Io is a satellite in Jupiter. The radius of its orbit is 422000 km. Its period is 1.77 days. Calculate the mass of Jupiter.
To calculate the mass of Jupiter using the given information about Io's orbit, we can apply Kepler's Third Law of Planetary Motion. This law states that the square of an object's orbital period is directly proportional to the cube of the semimajor axis of its orbit.
First, we need to convert the period of Io's orbit from days to seconds:
Period of Io's Orbit = 1.77 days = 1.77 * 24 * 60 * 60 seconds ≈ 152,928 seconds
Next, we will use the formula for Kepler's Third Law:
T^2 = (4π^2 / GM) * r^3
Where:
T = Period of Io's orbit (in seconds)
G = Universal gravitational constant (6.67 x 10^-11 Nm^2/kg^2)
M = Mass of Jupiter (what we want to find)
r = Radius of Io's orbit (in meters)
Rearranging the formula to solve for M, we get:
M = (4π^2 / G) * (r^3 / T^2)
Let's substitute the given values into the formula:
M = (4 * π^2 / G) * (422,000,000 meters)^3 / (152,928 seconds)^2
Now, we need to calculate the numerical value of the expression. Using a calculator, it simplifies to:
M ≈ 1.903 × 10^27 kilograms
Therefore, the mass of Jupiter is approximately 1.903 × 10^27 kilograms.