Question

If you want to launch a probe that orbits the Sun with a period of 4.00 years, what should be its orbital radius? Give your answer in astronomical units. The astronomical unit AU is defined as the mean distance from the Sun to the Earth 1AU=1.50Ă—1011m. Express your answer to three significant figures. Is the orbital speed of this probe greater than, less than, or equal to the orbital speed of the Earth?

Answer 1

As per Kepler's law we know that

$\frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3}$

here we know that

Time period of earth around sun is 1 year while its orbital radius around sun is 1 AU

Now here we know that another planet has time period T = 4 years and its orbital radius is R2

now from above equation we will have

$\frac{1^2}{4^2} = \frac{1^3}{R^3}$

$\frac{1}{16} = \frac{1}{R^3}$

$R^3 = 16$

$R = 2.52 AU = 3.78 \times 10^{11} m$

so above is the orbital radius