Callisto, one of Jupiter's moons, has an orbital period of 16.69 days and its distance from Jupiter is 1.88*10^6 km. What is Jupiter's mass?

Answer: Mass of Jupiter ~ = 1.89 * 10^23 kg

Explanation:

Given:

Period P = 16.69days * 86400s/day = 1442016s

Radius of orbit a = 1.88*10^6km * 1000m/km

r = 1.88 * 10^9 m

Gravitational constant G = 6.67*10^-11 m^3 kg^-1 s^-2

Applying Kepler's third law, which is stated mathematically as;

P^2 = (4π^2a^3) / G (M1+M2) ... 1

Where M1 and M2 are the radius of Jupiter and callisto respectively.

Since M1 >> M1

M1+M2 ~ = M1

Equation 1 becomes;

P^2 = (4π^2a^3) / G (M1)

M1 = (4π^2a^3) / GP^2 ... 3

Substituting the values into equation 3 above

M1 = (4 * π^2 * (1.88 * 10^9) ^3) / (6.67*10^-11 * 1442016^2)

M1 = 1.89 * 10^27 kg

The Jupiter's mass is approximately 1.89*10²⁷ kg.

Explanation:

The only force acting on Calisto while is rotating around Jupiter, is the gravitational force, as defined by the Newton's Universal Law of Gravitation:

Fg = G*mc*mj / rcj²

where G = 6.67*10⁻¹¹ N*m²/kg², mc = Callisto's mass, mj = Jupiter's mass, and rcj = distance from Jupiter for Callisto = 1.88*10⁹ m.

At the same time, there exists a force that keeps Callisto in orbit, which is the centripetal force, that actually is the same gravitational force we have already mentioned.

This centripetal force is related with the period of the orbit, as follows:

Fc = mc * (2*π/T) ²*rcj.

In order to be consistent in terms of units, we need to convert the orbital period, from days to seconds, as follows:

T = 16.69 days * 86,400 (sec/day) = 1.44*10⁶ sec.

We have already said that Fg = Fc, so we can write the following equality:

G*mc*mj / rcj² = mc * (2*π/T) ²*rcj

Simplifying common terms, and solving for mj, we get:

mj = 4*π² * (1.88*10⁹) ³m³ / ((1.44*10⁶) ² m²*6.67*10⁻11 N*m²/kg²)

mj = 1.89*10²⁷ kg.