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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.

Answer and Explanation:

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.


Learn more about this topic:

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Kepler's Three Laws of Planetary Motion

from Basics of Astronomy

Chapter 22 / Lesson 12
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