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?
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.
To determine the rate at which the pulses are emitted in the rocket's frame of reference, we need to understand how time dilation affects the pulse rate.
Time dilation is a phenomenon predicted by Einstein's theory of relativity. According to this theory, time appears to pass slower for objects in motion relative to an observer at rest.
Let's calculate the time dilation factor, γ, for both the pulsar and the rocket.
The time dilation factor is given by the equation:
γ = 1 / √(1 - v^2/c^2)
Where:
v is the velocity of the object relative to the speed of light (c).
For the pulsar:
v = 0.975c
γ_pulsar = 1 / √(1 - 0.975^2)
γ_pulsar ≈ 4.149
For the rocket:
v = 0.999c
γ_rocket = 1 / √(1 - 0.999^2)
γ_rocket ≈ 22.366
Now, we know that the pulsar emits 20.0 pulses per second in its own frame of reference. To find the rate at which the pulses are emitted in the rocket's frame of reference, we can use the time dilation factor, γ:
Rate in rocket's frame = Rate in pulsar's frame / γ
Rate in rocket's frame = 20.0 pulses per second / γ_pulsar
Rate in rocket's frame = 20.0 pulses per second / 4.149
Rate in rocket's frame ≈ 4.819 pulses per second
Therefore, in the rocket's frame of reference, the rate at which the pulses are emitted would be approximately 4.819 pulses per second.