Sir Isaac Newton, born the year Galileo died (1642), discovered the key laws of physics that are required to complete the picture of planetary motion. He also developed an entire new branch of mathematics called calculus to help him explain what he saw.
Newton’s Law of Gravity:
It is claimed that Newton saw an apple fall from a tree and suddenly realized that the same force might keep the moon in orbit. He realized that orbiting the earth is just a case where the object’s velocity is so high that it ‘falls around the earth.
Newton realized that for an object to move along an elliptical or circular path, the force that causes the object to change direction constantly instead of moving in a straight line is gravity.
Newton was able to take his law of gravity,
and derive Kepler’s three laws and make them more general so that they could be applied anywhere in the universe, not just in the solar system. The details are a bit complicated: Newton invented calculus to carry out the proofs.
The variation in the strength of the gravitational force with distance explains why planets move faster when closer to the sun – the gravitational force is greater and so a planet is accelerated when it moves closer. The solution for orbits in an inverse-distance-squared force yields ellipses, and with the right period-distance relation.
With Newton, we finally have the complete solution to the problem of the motions of the planets. Based on a few simple and widely applicable physical laws (and any amount of mathematics that is necessary!) we can show exactly how and why the planets move.