# Does the gravitational attraction near the surface of dense celestial objects diverge from inverse square?

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Does the gravitational attraction near the surface of dense celestial objects (neutron star, white dwarf itc) diverge (to infinity) from inverse square?

This question is inspired by the similarity between EM and gravity (inverse square force.). The paper by John Lekner here (doi:10.1098/rspa.2012.0133) shows that there is an electrostatic attraction between charged spheres no matter the polarities of the charges and that it diverges at close separation until electrical short for almost all charge ratios. I am wondering if there is a similar kind of gravitational inverse square divergence for anything other than a black hole.

Actually make this for a black hole as well, although I know that a black hole is not thought to have a normal surface.

Leckner's paper deals with the effect of induced polarization on the spheres. Electrons are redistributed, making the force different from what one would expect. The gravitational counterpart is tidal distortion: since the gravitational field is non-radial when you have two heavy masses close to each other, matter will move to make the surface an equipotential surface. This means that the gravitational acceleration at the surfaces will not be constant at all locations.

Doing an analytic solution of how two ellipsoids attract each other and deform seems to be tractable (e.g. see this question) but algebraically very tedious and likely involves lots of special functions. See the addendum below for an approximate numeric model.

Black holes produce another complication: since spacetime nearby is curved and expanded the meaning of the distance in the inverse square law becomes problematic. The Paczyński-Wiita potential is an approximation of the potential, and it deviates from the $$U=-GM/r$$ as $$U_{PW}=-GM/(r-R_S)$$ (where $$R_S$$ is the Schwarzschild radius). It makes the force increase faster than the classical potential as we approach $$r=R_S$$.

Addendum: I did a numeric exploration of the force between two ellipsoidal, self-gravitating masses with centres of mass separated by a given distance. To find the shape I started with spheres and adjusted the semi-major axis (while preserving volume) so the potential along the surface became more equal at the poles. After a few iterations this gives a self-consistent shape. Then I calculated the force (the derivative of the potential) due to this shape on the other mass.

The result is indeed that the force increases faster than $$1/r^2$$ as the bodies approach each other, since they elongate and eventually merge (a bit before this they will deviate from my ellipsoidal assumption). If one multiplies the force by the squared distance the product should be constant for pure $$1/r^2$$ forces, but it starts to increase as they approach enough. Note that this is a non-rotating model: with rotation the numbers will change and the ellipsoids will become tri-axial, but I suspect the qualitative behaviour stays the same.

## Newton's law of universal gravitation

Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. [note 1] The publication of the theory 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] [3]

This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. [4] It is a part of classical mechanics 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, 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 mass attracts every other point mass by a force acting along the line intersecting the two points. The force is proportional to the product of the two masses, and inversely proportional to the square of the distance between them. [5]

The equation for universal gravitation thus takes the form:

where F is the gravitational force acting between two objects, m1 and m2 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 theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. [6] 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 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 since been superseded by Albert Einstein's theory of general relativity, but it 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).

## Does the gravitational attraction near the surface of dense celestial objects diverge from inverse square? - Astronomy

(1) The force resulting from combining gravitation with centrifugal force, where gravitation is the force exerted by the mass of the Earth and the centrifugal force is the apparent force caused by the rotation of the Earth.

(2) The acceleration due to the force defined in (1).

The terms "gravitation" and "gravity" are sometimes used as if they were synonyms, but in geodesy they are not. The gravitational force is solely due to the attraction of masses, as described by Newton’s Law.

But Newton’s Law doesn’t accurately predict accelerations when two bodies are spinning together, as is the case of any object in contact with the Earth’s surface.

When this happens, we define gravity as the sum of gravitation plus a centrifugal force (&alpharotation) due to the spin of the Earth. (The centrifugal force is an “apparent force.” See the panel below for more information.)

#### How Can A Force Be An “Apparent Force”?

An “apparent force”, also sometimes called a “fictitious force” or “inertial force,” is a hidden force that comes from the movement of the frame of reference of the observer.

An event can appear to have different forces at work depending on the point of observation. For example, a person who is walking from one end of a moving bus to the other appears to be walking in a straight line from the perspective of another passenger.

However, if the bus is going around a curve, from the point of view of an observer outside the bus, that person would appear to be following a curved path.

As the bus follows the curve, the person walking will feel a force pushing them towards the outside of the bus. This is an apparent force that comes from the mass of the person trying to continue moving in a straight line, while the bus changes direction. This apparent force due to a rotation of the reference frame is called a centrifugal force.

