# GRAVITY EXPLAINS THE MOTIONS OF THE PLANETS

Gravity is the name given to the force of attraction between any two masses. It is the force that attracts all objects to Earth, giving them weight. It draws objects downward, toward the center of Earth. If the object were on the moon, a much smaller mass than Earth, the force would be six times less and its weight would be one sixth of its weight on Earth. English physicist, astronomer, and mathematician Isaac Newton was the first person to realize that gravity is a universal force, acting on all objects, and that it explains the movement of planets.

“To myself I am only a child playing on the beach, while vast oceans of truth lie undiscovered before me.” Isaac Newton

## Describing orbits

The shapes of the orbits of the planets were already well-known in Newton’s time, based on the three laws of planetary motion introduced by Johannes Kepler. Kepler’s first law stated that these orbits were ellipses, with the sun at one focus of each ellipse. The second law described the way that planets moved along their orbits more quickly when they were close to the sun than when they were farther away. The third law described the relation between the time taken to complete one orbit and the distance from the sun: the time taken for one orbit, squared, was equal to the cube of the average distance between the planet and the sun. For instance, Earth goes around the sun in one year, while Jupiter is 5.2 times farther away from the sun than Earth. 5.2 cubed equals 140, and the square root of 140 gives the correct figure for one Jupiter year: 11.86 Earth years.

However, although Kepler had correctly discovered the shapes and speeds of planetary orbits, he did not know why the planets moved as they did. In his 1609 book Astronomia Nova, he suggested that Mars was being carried around its orbit by an angel in a chariot. A year later, he had changed his mind, suggesting that the planets were magnets and were being driven around by magnetic “arms” extending from the spinning sun.

## Newton’s insight

Before Newton, several scientists, including Englishman Robert Hooke and Italian Giovanni Alfonso Borelli, suggested that there was a force of attraction between the sun and the individual planets. They also stated that the force decreased with distance. On December 9, 1679, Hooke wrote to Newton saying that he thought the force might decrease as the inverse square of distance.

However, Hooke did not publish the idea and did not possess the mathematical skills to fully demonstrate his proposition. By contrast, Newton was able to prove rigorously that an inverse square law of attractive force would result in an elliptical planetary orbit.

Newton used mathematics to demonstrate that, if the force of attraction (F) between the sun and the planets varied precisely as an inverse square of the distance (r) between them, this fully explained the planetary orbits and why they follow Kepler’s three laws. This is written mathematically as F ∝ 1/r2. It means that doubling the distance between the objects reduces the strength of the attractive force to a quarter of the original force.

## The Great Comet

Newton was a shy, reclusive man, and reluctant to publish his breakthrough. Two things forced his hand. The first was the Great Comet of 1680, and the second was the astronomer Edmond Halley.

The Great Comet of 1680 was the brightest comet of the 17th century—so bright that for a short time it was visible in the daytime. Two comets were seen: one that was approaching the sun in November and December 1680; and another that was moving away from the sun between late December 1680 and March 1681. As with all comets at the time, its orbit was a mystery, and the two sightings were at first not widely recognized as the same object. Astronomer John Flamsteed suggested that the two sightings might be of the same comet, which had come from the outer edge of the solar system, swung around the sun (where it was too close to the sun to be seen), and moved out again.

Halley was fascinated by the mysterious form of cometary orbits, and traveled to Cambridge to discuss the problem with his friend Newton. Using his law that related force to acceleration and his insistence that the strength of the force varied as the inverse square of distance, Newton calculated the parameters of the comet’s orbit as it passed through the inner solar system. This breakthrough intrigued Halley so much that he went on to calculate the orbits of 24 other comets, and to prove that one comet (Halley’s comet) returned to the sun around every 76 years. Perhaps more importantly, Halley was so impressed by Newton’s work that he strongly encouraged him to publish his findings. This resulted in the book Philosophiae Naturalis Principia Mathematica, published in Latin on July 5, 1687, in which Newton describes his laws of motion, his gravitational theory, the proof of Kepler’s three laws, and the method he used to calculate a comet’s orbit.

