Io and Europa are two of Jupiter's many moons. The mean distance of Europa from Jupiter is about twice as far as that for Io and Jupiter. By what factor is the period of Europa's orbit longer than that of Io's?
To find the factor by which the period of Europa's orbit is longer than that of Io's, we need to compare their orbital periods.
The orbital period of a moon is determined by the square root of its mean distance from the planet cubed, divided by the mass of the planet. It can be represented by the formula:
T = √(r³ / M)
Where:
T = Orbital period of the moon
r = Mean distance of the moon from the planet
M = Mass of the planet
Given that the mean distance of Europa from Jupiter is about twice as far as that for Io and Jupiter, we can say:
r(Europa) = 2 * r(Io)
So, the ratio of the mean distances can be expressed as:
r(Europa) / r(Io) = 2
Now, substituting these values into the orbital period formula:
T(Europa) = √(r(Europa)³ / M)
T(Io) = √(r(Io)³ / M)
Dividing the equation for Europa's orbital period by Io's orbital period:
T(Europa) / T(Io) = √((r(Europa)³ / M) / (r(Io)³ / M))
T(Europa) / T(Io) = √((r(Europa)³ / r(Io)³))
T(Europa) / T(Io) = √(8)
T(Europa) / T(Io) = 2
Therefore, the period of Europa's orbit is longer than that of Io's by a factor of 2.