We introduce the acceleration due to the centrifugal force (&alpharotation) to account for the fact that everything attached to the Earth is spinning with the Earth. We commonly think of gravity and gravitation as positive and directed toward the center of Earth. Since the centrifugal force always points away from Earth’s axis of rotation, &alpharotation is either negative or zero at all places on earth.

The formula for calculating centrifugal force is:

where λgeoscentric is geocentric latitude, ω is the average rotation rate of the Earth, and Re is the equatorial radius.

#### Geodetic versus Geocentric Latitude

Geocentric latitude is the angle between the equatorial plane and a line from a location on Earth’s surface to the geometric center of Earth. Geodetic latitude is the angle between the equatorial plane and a line from a location on Earth’s surface perpendicular to the ellipsoid.

We use geodetic latitude for mapping and navigation, and it is provided by our GPS receivers. The conversion between geocentric and geodetic latitude is:

where f is the flattening ratio:

and Re is the equatorial radius and Rp is the polar radius.

We often call the force of gravity experienced by an object its “weight,” which is its mass times the acceleration due to gravity (as in Equation 3). The weight of an object on Earth is always smaller than the force of gravitation alone because the centrifugal force is reducing the effect of gravitation. Put another way, if Earth were not spinning then everything on Earth’s surface would weigh more.

How much more? The maximum centrifugal force for the equator on the surface of the GRS 80 ellipsoid is 0.035 m/s 2 and the force of gravitation there is 9.82 m/s 2 (see Solution 2 for Equation 5). That’s 280 times smaller than the force of gravitation!

Caveat: The only exception to these statements is that the centrifugal force is zero at the exact locations of the North and South poles, so gravity is equal to gravitation there.

### 2. Properties of Gravitation and Gravity » 2e. Gravity versus Gravitation on an Ellipsoidal Earth » Review Questions

#### Question

The force of gravitation is defined by which properties of the objects it affects? (Choose all that apply)

The correct answers are a, d.

The force of gravitation is defined as:

Equation 1: Newton’s Law of Universal Gravitation

Where G is the Universal Gravity Constant, m1 and m2 are the objects’ masses, and r 2 is the distance between the objects, squared.

Answer b is incorrect because magnetization does not affect the gravitational force exerted on an object. Answer c is incorrect because the rotation speed only affects gravity, which is the combination of gravitation and the centrifugal force of the rotating object (Equation 9).

#### Question

Earth’s shape is best approximated by a sphere. (True or False)

The Earth’s shape is best approximated by an ellipsoid that is flattened at the poles and bulges at the equator. For the GRS 80 ellipsoid, Earth’s polar radius is 6,356,752.3141 m and Earth’s equatorial radius is 6,378,137 m.

#### Question

Since the centrifugal force always points away from Earth’s axis of rotation, the force is either negative or zero at all places on earth. The centrifugal force acts as a small force trying to throw objects off of the spinning planet’s surface.

Therefore, since gravity is the sum of gravitation (positive, directed toward the center of the Earth) and centrifugal force (negative, directed away from the center of Earth), the centrifugal force causes objects to weigh less than if the Earth weren’t spinning.

### 3. Understanding the Components of Earth’s Gravity Field

In reality, Earth’s gravity field is far more complex than what we have calculated (9.82 m/s 2 ) for the surface of a theoretical Earth that is spherical and homogenous (i.e., made of the same material throughout). The complexity of the actual Earth comes from four major components, which we will discuss in this section:

1. Its ellipsoidal shape
2. Its irregular surface
3. Its heterogeneous density (i.e. made of different materials throughout)
4. Movement of mass within the Earth system

### 3. Understanding the Components of Earth’s Gravity Field » 3a. Component 1: Earth’s Ellipsoidal Shape

In 1929, an Italian researcher derived a simple equation to describe the magnitude of the acceleration due to gravity on the surface of any ellipsoid, called “normal gravity.” The Somigliana-Pizetti equation (see below) shows that normal gravity depends on latitude and the chosen ellipsoid. In fact, it depends only on the absolute value of your latitude—meaning that normal gravity at 30 degrees North latitude is equal to normal gravity at 30 degrees South latitude, 55 degrees N latitude is equal to 55 degrees S latitude, and so on for all North and South latitudes!

#### Somigliana-Pizetti equation

This simple but very accurate equation is called the Somigliana-Pizetti equation, where an ellipsoid is first chosen to define all the constants:

where γ0 is normal gravity at the surface of the ellipsoidal Earth, γe is normal gravity at the equator on the surface, λgeodetic is geodetic latitude, e 2 is the first eccentricity squared (Equation 10b, below), and:

where γe is normal gravity at the equator on the surface, γp is normal gravity at the poles on the surface, Re is the radius at the equator, Rp is the radius at the poles.

