lew@ihuxr.UUCP (Lew Mammel, Jr.) (10/15/83)
Kee Hinckley suggested the problem of determining the acceleration of a photon sail ship immersed in a constant intensity light beam. This can be treated quantitatively to obtain a simple result. Consider the frame of reference in which the sail is initially at rest. We need to calculate the rate of momentum transfer of the photons to the sail. This is given by: I * (c-v) * delta(v) I*(c-v) is the rate that photons are reflected from the sail, where I is proportional to the intenstiy of the beam. More precisely, I is the product of the number density of photons in the beam with the area of the sail. The factor delta(v) gives the change in momentum of each reflected photon. delta is initially 2*p, where p (=h/lambda) is the momentum of the photons in the beam; 2*p = p - (-p). As the sail picks up speed, the reflected photons are red shifted. We can calculate the momentum of the reflected photons in the following way. First, calculate the momentum of the incident photons in the sail frame. Second reverse the sign of the momentum (reflection from sail which is stationary in this frame.) Third, calculate the momentum of the reflected photons in the rest frame. If p is the initial momentum, these steps yield: 1) gamma*(1-beta)*p /* red shift */ 2) -gamma*(1-beta)*p /* reflect */ 3) -gamma^2*(1-beta)^2*p /* red shift again */ ... this gives delta(v) = 2*p/(1+beta) The momentum of the sail is P=gamma*m*v and we can proceed with: dP/dt = gamma^2 * m * dv/dt = 2*p*I*(c-v)/(1+v/c) dv/dt = (2*p*I*c/m) * (1-v/c)^2 v = c*t/(t+T) ; T = m/(2*p*I) Watch out for the constant of integration on that last step! This is a nice simple result, but note that no tricks are necessary to keep v < c. Since the momentum goes to infinity as v goes to c, any finite rate of increase of momentum will cause no problem. Lew Mammel, Jr. ihuxr!lew