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?
TEu/TIo =
I know that Europa's period is twice Io's but I'm struggling in how to present it properly.
whats your awnser
I know that orbital period equation is T^2=4pi^2r^3/GM and that Europa's period is twice Io's but otherwise I'm pretty stumped.
since T^2 is proportional to r^3,
T is proportional to √(r^3)
so if r is doubled, T grows by a factor of √8
To find the factor by which the period of Europa's orbit is longer than that of Io's, you can use the concept of the orbital period being proportional to the mean distance raised to the 3/2 power.
Let's denote the mean distance of Io from Jupiter as DIo, the mean distance of Europa from Jupiter as DEu, and the periods of Io and Europa as TIo and TEu, respectively.
Given that the mean distance of Europa from Jupiter is about twice as far as that for Io (DEu = 2*DIo), we can write:
TEu / TIo = (DEu^3/2) / (DIo^3/2)
Since DEu = 2*DIo, we can substitute it into the equation:
TEu / TIo = [(2*DIo)^3/2] / (DIo^3/2)
Simplifying the equation:
TEu / TIo = (2^3/2 * DIo^3/2) / (DIo^3/2)
TEu / TIo = 2^3/2
So, the factor by which the period of Europa's orbit is longer than that of Io's is (√2)^3, which simplifies to 2^3/2, or 2*√2, approximately equal to 2.83.
Therefore, we can write:
TEu / TIo = 2.83