The first eccentricity squared (e 2 ) is:

where Re is the radius at the equator and Rp is the radius at the poles.

For the GRS80 ellipsoid γe =9.7803267715 m/s 2 and γp = 9.8321863685 m/s 2 .

Think of normal gravity as what gravity would be if the Earth were a simple, homogenous ellipsoid. Of course, Earth’s real gravity field is different from this theoretical field. Gravity fluctuates around the expected norm due to terrain, rock densities, and environmental influences. These major influences on Earth’s gravity field are detailed in the next three sections.

Note: Normal gravity can be calculated at any height above the ellipsoid’s surface, but much more complex calculations are needed to do that. For simplicity, here we consider only normal gravity on the ellipsoid surface and account for height as a separate correction, although this is the slightly less accurate calculation.

### 3. Understanding the Components of Earth’s Gravity Field » 3b. Component 2: Earth’s Irregular Surface— Vertical Differences in Gravity

The true Earth surface topography varies in height with respect to the ellipsoid. Any distance measured perpendicularly from the ellipsoid to a point is known as that point’s ellipsoidal height.

Remember, gravity is the sum of the gravitational and centrifugal forces. Since gravitation is related to the distance from the center of Earth and the centrifugal force varies with the distance from the rotation axis, we would expect gravity and heights to be related—and they are. We measure significant gravity variations from the deep oceanic trenches to the highest mountains.

The change in gravity with respect to change in ellipsoidal height (called the “gravity gradient”) is often roughly approximated with a “free-air correction”:

where is change in gravity over change in height, h is the ellipsoidal height, and 1 mGal is 1 x10-5 m/s 2 .

However, this approximation is not very accurate because it neglects Earth’s curvature and other factors. This simplification would cause large errors when calculating a height correction for any point more than a few meters from the Earth’s surface. See below for a more accurate gravity calculation.

#### Gravity Variation with Height over an Ellipsoid

When calculating the change in gravity due to height for locations that are more than a few meters from the Earth’s surface, we would use the more accurate formula:

where is change in gravity over change in height, γ0 is normal gravity at the surface of the ellipsoidal Earth at this location (calculated with Equation 10), Re is the radius at the equator, f is the ellipsoidal flattening, m is the geodetic parameter (see Equation 13, below), λgeodetic is geodetic latitude, and h is the ellipsoidal height.

The geodetic parameter (m) is defined as:

where ω is the average rotation rate of the Earth, Re is the radius at the equator, Rp is the radius at the poles, and GM is Earth’s Gravitational Constant.

For more information, see Featherstone and Dentith (1997), Damiani (2013), and Moritz (1980).

### 3. Understanding the Components of Earth’s Gravity Field » Exercise 1: Space Jump Gravity Calculation Exercise

Now let’s explore how gravitation changes with latitude and altitude with an interactive example.

On 24 October 2014, Alan Eustace set a record for the world’s highest altitude free-fall jump. When he was on the ground he was affected by gravity. But as soon as he left the surface, he was affected only by gravitation. (We will ignore the rotation forces caused by the atmosphere and just consider the solid-Earth effects on Mr. Eustace).

Wearing a custom space suit, he dangled from the bottom of a specially-designed balloon that rose to an altitude of over 135,000 feet (41 km). From there, Mr. Eustace released his tether and fell back to Earth. If another space jumper were to repeat Mr. Eustace’s feat from anywhere in the world, releasing herself/himself from any maximum altitude between 10 and 120 km, what would her/his gravitational acceleration and her/his weight be at the moment she/he released the tether from their balloon?

To answer this question, we will need to apply Equation 10, then Equation 7, and then Equation 11 to calculate the gravitational acceleration of the jumper at any height above the ellipsoid and at any latitude. In the following exercise, the calculations are done for you as you choose different altitudes and latitudes for a space jumper. Remember, the space jumper’s weight is simply her/his mass times the gravitational acceleration (Equation 3).

Calculating a space jumper’s weight at various latitudes and altitudes

Since the space jumper’s mass is 100 kg, we can calculate his weight at the surface to be 100 kg x 9.8 m/s 2 = 980 Newtons (N), which is equivalent to approximately 220 lbs.

Use the tool on the Calculator tab to determine the space jumper’s weight and acceleration at different altitudes and latitudes. Slide the jumper up or down in altitude and type in different values for the latitude.

Type in the correct missing weight and gravitational acceleration values to see the pattern of how weight and acceleration change for each location. Finally, answer the follow-up question and click Done when complete.

