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The period of planet X is 0.62 Earth years and that of planet Y is 1.9 Earth years. Which of the...

Question:

The period of planet X is {eq}0.62 {/eq} Earth years and that of planet Y is {eq}1.9 {/eq} Earth years. Which of the following is true?

a) Planet Y is closer to the sun than planet X

b) Planet Y is closer to the sun than Earth

c) The Earth is closer to the sun than planet Y

d) The Earth is between planet X and planet Y.

Kepler's Third Law of Planetary Motion:

The orbital period of the earth around the sun is {eq}1 {/eq} year, i.e., {eq}365.25 {/eq} days. The square of the orbital period or the time required to cover the circumference of the orbit around the sun is directly proportional to the cube of the size of the circular orbit of the planet. Mathematically:

$$\boxed {T^2 \propto r^3} $$

where:

  • {eq}T {/eq} is the orbital period of the planet
  • {eq}r {/eq} is the orbital size of the planet

Answer and Explanation:


Given data:

  • {eq}T_E {/eq} is the orbital period of the earth
  • {eq}T_X=0.62 T_E {/eq} is the orbital period of the planet X
  • {eq}T_Y=1.9 T_E {/eq} is the orbital period of the planet Y
  • {eq}r_E {/eq} is the radius of rotation of the earth
  • {eq}r_X {/eq} is the radius of rotation of the planet X
  • {eq}r_Y {/eq} is the radius of rotation of the planet Y


Using Kepler's third law, we have:

{eq}T^2 \propto r^3 \\[0.3cm] or \\[0.3cm] r \propto T^{\frac{2}{3}} {/eq}

Using this expression, we will compare the orbital size of the earth and planet X:

{eq}\begin{align} \dfrac{r_X}{r_E}&=\left (\dfrac{T_X}{T_E} \right )^{\dfrac{2}{3}} \\[0.3cm] \implies \dfrac{r_X}{r_E}&=\left (\dfrac{0.62T_E}{T_E} \right )^{\dfrac{2}{3}} \\[0.3cm] \implies r_X&=0.727r_E \ \ \ \ \ \ ....(i) \end{align} {/eq}

Similarly, we will compare the orbital size of the earth and planet Y:

{eq}\begin{align} \dfrac{r_Y}{r_E}&=\left (\dfrac{T_Y}{T_E} \right )^{\dfrac{2}{3}} \\[0.3cm] \implies \dfrac{r_Y}{r_E}&=\left (\dfrac{1.9T_E}{T_E} \right )^{\dfrac{2}{3}} \\[0.3cm] \implies r_Y&=1.53 r_E \ \ \ \ \ \ ....(ii) \end{align} {/eq}

From {eq}(i) {/eq} and {eq}(ii) {/eq}:

{eq}\boxed {r_X<r_E<r_Y} {/eq}

Therefore, it is clear that planet X is closest to the sun, and planet Y is farthest from the sun, and the earth is in between the planet X and planet Y.

So, option (d) is correct.



Learn more about this topic:

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Kepler's Three Laws of Planetary Motion

from Basics of Astronomy

Chapter 22 / Lesson 12
47K

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