The planet Jupiter has a satellite, Io, which travels in an orbit of radius 4.220×108 m with a period of 1.77 days. Calculate the mass of Jupiter from this information.
To calculate the mass of Jupiter using the information given, we can make use of Kepler's third law of planetary motion, which states that the square of the orbital period of a planet is proportional to the cube of its average distance from the sun (or any other central body).
Here's how you can calculate it step by step:
Step 1: Convert the period of Io's orbit to seconds.
The period is given as 1.77 days. Since there are 24 hours in a day and 60 minutes in an hour, we can multiply:
1.77 days × 24 hours/day × 60 minutes/hour × 60 seconds/minute = 152,496 seconds.
Step 2: Calculate the gravitational constant.
The gravitational constant, denoted as G, has a value of 6.67430 × 10^-11 m^3/kg/s^2.
Step 3: Calculate the mass of Jupiter.
Using the formula for the square of a period, we have:
T^2/R^3 = (4π^2/G)(M)
Where T is the period of Io's orbit, R is the orbital radius of Io, G is the gravitational constant, and M is the mass of Jupiter.
Rearranging the equation to solve for M, we have:
M = (T^2 × R^3) / (4π^2/G)
Plugging in the values, we get:
M = (152,496^2 × 4.220×10^8^3) / (4π^2 × 6.67430 × 10^-11 )
Evaluating this expression will give you the mass of Jupiter.