# If the Earth was at one half of its present distance from the Sun, how many days will be there in...

## Question:

If the Earth was at one half of its present distance from the Sun, how many days will be there in a year?

## Kepler's Law of Planetary Motion:

Planetary Motion is governed by the Newton's Laws of Motion and the gravitational force, which, to a great extent, obeys Newton's Law of Universal Gravitation. As a result, the planetary motion obeys three Kepler's Laws:

• All planets orbit the Sun along elliptical orbits, with the Sun located at on the foci of the ellipse.
• The area swept by the radius-vector of the planet per unit time is constant;
• The ratio of squares of periods of two planets equals the ratio of cubes of the major semi-axes of their orbits.

The present distance of the Earth from the Sun is called 'Astronomical Unit' and is denoted as AU. So the present distance of the Earth from the Sun is {eq}1 \ AU {/eq}.

Using Kepler's Third Law, we can write:

{eq}\left (\dfrac {T}{1 \ year} \right )^2 = \left (\dfrac {0.5 \ AU}{1 \ AU} \right )^3 {/eq}

Solving for the new period, we obtain:

{eq}T = 1 \ year \times \left (\dfrac {0.5 \ AU}{1 \ AU} \right )^{3/2} \approx \boxed {0.35 \ year = 129 \ days} {/eq}

Therefore, the year will be 129 days. 