Estimate the length of the neptunian year using the fact that the earth is 1.50Ã108km from the sun on average.

We know that,

Neptune is 4.5*10^9 km from the sun

And given that,

Earth is 1.5*10^8km from sun

Then,

Let P be the orbital period and

Let a be the semi-major axis

Using Keplers third law

Then, the relation between the orbital period and the semi major axis is

P² ∝ a³

Then,

P² = ka³

P²/a³ = k

So,

P (earth) ²/a (earth) ³ = P (neptune) ² / a (neptune) ³

Period of earth P (earth) = 1year

Semi major axis of earth is

a (earth) = 1.5*10^8km

The semi major axis of Neptune is

a (Neptune) = 4.5*10^9km

So,

P (E) ²/a (E) ³ = P (N) ² / a (N) ³

1² / (1.5*10^8) ³ = P (N) ² / (4.5*10^9) ³

Cross multiply

P (N) ² = (4.5*10^9) ³ / (1.5*10^8) ³

P (N) ² = 27000

P (N) = √27000

P (N) = 164.32years

The period of Neptune is 164.32years