# # Plot the velocity & stuff during the trip from Earth to AC # # # set terminal x11 0 # set output # We need Earth time, at the traveler's location, as a function of tau. # Amazingly, in that thick haze of algebra, we never computed that, # and it looks a bit messy. So we'll do it the parametric way instead. # set parametric set key left set trange [0:11.32] ##set xrange [0:7.04] set xrange [0:11.32] set yrange [*:*] set grid set title "View Seen via Telescope, From Earth" set xlabel "Earth Time (on Clock Next to Telescope)" # 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 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<(tf/2) ? v0(t) : v1(t) # 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) # 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) : dx1(t) # Gamma Gamma0(t) = 1/sqrt(1 - v(t)**2) Gamma(t) = (t <= tf) ? Gamma0(t) : Gamma0(t - tf) # Photon arrival time as a function of ship's Earth time coordinate # Outbound t_a0(t) = exp(tau(t)) - 1 t_a1(t) = tf + xf - (exp(tauf - tau(t)) - 1) # # Returning t_a2(t) = xf - (sqrt(1 + t**2) - t - 1) t_a3(t) = t + sqrt(tc(t)**2 + 1) - 1 # # Combined t_a(t) = (t <= tt) ? t_a0(t) : \ (t <= tf) ? t_a1(t) : \ (t <= tf+tt) ? tf + t_a2(t - tf) : \ tf + t_a3(t - tf) # Photon emission time in ship's frame, as a function of Earth time # Outbound it's just tau(t). # Returning, it's just tau(t) + tauf tau_e(t) = (t <= tf) ? tau(t) : tauf + tau(t - tf) # Rate of clock seen in telescope versus clock rate on Earth # Outbound dtau_dt0(t) = 1/(Gamma0(t) * (1 + v(t))) # Returning dtau_dt1(t) = 1/(Gamma0(t) * (1 - v(t))) # Combined dtau_dt(t) = (t <= tf) ? dtau_dt0(t) : dtau_dt1(t - tf) set xtics 1 set ytics 1 plot \ [t = ] \ t_a(t), tau(t) with lines lw 2, \ t_a(t), dtau_dt(t) with lines lw 2, \ t_a(t), t, \ t_a(t), t_a(t), \ t_a(t), Gamma(t)