In his book, Newton stressed that his law was universal—gravity affects everything in the universe, regardless of distance. It explained how an apple fell on his head in the orchard of Woolsthorpe where his mother lived, the tides in the seas, the moon orbiting Earth, Jupiter orbiting the sun, and even the elliptical orbit of a comet. The physical law that made the apple fall in his yard was exactly the same as the one that shaped the solar system, and would later be discovered at work between stars and distant galaxies. Evidence was all around that Newton’s law of gravitation worked. It not only explained where planets had been, but also made it possible to predict where they would go in the future.

## Constant of proportionality

Newton’s law of gravitation states that the size of the gravitational force is proportional to the masses of the two bodies (m1 and m2) multiplied together and divided by the square of the distance, r, between them. It always draws masses together and acts along a straight line between them. If the object in question is spherically symmetrical, like Earth, then its gravitational pull can be treated as if it were coming from a point at its center. One final value is needed to calculate the force—the constant of proportionality, a number that gives the strength of the force: the gravitational constant (G).

Newton’s law of universal gravitation shows how the force produced depends on the mass of the two objects and the square of the distance between them.

## Measuring G

Gravity is a weak force, and this means that the gravitational constant is rather difficult to measure accurately. The first laboratory test of Newton’s theory was made by the English aristocrat scientist Henry Cavendish in 1798, 71 years after Newton’s death. He copied an experimental system proposed by the geophysicist John Michell and successfully measured the gravitational force between two lead balls, of diameters 2 and 12 in (5.1 and 30 cm). Many have tried to refine and repeat the experiment since. This has led to a slow improvement in the accuracy of G. Some scientists suggested that G changed with time. However, recent analysis of type 1a supernovae has shown that, over the last nine billion years, G has changed by less than one part in 10 billion, if at all. The light from distant supernovae was emitted nine billion years ago, allowing scientists to study the laws of physics as they were in the distant past.

“Nature and Nature’s laws lay hid in night: God said, “Let Newton be!” and all was light.” Alexander Pope

## Seeking meaning

Like many of the scientists of his time, Newton was deeply pious and sought a religious meaning behind his observations and laws. The solar system was not regarded as a random collection of planets, and the sizes of the specific orbits were thought to have some specific meaning. For example, Kepler had sought meaning with his notion of “the music of the spheres.” Building on ideas first put forward by Pythagoras and Ptolemy, Kepler suggested that each planet was responsible for an inaudible musical note that had a frequency proportional to the velocity of the planet along its orbit. The slower a planet moved, the lower the note that it emitted. The difference between the notes produced by adjacent planets turned out to be well-known musical intervals such as major thirds.

There is some scientific merit behind Kepler’s idea. The solar system is about 4.6 billion years old. During its lifetime, the planets and their satellites have exerted gravitational influences on each other and have fallen into resonant intervals, similar to the way musical notes resonate. Looking at three of the moons of Jupiter, for every once that Ganymede orbits the planet, Europa goes around twice and Io four times. Over time, they have been gravitationally locked into this resonance.

## The three-body problem

The solar system as a whole has fallen into similar resonant proportions to Jupiter’s moons. On average, each planet has an orbit that is about 73 percent larger than the planet immediately closer to the sun. Here, however, there appears a difficult mathematical problem, and one that Newton had grappled with. The movement of a low-mass body under the gravitational influence of a large-mass body can be understood, and predicted. But when three bodies are involved, the mathematical problem becomes exceedingly difficult.An example of a three-body system is the moon-Earth-sun. Newton thought about this system but the mathematical difficulties were insurmountable, and human knowledge of where the moon will be in the distant future is still very limited. Variations in the orbit of Halley’s comet are another indicator of the influence of the gravitational fields of the planets operating in addition to the gravitation of the sun. Recent orbits have taken 76.0, 76.1, 76.3, 76.9, 77.4, 76.1, 76.5, 77.1, 77.8, and 79.1 years respectively due to the combined gravitational influence of the sun, Jupiter, Saturn, and other planets on the comet.

