Re: Gravity Inside A Cylinder

From: Bruce Scott TOK (Use-Author-Supplied-Address-Header_at_[127.1)
Date: 08/11/04 

Date: Wed, 11 Aug 2004 20:53:30 +0200 (MEST)

I wrote:

[Niven, Ringworld]
|> He came back in the sequel precisely because the force is zero if the
|> sun is in the plane of the ring. (In this case you use the limit
|> opposite to infinite length... assume L << R)

As usual it helps to actually do the calculation. With the help of
Binney and Tremaine (Ch 2 Sec 6) I find assuming a zero thickness ring
of mass M and radius R in the plane z = 0 centred at r = 0,

   Phi(r,z) = -(G M) \int_0^\infty dk e^(-k |z|) J_0(kr) J_0(kR)

where J_0 is the Bessel function of order zero.

Using a Bessel function identity this becomes an elliptic integral,

   Phi(r,z=0) = -(G M) \int dk J_0(kr) J_0(kR)

        = -(2/pi) (G M/R) \int_0^(pi/2) ds/sqrt(1 - a^2 sin^2 s)

for r < R with a = r/R, and

        = -(2/pi) (G M/r) \int_0^(pi/2) ds/sqrt(1 - a^2 sin^2 s)

for r > R with a = R/r. This is negative everywhere, bounded at r = 0,
zero at r = \infty, and blowing up to minus \infty at r = R. Hence, as
Steve Gray notes:

|> True, but that's not the only reason that the Ringworld needed
|> corrections. The net force in the system is zero when the sun is located
|> at the exact center, but the system is dynamically unstable. If things
|> drift a bit there's no force to automatically bring it back to
|> equilibrium. It's that realization that caused Galileo (Newton?) to
|> conclude that the rings of Saturn are not solid.

and John Morriss clarified:

|> It's worse than that!
|>
|> Once the ring drifts off centre, there is a net force driving the
|> off-centre motion, The force increases as the eccentricity
|> increases...

..there is a net gravitational force pushing the sun towards the ring
for all positions within the ring plane except the center. The
structure of the potential at r = 0 in the plane z = 0 is that of a
hilltop. The instability is indeed dynamical. 

-- 
cu,
Bruce
drift wave turbulence:  http://www.rzg.mpg.de/~bds/

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