The largest moon in the solar system is Ganymede, a moon of Jupiter. Ganymede orbits at a distance of 1.07 X 10^9 m from the center of Jupiter with an orbital period of about 6.18 X 10^5 s. Using this information, find the mass of Jupiter.
Well, calculating the mass of Jupiter is not rocket science... Well, actually, it kinda is. But fear not, I'm here to help.
To find the mass of Jupiter, we'll use Newton's Form of Kepler's Third Law. Because, hey, who doesn't love some good old Kepler?
The formula goes like this:
M = (4π²r³) / (GT²)
Where:
M = Mass of Jupiter
r = Distance between Jupiter and Ganymede
T = Orbital period of Ganymede
G = Universal gravitational constant
Now, let's just plug in the numbers and see what we get, shall we?
M = (4π² * (1.07 x 10^9)³) / (6.67430 x 10^-11 * (6.18 x 10^5)²)
After some number crunching, we get:
M ≈ 1.898 x 10^27 kg
So, the mass of Jupiter is approximately 1.898 x 10^27 kilograms. Quite a hefty planet, if you ask me. Hope that helps!
To find the mass of Jupiter, we can use Kepler's Third Law of Planetary Motion, which relates the orbital period and the distance from the center of the planet.
The equation for Kepler's Third Law is:
T^2 = (4π^2 / GM) * r^3
Where:
T is the orbital period,
G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2),
M is the mass of the planet, and
r is the distance from the center of the planet.
Let's plug in the given values:
T = 6.18 × 10^5 s
r = 1.07 × 10^9 m
(6.18 × 10^5)^2 = (4π^2 / (6.67430 × 10^-11)M ) * (1.07 × 10^9)^3
Solving for M:
M = (4π^2 / (6.67430 × 10^-11)) * ((1.07 × 10^9)^3 / (6.18 × 10^5)^2)
Calculating this expression gives:
M ≈ 1.90 × 10^27 kg
Therefore, the mass of Jupiter is approximately 1.90 × 10^27 kg.
To find the mass of Jupiter, we can use Kepler's Third Law of planetary motion, which states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
First, let's calculate the semi-major axis of Ganymede's orbit. The semi-major axis is the average distance from Ganymede to Jupiter, which is given as 1.07 × 10^9 m.
Next, let's calculate the orbital period of Ganymede, which is given as about 6.18 × 10^5 s.
Now, using Kepler's Third Law, we can set up the following equation:
(T^2) / (R^3) = (4π^2) / (G * M)
Where T is the orbital period, R is the semi-major axis, G is the gravitational constant, and M is the mass of Jupiter.
Rearranging the equation to solve for M:
M = (T^2 * R^3) / (4π^2 * G)
To find the mass of Jupiter, we need to substitute the values into the equation:
T = 6.18 × 10^5 s
R = 1.07 × 10^9 m
G = 6.67 × 10^-11 m^3 kg^-1 s^-2 (gravitational constant)
Substituting the values:
M = (6.18 × 10^5 s)^2 * (1.07 × 10^9 m)^3 / (4π^2 * (6.67 × 10^-11 m^3 kg^-1 s^-2))
Simplifying the equation:
M = 1.53 × 10^24 kg
Therefore, the mass of Jupiter is approximately 1.53 × 10^24 kg.