A pulsar is a stellar object that emits light in short bursts. Suppose a pulsar with a speed of 0.975c approaches Earth, and a rocket with a speed of 0.999c heads toward the pulsar (both speeds measured in Earth’s frame of reference). If the pulsar emits 20.0 pulses per second in its own frame of reference, at what rate are the pulses emitted in the rocket’s frame of reference?

Answer 1

Velocity of pulsar in the rocket's reference frame is:

v = (0.975 + 0.999)/(1+0.975*0.999) c

= 0.99998733552 c

We then need to compute the gamma factor at this speed. It is convenient to write the speed as (1-u) c. If you compute u using the above figure for v, you'll get a big loss in the number of significant digits. To prevent deal with this problem, you should compute u as follows:

u = 1-v/c =

1- (0.975 + 0.999)/(1+0.975*0.999) =

(1 + 0.975*0.999 - 0.975 -0.999) /(1+0.975*0.999) =

(1 -0.999 + 0.975*(0.999 - 1)) /(1+0.975*0.999) =

10^(-3) (1-0.975)/ (1+0.975*0.999) =

10^(-3)* 0.025/ (1+0.975*0.999) =

1.2664479933*10^(-5)

You can then comute the gamma factor as follows.

gamma(v) = 1/sqrt[1-(v/c)^2] =

1/sqrt[(1-v/c)(1+v/c)] =

1/sqrt[u(2-u)] = 198.69763

So, the pulses arrive approximately once every 9.935 seconds.