Spacecraft drives

This is a repost of a message I saved from the sci.space newsgroup that seems
relevent to the discussion of spacecraft drives.


- ------- Forwarded Message

From: [email removed] (Henry Spencer)
Newsgroups: sci.space
Subject: constant-boost fusion spaceships, power requirements of
Message-ID: [email removed]
Date: 10 Apr 89 02:34:50 GMT
References: [email removed] [email removed] [email removed] [email removed] [email removed] [email removed] [email removed] [email removed] [email removed]
Organization: U of Toronto Zoology
Lines: 100

In article [email removed] [email removed] (Mark A. Sobolewski) writes:
>><....the power requirements are not merely high, but staggering, if
>
>    For the record... I get 1 gigawatt for a 20 kilo-ton (metric) mass
>accelerating at 1g.  (P=E/t, delta(v)=g*delta(t) (1 second),E=0.5mv^2,
>20,000,000kg)  (minimum power consumption of course).

Ah, but that's for acceleration from 0 to (roughly) 10 m/s.  Do the same
computation for 1000 to 1010 m/s and you get about 10 terawatts.  Yet
obviously the same rocket hardware would do either job using the same
amount of fuel and energy.  Clearly you're not measuring what you think
you are.  What you are measuring is the amount of energy needed to do
that bit of accelerating by pushing on a fixed object.  That is, with
a catapult rather than a rocket.  The math is right, but you have to
apply it to what you're actually pushing on, which for a rocket is its
exhaust jet.

>    While not a very large ship....  It still has enough room...

Actually it's pretty sizable, I'd say.  That's the size of an Invincible-
class aircraft carrier.  For solar-system work we can build them a lot
smaller.  However, let's assume that mass for convenience.  Now, the
power requirement to expel a jet with a mass flow of (dm/dt) and an
exhaust velocity of v_exh is 0.5 * (dm/dt) * v_exh^2.  So we need to
figure out the mass flow and exhaust velocity.

Mass flow depends on how long a mission we want to fly and how much
fuel we want to carry.  Let's say we're going to the asteroid belt
and back.  In very round numbers, that's a round trip of 1e9 km.
For continuous acceleration (i.e. one half of each leg of the trip),
time = sqrt(2*distance/acceleration) = sqrt(2*250e9/10) = 224000.
That's a quarter of the trip, so the whole thing is 900000 seconds,
about 10 days.  As for fuel, if we don't want the fuel tank to
dominate the whole ship, a fuel fraction of maybe 10% is tops.
So that's 2e6 kg of fuel, giving a fuel flow of about 2 kg/s.  Clearly
fuel pumps, at least, will not be a problem.

Now, what about exhaust velocity?  There are roundabout ways to
compute that, but thrust = (dm/dt) * v_exh due to conservation of
momentum, so we can do it fairly directly.  Thrust to push 20e6 kg
along at 1 gee will be about 200e6 newtons.  (We assume a constant
thrust; in practice one would throttle back a bit as the ship burned
off fuel, to keep acceleration constant, but that complicates the
arithmetic no end and does not affect the result much.)  So v_exh =
200e6/2 or about 100e6 m/s.  The numerate reader will have gone
"gulp!" on seeing that number, since it is about 1/3 of the speed
of light.  Clearly spaceship Invincible's engines are going to be
a pretty remarkable piece of engineering.  But we press on...

We now have the numbers we need.  Assuming 100% efficiency and no
losses whatever, spaceship Invincible's fusion generator output needs
to be 0.5 * 2 * 100e6^2 = 10 petawatts.  The numerate reader is
having a terrible coughing fit at this point.  For the innumerate:
a petawatt is a million gigawatts, and the entire power output of Earth,
right down to cooking fires, is a small fraction of *one* petawatt.
Clearly, Invincible's generators are, um, impressive pieces of
engineering in their own right.

Clearly 100% efficiency is optimistic.  Let's say it's only 99.99%.
Then Invincible's cooling system has to dissipate a mere terawatt
of heat.  (The numerate reader's coughing fit returns.  A terawatt
roughly equals the electrical power generation of North America.)
The easiest way of cooling something is with flowing fluid; radiating
heat into space is much more hassle.  Assume we use water; it's cheap
and available in quantity, and it's a good coolant.  Melting from
ice and boiling to steam both introduce hassles, so let's say our
input is water just above freezing and output is water just below
boiling.  That's a temperature rise of about 100 C.  The heat capacity
of water is roughly 4200 joules/kg/C, so each kilo absorbs 420e3 J.
So to carry away 1e12 W, we need about 2400 tons per second.  Oops.
Spaceship Invincible has a *bad* cooling problem.

What's more, the numerate reader, having recovered from his coughing,
observes that since E=mc^2, boosting the fuel to 1/3 of the speed of
light requires converting about 5% of its mass to energy (we speak
sloppily here for clarity; physicist readers will please keep quiet).
No wonder the power requirement was massive.  The n.r. goes on to
observe that fusion simply doesn't convert that big a fraction.
Invincible's power requirement just cannot be met by fusion, period.
Antimatter might do; Invincible would need to burn about 50 grams
per second (remember that each gram annihilates its own mass in
normal matter).  Optimistic fans of antimatter propulsion think that
straightforward application of current technology might perhaps get
antimatter prices down to $1M/mg.  Invincible's operating costs will
then be fifty billion dollars per second.  I don't think even DoD
can afford to fly this spaceship.

I could go on, computing things like radiation output from the power
system, size of heat radiators, etc., but the numbers all come out to
be preposterous.  Not just difficult, but beyond current abilities by
so many orders of magnitude that straightforward extrapolation of
today's technology won't do -- breakthroughs will probably be needed.

The moral of the story is, don't get intoxicated by the possibility of
easy fusion.  It may give us the solar system, but it will not give us
1G constant-boost spaceships.  They are fundamentally hard to build;
still better technology will be needed.
- - --
Welcome to Mars!  Your         |     Henry Spencer at U of Toronto Zoology
passport and visa, comrade?    | [email removed] [email removed]

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