodies may refer to:
* The plural of body
* ''Bodies'' (2004 TV series), BBC television programme
* Bodies (upcoming TV series), an upcoming British crime thriller limited series
* "Bodies" (''Law & Order''), 2003 episode of ''Law & Order''
* ...
.
In this way, it can be shown that an object with a spherically symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its center.[Proposition 75, Theorem 35: p. 956 – I.Bernard Cohen and Anne Whitman, translators: ]Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
, ''The Principia'': Mathematical Principles of Natural Philosophy
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Preceded by ''A Guide to Newton's Principia'', by I.Bernard Cohen. University of California Press 1999 (This is not generally true for non-spherically-symmetrical 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 ''r''0 from the center of the mass distribution:
* The portion of the mass that is located at radii causes the same force at the radius ''r''0 as if all of the mass enclosed within a sphere of radius ''r''0 was concentrated at the center of the mass distribution (as noted above).
* The portion of the mass that is located at radii exerts ''no net'' gravitational force at the radius ''r''0 from the center. That is, the individual gravitational forces exerted on a point at radius ''r''0 by the elements of the mass outside the radius ''r''0 cancel each other.
As a consequence, for example, within a shell of uniform thickness and density there is ''no net'' gravitational acceleration anywhere within the hollow sphere.
Vector form
Newton's law of universal gravitation can be written as a vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.
where
* F21 is the force applied on object 2 exerted by object 1,
* ''G'' is the gravitational constant,
* ''m''1 and ''m''2 are respectively the masses of objects 1 and 2,
* , r21, = , r2 − r1, is the distance between objects 1 and 2, and
* is the unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
from object 1 to object 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 F12 = −F21.
Gravity field
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
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
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 r12 and ''m'' instead of ''m''2 and define the gravitational field g(r) as:
:
so that we can write:
:
This formulation is dependent on the objects causing the field. The field has units of acceleration; in SI, this is m/s2.
Gravitational fields are also conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
; 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
:
If ''m''1 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
:
the gravitational field is on, inside and outside of symmetric masses.
As per Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
, field in a symmetric body can be found by the mathematical equation:
:
where is a closed surface and is the mass enclosed by the surface.
Hence, for a hollow sphere of radius and total mass ,
:
For a uniform solid sphere of radius and total mass ,
:
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 and are both much less than one, where is the gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
, is the velocity of the objects being studied, and is the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in vacuum.
For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since
:
where is the radius of the Earth's orbit around the Sun.
In situations where either dimensionless parameter is large, then
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
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
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
.
Einstein's solution
The first two conflicts with observations above were explained by Einstein's theory of general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, in which gravitation is a manifestation of curved spacetime
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
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
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
of a body in free fall
In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on ...
is due to its world line
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from c ...
being a geodesic of spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
.
Extensions
In recent years, quests for non-inverse square terms in the law of gravity have been carried out by neutron interferometry.
Solutions of Newton's law of universal gravitation
The ''n''-body problem is an ancient, classical problem[Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the ''n''-body problem, especially Ms. Kovalevskaya's ~1868–1888, twenty-year complex-variables approach, failure; Section 1: The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics (Chapter 1, ''the motion of a rigid body about a fixed point'' (Euler and Poisson ''equations''); Chapter 2, ''Mathematical Exterior Ballistics''), good precursor background to the ''n''-body problem; Section 2: Celestial Mechanics (Chapter 1, ''The Uniformization of the Three-body Problem'' (Restricted Three-body Problem); Chapter 2, ''Capture in the Three-Body Problem''; Chapter 3, ''Generalized n-body Problem'').] of predicting the individual motions of a group of celestial object
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often u ...
s interacting with each other gravitationally. Solving this problem — from the time of the Greeks and on — has been motivated by the desire to understand the motions of the Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s and the visible stars. In the 20th century, understanding the dynamics of globular cluster
A globular cluster is a spheroidal conglomeration of stars. Globular clusters are bound together by gravity, with a higher concentration of stars towards their centers. They can contain anywhere from tens of thousands to many millions of membe ...
star systems became an important ''n''-body problem too.[See References sited for Heggie and Hut. This Wikipedia page has made their approach obsolete.] The ''n''-body problem in general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
is considerably more difficult to solve.
The classical physical problem can be informally stated as: ''given the quasi-steady orbital properties'' (''instantaneous position, velocity and time'') ''of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times''.
The two-body problem
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
has been completely solved, as has the restricted three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
.[A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary ''n'' can be approximated via ]Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, but in practice such an infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the ''n''-body problem may be solved using numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book Gravitational ''N''-body Simulations listed in the References.
See also
* Bentley's paradox
* Gauss's law for gravity
In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux ( surface inte ...
* Jordan and Einstein frames
Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan River ...
* Kepler orbit
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
* Newton's cannonball
Newton's cannonball was a thought experiment Isaac Newton used to hypothesize that the force of gravity was universal, and it was the key force for planetary motion. It appeared in his posthumously published 1728 work ''De mundi systemate'' (als ...
* Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
* Social gravity
Social gravity is a term relating social structure and socioeconomics to the theory of Newtonian gravity.
Political and commercial theory
The term has been used to describe the system of commercial and political influence. For instance, in ''The A ...
* Static forces and virtual-particle exchange
Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces ...
Notes
References
External links
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Newton‘s Law of Universal Gravitation Javascript calculator
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