Question

The mean distance of an asteroid from the Sun is 2.98 times that of Earth from the Sun. From Kepler's law of periods, calculate the number of years required for the asteroid to make one revolution around the Sun. Number Enter your answer in accordance to the question statement Entry field with incorrect answer Units Choose the answer from the menu in accordance to the question statement Entry field with correct answer.

Answer 1

Answer:

The asteroid requires 5.14 years to make one revolution around the Sun.

Explanation:

Kepler's third law establishes that the square of the period of a planet will be proportional to the cube of the semi-major axis of its orbit:

$T^{2} = a^{3}$ (1)

Where T is the period of revolution and a is the semi-major axis.

In the other hand, the distance between the Earth and the Sun has a value of $1.50x10^{8} Km$. That value can be known as well as an astronomical unit (1AU).

But 1 year is equivalent to 1 AU according with Kepler's third law, since 1 year is the orbital period of the Earth.

For the special case of the asteroid the distance will be:

$a = 2.98(1.50x10^{8}Km)$

$a = 4.47x10^{8}Km$

That distance will be expressed in terms of astronomical units:

$4.47x10^{8}Km.\frac{1AU}{1.50x10^{8}Km}$ ⇒ $2.98AU$

Finally, from equation 1 the period T can be isolated:

$T = \sqrt{a^{3}}$

$T = \sqrt{(2.98)^{3}}$  

$T = \sqrt{26.463592}$

$T = 5.14AU$

Then, the period can be expressed in years:

$5.14AU.\frac{1yr}{1AU} ⇒ 5.14 yr$

$T = 5.14 yr$

Hence, the asteroid requires 5.14 years to make one revolution around the Sun.