What Is Universal Gravitation?
Newton's inverse-square law: how a single equation explained falling apples, orbiting planets, ocean tides, and the return of Halley's comet.
What Is Universal Gravitation?
Newton’s law of universal gravitation, published in 1687 in the Philosophiæ Naturalis Principia Mathematica, is one of the most consequential equations in this history. In its modern form, it states that every particle of matter in the universe attracts every other particle with a force directed along the line joining them, with magnitude proportional to the product of their masses and inversely proportional to the square of the distance between them: F = G m₁ m₂ / r² where F is the magnitude of the gravitational force, m₁ and m₂ are the masses of the two particles, r is the distance between their centres, and G is the universal gravitational constant. This single equation, with the three laws of motion, explained the fall of an apple, the orbit of the Moon, the tides, the trajectories of comets, the precession of the equinoxes, and the slight wobbles in the Earth’s motion. It is the cornerstone of classical celestial mechanics and, with modifications, the foundation of the modern understanding of gravity. This page is an article in the Laws of Motion and Universal Gravitation, under the Major Discoveries. It assumes a general familiarity with the broad context, which is sketched in the section overview. For the three laws of motion that go with this law of force, see What Are Newton’s Three Laws of Motion?. For the biographical and historical context, see the Isaac Newton and the discussion in Newton and Gravity. For the planetary laws that the gravitational law was used to derive, see Kepler’s Three Laws.
What the Law Says
The law of universal gravitation has three parts. First, gravity is universal: it acts between any two masses, anywhere in the universe, regardless of what they are made of. There is no “celestial gravity” different from “terrestrial gravity”; the same force that pulls a stone to the ground pulls the Moon toward the Earth. Second, gravity is attractive: the force on each of two bodies is directed toward the other. There is no gravitational repulsion in Newton’s theory. The two bodies fall toward each other. Third, the magnitude of the force falls off as the square of the distance. If you double the distance between two masses, the force between them becomes a quarter of what it was. If you triple the distance, the force becomes a ninth. This inverse-square dependence is what makes planetary orbits ellipses with the Sun at one focus, and it is what gives the familiar 1/r² law of gravitational attraction. The proportionality constant G in the equation is the same for every pair of masses in the universe. It was not measured by Newton; the first reliable measurement was made by Henry Cavendish in 1798, using a torsion balance to detect the tiny gravitational attraction between lead spheres. Modern measurements give a value of about 6.674 × 10⁻¹¹ N·m²/kg².
The Unification of Celestial and Terrestrial Physics
The most important consequence of the law of universal gravitation is that it dissolves the ancient distinction between the physics of the heavens and the physics of the Earth. In the Aristotelian cosmos, the celestial realm was made of aether and moved in perfect circles; the terrestrial realm was made of the four elements and moved in straight lines. The two realms obeyed different laws. In the Newtonian cosmos, both realms are made of ordinary matter (extended in the eighteenth and nineteenth centuries to include the aether-free void) and both obey the same three laws of motion and the same law of gravitation. A stone dropped from a tower falls to the ground because the Earth attracts it. The Moon orbits the Earth because the Earth attracts it. The Earth orbits the Sun because the Sun attracts it. The comets return on hyperbolic or highly elliptical orbits because the Sun attracts them. The ocean tides rise and fall because the Moon and Sun attract the water. All of these are instances of the same equation. Newton himself was keenly aware of the philosophical significance of the unification. In a famous passage in the Principia, he wrote that “the same laws therefore are observed by the moon in its motion round the earth, by the planets in their motions round the sun, by the satellites of Jupiter and Saturn in their motions round those planets, and by the seas in their rise and fall on the earth; and therefore this gravity, by which the moon is kept in its orbit, extends itself to all bodies, and is proportional to the quantity of matter in each.”
