An elementary particle travels at \(0.840 c_{0}\) across a solar system that has a diameter of \(8.14 \times 10^{12} \mathrm{~m}\) (both measurements in the reference frame \(S\) of the solar system).

(a) What value does an observer in reference frame \(S\) measure for the time interval needed for the particle to cross the solar system?

(b) What time interval is needed in the reference frame \(\mathrm{P}\) of the particle?

(c) What is the proper length of the diameter? 

(d) What is the proper time interval for the particle's trip across the system?

(e) What is the diameter of the solar system in reference frame P?