# Suppose the Sun were somehow replaced by a star with five times as much mass. How long would the...

## Question:

Suppose the Sun were somehow replaced by a star with five times as much mass. How long would the Earth year last in this last case? (hint: Newton s version of Kepler s 3rd Law)

## Kepler's Third Law

Kepler's planetary laws are very important in astrophysics and mechanics. The third law gives the relationship between time period of a planet and distance between sun and the planet. It states that the square of the time period of a planet is proportional to the cube of the distance between Sun and the planet. This law is applicable to all the planets in the solar system.

According to Kepler's law, the time period of a satellite is given by:

{eq}\displaystyle{ T=\sqrt{\frac{4\pi^2r^3}{GM}} } {/eq}

Where,

• {eq}r {/eq} is the average distance between Sun and the planet,
• {eq}G {/eq} is the gravitational constant
• {eq}M {/eq} is the mass of Sun.

We know that the time period of Earth is 365\ days.

That is {eq}T=365 {/eq} days

When the mass of Sun is replaced with 5 times massive star, The time period will be,

{eq}\displaystyle{ T'=\sqrt{\frac{4\pi^2r^3}{G\times5M}}=\frac{1}{\sqrt5}\times T=\frac{1}{\sqrt5}\times365\ days=163\ days } {/eq}

When mass of Sun increases the time period of the planet decreases. 