## Newton's law of universal gravitation

Newton's law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.[note 1] This is a general physical law derived from empirical observations by what Isaac Newton called induction.[2] It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. (When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him see the History section below.)

In modern language, the law states: Every point mass attracts every single other point mass by a force pointing 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.[3] The first test of Newton's theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798.[4] It took place 111 years after the publication of Newton's Principia and 71 years after his death.

Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of 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 electrostatic constant in place of the gravitational constant.

Newton's law has since been superseded by Einstein's theory of general relativity, but it 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 precision, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the sun).

A recent assessment (by Ofer Gal) about the early history of the inverse square law is "by the late 1660s", 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". The same author does credit Hooke with a significant and even seminal contribution, but he treats Hooke's claim of priority on the inverse square point as uninteresting since several individuals besides Newton and Hooke had at least suggested it, and he points instead to the idea of "compounding the celestial motions" and the conversion of Newton's thinking away from "centrifugal" and towards "centripetal" force as Hooke's significant contributions.
Plagiarism dispute

In 1686, when the first book of Newton's Principia was presented to the Royal Society, 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's contemporary report) Hooke agreed that "the Demonstration of the Curves generated thereby" was wholly Newton's.[5]

In this way the question arose as to what, if anything, Newton owed to Hooke. This is a subject extensively discussed since that time and on which some points continue to excite some controversy.
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 on 21 March 1666 a paper "On gravity", "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".[6] 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] "they do also attract all the other Celestial Bodies that are within the sphere of their activity"[7] 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 clearly 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.[8] 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 [of attraction] are 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").[6] 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 Supposes Reciprocall to the Distance."[9] (The inference about the velocity was incorrect.[10])

Hooke's correspondence of 1679-1680 with Newton mentioned not only 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".[11]
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.[12] Newton also pointed out and acknowledged prior work of others,[13] including Bullialdus,[14] (who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and Borelli[15] (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.[16]

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."[17]

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.[18] In addition, Newton had formulated in Propositions 43-45 of Book 1,[19] 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 accurately 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 arrived by 1669 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.[20] 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.[21] This background shows there was basis for Newton to deny deriving the inverse square law from Hooke.

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.[22] 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 . "[13]

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).[23] 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.[24] 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).[25][26]

About thirty years after Newton's death in 1727, Alexis Clairaut, 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".[27][28]

In modern language, the law states the following:
Every point mass attracts every single other point mass by a force pointing 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:[3]
Diagram of two masses attracting one another

F is the force between the masses
G is the gravitational constant ( (6.673×10^11 N · (m/kg)^2) )
(m_1 ) is the first mass
(m_2 ) is the second mass
r is the distance between the centers of the masses.

Assuming SI units, F is measured in newtons (N), m1 and m2 in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6.674×10−11 N m2 kg−2.[29] 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.[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'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
Gravitational field strength within the Earth.
Gravity field near earth at 1,2 and A.

If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which 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 centre.[3] (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:[30]

The portion of the mass that is located at radii r < r0 causes the same force at 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 distance r0 from the center. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at r0, cancel each other out.

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. The gravity of the Earth may be highest at the core/mantle boundary.
Vector form
Field lines drawn for a point mass using 24 field lines
Gravity field surrounding Earth from a macroscopic perspective.
Gravity field lines representation is arbitrary as illustrated here represented in 30x30 grid to 0x0 grid and almost being parallel and pointing straight down to the center of the Earth
Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines can be approximated as being parallel and pointing straight down to the center of the Earth

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.

( mathbf_ <12>) is the force applied on object 2 due to object 1,
G is the gravitational constant,
( m_1 ) and (m_2 ) are respectively the masses of objects 1 and 2,
( |r_<12>| = |r_2 − r_1| is the distance between objects 1 and 2, and
( mathbf>_ <12> stackrel><=> frac_2 - mathbf_1>_2 - mathbf_1vert> ) is the unit vector from object 1 to 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.

Gravitational field
Main article: Gravitational field

The gravitational field is a vector field that describes the gravitational force which 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 generalization of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). For 2 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:

( mathbf( 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

( mathbf(mathbf) = - abla 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

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

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

where ( partial V )is a closed surface and M_ is the mass enclosed by the surface.

