Legend

 ac = centripetal acceleration v = velocity r = radius Fc = centripetal force m = mass Fg = force of gravity G = universal constant

Kepler's Laws

 Kepler's three laws of planetary motions are:1) The planets move in elliptical orbits with the Sun at one focus (and nothing at the other). 2) Any planet moves in such a way that a line drawn from the Sun to its center sweeps out equal areas in equal time intervals. 3) The ratio of the average distance from the Sun cubed to the period squared is the same constant value for all planets.  That is: r3/T2 = C Kepler's solar constant C has it simplest value, 1.00.  Which gives: r3 = T2 A line drawn from the Earth to the Sun for each month.  The areas between lines are equal.  With the knowledge of gravity, we know that the greater the distance between the two objects is, the greater the force, as result, greater the speed.  We see in the picture that as January comes, it's orbital speed is greater than July, since it's closest to the Sun.  And vice-versa for the month July. After twenty years of work, Kepler found the relationship between a planet's orbital period (T), the time it takes to travel once around the Sun, and its average distance from the Sun (r).   Which gives us the above equation.Now imagine the change in time is infinitely small, as the number of intervals becomes infinitely many, until the distinct impulses becomes a smooth continuous center seeking force.  As established before in circular motion.  Under the influence of a centripetal force, an object moves about a center pivot sweeping out equal areas of areas in equal times.  This is extremely similar to Kepler's 2nd law.  Thus, the mystery behind Kepler's Second Law is reveled: gravity is the centripetal force.   Hence: Fg = Fc and, Fg = Gm1m2/r2 Fc = mac = mv2/r then, Fg = Gm1m2/r2 = mv2/r = Fc Since 2r, is the circumference, dividing it by T would obtain the orbital speed.  Then, substituting for v and canceling out m: Gm1/r2 = (2r/T)2/r Rearranging it, r3/T2 = Gm1/42 Which is Kepler's Third law.
 Terrestrial Gravity Orbits