|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.
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.
Fg = Fc
Fg = Gm1m2/r2
Fc = mac = mv2/r
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
r3/T2 = Gm1/42
Which is Kepler's Third law.