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(Re:)^4 starship-design: The Size of the Problem

While I haven't had time to do a more complete write-up, I thought I
would also mention an interesting corollary to Rex's analysis of the
energy requirements of beaming power to accelerate a relativistic
spacecraft.  Not only is a large amount of power required, but the
beaming equipment must be capable of (typically) output at a rate that
can be over two orders of magnitude larger than is needed to accelerate
the spacecraft at the start of the trip.

I'm going to present some of the math without proof or demonstration at
this time, but I'm sure it will be interesting fodder for discussion
(either because Timothy or Rex will find any mistakes I might have made
or because it shows another facet of difficulty to the problem of
beaming power).

I've recently been working on the physics of light signals between a
"stationary" object and an object undergoing relativistic acceleration
relative to it.  Consider an object undergoing uniform accleration
relative to itself; its frame position at its proper time t1 is:

[ t x ] = [ 1/a * sinh(a * t1)
            1/a * cosh(a * t1) ]

At time t1 = 0, its position is [ 0 1/a ] (note again that for
simplicity I am using geometrized units where c = 1 and acceleration has
units 1/s (acceleration is fraction of c per unit time)).  Consider an
object beaming power to the object to accelerate it that also starts at
that position, so that at time t=0 it coincides with the accelerated
object at its proper time t1=0.  If energy (light) from the beamer is
emitted at time t, then the time t1 at which the accelerated object
receives the energy is:

t1 = -(ln(1 - a*t))/a

Note that this is an asymptotic relationship -- as the frame time t of
the beamer approaches 1/a, the object proper time t1 approaches
infinity.  This consequently means that the beamer must send energy for
any possible trip within a time 1/a, no matter how far the acclerated
object goes, and that the rate at which power is sent increases
asymptotically to infinity as t approaches 1/a.  The relative rate of
time passage between the beamer and the accelerated object at frame time
t has the relationship

dt1 / dt = 1/(1 - a*t)

In the case where a = 9.8 m/s^2 (or in geometrized units, 3.267e-8 c/s),
the asymptote is reached within about year of beamer time (3.06e8 s).
The good news is that to boost an object at 1 g up to its turnaround
point and then provide deceleration power to its destination, you beam
power for no more than two years, no matter how far away you send the
object.  The bad news is that at the turnaround point you are beaming
some large multiple of the power needed to keep the object accelerating
at 1 g at the beginning and end of the trip, because of the relative
rate of time lapse between the beamer and the accelerated object.

In fact, given the relationship between t1 and t, we can characterize
just what this multiple is based on halfway trip time of the object.
Solving t1 = -(ln(1 - a * t))/a for t, we get:

t = (1 - e^(-a*t1))/a

Substituting into 1/(1 - a*t), we get:

dt1 / dt = e^(a*t1)

In other words, the maximum power output at turnaround is exponentially
related to trip time for the object.

Perhaps the worse news is that because the relative rate of time lapse
is asymptotic, even the schemes proposed for exponentially
self-reproducing power generation equipment ultimately run up against
the asymptotic limit; the asymptotic relationship always reaches a point
where it is growing faster than the exponential function.

Hopefully this makes sense to at least some of you.  I hope to have time
to cover more of the background soon, as I'm sure this is confusing
without it.  Timothy knows I've been working on analysis of accelerated
objects in