# Mutual Attraction

## Brian Koberlein

27 March 2013

Medieval astronomy was dominated by the writings of Aristotle. Aristotle divided motion into earthly lines and heavenly circles, so the planets must surely move about us in perfect circles. Astronomers soon learned this wasn’t true, but the physics of Aristotle was so deeply rooted in the minds of scholars that astronomers imposed circular motion upon the heavens for a thousand years. When a single circle could not describe the motion of a planet, they placed circles upon circles (known as epicycles), each rotating just so, to match a planet’s motion. As more precise measurements of the planets were made, more epicycles were needed.

Then in the early seventeenth century Johannes Kepler published three simple rules that described the motion of the planets. They are now known as Kepler’s laws of planetary motion. Kepler did not use circles to move the planets. He allowed them to move in a more general shape, known as an ellipse. What made Kepler’s approach so radical is that an ellipse is neither a circle nor a line. It is a geometric form that connects the two, unifying earthly and heavenly motion. Kepler’s theory was the first step toward modern astrophysics, giving us an accurate description of planetary motion. But Kepler’s laws were still merely a description of motion. Kepler gave us form, but not function.

It was Isaac Newton who gave us the mechanism. In the late seventeenth century Newton published his Principia, which described a world governed by a simple set of rules for forces and motion. The equation below is one of these rules, and is known as Newton’s law of gravity. In it F represents the force between two bodies (the subscript G just denotes it is a gravitational force), the M’s are the masses of the two bodies, R is the distance between them, and G is a number known as the gravitational constant. What the equation says is that bodies are drawn to each other through gravitational attraction. The strength of their attraction is greater if they are close together, and lesser if they are more distant. This force of attraction exists between any two bodies. Between Sun and planet, between Earth and moon, and between me and you.

Newton’s triumph was that he could use his rules to explain why the planets moved in ellipses. They didn’t move in ellipses just because, they were driven to move by forces that followed simple rules. Rules you could test here on Earth. It is hard to overstate the effect Newton’s work had on our view of the universe. At the beginning of the 1600s the universe was one of epicycles and celestial spheres. By the end that century the universe was driven by fundamental physical laws we could prove and understand.

One thing Newton couldn’t do was determine the value of his gravitational constant. The only gravitational forces he could observe were between the planets Moon and Sun, and no one had any idea what their masses were. Without them, the value of G couldn’t be determined. A solution wasn’t found until 1797 when Henry Cavendish devised a clever experiment. He placed lead balls in wooden frame suspended by a thin wire that was free to twist. He then placed larger lead balls near the frame. By measuring just how much the frame twisted, Cavendish could measure the gravitational attraction between masses, and thereby determine the value of G. This experiment is now known as the Cavendish experiment, but it could also be called “weighing the heavens.” With the gravitational constant known, astronomers could observe the motions of the Sun and planets to determine their mass. It is a technique we still use today to measure the mass of stars, planets, and even galaxies.

There is, however, a mysterious consequence of Newton’s equation. The force of gravity is always attractive, and the closer two bodies are the stronger their attraction. It would seem then that if large enough masses got close enough together the gravitational attraction would be so strong that the objects would be crushed under their own weight. Gravity would pull ever stronger, squeezing the objects more and more, making them smaller and smaller until they finally collapsed into a single, infinitely dense point. A gravitational singularity.

This was such a bizarre idea that astronomers long thought it was impossible. Surely there must be some unknown physical mechanism that would prevent singularities. But in the early 1900s, Einstein’s theory of general relativity was confirmed, and the singularity problem became more severe. In essence Einstein combined Newton’s gravity with relativity. If you remember from yesterday mass and energy are connected. This means the energy of gravitational attraction is itself gravitationally attractive. Put simply, not just mass, but gravity itself is heavy. Put enough mass in a small enough volume, and it will collapse under its own gravitational weight. Einstein’s theory made gravitational singularities inevitable. Near such a singularity the gravitational attraction is so strong that nothing can escape its pull, not even light, which is why they are now known as black holes.

In 1974, radio astronomers discovered an intense energy source at the center of our galaxy. Named Sagittarius A*, it appeared to be a large black hole. By the dawn of the twenty-first century, astronomers were able to observe stars orbiting this galactic black hole. The motions of these stars follow the ellipses of Kepler, driven by Newton’s gravity. By observing their motions, and with the equation above, we can determine the black hole’s mass. In the center of our galaxy, just 27,000 light years away, is a black hole with a mass of more than four million Suns.

Newton’s equation gave us the mechanism behind the motion of the planets. It tells how we are connected to everything in the universe through mutual attraction.

It has also revealed the gravitational dragon that rests at the heart of our galaxy.

Next time: How a beam of light overturned 300 years of physics, and changed our view of the universe. Part 3, coming tomorrow.