Question

The planet earth is 1.5 10^11 from the sun and orbits the sun in one year. The planet
Pluto takes 248 years to orbit the sun. How far is Pluto from the sun?

Answer 1

Answer on Question #42080, Physics, Mechanics | Kinematics | Dynamics

The planet earth is $1.5 \times 10^{11} \, \text{m}$ from the sun and orbits the sun in one year. The planet Pluto takes 248 years to orbit the sun. How far is Pluto from the sun?

Solution:

Given:

$$a _ {E a r t h} = 2 1. 5 \cdot 1 0 ^ {1 1} \mathrm {m}$$ $$P _ {P l u t o} = 2 4 8 \text { years}$$ $$a _ {P l u t o} =?$$

The third Kepler law captures the relationship between the distance of planets from the Sun, and their orbital periods.

"The square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit."

Mathematically, the law says that the expression $P^2 / a^3$ has the same value for all the planets in the solar system, where $P$ is period and $a$ is distance from the Sun.

$$\frac {P _ {E a r t h} ^ {2}}{a _ {E a r t h} ^ {3}} = \frac {P _ {P l u t o} ^ {2}}{a _ {P l u t o} ^ {3}}$$ $$a _ {P l u t o} = a _ {E a r t h} ^ {3} \sqrt {\frac {P _ {P l u t o} ^ {2}}{P _ {E a r t h} ^ {2}}}$$ $$a _ {P l u t o} = 1. 5 \cdot 1 0 ^ {1 1} \sqrt {\frac {2 4 8 ^ {2}}{1 ^ {2}}} = 1. 5 \cdot 1 0 ^ {1 1} \cdot 3 9. 4 7 3 = 5 9. 2 \cdot 1 0 ^ {1 1} \mathrm {m}$$

Answer. $a_{Pluto} = 59.2 \cdot 10^{11} \, \text{m}$

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