What is the mass of Jupiter as determined from Ganymede's orbit?
Determining the mass of Jupiter from Ganymede's orbit involves using Kepler's third law of planetary motion. This law relates the orbital period of a moon or satellite to the mass of the planet it revolves around. Here's how you can calculate it:
1. Find the orbital period of Ganymede: The orbital period is the time it takes for Ganymede to complete one orbit around Jupiter. This information can be obtained from astronomy databases or scientific literature. The orbital period of Ganymede is approximately 7.16 Earth days.
2. Convert the orbital period to seconds: To use the equation for Kepler's third law, you need the orbital period in seconds instead of days. There are 86,400 seconds in a day, so multiply the orbital period (in days) by 86,400 to get the orbital period in seconds.
3. Determine the distance between Ganymede and Jupiter: Measure or look up the average distance between Ganymede and Jupiter. This value is also available in astronomical databases, and it is approximately 1,070,400 kilometers.
4. Apply Kepler's third law: Kepler's third law states that the square of the orbital period is proportional to the cube of the average distance between the two objects. The equation is T^2 = (4π^2/GM) × R^3, where T is the orbital period in seconds, G is the gravitational constant, M is the mass of Jupiter, and R is the distance between Ganymede and Jupiter.
Rearrange the equation to solve for the mass of Jupiter:
M = (4π^2/G) × (R^3/T^2)
5. Plug in the values and calculate: Substitute the known values into the equation and perform the calculation. The gravitational constant (G) is approximately 6.67430 × 10^-11 m^3/kg/s^2.
M = (4π^2 / (6.67430 × 10^-11)) × ((1,070,400 × 10^3)^3 / (7.16 × 86,400)^2)
After evaluating this expression, you will obtain the mass of Jupiter as determined from Ganymede's orbit.