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*To*: stevev@efn.org*Subject*: Reply to your letter of 9/01*From*: DotarSojat@aol.com*Date*: Thu, 5 Sep 1996 15:21:19 -0400*cc*: T.L.G.vanderLinden@student.utwente.nl

Hi Steve Thanks for the kind words about my derivation of 9/01. Use any or all of it as you see fit. It was your idea in the begin- ning. >I wonder what the worldline would look like for an object that >is accelerated by a constant-output emitter? In other words, >the emitter would send constant output power, meaning the re- >ceiver would experience gradually decreasing received power >and acceleration as its proper time increases. For a power, Pe, sent out by an emitter, the power received by a sail (ignoring inverse-square effects) receding at a veloc- ity, beta lt-yr/yr, is Pr = Pe * sqrt[(1 - beta)/(1 + beta)] ... Doppler shift = Pe * gamma * (1 - beta) ... gamma = 1/sqrt(1 - beta^2) = Pe * [cosh(theta) - sinh(theta)] ... gamma = cosh(theta); beta = tanh(theta) (definition of velocity parameter, theta) = Pe * exp(-theta) ... using exp forms of hyp functions. (I believe this expression, with theta = a * t' for constant a, is the source of the logarithmic dependence you have been citing.) The velocity-parameter equation of motion for a thrust, T = Pr/c, applied to a mass, M, is T = M * d(theta)/dt' = Pr/c = Pe * exp(-theta) ... c = 1 lt-yr/yr, which gives the "differential equation" exp(theta) * d(theta) = (Pe/M) * dt' . Integrating from theta = 0 at t' = 0, with Pe constant, gives [exp(theta) - 1] = Pe * t'/M , so the desired description of motion should be theta = ln[(Pe * t'/M) + 1] . For one space dimension, dx/dt' = u = sinh(theta) dt/dt' = gamma = cosh(theta) . Both of these should be integrable (integrands of the form exp[ln()], using the exponential forms of the hyperbolic func- tions), to give the worldline. I hope this is the answer you were looking for. Regards, Rex

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