Robert J. Bradbury wrote:
> > You are suggesting that an optimum is m1 = 0 and m2 = M, putting all
> > the metal at one star so as to get every last photon and reradiate at near
> > 3K, while completely leaving the other star alone. This requires that
> > P''(m) > 0 on average, with *increasing*, not diminishing, returns.
>
> No. I'm suggesting that there are always decreasing *local* optima.
> If there were *no* costs to transfering material (or information) from
> 1 to 2, then doing that would make sense. But if the costs of the
> material/information transfer *exceed* the cost of local manufacture
> then it makes sense to reject remote information/material (if you
> have to pay for it).
But mass transport costs make the puzzle worse! Let n1 and n2 be the initial metal amounts at the two stars. The problem then is: max
P(m1) + P(m2) such that m1 + m2 = n1 + n2
m1,m2 - T*abs(m1-n1) and m1 >=0, m2 >=0
(Note that abs(m1-n1) = abs(m2-n2).) Assume without loss of generality that n1 > n2. If P(m) has diminishing returns (P''(m) < 0), then when transport costs are very high, m1 = n1 and m2 = n2, so nothing moves. And when transport costs are zero, m1 = m2 = (n1 + n2)/2. And for intermediate transport costs n2 is in the range [n2,(n1+n2)/2]. So we should see at least P(n2) power intercepted from star 2.
So in this model the only way to explain all those stars we see is to say they all have very little metal near them, so little that even when it is all used to intercept starlight, less than 1% of the light is intercepted. This isn't true of our system, and I doubt it is true of a great many.