From Gravity to Kepler’s Laws
One of the great triumphs of the Principia was the derivation of Kepler’s three laws of planetary motion from the inverse-square law of gravitation combined with Newton’s three laws of motion. The derivations are outlined briefly here; the full mathematical details are available in standard texts and in the Kepler’s Three Laws page. The derivation of Kepler’s first law — that the orbit of a planet is an ellipse with the Sun at one focus — uses the geometry of conic sections. If a body moves under a central force directed toward a fixed point and varying as the inverse square of the distance, the body’s orbit is a conic section: an ellipse, a parabola, or a hyperbola, depending on the total energy. For a bound orbit, the relevant conic is the ellipse, and the Sun sits at one focus of that ellipse. This is exactly Kepler’s first law. The derivation of Kepler’s second law — that the line joining a planet to the Sun sweeps out equal areas in equal times — is more general: it holds for any central force, not just an inverse-square one. The law is a consequence of the conservation of angular momentum, which in turn follows from Newton’s second law and the third law. If the force on a planet is always directed toward the Sun, the torque on the planet (about the Sun) is zero, so its angular momentum is conserved, and the area swept out per unit time is constant. The derivation of Kepler’s third law — that the square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit — uses the inverse-square law specifically. For a circular orbit of radius r and period T, the centripetal force needed to keep the planet in its orbit is m v² / r = 4π² m r / T². Setting this equal to the gravitational force G M m / r² gives T² = 4π² r³ / (G M), which is Kepler’s third law. For elliptical orbits, the law holds approximately when the ellipse is nearly circular, and exactly with the semi-major axis in place of the radius. The full mathematical details, including the derivations from Newton’s laws and the geometry of conic sections, are discussed in the Isaac Newton, in The Principia Mathematica, and in Newton and Gravity.
The Explanation of the Tides
The law of universal gravitation also gave the first quantitative account of the tides. Newton showed, in the Principia, that the tides are caused by the differential gravitational pull of the Moon (and, to a lesser extent, the Sun) on different parts of the Earth. The water on the side of the Earth facing the Moon is pulled toward the Moon more strongly than the solid Earth, and the water on the far side is pulled less strongly, producing two tidal bulges. As the Earth rotates through these bulges, any given point on the coast experiences two high tides and two low tides per day. The explanation of the tides was striking because it connected an astronomical cause (the position of the Moon) to an everyday terrestrial phenomenon (the rise and fall of the sea) through a single physical law. The success of the tidal theory was, in the eighteenth century, one of the most persuasive demonstrations of the power of the Newtonian synthesis.
Halley’s Comet and the Predictive Power of the Theory
In 1705, Edmond Halley, using Newton’s theory, computed the orbits of twenty-four bright comets that had been observed between 1337 and 1698. He noticed that the comets of 1531, 1607, and 1682 had very similar orbital elements and periods of about seventy-five to seventy-six years. He predicted that the comet would return in 1758. When the comet did reappear, on Christmas night of 1758 (just before its perihelion passage in March 1759), it was the first successful prediction of a comet’s return based on gravitational theory. The comet was subsequently named after Halley, and the prediction became one of the most celebrated demonstrations of the predictive power of Newtonian gravitation. The comet has since been identified with earlier apparitions, including a possible observation in 240 BC and a definite appearance in 1066 depicted in the Bayeux Tapestry.
Limits of the Newtonian Theory
The Newtonian theory of gravitation is not the last word on gravity. Two important limitations emerged in the nineteenth and twentieth centuries. The first is the anomalous precession of Mercury’s perihelion. Newtonian theory predicts that the orbit of Mercury precesses by about 531 arcseconds per century due to planetary perturbations, mainly from Venus, the Earth, and Jupiter. The observed precession is about 574 arcseconds per century, leaving an unexplained 43 arcseconds per century. This discrepancy was explained in 1915 by Albert Einstein’s general theory of relativity, in which gravity is not a force but a curvature of spacetime produced by mass-energy. Mercury’s orbit is the classic test case for the new theory. The second is the equality of inertial and gravitational mass. In Newtonian physics, the fact that all bodies fall at the same rate in a gravitational field is an empirical coincidence: it is not derived from any deeper principle. In Einstein’s general relativity, it is a fundamental postulate: gravity is geometry, and all bodies follow the same geodesics of spacetime regardless of their mass. Eötvös experiments and modern satellite tests have confirmed the equality to extremely high precision. Despite these limitations, the law of universal gravitation remains an extraordinarily accurate approximation for nearly all practical purposes. It is the law used to plan interplanetary missions, to predict eclipses centuries in advance, to model the orbits of binary stars, and to estimate the masses of galaxies from the motions of stars within them. The fact that a single equation, written in the 1680s, continues to do this work is one of the most remarkable facts in this history.