# Newton's law of gravitation

The equivalent forces of attraction of two masses

The Newton's law of universal gravitation is a physical law of classical physics , according to which each ground point to every other point mass with an attractive gravitational force acts. This gravitational force is directed along the line connecting the two mass points and its strength is proportional to the product of the two masses and inversely proportional to the square of their distance.

Newton's law of gravity is one of the fundamental laws of classical physics. It was established by Isaac Newton in his 1687 work Philosophiae Naturalis Principia Mathematica . With this Newton succeeded in the context of the classical mechanics which he founded at the same time, the first joint explanation for the force of gravity on the earth, for the moon orbit around the earth and for the planetary movement around the sun. The Newtonian theory of gravity explains these and other gravitational-related phenomena such as the tides on the earth and orbital disturbances of the moon and planets with great accuracy. Remaining discrepancies were not cleared up until the beginning of the 20th century by the general theory of relativity developed by Albert Einstein .

## history

Newton's first, more intensive occupation with the physical description of the planetary orbits and the role of gravity, which took place in his annus mirabilis 1665/66, partly contained the concept of a quadratically decreasing force of gravity. Newton did not justify this, however, or made incorrect assumptions, in particular not yet from the idea of ​​the universal (i.e. extraterrestrial) effect of gravity.

From 1678 on, Newton, in collaboration with Hooke and Flamsteed , worked intensively on mechanics, in particular with Kepler's laws. In an exchange of letters with Newton, Hooke mentioned his theory of planetary motion, in which there was talk of an attraction that decreases with distance; in Newton's answer, it assumed constant gravity. This correspondence was the starting point for Hooke's later accusation of plagiarism to Newton. Newton admitted that Hooke had led him on the right path: both the idea that the orbital ellipse stems from a gravitational force that decreases (with the square of the distance from a focal point) came from Hooke as well as the idea that this concept too is applicable to planetary motions. Hooke's suggestion of decreasing gravity, however, was based on intuition and not - as with Newton - on observation and logical deduction.

Newton published his preliminary results in 1684 under the title De Motu Corporum . Building on this, he laid the foundations of classical mechanics in his three-volume work Philosophiae Naturalis Principia Mathematica ( Mathematical Foundations of Natural Philosophy ) in 1687 . In it Newton formulated Newton's three laws of motion and the law of gravity, the latter, however, not in the concise form as given at the beginning of this article, but distributed over several sections. He justified the laws in detail using the geometrical form of the infinitesimal calculus created by him for the first time . The third part of the work, entitled About the World System, deals with the application of the new laws to the actual movements of celestial bodies, whereby Newton compares his calculations with a large number of measurement data from other naturalists and in this way proves the correctness of his theoretical derivations.

In 1797, Henry Cavendish was the first to succeed in an experiment with a sensitive rotary balance to experimentally measure the mutual attraction of two bodies of known mass, as follows from Newton's law of gravitation. The measuring device is similar to the torsion balance with which Charles Augustin de Coulomb examined electrostatic attraction and repulsion in 1785; it was originally designed by the geologist John Michell . To prove gravitation, Cavendish had to rule out the influence of the smallest disturbances, for example he operated his experiment from another room and took the readings with a telescope.

In the explicit form used today, the law of gravity was not formulated by Newton himself, but only in 1873, i.e. 200 years later, by Alfred Cornu and Jean-Baptist Baille . Until then, Newton's law of gravitation had only been used in its original form; H. in the form of proportionalities , and without definition of a "gravitational constant". ${\ displaystyle F \ propto m_ {1} \, \ F \ propto m_ {2} \, \ F \ propto r ^ {- 2}}$

Newton's law of gravitation made it possible to calculate the positions of the planets much more precisely than before. The positions calculated according to Ptolomew or Copernicus often deviated by (this corresponds to 1/3 moon diameter) from the observations, which calculated according to Kepler's laws by up to . On the other hand, with the help of Newtonian celestial mechanics , it was possible to attribute these deviations , known as orbital disturbances , to the attraction of the other planets. In the case of Uranus , it was even concluded that there was a previously unknown planet, Neptune , whose approximate position was first calculated by Urbain Le Verrier from the exact values ​​of the orbital disturbance. Shortly afterwards, Johann Gottfried Galle discovered the new planet at a distance of only one degree from the forecast. However, the later discovered perihelion of the orbit of Mercury could only be explained to about 90% with the same method. For the full explanation, the general theory of relativity first had to be developed. This much more comprehensive theory contains Newton's law of gravitation as the limiting case that only applies to sufficiently small mass densities and velocities. ${\ displaystyle 10 ^ {\ prime}}$${\ displaystyle 1 ^ {\ prime}}$

