lew@ihuxr.UUCP (10/18/83)
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