Question

The mass of the Sun is 2 × 1030 kg, and the mass of Saturn is 5.68 × 1026 kg. The distance between Saturn and the Sun is 9.58 AU. Veronica is solving the following equation to calculate the orbital period of Saturn, but there is an error in the equation. T = What should Veronica change to correct the equation?
change the position of 2 x 1030 kg and 9.58 AU
change 2 x 1030 kg to 5.68 x 1026 kg
change the square root to a cube root
change 9.58 AU to the distance in meters

Answer 1

The correct answer is D.

Answer 2

The formula for computing the orbital time period of a body is given as:

T² = 4π²r³ / GM

where T is the time period, r is the distance between the two bodies, G is the gravitational constant and M is the mass of the body that is being orbited. If we compute this time using SI units, the working is:

9.58 AU is 1.43 x 10¹² meters

T = √[(4*π²*(1.43 x 10¹²)³) / (6.67 × 10⁻¹¹ * 2 x 10³⁰)]
T = 9.30 x 10⁸ seconds which is approximately 29 years


Using the astronomical units, distance is in astronomical units and the mass is in solar masses. In these conditions, the ratio:
4π²/G = 1 so

T² = a³ (since the solar mass of the sun is 1)

T = √(9.58)³
T = 27 years