Newton's law of universal gravitation describesgravity as aforce by stating that everyparticle attracts every other particle in the universe with a force that isproportional to the product of their masses andinversely proportional to the square of the distance between their centers of mass. Separated, spherically symmetrical objects attract and are attractedas 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.[1][2]
The equation for universal gravitation thus takes the form:whereF is the gravitational force acting between two objects,m1 andm2 are the masses of the objects,r is the distance between thecenters of mass, andG is thegravitational constant, (6.674×10−11 m3⋅kg−1⋅s−2).
The first test of Newton's law of gravitation between masses in the laboratory was theCavendish experiment conducted by the British scientistHenry Cavendish in 1798.[4] It took place 111 years after the publication of Newton'sPrincipia and approximately 71 years after his death.
Newton's law of gravitation resemblesCoulomb's law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both areinverse-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Newton's law was later superseded byAlbert Einstein's theory ofgeneral relativity, but the universality of the gravitational constant is intact and the law still continues to be used as an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 asMercury's orbit around theSun).
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 asAristotle who thought that rocks fall to the ground because seeking the ground was an essential part of their nature.[5]
Around 1666Isaac Newton developed the idea that Kepler's laws must also apply to the orbit of the Moon around the Earth and then to all objects on Earth. 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. His calculations of the Moon orbit time was within 16% of the known value. 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.[7]: 201
In 1687 Newton published hisPrincipia which combined hislaws of motion with new mathematical analysis to explain Kepler's empirical results.[6]: 134 Newton's formulation was later condensed into the inverse-square law:whereF is the force,m1 andm2 are the masses of the objects interacting,r is the distance between the centers of the masses andG is thegravitational constant6.674×10−11 m3⋅kg−1⋅s−2.[8] WhileG is also calledNewton's constant, Newton did not use this constant or formula, he only discussed proportionality.[9]: 31 That was sufficient to show that the gravity of the Earth on the Moon is the same as the gravity of the Earth on an apple: Using the values known at the time, Newton was able to verify this form of his law. The value ofG was eventuallymeasured byHenry Cavendish in 1797.[9]: 31
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."[11][12]: 26
Newton's 1713General Scholium in the second edition ofPrincipia explains his model of gravity, translated in this case bySamuel Clarke:
I have explained the Pharnomena of the Heavens and the Sea, by the Force of Gravity; but the Cause of Gravity I have not yet assigned. It is a Force arising from some Cause, which reaches to the very Centers of the Sun and Planets, without any diminution of its Force: And it acts, not proportionally to the Surfaces of the Particles it acts upon, as Mechanical Causes use to do; but proportionally to the Quantity of Solid Matter: And its Action reaches every way to immense Distances, decreasing always in a duplicate ratio of the Distances. But the Cause of these Properties of Gravity, I have not yet found deducible from Pharnomena: And Hypotheses I make not.[13]: 383
The last sentence is Newton's famous[13] and highly debated[14]Latin phraseHypotheses non fingo. In other translations it comes out "I feign no hypotheses".[15]
r is the distance between the centers of the masses.
Error plot showing experimental values forG
AssumingSI units,F is measured innewtons (N),m1 andm2 inkilograms (kg),r in meters (m), and the constantG is6.67430(15)×10−11 m3⋅kg−1⋅s−2.[8] The value of the constantG was first accurately determined from the results of theCavendish experiment conducted by theBritish scientistHenry Cavendish in 1798, although Cavendish did not himself calculate a numerical value forG.[4] 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'sPrincipia and 71 years after Newton's death, so none of Newton's calculations could use the value ofG; instead he could only calculate a force relative to another force.
Gravitational field strength within the EarthGravity field near the surface of the Earth – an object is shown accelerating toward the surface
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 entailsintegrating the force (in vector form, see below) over the extents of the twobodies.
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.[16] (This is not generally true for bodies that are not spherically symmetrical.)
For pointsinside a spherically symmetric distribution of matter, Newton'sshell 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 distancer0 from the center of the mass distribution:[17]
The portion of the mass that is located at radiir <r0 causes the same force at the radiusr0 as if all of the mass enclosed within a sphere of radiusr0 was concentrated at the center of the mass distribution (as noted above).
The portion of the mass that is located at radiir >r0 exertsno net gravitational force at the radiusr0 from the center. That is, the individual gravitational forces exerted on a point at radiusr0 by the elements of the mass outside the radiusr0 cancel each other.
As a consequence, for example, within a shell of uniform thickness and density there isno net gravitational acceleration anywhere within the hollow sphere.
Gravity field surrounding Earth from a macroscopic perspective
Newton's law of universal gravitation can be written as avectorequation 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 body 2 exerted by body 1,
It can be seen that the vector form of the equation is the same as thescalar form given earlier, except thatF is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen thatF12 = −F21.
Thegravitational field is avector 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 thegravitational 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 writer instead ofr12 andm instead ofm2 and define the gravitational fieldg(r) as:so that we can write:
This formulation is dependent on the objects causing the field. The field has the dimension of acceleration; in theSI, its unit is m/s2.
Gravitational fields are alsoconservative; 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 fieldV(r) such that
Ifm1 is a point mass or the mass of a sphere with homogeneous mass distribution, the force fieldg(r) outside the sphere is isotropic, i.e., depends only on the distancer from the center of the sphere. In that case
As perGauss's law, 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,
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 thegravitational potential, is the velocity of the objects being studied, and is thespeed of light in vacuum.[19] For example, Newtonian gravity provides an accurate description of the Earth/Sun system, sincewhere is the radius of the Earth's orbit around the Sun.
In situations where either dimensionless parameter is large, thengeneral 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.
Newton's theory does not fully explain theprecession of the perihelion of the orbits of the planets, especially that of Mercury, which was detected long after the life of Newton.[20] There is a 43arcsecond 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 angulardeflection 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.[citation needed] 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 ofdark matter.
The first two conflicts with observations above were explained by Einstein's theory ofgeneral relativity, in which gravitation is a manifestation ofcurved 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 afictitious force resulting from thecurvature of spacetime, because thegravitational acceleration of a body infree fall is due to itsworld line being ageodesic ofspacetime.
The problem of predicting the motion ofn objects subject to gravity is known as then-body problem. Thetwo-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 thethree-body problem, for which several solutions for particular cases are known, for example those giving rise to theLagrange points.[citation needed]
^Isaac Newton: "In [experimental] philosophy particular propositions are inferred from the phenomena and afterwards rendered general by induction":Principia, Book 3,General Scholium, at p. 392 in Volume 2 of Andrew Motte's English translation published 1729.
^abHesse, Mary B. (2005).Forces and fields: the concept of action at a distance in the history of physics. Mineola, New York: Dover.ISBN978-0-486-44240-2.
^Max Born (1924),Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and the Earth.)
