Newton's law of universal gravitation

Modern form

In modern language, the law states the following:

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. 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. 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.

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. (This is not generally true for bodies that are not spherically symmetrical.)

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:

• 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.

Vector form

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. $${\displaystyle {\mathbf{F}}_{21}=-G\frac{m_1{m}_2}{{|{\mathbf{r}}_{21}|}^2}{\hat{\mathbf{r}}}_{21}=-G\frac{m_1{m}_2}{{|{\mathbf{r}}_{21}|}^3}{\mathbf{r}}_{21}}$$ where

• F₂₁ is the force applied on body 2 exerted by body 1,
• G is the gravitational constant,
• m₁ and m₂ are respectively the masses of bodies 1 and 2,
• r₂₁ = r₂ − r₁ is the displacement vector between bodies 1 and 2, and
• ${\displaystyle {\hat{\mathbf{r}}}_{21}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{{\mathbf{r}}_{\mathbf{2}}\mathbf{-}{\mathbf{r}}_{\mathbf{1}}}{|{\mathbf{r}}_{\mathbf{2}}\mathbf{-}{\mathbf{r}}_{\mathbf{1}}|}}$ 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 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 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: $${\displaystyle \mathbf{g}(\mathbf{r})=-G\frac{m_1}{{|\mathbf{r}|}^2}\hat{\mathbf{r}}}$$ 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 the dimension of acceleration; in the SI, its unit 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 $${\displaystyle \mathbf{g}(\mathbf{r})=-\mathrm{\nabla }V(\mathbf{r}).}$$

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 $${\displaystyle V(r)=-G\frac{m_1}{r}.}$$

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

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

where ${\displaystyle \mathrm{\partial }V}$ 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 ${\displaystyle M}$, $${\displaystyle |\mathbf{g}\mathbf{(}\mathbf{r}\mathbf{)}|=\{\begin{array}{ll}0,& \text{if }r<R\\ \\ {\displaystyle \frac{GM}{r^2}},& \text{if }r\ge R\end{array}}$$

For a uniform solid sphere of radius ${\displaystyle R}$ and total mass ${\displaystyle M}$, $${\displaystyle |\mathbf{g}\mathbf{(}\mathbf{r}\mathbf{)}|=\{\begin{array}{ll}{\displaystyle \frac{GMr}{R^3}},& \text{if }r<R\\ \\ {\displaystyle \frac{GM}{r^2}},& \text{if }r\ge R\end{array}}$$

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 ${\displaystyle \varphi /c^2}$ and ${\displaystyle (v/c{)}^2}$ are both much less than one, where ${\displaystyle \varphi }$ is the gravitational potential, ${\displaystyle v}$ is the velocity of the objects being studied, and ${\displaystyle c}$ is the speed of light in vacuum. For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since $${\displaystyle \frac{\varphi }{c^2}=\frac{G{M}_{\mathrm{s}\mathrm{u}\mathrm{n}}}{r_{\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}}{c}^2}\sim {10}^{-8},{\left(\frac{v_{\mathrm{E}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}}}{c}\right)}^2={\left(\frac{2\pi r_{\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}}}{(1\mathrm{y}\mathrm{r})c}\right)}^2\sim {10}^{-8},}$$ 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.

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.