Question

Find the mass of Jupiter based on the fact that Io, its innermost moon, has an average orbital radius of 421,700 km and a period of 1.77 days.

a) 1.898×10^27 kg
b) 1.898×10^28 kg
c) 1.898×10^29 kg
d) 1.898×10^30 kg

Answer 1

Using Kepler's third law and the given values for Io's orbital period and radius, the mass of Jupiter can be calculated and is found to be approximately 1.898×10²⁷ kg, which aligns with option a).

Step-by-step explanation:

To find the mass of Jupiter using the orbital characteristics of its moon Io, we can apply Kepler's third law of planetary motion. This law states that the square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis (r) of its orbit, with the proportionality constant being related to the mass (M) of the central body (in this case, Jupiter) and the gravitational constant (G).

The formula derived from Kepler's third law is:
T² = (4π²/(G*M)) · r³

Given the values:
T = 1.77 days (converted to seconds: 1.77 * 86400 = 152928 seconds),
r = 421700 km (converted to meters: 421700000 m),
and G = 6.67430 × 10⁻¹± m³·kg⁻¹·s⁻² (the gravitational constant),
we can solve for M (the mass of Jupiter).

Reorganizing the formula to solve for M, we get:
M = (4π² · r³) / (G · T²)

Substituting the known values:
M = (4π² · (421700000 m)³) / (6.67430 × 10⁻¹± m³·kg⁻¹·s⁻² · (152928 s)²)

After calculating, we find that the mass of Jupiter is approximately 1.898×10²⁷ kg, which is the closest to option a) 1.898×10²⁷ kg.