Question

A new planet is discovered beyond Pluto at a mean distance to the sun of 4004 million miles. Using Kepler's third law, determine an estimate for the time T to travel around the sun in an orbit.

Answer 1

Answer:

103239.89 days

Explanation:

Kepler's third law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

                               a³ / T² = 7.496 × 10⁻⁶  (a.u.³/days²)

where,

a is the distance of the semi-major axis in a.u

T is the orbit time in days

Converting the mean distance of the new planet to astronomical unit (a.u.)

                       1 a.u = 9.296 × 10⁷ miles        

                                      $\frac{4004 * 10^{6}}{9.296 * 10^{7}} = 43.07\ a.u.$

Substituting the values into Kepler's third law equation;

                                    $\frac{(43.07)^{3}}{T^{2}} = 7.496 * 10^{-6}$  

                                    $T^{2} = \frac{(43.07)^{3}}{7.496 * 10^{-6}}$ (days)²

                                    $T^{2} = \sqrt{\frac{(43.07)^{3}}{7.496 * 10^{-6}}}$

                                    T = 103239.89 days

An estimate time T for the new planet to travel around the sun in an orbit is 103239.89 days