# How long would a year be if the distance of the earth from the sun is 20% more?

## Question:

How long would a year be if the distance of the earth from the sun is 20% more?

## Period of a body in a circular orbit

If a body is in a circular orbit with radius {eq}r {/eq} around a more massive body with mass {eq}M {/eq}, then its period is

{eq}T = \dfrac{2\pi r^{3/2}}{\sqrt{GM}}, {/eq}

where {eq}G = 6.67 \times 10^{-11} \ \text{N} \ \text{m}^2/\text{kg}^2 {/eq} is the gravitation constant.

The period {eq}T {/eq} of earth around the sun with radius {eq}r {/eq} is {eq}T = 365.4 \text{ days} {/eq}, i.e.

{eq}T = \dfrac{2\pi r^{3/2}}{\sqrt{GM}} = 365.4 \text{ days}, {/eq}

where {eq}M {/eq} is the mass of sun. Now, if {eq}r {/eq} becomes 20% more, i.e. {eq}r \rightarrow r' = (1+0.20)r = 1.20 r {/eq}, then the new period {eq}T' {/eq} is

{eq}T' = \dfrac{2\pi (r')^{3/2}}{\sqrt{GM}} = \dfrac{2\pi (1.20 r)^{3/2}}{\sqrt{GM}} = 1.20^{3/2} \dfrac{2\pi r^{3/2}}{\sqrt{GM}} = 1.20^{3/2} T, {/eq}

{eq}T' = (1.20)^{3/2}(365.4 \text{ days}) = 480.3 \text{ days}. {/eq}

Hence, the year, which is originally 365.4 days, will be 480.3 days long when the distance from earth to sun increased by 20%.