You have discovered the planet Zoltron. It is 16.0 times farther from the Sun than Earth (16.0 AU). How long does it take this planet to go once around the Sun?

Answer 1

Hi Phebe,

So this is a classic Kepler's Third Law problem, which relates the orbital period of a body to it's distance from the force center. Mathematically,

T² = [4*pi²/(GM)] * a³

^{where T is the period, M is the mass of the sun, a is the distance from the sun to the the planet and G is the universal gravitational constant.}

Let's let the subscripts e stand for Earth and z stand for Zoltron. In addition, let's let the quantity in brackets be K and note that it is the SAME for all planets! Kepler's law for both planets:

T_e² = K*a_e³ for Earth

and

T_z² = K*a_z³ for Zoltron

Dividing the equation for Zoltron by the equation for the Earth gives,

T_z²/T_e² = a_z³/a_e³ (the constant K cancels)

We need to solve for T_z. Doing so gives,

T_z² = T_e² * (a_z³/a_e³)

We know that a_z = 16 AU but we also know that a_e = 1 AU so that a_z = 16*a_e . Also, T_e = 1 (year)

Therefore, T_z² = 1² * 16³. Taking the square root of both sides gives, T_z = 64 years (Earth years).