Reply to your letter of 9/01

[DotarSojat@aol.com]

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