Universal Gravitation

Discussion

the comet

Isaac Newton was born on Christmas Day, 1642 in the village of Woolsthorpe (near Grantham), Lincolnshire, England. In 1661 he enrolled in Trinity College, Cambridge University (about midway between Woolsthorpe and London) where he studied mathematics. In 1665 the Black Plague made it's way to England forcing the closure of Trinity and sending Newton home to Woolsthorpe for a year or two. It was during this time that he formulated most of his important contributions to mathematics and physics including the binomial theorem, differential calculus, vector addition, the laws of motion, centripetal acceleration, optics, and universal gravitation. Upon returning to Cambridge, Newton was made a professor of mathematics and then proceeded to do what professors still do to this day — teach and publish. Most of these papers Newton submitted for publication were on optics, especially on the theory of colors. Then, eighteen years later in 1684, EdmondHalley (1656–1742) came to Newton with a problem he thought Newton might be able to solve.

Comets are astronomical objects that are visible to the unaided for only a month or so. They were a serious problem for early astronomers as they would appear without warning, hang around in the sky for awhile, and then disappear never to be seen again. Halley was studying historical records of cometary appearances when he noticed four comets with nearly the same orbit separated in time by approximately 76 years. He reasoned that the comets of 1456, 1531, 1607, and 1682 were sightings of a single comet and that this comet would reappear in the winter of 1758. When it did as predicted, sixteen years after his death, it became known as Comet Halley. It should be noted that Halley did not discover the comet that bears his name, he was just the one who identified it as a celestial body with a definite period in orbit around the sun. Halley's comet has probably been seen since the dawn of civilization when humans first looked up and they sky and wondered how it all worked. Historical records from India, China, and Japan record its appearance as far back as 240 BCE (with one appearance not recorded). Its most recent appearances were in 1833, 1909, and 1985 and its next will be in 2061.

Halley also noticed that the comet described an orbit around the sun that was in accordance with Kepler's laws of planetary motion; namely, that the orbit was an ellipse (albeit a highly elongated one) with the sun at one focus and that it obeyed the harmonic law (r3 ∝ T2) as if it was another planet in our solar system. Halley asked Newton in 1684 if he had some idea why the planets and this comet obeyed Kepler's laws; that is, if he knew the nature of the force responsible. Newton replied that he had indeed solved this problem and "much other matter" pertaining to mechanics eighteen years earlier but hadn't told anyone about it. He then proceeded to rummage around looking for his notes from the plague years, but could not find them. Halley persuaded Newton to compile everything he ever knew on mechanics and offered to pay the costs so that his ideas might be published.

In 1687, after eighteen months of effectively non stop work, Newton published Philosophiæ Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy). Probably the single most important book in physics and possibly the greatest book in all of science, it is almost always just known as the Principia. It contains the essence of the concepts presented in the chapters on mechanics in every subsequent physics textbook, including this one. Probably the only important concept it misses is energy, but everything else is there: force, mass, acceleration, inertia, momentum, weight, vector addition, projectile motion, circular motion, satellite motion, gravitation, tidal forces, the precession of the equinoxes….

In 1684 Dr Halley came to visit him at Cambridge, after they had been some time together, the Dr asked him what he thought the curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distance from it. Sr Isaac replied immediately that it would be an Ellipsis, the Doctor struck with joy & amazement asked him how he knew it, why saith he I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, Sr Isaac looked among his papers but could not find it, but he promised him to renew it; & then to send it him

Abraham de Moivre

De motu corporum in gyrum (On the motion of bodies in orbit) is the (presumed) title of a manuscript by Isaac Newton sent to Edmond Halley in November 1684.

the law

The Principia contains in it the unification of terrestrial and celestial gravitation. The acceleration due to gravity described by Galileo and the laws of planetary motion observed by Kepler are different aspects of the same thing. There is no terrestrial gravitation for earth and no celestial gravitation for the planets, but rather a universal gravitation for everything.

  1. Every object in the universe attracts every other object in the universe with a gravitational force.
  2. The magnitude of the gravitational force between two objects is …
    1. directly proportional to the product of their masses and
    2. inversely proportional to the square of the separation between their centers

Newton's law works since we live in a universe with three spatial dimension. As gravity extends out into space it spreads itself thinner and thinner, covering an area that expands as the square of the distance from the source. If space wasn't three-dimensional, Newton's law wouldn't work.

Although space appears three dimensional, there's no obvious reason why it has to be. Some advanced theories seem to suggest that there may be additional spatial dimensions. The reason we haven't seen them is that they're curled up rather tightly. If they exist, it should be possible to find deviations in the force of gravity from Newton's inverse square law at extremely small distances. Testing for these deviations is quite difficult. The best experiments currently (2001) show that the inverse square law holds down to 218 μm (2.18 × 10−4 m). Since the size of these hidden dimensions is thought to be on the order of 10−35 m, we've still got a long ways to go.

the moon

The earth-moon separation is approximately sixty times greater than the radius of the earth. The acceleration due to gravity at this distance is 1/3600 the acceleration due to gravity at the surface of the earth.

the apple

Isaac Newton entered Trinity College at the University of Cambridge in 1661. He received his bachelor of arts degree in 1665 as the Great Plague was sweeping through London. The University of Cambridge closed as a precaution and Newton fled to his family's farm in Lincolnshire 90 km (60 miles) to the north. In the summer of 1666, Newton began work on his theory of universal gravitation. A bit more than twenty years later, the final theory was released to the public as a part of his grand tome Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). Apples were not a part of the discussion.

