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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.

Answer and Explanation:

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.


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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