# # Plot the velocity & stuff during the trip from Earth to AC # # set terminal x11 0 # set output set parametric set key left set trange [0:11.32] set xrange [0:7.04] set yrange [*:*] set grid set title "From Earth to Alpha Centaurus and back, In the Traveler's MCRF" set xlabel "Ship's Time" # Earth time at turnover and arrival tt = 2.83 tf = 5.66 # Ship time at turnover and arrival taut = 1.76 tauf = 3.52 # Distance at turnover and arrival xt = 2 xf = 4 # Complemented time, for convenience tc(t) = tf - t # Velocity before and after turnover, and combined v0(t) = t/sqrt(1 + t**2) v1(t) = tc(t)/sqrt(1 + tc(t)**2) v(t) = (t < tt) ? v0(t) : \ (t < tf) ? v1(t) : \ (t < tf + tt) ? -v0(t - tf) : \ -v1(t - tf) # Ship time before and after turnover, and combined tau0(t) = log(t + sqrt(t**2 + 1)) tau1(t) = tauf - tau0(tf - t) tau(t) = (t <= tt) ? tau0(t) : \ (t <= tf) ? tau1(t) : \ (t <= tf + tt) ? tauf + tau0(t-tf) : \ tauf + tau1(t-tf) # Gamma Gamma0(t) = 1/sqrt(1 - v(t)**2) Gamma(t) = (t <= tf) ? Gamma0(t) : Gamma0(t - tf) # Ship location before and after turnover, and combined dx0(t) = sqrt(t**2 + 1) - 1 dx1(t) = xf - dx0(tc(t)) distance(t) = (t <= tt) ? dx0(t) : \ (t <= tf) ? dx1(t) : \ (t <= tt + tf) ? xf - dx0(t - tf) : \ xf - dx1(t - tf) # Earth time, simultaneous with the ship's "now", in the MCRF ts0(t) = t/Gamma(t) ts1(t) = tf - tc(t) * (xf + 1)/Gamma(t) ts2(t) = t - (distance(t)/Gamma(t)) * v(t) * Gamma(t) t_simultaneous(t) = (t < tt) ? ts0(t) : (t <= tf) ? ts1(t) : ts2(t) # Difference between Earth-simultaneous time and Earth-ship time ts_diff(t) = t_simultaneous(t) - t # Derivative of Earth time wrt Ship's time dtdT0(t) = 1/(1 + t**2) dtdT1(t) = 5/(1 + (tf - t)**2) dtdT_pre_tf(t) = (t