Jump ahead to 1726. Sir Isaac Newton was a legend nearing the end of his life. He had dinner with a friend, William Stukeley, and they sat in a garden afterwards and talked about many things. Newton was 83 years old at the time when he recalled an event that took place 60 years earlier. This is the story as Stukeley tells it (using his original spelling, capitalization, and punctuation).

on 15 April 1726 I paid a visit to Sir Isaac, at his lodgings in Orbels buildings, Kensington: din'd with him … after dinner, the weather being warm, we went into the garden, & drank thea under the shade of some appletrees, only he, & myself. amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. "why should that apple always descend perpendicularly to the ground," thought he to him self: occasion'd by the fall of an apple, as he sat in a contemplative mood: "why should it not go sideways, or upwards? but constantly to the earths centre? assuredly, the reason is, that the earth draws it. there must be a drawing power in matter. & the sum of the drawing power in the matter of the earth must be in the earths center, not in any side of the earth. therefore dos this apple fall perpendicularly, or toward the center. if matter thus draws matter; it must be in proportion of its quantity. therefore the apple draws the earth, as well as the earth draws the apple."

William Stukeley — 1752

Another variation of the apple story was recorded by Newton's assistant at the Royal Mint (and also his nephew-in-law), John Conduitt.

In the year [1666] he retired again from Cambridge on acct of the plague to his mother [in] Lincolnshire & whilst he was musing in a garden it came into his thought that the same power of gravity (wch made an apple fall from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much farther than was usually thought — Why not as high as the Moon said he to himself & if so that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating … & found it perfectly agreable to his Theory —

John Conduitt — ca. 1728

Newton himself never wrote anything about apples. He was more interested in the motion of the moon as a means to test his theory.

In the same year [1666] I began to think of gravity extending to the orb of the moon, and having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere, from Kepler's rule of the periodical times of the planets being in a sesquilaterate proportion of their distances from the centres of their orbs I deduced that the forces which keep the Planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth, and found them to answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since.

Isaac Newton — ca. 1715

When Newton was asked how he discovered the law of universal gravitation, his reply was …

[I]f I have done ye publick any service this way 'tis due to nothing but industry & a patient thought.

Isaac Newton — 1692

the formula

Fg = − Gm1m2 
r2
g = −  Gm
r2
object mass (kg) radius (km) g (m/s2) g (g)
sun 1.99 × 1030 696,000 270 28
mercury 3.30 × 1023 2,440 3.7 0.38
venus 4.87 × 1024 6,050 8.9 0.90
earth 5.97 × 1024 6,380 9.8 1.0
moon 7.36 × 1022 1,740 1.6 0.17
mars 6.42 × 1023 3,400 3.7 0.38
jupiter 1.90 × 1027 71,500 25 2.5
saturn 5.69 × 1026 60,300 10 1.1
uranus 8.68 × 1025 25,600 8.9 0.90
neptune 1.02 × 1026 24,800 11 1.1
pluto 1.30 × 1022 1,180 0.62 0.064
white dwarf star ~ 1 solar mass ~ 1 earth radius 3,000,000 300,000
neutron star 2 ~ 3 solar masses ~ 10 1013 1012
stellar black hole > 3 solar masses 9 5 × 1012 5 × 1011
supermassive black hole 105 ~ 109 solar masses 105 ~ 109 108 ~ 104 107 ~ 103
Gravitational field strength (acceleration due to gravity)
Note: The gravitational field strength for black holes were calculated on the surface of the event horizon (Schwarzschild radius). Gravity within the event horizon may approach infinity. Also note that gravitational field strength at the event horizon decreases as the mass of a black hole increases.

the constant

Cavendish experiment

The Great Pyramid is so massive that a plumb line will not hang straight down when near the pyramid but will swing toward the structure. Cf. Tompkins, Secrets of the Great Pyramids, pp. 84-85, where Tompkins, discussing the measurements taken by Piazzi Smyth, writes "To obtain the correct latitude of the Great Pyramid without having his plumb line diverted from the perpendicular by the attraction of the huge bulk of the Pyramid, Smyth made his observations from the very summit; there the Pyramid's pull of gravity would be directly downward". Tompkins, Peter. Secrets of the Great Pyramid (New York: Harper Collins, 1971).

the critics

Action at a distance. Newton's reply to these criticisms was basically, "I don't care. The theory works."

Rationem vero harum gravitatis proprietatum ex phænomenis nondum potui deducere, & hypotheses non fingo…. Et satis est quod gravitas revera existat, & agat secundum leges a nobis expositas, & ad corporum cælestium & maris nostri motus omnes sufficiat.   I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses…. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our seas.

beyond that …

extra dimensions

A (modified) quote from The Physics Teacher that will be paraphrased. "An essential ingredient for black hole production at the LHC (Large Hadron Collider) is the existence of extra dimensions. A black hole is a region of intense gravitational field creating conditions that are contrary to what we observe about gravitational forces to be in our everyday world. The presence of extra dimensions guarantees the extra strength of gravity needed to produce black holes. When protons collide at the LHC, they come so close to each other that they essentially "see" the extra dimensions (where gravity is strong) and black hole formation may be possible. If this is the case, then the extra dimensions must be ~10−14 m in size. "

Three dimensional space

F ∝  1
r2

n-dimensional space

F ∝  1
rn − 1

Gravitational force in n > 3 dimensional space increases more rapidly at small distances than in 3D space. Dimensions greater than 3 are assumed to be small, and therefore are only revealed at small separations.