Question

the planet mars takes 1.88 years to complete on revolution around the sun. the mean distance of the earth from the sun is 1.5x1o*8 km . calculate that of planet of mars

Answer 1

Answer on Question 73123, Physics, Astronomy, Astrophysics

Question:

The planet Mars takes 1.88 years to complete on revolution around the Sun. The mean distance of the Earth from the Sun is $1.5 \cdot 10^{8} \, \text{km}$. Calculate that distance of planet of Mars.

Solution:

We can find the mean distance of the Mars from the Sun from the third Kepler’s law of planetary motion. It states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis (mean distance) of its orbit:

$$\frac {P _ {M} ^ {2}}{a _ {M} ^ {3}} = \frac {P _ {E} ^ {2}}{a _ {E} ^ {3}},$$

here, $P_{M}$ is the orbital period of the Mars, $P_{E}$ is the orbital period of the Earth, $a_{M}$ is the mean distance of the Mars from the Sun, $a_{E}$ is the mean distance of the Earth from the Sun.

Then, from this formula we can find the mean distance of the Mars from the Sun:

$$a _ {M} ^ {3} = a _ {E} ^ {3} \frac {P _ {M} ^ {2}}{P _ {E} ^ {2}},$$ $$a _ {M} = \sqrt [ 3 ]{a _ {E} ^ {3} \frac {P _ {M} ^ {2}}{P _ {E} ^ {2}}} = \sqrt [ 3 ]{(1.5 \cdot 10 ^ {8} \, \text{km}) ^ {3} \cdot \frac {(1.88 \, \text{year}) ^ {2}}{(1.0 \, \text{year}) ^ {2}}} = 2.28 \cdot 10 ^ {8} \, \text{km}.$$

Answer:

$$a _ {M} = 2.28 \cdot 10 ^ {8} \, \text{km}.$$

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