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*To*: starship-design@lists.uoregon.edu*Subject*: starship-design: Motion of sail driven by constant-power beam*From*: DotarSojat@aol.com*Date*: Wed, 11 Sep 1996 02:56:25 -0400*Reply-To*: DotarSojat@aol.com*Sender*: owner-starship-design

Hi all Steve has written, on 9/1 to me (and Timothy), >I wonder what the worldline would look like for an object that >is accelerated by a constant-output emitter? In other words, >the emitter would send constant output power, meaning the re- >ceiver would experience gradually decreasing received power >and acceleration as its proper time increases. Kevin has written, on 9/4 to the Group, >How about a mission which has a constant beam power, the >acceleration would drop off toward the turnaround point. In >this case, the crew would start off with earth-like gravity, >and towards the middle of the trip, the gravity would be more >lunar-like. ... The advantage would be simplified beaming >requirements, and the disadvantage would be a slightly longer >flight time. > >Questions: >What would the top speed relative to Earth be? >What is the total trip time. (crew time?) >How much of this time is spent at less than 1/2 G?] DERIVATION For a power, Pe, sent out by an emitter, the power received by a sail (ignoring inverse-square effects) that is receding at an apparent velocity, beta lt-yr/yr, is Pr = Pe * sqrt[(1 - beta)/(1 + beta)] ....Doppler shift = Pe * gamma * (1 - beta) ....gamma = 1/sqrt(1 - beta^2) = Pe * [cosh(theta) - sinh(theta)] ....gamma = cosh(theta); beta = tanh(theta) (definition of velocity parameter, theta) = Pe * exp(-theta) ....using exp forms of hyp functions. (I believe this relation, with theta = a * t' for constant a, is the source of the logarithmic time dependence introduced by Steve in his email of 8/20 to the Group.) The velocity-parameter equation of motion for a thrust, T = Pr/c, applied to a mass, M, is T = M * d(theta)/dt' = Pr/c = Pe * exp(-theta) ....c = 1 lt-yr/yr, which gives the "differential equation" exp(theta) * d(theta) = (Pe/M) * dt' . Integrating from theta = 0 at t' = 0, with Pe constant, gives [exp(theta) - 1] = Pe * t'/M , so the description of the motion of the sail, in terms of the dependence of the velocity parameter, theta, on ship time, t', is theta = ln[(Pe * t'/M) + 1] . At the beginning of the flight, when thrust = To, acceleration = ao and the received power, Pr, equals the emitted power, Pe, To = M * ao = Pe/c , which leads to ao = Pe/(c * M) --> Pe/M (for c = 1 lt-yr/yr), which makes the description of the motion of the sail theta = ln(ao * t' + 1) . For one space dimension, dx/dt' = u = sinh(theta) = 0.5 * [exp(theta) - exp(-theta)] = 0.5 * [(ao * t' + 1) - 1/(ao * t' + 1)] . Integrating this from x = 0 at t' = 0 gives x = 0.5 * (0.5 * ao * t'^2 + t' - theta/ao) . The apparent (Earth) time, t, for the ship time, t', is obtained from dt/dt' = gamma = cosh(theta) = 0.5 * [exp(theta) + exp(-theta)] = 0.5 * [(ao * t' + 1) + 1/(ao * t' + 1)] . Integrating this from t = 0 at t' = 0 gives t = 0.5 * (0.5 * ao * t'^2 + t' + theta/ao) . The Earth time of emission, te, of the energy that arrives at the sail at t' is simply (for c = 1) te = t - x . The proper velocity, u, is given by u = sinh(theta) . The instantaneous proper acceleration, a, is given by the velocity-parameter equation of motion-- a = c * d(theta)/dt' = ao/(ao * t' + 1) (for c = 1). RESULTS Putting these relations together in the Fortran program COPOBM.FOR, which is appended, gives the following values of theta, distance, proper velocity, instantaneous acceleration, Earth time for t' and Earth time of emission for reception at t', as a function of ship time, t', for an initial acceleration of 1 g, Tship Theta Dist Prop Vel Accel TEarth Temit (yr) (rad) (lt-yr) (lt-yr/yr) (g) (yr) (yr) 0.0 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.5 0.4162 0.1130 0.4283 0.6595 0.5161 0.4031 1.0 0.7092 0.4146 0.7702 0.4920 1.1016 0.6870 1.5 0.9355 0.8776 1.0781 0.3924 1.7838 0.9062 2.0 1.1200 1.4900 1.3693 0.3263 2.5748 1.0848 2.5 1.2756 2.2453 1.6509 0.2793 3.4809 1.2356 3.0 1.4103 3.1399 1.9266 0.2441 4.5059 1.3660 3.5 1.5290 4.1712 2.1983 0.2168 5.6522 1.4810 4.0 1.6350 5.3377 2.4673 0.1949 6.9215 1.5837 4.5 1.7309 6.6382 2.7343 0.1771 8.3148 1.6766 5.0 1.8184 8.0718 2.9999 0.1623 9.8332 1.7613 For an example trip to a star 4.4906 (= 2 * 2.2453) lt-yr from Earth, the ship would accelerate for 2.5 ship years, reach a maximum proper velocity* of 1.6509 lt-yr/yr at an apparent (Earth) time of 3.4809 yr, and at turnover receive power emit- ted from Earth at 1.2356 yr after the departure date, giving an instantaneous acceleration of 0.2793 g. __________ *For a proper velocity, u, the apparent velocity, v, is given by v = u/sqrt(1 + u^2) . __________ These results seem to confirm Kevin's intuitive estimates regard- ing acceleration levels. They also substantiate Steve's conclu- sion that the time of emission is limited (even for a constant- output emitter); from the table above, the duration of emission of the radiation accelerating a sail half way to a destination more than 16 lt-yr away (the last entry) is only about a year and three quarters. The deceleration phase (here assumed without justification to be a mirror image of the acceleration phase) needs to be addressed in a separate discussion. Timothy has already put a lot of thought into it. (Note: This exercise may turn out to be purely academic because the inverse-square effects, without unforeseeable advances in focusing abilities, would be much larger.) Rex ADDENDUM PROGRAM COPOBM !9/7/96 101 FORMAT(2X, 21H Initial Accel (g) = ) 102 FORMAT(1X, 6H Tship, 3X, 6H Theta, 5X, 5H Dist, 2X, & 9H Prop Vel, 3X, 6H Accel, 2X, 7H TEarth, 2X, 6H Temit) 103 FORMAT(3X, F4.1, 3X, F6.4, 3X, F7.4, 4X, F7.4, 3X, F6.4, & 3X, F6.4, 2X, F6.4) 1 CONTINUE WRITE(*,101) READ(*,*) AGO IF(AGO .EQ. 0.) GO TO 99 AO = 1.0324 * AGO WRITE(*,102) DO 10 IT = 1, 11 FT = IT - 1 TIM = 0.5 * FT ARG = AO * TIM + 1. THET = LOG(ARG) DIST = 0.5 * (0.5 * AO * TIM*TIM + TIM - THET/AO) PVEL = 0.5 * (ARG - 1./ARG) ACC = AGO/ARG TAPP = 0.5 * (0.5 * AO * TIM*TIM + TIM + THET/AO) TEM = TAPP - DIST WRITE(*,103) TIM, THET, DIST, PVEL, ACC, TAPP, TEM 10 CONTINUE GO TO 1 99 STOP END

**Follow-Ups**:**starship-design: Motion of sail driven by constant-power beam -- Excellent work Rex!***From:*"Kevin \"Tex\" Houston" <hous0042@maroon.tc.umn.edu>

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