# starship-design: Back from the wed

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Gee - can't a guy go get married these days without coming home to
hundreds of emails??

I have to confess I didn't read them all in depth, but I liked the
recent idea of the Fuel/Sail, building the sail out of fusion materials.
It got me thinking about using the sail for reaction mass for slowing
down, and our months-old discussion about momentum/energy (P/E) ratios
came to mind.  (For all I know this has been discussed at length, so
forgive me if I'm going over old ground)  Photons have a very
inefficient P/E, compared to massive particles, and I started thinking
about catching beamed photons, and then transferring the energy into a
massive particle, and shooting it forward.  Assuming no energy is lost,
the massive particle would have a higher P/E, and therefore you would
lose more momentum to the particle (for a given energy) than you would
gain from catching the photon and using it in the first place.

The key equation is: E^2 = (K + mc^2)^2 = P^2 c^2 + M^2 c^4

This holds for both photons (M=0) and massive particles, where E is the
total energy (including rest mass) and K is the kinetic energy (really

This simplifies to P^2 = K^2 / c^2 + 2 K M

At this point I'm going to define E as the kinetic energy, and we can
get replace those K's:

P = [(E^2 / c^2) + 2 E M]^0.5

So if you "catch" some beamed energy (say 'E' joules) you gain a
momentum of E/c.  But if you can convert that energy to kinetic energy,
and accelerate part of the sail forward, you can get a net backwards
momentum out of the whole deal.  If you want to have a net momentum loss
of 2E/c (the same as your momentum gain for a reflecting sail), it turns
out that every particle you accelerate forward must have a kinetic
energy equal to exactly 1/4 of its rest mass, so M = 4E/c^2, and P = -
3E/c (backwards).  Adding this to the E/c you gained from catching the
beamed energy in the first place gives you a momentum loss of exactly -
2E/c; the same as you momentum gain during the accleration portion of
the journey.

Of course, things get more complicated once the ship starts losing mass.
Fortunately, it works somewhat in our favor; less momentum is needed to
slow the ship down when it's lighter.  Turns out that accelerating a
"mere" half of the total ship mass with the M/E=4/c^2 mass/energy ratio
will slow the ship down from a third of lightspeed.  Faster trips will
require a larger Sail Mass / Ship Mass ratio.  (I didn't do this
relativistically, but I know some of you have those equations
somewhere...)

The biggest problem with losing mass (besides whittling away the sail
without destroying it) will be that the sail will get too small too
fast:  we won't be able to "catch" all the energy once we start using up
the sail as reaction mass.  The amount of energy we need will drop in
proportion to the mass of the ship as a whole, but the amount of energy
we can catch (due to sail area) will drop faster, as the mass of the
sail alone.

However, given that we are talking about mass ratios of 400, these two
masses are pretty much identical; with a sail that big the two values
drop should drop pretty much at the same rate.

So now the deceleration problem simply involves converting light energy
to particle energy.  Clearly the parabola-shaped sail will reflect the
beamed power to an onboard engine.  But not only do you have to have a
decent conversion efficiency, but we're talking about huge amounts of
energy we have to manipulate.  Even if the "energy storage time" between
receiving the power and shooting it off is very small, the net power the
engine would have to handle would be terrible.

Clearly a "passive-engine" scenario like the plasma mirror would be
simplest, but we have to give a lot of photons to each proton to get the
energy/mass ratio to come out right.  Each proton will need 230GeV
imparted to it, and it seems unlikely we can give a proton that much
energy from a microwave beam without an "active" engine, where the
energy distribution is manipulated by an accelerating structure.  For
mm-sized microwaves, this is 10^15 photons per proton!  Perhaps starting
off with a lower proton energy would be okay; if you transfer all the
energy, the slower the protons the more you slow down, paradoxically
enough.  The problem with slow protons, though, is that you need a LOT
of them to do the job of a few fast ones.  If you shoot off too much
mass too slow you'll run out of sail material before you can slow down
enough...

Run with this, someone...

Ken Wharton

"No brakes?  Well, no point in steering now"
(Bob MacKenzie, Strange Brew)

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