A short check on Kepler's 3rd Law: P2 α a3
We have just found out that the orbital period of Mercury is 88 days,
and Venus is 225 days. We can use Kepler's 3 rd law to figure out their
distances from the Sun. First, we have to recall that the “α” in that
equation means “proportional to”. There is a constant in that equation
to make the formula actually work:
P2 = k a3
We could go find out what “k” actually is, but there is a better way,
simply use the ratio of two equations, one where you know “P” and
“a”:
PP2 = k ap3
---------------------------
PE2 = k aE3
The subscript “E” is for the Earth. We know the orbital period of the
Earth (PE = 1 year = 365 day) and we know its distance from the sun
(aE = 1 AU), so the equation becomes:
PP2/12 = ap3 /13 (the “k” divides out!)
For Mercury: (88/365)2 = 0.0581 = a3 so a = (0.0581)1/3 = 0.387AU
For Venus: (225/365)2 = 0.380 = a3 , a = 0.72 AU
As you can see, these are the actual numbers for the mean distances
from the Sun for Mercury and Venus.