From: pksuk@mothra.syr.EDU (Peter K. Suk)
Subject: Re: Weapons systems?
Followup-To: rec.arts.sf.science
Summary: Point well taken.
Keywords: high-G space weapons
Organization: Syracuse University, Syracuse, NY
Date: Sat, 21 Nov 92 11:47:16 EST

In article <1992Nov18.141715@gracie.IntelliCorp.COM> 
treitel@gracie.IntelliCorp.COM (Richard Treitel) writes:

>In article <1992Nov17.005834.11243@newstand.syr.edu>, 
>pksuk@mothra.syr.EDU (Peter K. Suk) writes:
>
>|> A ship that is capable of a certain acceleration 
>|> has a certain safe distance from which it cannot be hit _at all_, due to
>|> the fact that information doesn't propagate faster than light.  Given
>|> enough distance and accelerating ability, the ship can make sure that
>|> you will never know where it is precisely enough to hit.  
>
>Ever hear of saturation bombing?  I'd say your physics is getting ahead of
>your military savvy.

Point well taken.  However, such a limit does exist for single
unguided projectiles, hitting with 100% certainty.  This can be used
as a measure of defensive capability, which is what I was thinking of
in the first place.  In addition, we can get equations giving us an
idea of how difficult 'saturation bombing' gets for targets of a given
size, capable of a certain acceleration.

We'll make some simplifying assumptions:  All spaceships are spheres.
Projectiles are unguided and undetectable.  Firing is always detected and
the detection signal travels at c.  These assumptions are NOT meant to
describe what ships would be like.  We use them here to get at certain 
general limits and relationships.

If we take the basic X = V0(t) + (1/2)a(t^2) and let the relative velocity
of the ships start out to be zero, then we get:

	Rs = (1/2)a(t^2)  where Rs = radius of target ship
			    "   a  = max acceleration of target ship
			    "   t  = amount of time the ship has to dodge

Using a little basic algebra, we rearrange to get an equation for t:
(The time it takes for the ship to move its radius from a 'standstill.')

	t = sqrt( 2(Rs/a) )

And the time that the ship has to dodge:

	T = (d/Vw) + (d/c) where d  = distance between the ships
			     "   Vw = velocity of weapon (projectile)
			     "   c  = the speed limit we all know & love

So if we set the two equations equal and solve for d:

	d = sqrt( 2(Rs/a) ) / ( 1/Vw - 1/c )

Of course, this is only the limit for guaranteed hits.  For single projectiles
with the ship past the safe distance, we can get a probability for hits with
the ship at some distance d:

	<Prob hit> = Rs^2 / ( Rs + (1/2)a (d/Vw - d/c)^2 )^2

We can guarantee a hit at a certain distance by simply saturating the
sphere around the ship's last known location.  Of course, since the target
can accelerate away from the gun as well, we need to make a pattern which
has smaller spaces towards the middle:

	   	       	     /
	    /	  	    /
	   /	   	   / ###
	  ###	   	  /  ###
	 /###	   	 /   ###
	*-###---   	*---------

But we can take the reciprocal of the probability as an approximation.
This gives us an idea of what a battle between a single large ship and
a bunch of little fighters would be like.

We can pin this down even more if we make the assumption of a maximum
force-gradient Grf for linear accelerators.  If we assume that linear
accelerators are also the drives, then we can establish ranges of
values for Grf where large ships will lose to X small ships, or vice
versa.  (We can get values for a and Vw for given masses and sizes of
the ships.  So acceleration and projectile velocity will be dependent
on the ship's spatial dimensions -- calculations forthcoming.)

(Again, the given assumptions are NOT meant to be a description of what
actual ships would be like.)
	
	--Peter Kwangjun Suk

(P.S.  Dairy Physics: "If we assume spherical cows..."   :)