How does Doppler affect the observed period of a moon of Jupiter?

[kp]

Question:
The variable period of a moon of Jupiter, which is the basis of measurement of the speed of light in Roemer's method (whoever he is), is regarded as the Doppler affect. The period of the orbital motion of one of Jupiter's moons is approximately 42.5 hours; the speed of light is 2.99e8 m/s. The orbital speed of the Earth about the Sun is 2.98e4 m/s. What is the maximum change in the period (in seconds) of this moon as observed from Earth? (hint: f = 1/T)

I've done doppler problems with two moving objects
but I can't seem to come up with an answer that makes sense.
when I use f = 1/T i get this small number that look like wavelength.

any tips would be appreciated, thanks

 

[Astronuc]

The first quantitative estimate of the speed of light was made in 1676 by Ole Rømer, who was studying the motions of Jupiter's satellite Io with a telescope. It is possible to time the revolution of Io because it is entering/exiting Jupiter's shadow at regular intervals. Rømer observed that Io revolved around Jupiter once every 42.5 hours when Earth was closest to Jupiter. He also observed that, as Earth and Jupiter moved apart, Io's exit from the shadow would begin progressively later than predicted. It was clear that these exit "signals" took longer to reach Earth, as Earth and Jupiter moved further apart, as a result of the extra time it took for light to cross the extra distance between the planets, which had accumulated in the interval between one signal and the next. Similarly, about half a year later, Io's entries into the shadow happened more frequently, as Earth and Jupiter were now drawing closer together. On the basis of his observations, Rømer estimated that it would take light 22 minutes to cross the diameter of the orbit of the Earth (that is, twice the astronomical unit); the modern estimate is closer to 16 minutes and 40 seconds.

from http://en.wikipedia.org/wiki/Speed_of_light#Measurement_of_the_speed_of_light

f = 1/T and $\omega$ = 2$\pi$f = v/r where v is the linear velocity and r is the radius of the orbit of an object with velocity v.

 

[kp]

Interesting info...

so..I need to find the velocity of the moon around Jupiter by it's period?

then use f' = f((1 -+ v(earth)/ c)/(1-+ v(moon)/c)) to find the change frequency?

 

[kp]

anybody...?

 

[Astronuc]

Sorry about not getting back to you.

I don't think the problem is concerned with the effect of frequency change, but rather about the relative time that an event is observed.

The period could be estimated by observing the moon at the same point in its orbit during successive periods. However the Earth is also in its orbit.

So the biggest change in the observed period occurs when the change in the Earth's relative position with respect to Jupiter changes the most during one of the moons periods.

In 42.5 hrs, the Earth moves 4.5594e+9 m (but that is a circular arc). Light moves at 2.99e8 m/s, so if the Earth move 4.5594e+9 m closer to Jupiter, one observes the subesequent appearance of the moon ~15.2 seconds sooner, just due to the relative motion of the earth. If the Earth is moving away, then the one would observe a successive appearance of the moon 15.2 seconds later.

But this applies to one period, and I did not account for any relative motion by Jupiter.