## Mathematical formulation

### Mass points

The amount of force between two mass points and in the distance is ${\ displaystyle m_ {1}}$${\ displaystyle m_ {2}}$${\ displaystyle r}$

${\ displaystyle F = G {\ frac {m_ {1} m_ {2}} {r ^ {2}}}}$

The size is the gravitational constant . The forces acting on the two masses have the same amount and each point to the other mass point (see illustration). In contrast to the mathematically similar Coulomb's law, Newton 's law of gravitation describes an always attractive force. ${\ displaystyle G}$

#### Vectorial

In vector form applies to the force acting on mass point 1 ${\ displaystyle {\ vec {F}} _ {1}}$

${\ displaystyle {\ vec {F}} _ {1} = G {\ frac {m_ {1} m_ {2}} {r ^ {3}}} {\ vec {r}} _ {12} = G {\ frac {m_ {1} m_ {2}} {| {\ vec {r}} _ {2} - {\ vec {r}} _ {1} | ^ {3}}} ({\ vec { r}} _ {2} - {\ vec {r}} _ {1})}$,

where and are the positions (position vectors ) of the two mass points. ${\ displaystyle {\ vec {r}} _ {1}}$${\ displaystyle {\ vec {r}} _ {2}}$

${\ displaystyle {\ vec {F}} _ {2}}$points to mass point 1 and is the opposite vector to :${\ displaystyle {\ vec {F}} _ {1}}$

${\ displaystyle {\ vec {F}} _ {2} = - {\ vec {F}} _ {1}}$.

If the mass point 1 is attracted by several mass points 2, 3 ..., n, the individual forces add up to the total force acting on mass point 1

${\ displaystyle {\ vec {F}} _ {1} = Gm_ {1} \ sum _ {i = 2} ^ {n} m_ {i} {\ frac {{\ vec {r}} _ {i} - {\ vec {r}} _ {1}} {| {\ vec {r}} _ {i} - {\ vec {r}} _ {1} | ^ {3}}}}$

#### Gravitational acceleration

According to Newton's second axiom, this results in an acceleration with the absolute value

${\ displaystyle a_ {1} = {\ frac {F_ {1}} {m_ {1}}}}$,

which is also called gravitational acceleration or gravitational field strength at the location of the mass (see gravitational field ). ${\ displaystyle m_ {1}}$

Two point masses and experience the accelerations at a distance in the absence of other forces due to Newton's law of gravitation: ${\ displaystyle m_ {1}}$${\ displaystyle m_ {2}}$${\ displaystyle r}$

${\ displaystyle a_ {1} = {\ frac {F} {m_ {1}}} = G {\ frac {m_ {2}} {r ^ {2}}} \ qquad {\ text {or}} \ qquad a_ {2} = {\ frac {F} {m_ {2}}} = G {\ frac {m_ {1}} {r ^ {2}}}}$

The mass attracts the mass and vice versa. Both masses are accelerated towards the common center of gravity . Seen from one of the bodies, the other moves with an acceleration that is the sum of the individual accelerations: ${\ displaystyle m_ {1}}$${\ displaystyle m_ {2}}$

${\ displaystyle a_ {1} + a_ {2} = G \, {\ frac {m_ {1} + m_ {2}} {r ^ {2}}}}$

If one of the masses is much smaller than the other, it is approximately sufficient to only consider the larger mass. The earth has much more mass than an apple, a person or a truck, so that it is sufficient for all these objects to insert the earth's mass in the equation for the acceleration. If they are in the same place, all three objects are accelerated equally strongly towards the center of the earth. They fall at the same speed and in the same direction. However, when looking at a binary star system, both star masses have to be considered because they are about the same size.

If an object changes only very slightly during the movement, the gravitational acceleration is practically constant, for example in the case of an object close to the earth's surface that falls only a few meters deep, i.e. vanishingly little compared to the earth's radius of r = approx. 6370 km. In a sufficiently small area, the gravitational field can therefore be viewed as homogeneous. If one cannot neglect the change in the gravitational force with the distance, it is possible to calculate, for example, the impact speed of a freely falling body with the help of integral calculus, i.e. H. about the gravitational potential${\ displaystyle r}$

${\ displaystyle \ Phi (r) = - {\ frac {Gm} {r}}}$.

