Short answers, pulled from the story.
Johannes Kepler published the first two laws of planetary motion in 1609 within a book titled Astronomia nova. This work contained findings derived from precise astronomical observations collected by Tycho Brahe.
The orbit of Mars takes an elliptical shape with the Sun located at one of the two foci rather than the geometric center. Kepler proved this after years of trying to fit data into a circular orbit model which failed to align.
It takes around 186 days for Earth to travel from the March equinox to the September equinox. In contrast, the period from the September equinox back to the March equinox takes only about 179 days due to the elliptical nature of the orbit.
Kepler published his third law of planetary motion in 1619 within a work titled Harmonice Mundi. The law states that the square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.
Isaac Newton demonstrated relationships like Kepler's in 1687 through his work Philosophiæ Naturalis Principia Mathematica. He showed that the acceleration of a planet points directly toward the Sun with magnitude inversely proportional to the square of the distance.