Circular and elliptical orbits

When R=a, the rotation periods of the bodies in these orbits are the same.

Kepler's laws, which were forever the basis of theoretical astronomy, were explained in Newton's mechanics, in particular in the law of universal gravitation.

Despite the fact that Kepler's laws were the most important step in understanding the motion of planets, they were still only empirical rules derived from astronomical observations; Kepler could not find a reason for these General laws for all planets. Kepler's laws were in need of theoretical justification.

And only Newton made a particular, but very important conclusion: there must be a connection between the centripetal acceleration of the moon and the acceleration of free fall on Earth. This relationship had to be established numerically and verified.

This is where Newton's reasoning differed from that of other scientists. Before Newton, no one was able to clearly and mathematically prove the law of gravity (force inversely proportional to the square of distance) and the laws of planetary motion (Kepler's laws).

Two of the greatest scientists, far ahead of their time, created a science called celestial mechanics, discovered the laws of motion of celestial bodies under the influence of gravity, and even if this were all their achievements, they would still enter the Pantheon of the great of this world.

As it happened, they didn't cross paths in time. Only thirteen years after Kepler's death was Newton born. Both were proponents of the Copernican heliocentric system.

After many years of studying the motion of Mars, Kepler experimentally discovered the three laws of planetary motion, more than fifty years before Newton discovered the law of universal gravitation. Not yet understanding why the planets move the way they do. It was a genius of foresight.

But Newton was using Kepler's laws to test his law of gravitation. All three of Kepler's laws are consequences of the law of gravitation. And Newton opened it. The results of Newton's calculations are now called Newton's law of universal gravitation, which we will discuss in the next Chapter.