Hence, for a hollow sphere of radius R and total mass M,

For a uniform solid sphere of radius R and total mass M,

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 φ/c2 and (v/c)2 are both much less than one, where φ is the gravitational potential, v is the velocity of the objects being studied, and c is the speed of light.[31] For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since

where rorbit 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.
Theoretical concerns with Newton's expression

There is no immediate prospect of identifying the mediator of gravity. Attempts by physicists to identify the relationship between the gravitational force and other known fundamental forces are not yet resolved, although considerable headway has been made over the last 50 years (See: Theory of everything and Standard Model). Newton himself felt that the concept of an inexplicable action at a distance was unsatisfactory (see "Newton's reservations" below), but that there was nothing more that he could do at the time.

Newton's theory of gravitation requires that the gravitational force be transmitted instantaneously. Given the classical assumptions of the nature of space and time before the development of General Relativity, a significant propagation delay in gravity leads to unstable planetary and stellar orbits.

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 of planet Mercury, which was detected long after the life of Newton.[32] 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 that is calculated by using Newton's Theory is only one-half of the deflection that is actually 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 to Newton's law of universal gravitation. Astrophysicists, however, explain this spectacular phenomenon in the framework of the Newton's laws, with the presence of large amounts of Dark matter.

The observed fact that the gravitational mass and the inertial mass is the same for all objects is unexplained within Newton's Theories. General Relativity takes this as a basic principle. See the Equivalence Principle. In point of fact, the experiments of Galileo Galilei, decades before Newton, established that objects that have the same air or fluid resistance are accelerated by the force of the Earth's gravity equally, regardless of their different inertial masses. Yet, the forces and energies that are required to accelerate various masses is completely dependent upon their different inertial masses, as can be seen from Newton's Second Law of Motion, F = ma.
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" which 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 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."[33]
Einstein's solution

These objections were explained by Einstein's theory of general relativity, in which gravitation is an attribute 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 due 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.
Extensions

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

( F = G frac + B frac , B a ) constant

attempting to explain the Moon's apsidal motion. Other extensions were proposed by Laplace (around 1790) and Decombes (1913):[34]

In recent years quests for non-inverse square terms in the law of gravity have been carried out by neutron interferometry.[35]
Solutions of Newton's law of universal gravitation
Main article: n-body problem

The n-body problem is an ancient, classical problem[36] 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.[37] 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)[38] of a group of celestial bodies, predict their interactive forces and consequently, predict their true orbital motions for all future times.[39]

The two-body problem has been completely solved, as has the Restricted 3-Body Problem.[40]
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It was shown separately that large, spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers.

Walter Lewin (October 4, 1999). Work, Energy, and Universal GravitatioT Course 8.01: Classical Mechanics, Lecture 11 (OGG) (videotape). Cambridge, MA USA: MIT OCW. Event occurs at 1:21-10:10. Retrieved December 23, 2010.
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.
- Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. Preceded by A Guide to Newton's Principia, by I.Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4
The Michell-Cavendish Experiment, Laurent Hodges
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 1674 statement in "An Attempt to Prove the Motion of the Earth from Observations" is available in online facsimile here.
Purrington, Robert D. (2009). The First Professional Scientist: Robert Hooke and the Royal Society of London. Springer. p. 168. ISBN 3-0346-0036-4., Extract of page 168
See page 239 in Curtis Wilson (1989), "The Newtonian achievement in astronomy", ch.13 (pages 233-274) in "Planetary astronomy from the Renaissance to the rise of astrophysics: 2A: Tycho Brahe to Newton", CUP 1989.
Page 309 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #239.
See Curtis Wilson (1989) at page 244.
Page 297 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #235, 24 November 1679.
Page 433 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #286, 27 May 1686.
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.
Bullialdus (Ismael Bouillau) (1645), "Astronomia philolaica", Paris, 1645.
Borelli, G. A., "Theoricae Mediceorum Planetarum ex causis physicis deductae", Florence, 1666.
D T Whiteside, "Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664-1684", Journal for the History of Astronomy, i (1970), pages 5-19 especially at page 13.
Page 436, Correspondence, Vol.2, already cited.
Propositions 70 to 75 in Book 1, for example in the 1729 English translation of the Principia, start at page 263.
Propositions 43 to 45 in Book 1, in the 1729 English translation of the Principia, start at page 177.
D T Whiteside, "The pre-history of the 'Principia' from 1664 to 1686", Notes and Records of the Royal Society of London, 45 (1991), pages 11-61 especially at 13-20.
See J. Bruce Brackenridge, "The key to Newton's dynamics: the Kepler problem and the Principia", (University of California Press, 1995), especially at pages 20-21.
See for example the 1729 English translation of the Principia, at page 66.
See page 10 in D T Whiteside, "Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664-1684", Journal for the History of Astronomy, i (1970), pages 5-19.
Discussion points can be seen for example in the following papers: N Guicciardini, "Reconsidering the Hooke-Newton debate on Gravitation: Recent Results", in Early Science and Medicine, 10 (2005), 511-517 Ofer Gal, "The Invention of Celestial Mechanics", in Early Science and Medicine, 10 (2005), 529-534 M Nauenberg, "Hooke's and Newton's Contributions to the Early Development of Orbital mechanics and Universal Gravitation", in Early Science and Medicine, 10 (2005), 518-528.
See for example the results of Propositions 43-45 and 70-75 in Book 1, cited above.
See also G E Smith, in Stanford Encyclopedia of Philosophy, "Newton's Philosophiae Naturalis Principia Mathematica".
The second extract is quoted and translated in W.W. Rouse Ball, "An Essay on Newton's 'Principia'" (London and New York: Macmillan, 1893), at page 69.
The original statements by Clairaut (in French) are found (with orthography here as in the original) in "Explication abregée du systême du monde, et explication des principaux phénomenes astronomiques tirée des Principes de M. Newton" (1759), at Introduction (section IX), page 6: "Il ne faut pas croire que cette idée . de Hook diminue la gloire de M. Newton", [and] "L'exemple de Hook" [serve] "à faire voir quelle distance il y a entre une vérité entrevue & une vérité démontrée".
Mohr, Peter J. Taylor, Barry N. Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP. 80..633M. doi:10.1103/RevModPhys.80.633. Direct link to value..
Equilibrium State
Misner, Charles W. Thorne, Kip S. Wheeler, John Archibald (1973). Gravitation. New York: W. H.Freeman and Company. ISBN 0-7167-0344-0 Page 1049.
- 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.)
- The Construction of Modern Science: Mechanisms and Mechanics, by Richard S. Westfall. Cambridge University Press. 1978
http://physicsessays.org/doi/abs/10.4006/1.3038751?journalCode=phes
http://journals.aps.org/prc/abstract/10.1103/PhysRevC.75.015501
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).
Quasi-steady loads refers to the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, a steady-state condition refers to a system's state being invariant to time otherwise, the first derivatives and all higher derivatives are zero.
R. M. Rosenberg states the n-body problem similarly (see References): Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move? Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces first before the motions can be determined.

