Question

Io, a satellite of Jupiter, orbits that planet in a nearly circular orbit. It has an orbital period of 1.77 days and an orbital radius of 4.22 x 10^5 km. From these data, determine the mass of Jupiter.

a) 1.89 x 10^27 kg
b) 2.42 x 10^26 kg
c) 3.15 x 10^24 kg
d) 4.58 x 10^28 kg

Answer 1

Using Kepler's third law of planetary motion and the given orbital data of Io, the mass of Jupiter is calculated to be approximately 1.9 × 10^27 kg, which corresponds to option a) 1.89 × 10^27 kg.

Step-by-step explanation:

To find the mass of Jupiter using the orbital data of Io, we can use Kepler's third law of planetary motion. Kepler's third law can be formulated as:

T^2 = (4π^2/GM)r^3,

where T is the orbital period of Io, r is the orbital radius, G is the gravitational constant (6.674×10^-11 N(m/kg)^2), and M is the mass of Jupiter.

Given that Io's orbital period (T) is 1.77 days (which is equal to 1.77 × 24 × 3600 seconds) and the orbital radius (r) is 4.22 × 10^5 km (which is equal to 4.22 × 10^8 meters), we can rearrange the formula to solve for M:

M = (4π^2/G)(r^3/T^2).

By plugging in the values for T and r, we can calculate the mass of Jupiter. After computation, the mass of Jupiter is found to be approximately 1.9 × 10^27 kg.

Therefore, option a) 1.89 × 10^27 kg is the correct answer, aligning with the actual mass of Jupiter.