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From: lew@ihuxr.UUCP
Newsgroups: net.physics
Subject: revised sail solution
Message-ID: <715@ihuxr.UUCP>
Date: Mon, 17-Oct-83 17:46:08 EDT
Article-I.D.: ihuxr.715
Posted: Mon Oct 17 17:46:08 1983
Date-Received: Tue, 18-Oct-83 21:35:01 EDT
Organization: AT&T Bell Labs, Naperville, Il
Lines: 43

I made a mistake in my "photon sail problem" solution. My analysis of
the momentum transfer due to reflection was correct (or at least I still
believe so) but I made a basic error in my integration. I stated that
the differential of the relativistic momentum was proportional to gamma
squared, actually it is proportional to gamma cubed. That is:

	d(gamma*v) = gamma^3 * dv

The correct integration is:

	t/T = int 0 to beta of (1-x)^-2 * (1-x^2)^-.5 * dx

That last factor is the extra factor of gamma. This ruins the cute solution,
but I could still solve the integral using my trusty CRC tables. The answer
is a little messy and I can't invert it to get beta(t/T), but this is just
math. There is an interesting physical interpretation of T, which scales time
regardless of the form of the solution. Remember that:

	T = m/(2*p*I)

where m is the mass of the sail ship, p is the momentum per photon,
and I is the number density of the photon beam times the area of the sail.
Replacing I by N*A and multiplying top and bottom by c^2, we have:

	T = m*c^2/A / (2*N*c * p*c)

The numerator is the rest energy per unit area of the sail, and the
denominator is twice the power per unit area of the photon beam.
(p*c is the energy of a photon with momentum p.) T will give the
time required to achieve relativistic speed from rest. For a beam
of 1 megawatt/meter2 and a sail of 1 gram/meter2, I get T = 1e8 sec,
which is about 3 years. Here is a table of the times required to reach
a few values of beta (according to my revised solution):

	beta	t/T

	.5	1.065
	.9	15.316
	.95	43.048
	.99	474.26
	.999	14917.6

	Lew Mammel, Jr. ihuxr!lew