Astronomers have proposed the existence of a ninth planet in the distant solar system. Its semi-major axis is suggested to be approximately 600 AU. If this prediction is correct, what is its orbital period in years

For this exercise we must use Kepler's laws, specifically the third law which is the application of the universal law of gravitation to Newton's second law

F = ma

G m M / r² = m a_c = m v² / r

G M / r = v²

the speed of the circular orbit is

v = 2π r / T

we substitute

G M / r = 4π² r² / T²

T² = (4π² / G M) r³

Kepler proved that this expression is the same if the radius is changed by the semi-major axis of an ellipse

T² = (4π² /GM) a³

the constant is worth

(4π² / GM) = 2.97 10⁻¹⁹ s² / m³

let's reduce the distance to SI units

AU is the distance from the Earth to the Sun

a = 600 AU = 600 AU (1.496 10¹¹ m / 1 AU)

a = 8.976 10¹³ m

T² = 2.97 10⁻¹⁹ (8.976 10¹³)³

T² = 21.4786 10²²

T = 4.63 10¹¹ s

Let's reduce to years

T = 4.63 10¹¹s (1 h / 3600s) (1 day / 24 h) (1 year / 365 days)

T = 1.4696 10⁴ years