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, but in practice such an infinite series 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, but these, too, are approximate solutions and again obsolete. See Sverre J. Aarseth's book Gravitational N-body Simulations listed in the References.

Feather & Hammer Drop on Moon on YouTube
Newton‘s Law of Universal Gravitation Javascript calculator

To calculate the gravitational force pulling the Earth and Moon together, you need to know their separation and the mass of each object.

### Distance

The Earth and Moon are approximately an average of 3.844*10 5 kilometers apart, center to center.

(Note that the orbit of the Moon around the Earth is not a true circle, so an average separation is used. This also means that the force of attraction varies.)

Since the units of G are in N-m 2 /kg 2 , you need to convert the units of R to meters.

### Mass of each object

Let M be the mass of the Earth and m the mass of the Moon.

M = 5.974*10 24 kg

m = 7.349*10 22 kg

### Force of attraction

Thus, the force of attraction between the Earth and Moon is:

F = GMm/R 2

F = (6.674*10 &minus11 N-m 2 /kg 2 )(5.974*10 24 kg)(7.349*10 22 kg)/(3.844*10 8 m) 2

F = (2.930*10 37 N-m 2 )/(1.478*10 17 m 2 )

F = 1.982*10 20 N

Note: Notice how all the units, except N, canceled out.

Attraction between Earth and Moon

### Result of force

This considerable force is what holds the Moon in orbit around the Earth and prevents it from flying off into space. Inward force of gravitation equals the outward centrifugal force from the motion of the Moon.

Also, the gravitational force from the Moon pulls the oceans toward it, causing the rising and falling tides, according to the Moon's position.

## Articles

Bartusiak, M. “A Beast in the Core.” Astronomy (July 1998): 42. On supermassive black holes at the centers of galaxies.

Disney, M. “A New Look at Quasars.” Scientific American (June 1998): 52.

Djorgovski, S. “Fires at Cosmic Dawn.” Astronomy (September 1995): 36. On quasars and what we can learn from them.

Ford, H., & Tsvetanov, Z. “Massive Black Holes at the Hearts of Galaxies.” Sky & Telescope (June 1996): 28. Nice overview.

Irion, R. “A Quasar in Every Galaxy?” Sky & Telescope (July 2006): 40. Discusses how supermassive black holes powering the centers of galaxies may be more common than thought.

Kormendy, J. “Why Are There so Many Black Holes?” Astronomy (August 2016): 26. Discussion of why supermassive black holes are so common in the universe.

