Let's go back in time to answer this question. We will be taking the path of going from Newton's Law of Gravitation, and seeing how this motivates Kepler's 3 Laws.
Newton's Law of gravitation states that there is a force between two particles that have mass, that pushes them towards each other.
Notice that the force between them gets much stronger as they get closer together. In the simulation below, try to simulate the following: - Make the particles twice as close to each other. What multiplier did the force increase by? - Make the particles thrice as close to each other. What multiplier did the force increase by?
Notice, that the amount the force change is proportional to the square of their distance. In the simulation below, try to simulate the following: - Make one of the particles twice as heavy. What multiplier did the force increase by? - Make the other particle three times as heavy. What multiplier did the force increase by?
The force is also proportional to each of their masses! Therefore, we can state the following: (Newton's law of gravitation) $$F = \frac{G m_1 m_2}{r^2}$$ for some constant G, which we call the gravitational constant. It has a value of $6.67430 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$, which is tiny! This is good, because you don't want everything to suddenly get attracted to each other.
Planets travel around the sun in elliptical orbits.
The area swept out by the planet over a period of time is constant.
The square of the planet's time period of orbit is directly proportional to the cube of its semimajor axis.