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Reply to your letter of 9/01
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-
>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
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.