Kruesi, L. “Secrets of the Brightest Objects in the Universe.” Astronomy (July 2013): 24. Review of our current understanding of quasars and how they help us learn about black holes.

Miller, M., et al. “Supermassive Black Holes: Shaping their Surroundings.” Sky & Telescope (April 2005): 42. Jets from black hole disks.

Nadis, S. “Exploring the Galaxy–Black Hole Connection.” Astronomy (May 2010): 28. Overview.

Nadis, S. “Here, There, and Everywhere.” Astronomy (February 2001): 34. On Hubble observations showing how common supermassive black holes are in galaxies.

Nadis, S. “Peering inside a Monster Galaxy.” Astronomy (May 2014): 24. What X-ray observations tell us about the mechanism that powers the active galaxy M87.

Olson, S. “Black Hole Hunters.” Astronomy (May 1999): 48. Profiles four astronomers who search for “hungry” black holes at the centers of active galaxies.

Peterson, B. “Solving the Quasar Puzzle.” Sky & Telescope (September 2013): 24. A review article on how we figured out that black holes were the power source for quasars, and how we view them today.

Tucker, W., et al. “Black Hole Blowback.” Scientific American (March 2007): 42. How supermassive black holes create giant bubbles in the intergalactic medium.

Voit, G. “The Rise and Fall of Quasars.” Sky & Telescope (May 1999): 40. Good overview of how quasars fit into cosmic history.

Wanjek, C. “How Black Holes Helped Build the Universe.” Sky & Telescope (January 2007): 42. On the energy and outflow from disks around supermassive black holes nice introduction.

## Gravitation in Action

The following time-lapse movies (about 30 seconds per frame) show the torsion balance responding to the gravitational field generated by two 740 gram competition pétanque balls. The picture at left shows the camera angle employed in both movies. In each, the movie begins with the bar stationary, in contact with one of the balls or the foam supporting it. The balls are then shifted to the opposite corners, where they attract the lead weights on the ends of the bar. The bar then turns, slowly at first and then with increasing speed as it is accelerated by the gravitational force growing as the inverse square of the decreasing distance between the masses. The bar bounces when it hits the stop on the other end, and finally, after a series of smaller and smaller bounces as the water brake dissipates its kinetic energy, comes to rest in contact with the closer ball or support. This is the lowest energy state, at which the bar will always arrive at the end of the experiment.

#### Movie 1

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#### Movie 2

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Pay no attention to the plastic robot ant&mdashshe's just curious. It's a long story.

### Early history

A recent assessment (by Ofer Gal) about the early history of the inverse square law is "by the late 1660s", 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". The same author does credit Hooke with a significant and even seminal contribution, but he treats Hooke's claim of priority on the inverse square point as uninteresting since several individuals besides Newton and Hooke had at least suggested it, and he points instead to the idea of "compounding the celestial motions" and the conversion of Newton's thinking away from "centrifugal" and towards "centripetal" force as Hooke's significant contributions.

### Plagiarism dispute

In 1686, when the first book of Newton's Principia was presented to the Royal Society, 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's contemporary report) Hooke agreed that "the Demonstration of the Curves generated thereby" was wholly Newton's. [4]

In this way the question arose as to what, if anything, Newton owed to Hooke. This is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy.

### 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 on March 21, 1666, a paper "On gravity", "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". [5] 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] "they do also attract all the other Celestial Bodies that are within the sphere of their activity" [6] 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 clearly 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. [7] 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 [of attraction] are 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"). [5] It was later on, in writing on 6 January 1679|80 [8] 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 Supposes Reciprocall to the Distance." [9] (The inference about the velocity was incorrect. [10] )

Hooke's correspondence of 1679-1680 with Newton mentioned not only 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". [11]

### 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. [12] Newton also pointed out and acknowledged prior work of others, [13] including Bullialdus, [14] (who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and Borelli [15] (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. [16]

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." [17]

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. [18] In addition, Newton had formulated in Propositions 43-45 of Book 1, [19] 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 accurately 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 arrived by 1669 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. [20] 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. [21] 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. [22] 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 . " [13]

### Modern 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). [23] 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. [24] 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). [25] [26]

About thirty years after Newton's death in 1727, Alexis Clairaut, 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". [27] [28]

In physics, Kaluza–Klein theory is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.

In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:

1. The laws of physics are invariant in all inertial frames of reference.
2. The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or observer.

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

In the special theory of relativity, four-force is a four-vector that replaces the classical force.

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872�), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.

General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the observed gravitational attraction between masses results from the warping of space and time by those masses.

