Signals sent during relativistic space travel

[Oijl]

Homework Statement

Suppose rocket traveler Amelia has a clock made on Earth. She flies to and back from a planet 12 light-years away (as measured from rest with respect to Earth) from Earth at a speed of 0.6c. Every year she sends a signal to Earth. How many signals does Earth receive by the time she gets back?

Homework Equations

f = f'*[sqrt((1-u/c)/(1+u/c))]
(relativistic Doppler shift)
L = Lo*sqrt(1-u^2/c^2)
(relativistic length contraction)

The Attempt at a Solution

The distance to the planet for Amelia is shorter than 12 light-years. It is 9.6 light-years. So she takes 16 years (as she measures them) to get there and 16 to get back.

On the way there, by the relativistic Doppler equation I know that Earth receives her signals at a rate of 0.5 signals every Amelia-year, and on her way back Earth receives her signals at a rate of 2 signals every Amelia-year.

There are 16 Amelia-years in the trip to the planet, and 16 in the trip from the planet. So the total number of signals is

16*0.5 + 16*2 = 8 + 32 = 40.

But that is wrong, because it should be 32 signals, if she sent one every year, since her trip lasted 32 years to her.

How am I doing this wrong?
(Thanks.)

 

[HallsofIvy]

You calculated the distance in Amelia's frame of reference but did not account for the slowing of time in her frame.

 

[Oijl]

Ta-Da! Thank you, Halls of Ivy, that was kind of you. That's a load off my mind.

 

[thelahana]

I'm dabbling over the same question but how would that time dilation effect be? Would it be affecting the 32 years time? or, since the Earth has a relativistic speed of 0.6c relative to Amelia, should we consider the effect on Earth's time?

Thanks