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?
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?