In general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically. Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

In theoretical physics, massive gravity is a theory of gravity that modifies general relativity by endowing the graviton with a nonzero mass. In the classical theory, this means that gravitational waves obey a massive wave equation and hence travel at speeds below the speed of light.

In theoretical physics, a scalar–tensor theory is a field theory that includes both a scalar field and a tensor field to represent a certain interaction. For example, the Brans–Dicke theory of gravitation uses both a scalar field and a tensor field to mediate the gravitational interaction.

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.

Scalar–tensor–vector gravity (STVG) is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario. The theory is also often referred to by the acronym MOG.

In general relativity, post-Newtonian expansions are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which express orders of deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher order terms can be added to increase accuracy, but for strong fields sometimes it is preferable to solve the complete equations numerically. This method is a common mark of effective field theories. In the limit, when the small parameters are equal to 0, the post-Newtonian expansion reduces to Newton's law of gravity.

In the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor that incorporates the energy–momentum of gravity. It allows the energy–momentum of a system of gravitating matter to be defined. In particular it allows the total of matter plus the gravitating energy–momentum to form a conserved current within the framework of general relativity, so that the total energy–momentum crossing the hypersurface of any compact space–time hypervolume vanishes.

Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition to Einstein's theory of general relativity. There have been many different attempts at constructing an ideal theory of gravity.

The two-body problem in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity and the motion of a planet orbiting its sun. Solutions are also used to describe the motion of binary stars around each other, and estimate their gradual loss of energy through gravitational radiation.

Newton–Cartan theory is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced by Élie Cartan and Kurt Friedrichs and later developed by Dautcourt, Dixon, Dombrowski and Horneffer, Ehlers, Havas, Künzle, Lottermoser, Trautman, and others. In this re-formulation, the structural similarities between Newton's theory and Albert Einstein's general theory of relativity are readily seen, and it has been used by Cartan and Friedrichs to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity, and by Jürgen Ehlers to extend this correspondence to specific solutions of general relativity.

f ( R ) is a type of modified gravity theory which generalizes Einstein's general relativity. f ( R ) gravity is actually a family of theories, each one defined by a different function, f , of the Ricci scalar, R . The simplest case is just the function being equal to the scalar this is general relativity. As a consequence of introducing an arbitrary function, there may be freedom to explain the accelerated expansion and structure formation of the Universe without adding unknown forms of dark energy or dark matter. Some functional forms may be inspired by corrections arising from a quantum theory of gravity. f ( R ) gravity was first proposed in 1970 by Hans Adolph Buchdahl. It has become an active field of research following work by Starobinsky on cosmic inflation. A wide range of phenomena can be produced from this theory by adopting different functions however, many functional forms can now be ruled out on observational grounds, or because of pathological theoretical problems.

In comparison with General Relativity, dynamic variables of metric-affine gravitation theory are both a pseudo-Riemannian metric and a general linear connection on a world manifold . Metric-affine gravitation theory has been suggested as a natural generalization of Einstein–Cartan theory of gravity with torsion where a linear connection obeys the condition that a covariant derivative of a metric equals zero.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

Horndeski's theory is the most general theory of gravity in four dimensions whose Lagrangian is constructed out of the metric tensor and a scalar field and leads to second order equations of motion. The theory was first proposed by Gregory Horndeski in 1974 and has found numerous applications, particularly in the construction of cosmological models of Inflation and dark energy. Horndeski's theory contains many theories of gravity, including General relativity, Brans-Dicke theory, Quintessence, Dilaton, Chameleon and covariant Galileon as special cases.

## Scaling in gravity [ edit | edit source ]

The gravitational constant G has been calculated as:

Thus the constant has dimension density −1 time −2 . This corresponds to the following properties.

Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the Earth.

Scaling of distances while keeping the masses the same (in the case of point masses, or by reducing the densities) gives similar orbits if distances are multiplied by 4, gravitational forces and accelerations are divided by 16, velocities are halved and orbital periods are multiplied by 8.

When all densities are multiplied by 4, orbits are the same gravitational forces are multiplied by 16 and accelerations by 4, velocities are doubled and orbital periods are halved.

When all densities are multiplied by 4, and all sizes are halved, orbits are similar masses are divided by 2, gravitational forces are the same, gravitational accelerations are doubled. Hence velocities are the same and orbital periods are halved.

In all these cases of scaling. if densities are multiplied by 4, times are halved if velocities are doubled, forces are multiplied by 16.

These properties are illustrated in the formula (derived from the formula for the orbital period)

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period. See also Kepler's Third Law.