Newton's law of universal gravitation

In modern language, the law states the following:

Every point mass attracts every single other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:[5]	Diagram of two masses attracting one another
${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }$ where: F is the force between the masses; G is the gravitational constant (6.674×10−11 m3⋅kg−1⋅s−2); m1 is the first mass; m2 is the second mass; r is the distance between the centers of the masses.

Error plot showing experimental values for G.

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.[34] The value of the constant G was first accurately determined from the results of the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798, although Cavendish did not himself calculate a numerical value for G.[6] 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.

Gravitational field strength within the Earth

Gravity 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 entails integrating the force (in vector form, see below) over the extents of the two bodies.

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.[5] (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 r₀ from the center of the mass distribution:[35]

• The portion of the mass that is located at radii r < r₀ causes the same force at the radius r₀ as if all of the mass enclosed within a sphere of radius r₀ was concentrated at the center of the mass distribution (as noted above).
• The portion of the mass that is located at radii r > r₀ exerts no net gravitational force at the radius r₀ from the center. That is, the individual gravitational forces exerted on a point at radius r₀ by the elements of the mass outside the radius r₀ 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.

Furthermore, inside a uniform sphere the gravity increases linearly with the distance from the center; the increase due to the additional mass is 1.5 times the decrease due to the larger distance from the center. Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than 2/3 of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary. Given this, the gravity of the Earth may be highest at the core/mantle boundary.

Gravity field surrounding Earth from a macroscopic perspective.

Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.

${\displaystyle \mathbf {F} _{21}=-G{m_{1}m_{2} \over {\vert \mathbf {r} _{21}\vert }^{2}}\,\mathbf {\hat {r}} _{21}}$

where

F₂₁ is the force applied on object 2 exerted by object 1,

G is the gravitational constant,

m₁ and m₂ are respectively the masses of objects 1 and 2,

|r₂₁| = |r₂ − r₁| is the distance between objects 1 and 2, and

${\displaystyle \mathbf {\hat {r}} _{21}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert }}}$ is the unit vector from object 1 to object 2.[36]

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.

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 r₁₂ and m instead of m₂ and define the gravitational field g(r) as:

${\displaystyle \mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}} }$

so that we can write:

${\displaystyle \mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} ).}$

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; 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

${\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}$

If m₁ 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

${\displaystyle V(r)=-G{\frac {m_{1}}{r}}.}$

the gravitational field is on, inside and outside of symmetric masses.

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

${\displaystyle \partial V}$ ${\displaystyle \mathbf {g(r)} \cdot d\mathbf {A} =-4\pi GM_{\text{enc}},}$

where ${\displaystyle \partial V}$ is a closed surface and ${\displaystyle M_{\text{enc}}}$ is the mass enclosed by the surface.

Hence, for a hollow sphere of radius ${\displaystyle R}$ and total mass ${\displaystyle M}$,

${\displaystyle |\mathbf {g(r)} |={\begin{cases}0,&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}$

For a uniform solid sphere of radius ${\displaystyle R}$ and total mass ${\displaystyle M}$,

${\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}$

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 ${\displaystyle \phi /c^{2}}$ and ${\displaystyle (v/c)^{2}}$ are both much less than one, where ${\displaystyle \phi }$ is the gravitational potential, ${\displaystyle v}$ is the velocity of the objects being studied, and ${\displaystyle c}$ is the speed of light in vacuum.[37] For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since

${\displaystyle {\frac {\phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8}}$

where ${\displaystyle r_{\text{orbit}}}$ 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.

Einstein's solution

Part of a series on
Spacetime
Special relativity General relativity
Spacetime concepts Spacetime manifold Equivalence principle Lorentz transformations Minkowski space
General relativity Introduction to general relativity Mathematics of general relativity Einstein field equations
Classical gravity Introduction to gravitation Newton's law of universal gravitation
Relevant mathematics Four-vector Derivations of relativity Spacetime diagrams Differential geometry Curved spacetime Mathematics of general relativity Spacetime topology

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 to 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.

Newton was the first to consider in his Principia an extended expression of his law of gravity including an inverse-cube term of the form

${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}+B{\frac {m_{1}m_{2}}{r^{3}}},\qquad B{\text{ a constant}}}$

attempting to explain the Moon's apsidal motion. Other extensions were proposed by Laplace (around 1790) and Decombes (1913):[39]

${\displaystyle F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\exp(-\alpha r)\qquad {\text{(Laplace)}}}$

${\displaystyle F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\left(1+{\alpha \over {r^{3}}}\right)\qquad {\text{(Decombes)}}}$

In recent years, quests for non-inverse square terms in the law of gravity have been carried out by neutron interferometry.[40]

The n-body problem is an ancient, classical problem[41] of predicting the individual motions of a group of celestial objects 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, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.[42] The n-body problem in general relativity 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)[43] of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.[44]

The two-body problem has been completely solved, as has the restricted three-body problem.[45]

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