Question

a) One of the moons of Jupiter, named Io, has an orbital radius of 4.22 108 m and a period of 1.77 days. Assuming the orbit is circular, calculate the mass of Jupiter. (b) The largest moon of Jupiter, named Ganymede, has an orbital radius of 1.07 109 m and a period of 7.16 days. Calculate the mass of Jupiter from this data. (c) Are your results to parts (a) and (b) consistent? Explain.

Answer 1

Answer:

According to Kepler's 3rd law.

It states that the orbital period, T is related to the distance, r as:

T² =  4 π²r³ /G  M

where G is the universal gravitational constant  =  6.673 × 10⁻¹¹ Nm²/kg²

Rearranging for M should give Jupiter's mass.

M =   4 π²r³/GT²

T= 1.77 days × 24 h/day × 60 min/h × 60 s/min = 1.53 × 10⁵ s

r = 4.22x10⁸ m

M = 4π² ((4.22 × 10⁸ m)³/(6.673 × 10⁻¹¹ Nm²/kg² x (1.53 × 10⁵ s)²)

M = 1.90 × 10²⁷kg

The mass of Jupiter is 1.90 × 10²⁷kg.

1.90 × 10²⁷kg

T= 7.16 days × 24 h/day × 60 min/h × 60 s/min = 6.19 × 10⁵s

r = 1.07x10⁹ m

M = 4π² ((1.07 × 10⁹ m)³/(6.673 × 10⁻¹¹ Nm²/kg² x (6.19 × 10⁵ s)²)

M = 1.90 × 10¹⁷kg

The mass of Jupiter is 1.90 × 10¹⁷kg.

THE RESULTS TO PART A and B ARE NOT CONSISTENT. The reason is because of the difference in radius of each satellites from Jupiter. i.e the farther away the moons, the smaller they become in space and the more the number of days to complete an orbit.