“I have not been able to discover the cause of these properties of gravity from phenomena, and I frame no hypotheses.” Isaac Newton

## Shaping the planets

While Newton searched for religious meaning in his scientific work, he could find none behind his theory of gravity. He did not discover the hand of God setting the planets in motion, but he had found a formula that shaped the universe.

The action of gravity is key to understanding why the universe looks as it does. For instance, gravity is responsible for the spherical shapes of the planets. If a body has sufficient mass, the gravitational force that it exerts exceeds the strength of the material of the body and it is pulled into a spherical shape. Astronomical rocky bodies, such as the asteroids between the orbits of Mars and Jupiter, are irregular in shape if they have a diameter of less than about 240 miles (380 km) (the Hughes-Cole limit).

Gravitation is also responsible for the size of the deviations from a sphere that can occur on a planet. There are no mountains on Earth higher than the 5.5 miles (8.8 km) of Mount Everest because the gravitational weight of a taller mountain would exceed the strength of the underlying mantle rock, and sink. On planets with lower mass, the weight of objects is less, and so mountains can be bigger. The highest mountain on Mars, for instance, Olympus Mons, is nearly three times as high as Everest. The mass of Mars is about one-tenth that of Earth, and its diameter is about half Earth’s. Putting these numbers into Newton’s formula for gravitation, this gives a weight on the surface of Mars of just over one-third that on Earth, which explains the size of Olympus Mons.

Gravity thus also shapes life on Earth by limiting the size of animals. The largest land animals ever were dinosaurs weighing up to 40 tons. The largest animals of all, whales, are found in the oceans, where the water supports their weight. Gravity is also responsible for the tides, which are produced because water bulges toward the sun and moon on the side of Earth nearer to them, and also bulges away from them on the other side where their gravitational pull is weaker. When the sun and moon are aligned, there is a high spring tide; when they are at right angles, there is a low neap tide.

“The motions of the comets are exceedingly regular, and they observe the same laws as the motions of the planets.” Isaac Newton

## Escape velocity

Gravity profoundly affects human mobility. The height a person can jump is determined by the gravitational field at ground level. Newton realized that the strength of gravity would affect the ease of travel beyond the atmosphere. To break free from Earth’s gravitational pull, it is necessary to travel at 25,020 mph (40,270 km/h). It is much easier to get away from less massive bodies
such as the moon and Mars. Turning the problem around, this escape velocity is also the minimum velocity that an incoming asteroid or comet can have when it hits Earth’s surface, and this affects the size of the resulting crater. Today, gravity is held to be most accurately described by the general theory of relativity proposed by Albert Einstein in 1915. This does not describe gravity as a force, but instead as a consequence of the curvature of the continuum of spacetime due to the uneven distribution of mass inside it. This said, Newton’s concept of a gravitational force is an excellent approximation in the vast majority of cases. General relativity only needs to be invoked in cases requiring extreme precision or where the gravitational field is very strong, such as close to the sun or in the vicinity of a massive black hole. Massive bodies that are accelerating can produce waves in spacetime, and these propagate out at the speed of light. The first detection of one of these gravitational waves was announced in February 2016.

## ISAAC NEWTON

Isaac Newton was born on a farm in Woolsthorpe, Lincolnshire, on December 25, 1642. After school in Grantham, he attended Trinity College Cambridge, where he became a Fellow and taught physics and astronomy. His book Principia set out the principle of gravity and celestial mechanics. Newton invented the reflecting telescope; wrote theses on optics, the prism, and the spectrum of white light; was one of the founders of calculus; and studied the cooling of bodies. He also explained why Earth was oblate (a squashed sphere) in shape and why the equinox moved, and formalized the physics of the speed of sound. He spent much time on biblical chronology and alchemy. Newton was at various times President of the Royal Society, Warden and Master of the Royal Mint, and member of parliament for Cambridge University. He died in 1727.