PART 1-Computing Jupiter’s mass with Jupiter’s moon Callisto. Let’s use our Eart

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PART 1-Computing Jupiter’s mass with Jupiter’s moon Callisto. Let’s use our Earth/Moon system for units. Yes, we can compute Jupiter’s mass, relative to the mass of Earth, with Jupiter’s moon Callisto. All we need to know is Callisto’s mean distance from Jupiter, or semi-major axis, in Lunar Distances (LD), and Callisto’s orbital period relative to the moon’s orbital period (sidereal month).
One Lunar Distance Unit (LD) = 384,400 kilometers (238,855 miles)
Moon’s orbital period = 27.322 days. This will be our Lunar Period unit.
Callisto’s mean distance from Jupiter is 1,882,700 kilometers (1,169,856 miles) and its orbital period is 16.689 days. Converting Callisto’s mean distance and orbital period into lunar figures:
a = Callisto’s mean distance = 4.898 lunar
p = Callisto’s orbital period = 0.611 lunar
We plug these numbers into the equation below. Voila! We have Jupiter’s mass in Earth masses.
Mass of Jupiter = a3/p2
Perform the equation below:



Computing Jupiter’s mass with Jupiter’s moon Io.We can double-check our answer, using Jupiter’s moon Io, whereby a = Io’s mean distance = 1.097 Lunar Distances (LD), and p = Io’s orbital period = 0.0648 lunar.
Mass of Jupiter = a3/p2
Perform equation below:



Galileo was the first person to observe those moons in the late 17th century. After many careful observations, he realized that the moons appeared to move east and west of Jupiter and he correctly interpreted this sideways motion as a one-dimensional, edge-on view of a two-dimensional circular orbit. Therefore, Galileo was the first person to recognize that not everything in the solar system was moving in circles around the Earth. His data, in conjunction with Kepler’s ideas about planetary orbits played a major role in the development of modern science, and these two ideas played a fundamental role in the Copernican Revolution, which has led people to believe that the Earth is not located at a special place in the universe.

Kepler’s third law of planetary motion, as revised by Isaac Newton (in his law of gravity), says that if a smaller object circularly orbits a much more massive object, the orbit’s size and time are related to the mass of the much more massive object in the following way:

(Radius of orbit)3 / (Time for one orbit)2 = (mass of large object)

Note: For this equation to be valid as written, you must measure the radius in terms of astronomical units, and you must measure the time in years. The output mass of the large object will be the mass of the object divided by the mass of the Sun. (If you use more standard units, such as meters, seconds, and kilograms, you must multiply a constant to one side of the equation.) Also note that this law is a direct consequence of the law of gravity, because gravity is what keeps objects in orbit around other objects.

Explanation / Answer

Given = Mass of Jupiter = a3/p2

Using our Earth/ Moon system for units and using Callisto moon
given a = Callisto’s mean distance = 4.898 lunar
p = Callisto’s orbital period = 0.611 lunar

Thus Mass of jupiter = (4.898)3/(0.611)2 = 314.75593 earth Masses

Now Computing Jupiter’s mass with Jupiter’s moon Io.
Mass of Jupiter = a3/p2
a = Io’s mean distance = 1.097 Lunar Distances (LD),
and p = Io’s orbital period = 0.0648 lunar.
Thus Mass of jupiter = (1.097)3/(0.0648)2 = 314.3908 earth Masses

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