# 2 reasons why it takes long for the outer planets to orbit around the sun?

Mar 9, 2018

1. The orbits of the outer planets is longer. So, even if the orbital speeds of the outer planets had been the same as the inner planets (they are not - see point 2), they would have taken more time to complete an orbit( $T \propto \frac{R}{v}$).
2. The gravitational force of the sun is weaker than on the inner planets due to the inverse square nature of the force- this means that the centripetal acceleration is smaller, leading to a smaller velocity : ${v}^{2} / R \propto \frac{1}{R} ^ 2 \implies v \propto {R}^{- \frac{1}{2}}$
$T \propto \frac{R}{R} ^ \left(- \frac{1}{2}\right) = {R}^{\frac{3}{2}} q \quad \setminus m b \otimes \left\{\mathmr{and}\right\} q \quad {T}^{2} \propto {R}^{3}$