Question

In 1610 Galileo made a telescope and used it to study the planet Jupiter. He discovered four moons. One of them was Ganymede. The mean radius of the orbit of Ganymede around Jupiter is 10.7 × 108m and the period of the orbit is 7.16 days. i) Determine the mass of Jupiter.

Answer 1

Answer:
`1.893(10)^(27)kg`

Step-by-step explanation:

This problem can be solved by the Third Kepler’s Law of Planetary motion, which states:

“The square of the orbital period of a planet is proportional to the cube of the semi-major axis (size) of its orbit”.

In other words, this law stablishes a relation between the orbital period
`T` of a body (moon, planet, satellite) orbiting a greater body in space with the size
`a` of its orbit.

This Law is originally expressed as follows:

`T^(2)=(4\pi^(2))/(GM)a^(3)` (1)

Where;

`T=7.16days=618624s` is the period of the orbit Ganymede describes around Jupiter

`G` is the Gravitational Constant and its value is
`6.674(10)^(-11)(m^(3))/(kgs^(2))`

`M` is the mass of Jupiter (the value we need to find)

`a=10.7(10)^(8)m` is the semimajor axis of the orbit Ganymede describes around Jupiter (assuming it is a circular orbit, the semimajor axis is equal to the radius of the orbit)

If we want to find
`M`, we have to express equation (1) as written below and substitute all the values:

`M=(4\pi^(2))/(GT^(2))a^(3)` (2)

`M=(4\pi^(2))/((6.674(10)^(-11)(m^(3))/(kgs^(2)))(618624s)^(2))(10.7(10)^(8)m)^(3)` (3)

Finally:

`M=1.8934(10)^(27)kg` This is the mass of Jupiter