Question

The mean distance of Neptune from the Sun is 30.07 AU (=astronomical unit, means Earth-Sun distance), that of Pluto 39.4 AU. Are these two numbers connected? (Hint: Derive the ratio of the orbital periods!)

Answer 1

These numbers behave like Kepler's 3rd law.

Write
D_n = 30.07
D_p = 39.4

The period of orbit of Neptune is
P_n = 164.79 years;
and Pluto's
P_p = 247.92 years

Now you can calculate and confirm that
\[ \left(\frac{P_p}{P_n}\right)^2 = \left(\frac{D_p}{D_n}\right)^3 \]

If the distance D were the length of the semi-major axis, this would exactly Kepler's 3rd law.