To find the mean orbital radius of Europa in AU, we can use the information given about Io and Jupiter.
First, let's find the period of revolution of Europa around Jupiter. The question states that Io revolves around Jupiter in 0.0048 sidereal years, and Europa has a period of rotation of 0.0097 sidereal years.
Since the period of rotation of a moon is equal to the period of revolution around the planet, we can say that the period of revolution of Europa is also 0.0097 sidereal years.
Now, let's use Kepler's Third Law, which states that the square of the period of revolution is proportional to the cube of the mean orbital radius.
We can set up the following equation:
(T1^2) / (R1^3) = (T2^2) / (R2^3)
Let's plug in the values we know:
(T1 = 0.0048 sidereal years for Io)
(R1 = 0.0028 AU for Io)
(T2 = 0.0097 sidereal years for Europa)
(R2 = ?)
0.0048^2 / 0.0028^3 = 0.0097^2 / R2^3
Simplifying the equation, we have:
R2^3 = (0.0097^2 * 0.0028^3) / 0.0048^2
Calculating the right side of the equation, we find:
R2^3 ≈ 0.00008874 AU^3
To find R2, we can take the cube root of both sides:
R2 ≈ ∛0.00008874 AU
R2 ≈ 0.0416 AU
Therefore, the mean orbital radius of Europa is approximately 0.0416 AU.