### Expansive bodies

Real bodies are not point masses, but have a spatial extension. Since the law of gravity is linear in the masses, the body can be broken down into small parts, the contributions of which, as shown in the previous section, can be added vectorially. When crossing the border to infinitely small parts, an integral results instead of a sum .

In this way it can be shown, among other things, that an object with spherically symmetrical mass distribution in the outer space has the same gravitational effect as if its entire mass were united in its center of gravity. Therefore, extended celestial bodies can be treated approximately as mass points. In the inside of an elliptical or spherical symmetrical homogeneous mass distribution, z. B. a hollow sphere , the gravitational force emanating from this mass is zero. From this it follows that at any distance from the center of a spherically symmetrical mass distribution, the gravitational force is generated precisely by the portion of the total mass that lies within a sphere with the radius . Newton proved this theorem (which is also called Newton's shell theorem) in his Philosophiae Naturalis Principia Mathematica . The theorem does not generally apply to bodies that are not elliptically symmetrical or to inhomogeneous mass distributions. It should also be noted that gravity has no counterforce, so it cannot be shielded. The actual gravitational field in a hollow sphere would therefore not be zero, since the gravitational forces of all other masses in the universe would naturally act inside - only the spherical shell itself would not contribute anything. ${\ displaystyle r}$${\ displaystyle r}$

## Limits of Theory

Although it is sufficiently accurate for practical purposes, Newton's law of gravitation is only an approximation for weak and time-independent gravitational fields. For strong fields, the more precise description by means of the general theory of relativity is used , from which the Poisson equation of the classical theory of gravitation and thus also Newton's law of gravitation can be derived directly, if one only assumes that the gravitation is a conservative field . The law is therefore often referred to today as the limiting case of small fields . The general theory of relativity also solves the problems of Newton's theory of gravity described here.

### Theoretical limits

• Newton's theory is an effective theory , which means it neither gives a cause for the force of gravity, nor does it explain how gravity can work over a distance. Many contemporaries, including Newton himself and also Leonhard Euler , rejected the possibility of a direct long-distance effect through empty space. In order to close this explanatory gap, the so-called Le Sage gravity was developed as a model, which however never really caught on.
• The Newtonian theory assumes that the gravitational effect spreads infinitely fast so that Kepler's laws are fulfilled. This leads to conflicts with the special theory of relativity . This requires that gravity also only spreads at the speed of light.
• The equivalence of inert and heavy mass is not explained in Newtonian mechanics.

• Newton's theory does not fully explain the perihelion of planetary orbits, particularly Mercury . In this case, the difference between the perihelion rotation calculated according to Newton's theory and the observed perihelion rotation is 43 arc seconds per century.
• In Newton's theory, whether light is deflected in the gravitational field or not depends on the nature of the light. If it is understood as an electromagnetic wave , there is no deflection. If, however, according to the corpuscle theory , it is understood as a particle with mass, then according to Newton's law of gravity a light deflection results , whereby a prediction can be made from the equation of motion that is independent of the mass and thus remains valid even in the limit of vanishing mass. However, this value is only half of the deflection actually observed. The measured value results correctly from the equations of general relativity.

## Individual evidence

1. Jürgen Ready: Isaac Newton's first moon test that wasn't! In: Communications from the German Geophysical Society, 1/2016
2. ^ Henry Cavendish: Experiments to determine the Density of the Earth (PDF) 1798 (English)
3. Clive Speake, Terry Quinn: The search for Newton's constant . In: Physics Today . tape 67 , no. 7 , 2014, p. 27 , doi : 10.1063 / PT.3.2447 .
4. ^ A. Cornu, J. Baille: Détermination nouvelle de la constante de l'attraction et de la densité moyennede la Terre . In: Comptes Rendus Hebd. Seances Acad. Sci . tape 76 , 1873, pp. 954 ( online [accessed April 3, 2019]).
5. Gearhart, CA: Epicycles, eccentrics, and ellipses: The predictive capabilities of Copernican planetary models . In: Archive for History of Exact Sciences . tape 32 , no. 3 , 1985, pp. 207-222 , doi : 10.1007 / BF00348449 .
6. James Lequeux: Le Verrier - Magnificent and Detestable Astronomer , Springer Verlag, 2013. p. 23
7. Thomas Bührke: Great moments of astronomy: from Kopernikus to Oppenheimer, Munich 2001, p. 150.
8. ^ Greiner, Walter .: Classical Mechanics 1: Kinematics and Dynamics of Point Particles Relativity . 8., revised. u. exp. Edition Harri Deutsch, Frankfurt 2008, ISBN 978-3-8171-1815-1 , p. 4 .