# Re: starship-design: Deceleration of sail pushed by constant-power beam

```At 2:46 AM 9/20/96, DotarSojat@aol.com wrote:
>Hi all
>
>In my note of 9/11, "Motion of sail driven by constant-power
>beam," I wrote--
>
>>The deceleration phase (here assumed without justification to
>>be a mirror image of the acceleration phase) needs to be
>>lot of thought into it.
>
>The deceleration phase is not a mirror image of the acceleration
>phase.  In the acceleration phase, the sail/ship is driven by
>the beam's radiation pressure alone; in the deceleration phase,
>the radiation pressure of the beam continues to provide a for-
>ward thrust, but the power of the beam is collected and convert-
>ed into a retro-thrust "jet" of, say, protons ejected forward to
>slow down the sail/ship.
>
>Timothy, in his 3/29 note, solved the problem of bringing the
>sail/ship to a halt with the "jet's" retro-thrust overcoming the
>beam's radiation pressure to give a constant deceleration level,
>allowing the beam power to take on any variable value required
>for the constant deceleration.  He found the optimum exhaust
>velocity that gives a minimum energy expenditure, but did not
>consider the requirement to bring the sail/ship to a halt in a
>given distance.
>
>This analysis considers a beam from a constant-output emitter
>with the same power that accelerated the sail/ship to a peak
>velocity half way to the destination star (per my 9/11 note).
>The analysis selects the exhaust velocity that brings the
>sail/ship to a halt in the remaining half of the distance.
>
>ANALYSIS
>
>Following the notation of my 9/11 note, the power radiated from
>the emitter is Pe, and the power received by the sail, ignoring
>inverse-square effects, is Pr.  The thrust exerted by the radia-
>tion pressure is Pr/c.  At the start of the acceleration phase
>when Pr = Pe, the mass of the sail/ship is Mo, the initial
>acceleration is ao, and the required emitted power is
>     Pe = T * c
>        = Mo * ao * c   .
>
>When the sail/ship has been accelerated to an apparent velocity,
>beta lt-yr/yr, having the velocity parameter, theta, from the
>relation: beta = tanh(theta), the received power is reduced by
>the Doppler shift according to
>     Pr = Pe * exp(-theta)   ,
>
>     Tb = Pr/c
>        = (Pe/c) * exp(-theta)
>        = Mo * ao * exp(-theta)   .
>
>The beam power is collected and converted with an efficiency,
>eta, to the power, Pex, of a retro-thrust exhaust "jet," , or
>     Pex = eta * Pr   .
>
>The exhaust-jet power is the rate of ejection of kinetic energy,
>or
>     Pex = (dM/dt') * (gexh - 1) * c^2   ,
>
>where (dM/dt') is the rate of ejection of propellant mass
>(made up of protons, say), and gexh is the jet's energy factor,
>(gamma) = 1/sqrt(1 - Vexh^2), derived from the jet's exhaust
>velocity, Vexh (lt-yr/yr).
>
>The thrust, Tex, of the exhaust jet is given by
>     Tex = (dM/dt') * gexh * Vexh * c   .
>
>Substituting the dM/dt' derived from the equation for Pex above
>gives
>     Tex = [Pex * gexh * Vexh * c]/[(gexh - 1) * c^2]
>         = eta * (Pr/c) * [factor]   ,
>
>where
>     [factor] = gexh * Vexh/(gexh - 1)   .
>
>Note that the exhaust-jet retro-thrust exceeds the radiation-
>pressure thrust, making deceleration of the sail/ship possible,
>when eta * [factor] is greater than 1.  For Vexh = 0.9, for
>example, the value of [factor] is 1.5954, and deceleration is
>possible only if the efficiency, eta, of conversion of received
>power to exhaust power is greater than 1/1.5954 = 0.6268.
>
>The acceleration, a, of the sail/ship (hopefully, negative) is
>given by
>     a = (Tb - Tex)/M
>
>where M is the mass of the sail/ship at the ship time, t'.
>
>The rate of change of the mass of the sail/ship as propellant is
>ejected is obtained from the Tex equation above--
>     dM/dt' = Tex/(gexh * Vexh * c)   .
>
>The rate of change of the velocity parameter is given by the
>velocity-parameter equation of motion--
>     d(theta)/dt' = a/c
>                  = (Tb - Tex)/(M * c)   .
>
>We thus have two simultaneous differential equations, with
>dM/dt' involving exp(-theta) through the dependence of Tex on
>Pr, and with d(theta)/dt' involving 1/M.  The coupling therefore
>is non-linear, and the method of solution I find in my math book
>is for linear simultaneous differential equations.  Not being a
>"mathochist" (one who enjoys suffering in the solution of higher
>math problems), I chose to integrate these equations numerically;
>see the appended Fortran program SAILTRIP.  The implemented
>difference equations are
>     M(n+1) = M(n) - [Tex(n)/(gexh * Vexh)] * [t'(n+1) - t'(n)]
> theta(n+1) = theta(n) + [(Tb(n) - Tex(n))/M(n)] *
>                                        [t'(n+1) - t'(n)]   .
>
>(The SAILTRIP program also includes the acceleration-phase
>calculation outlined in my 9/11 note, allowing the program to
>cover the whole trip from start to destination.)
>
>Results become consistent to better than three significant
>figures for time steps, [t'(n+1) - t'(n)], smaller than 0.01 yr,
>with insignificant computation time (about one second for the
>deceleration phase) for a time step of 0.001 yr.  The decelera-
>tion phase is repeated with trial values of the exhaust velocity
>until interpolation yields the desired deceleration distance
>within a tolerance of 0.0001 lt-yr.
>
>RESULTS
>
>Calculations with the SAILTRIP program were made for a trip to
>tau Ceti, whose distance was taken to be 11.9 lt-yr.  It was
>found that, with the constant-output emitter, the deceleration
>grows to exceed 1 g near the destination, where the thrust
>increases as the velocity and therefore the Doppler shift
>decrease, and where the mass of the sail/ship decreases as the
>propellant is depleted.  The deceleration is limited in the
>calculation to 1 g by the simple expedient of furling the sail.
>
>The calculated values of theta, distance, proper velocity, accel-
>eration, apparent (Earth) time, time of emission of radiation
>from Earth and mass ratio (ratio of initial mass to the instan-
>taneous mass), as functions of ship time, t', are given in the
>following table for the tau Ceti trip, for a conversion effic-
>iency of received power to exhaust power of 1.0.  Also stated are
>the values of the exhaust velocity, in lt-yr/yr, that gives the
>desired deceleration distance, with the corresponding kinetic
>energy, in MeV, of protons having that velocity, and of the final
>relative area of the furled sail.
>
> Tship   Theta   Dist    Prop Vel  Accel  TEarth   Temit  Mratio
>  (yr)   (rad) (lt-yr)  (lt-yr/yr)  (g)    (yr)     (yr)
>0.0000  0.0000  0.0000    0.0000  1.0000  0.0000  0.0000  1.0000
>0.5000  0.4162  0.1130    0.4283  0.6595  0.5161  0.4031  1.0000
>1.0000  0.7092  0.4146    0.7702  0.4920  1.1016  0.6870  1.0000
>1.5000  0.9355  0.8776    1.0781  0.3924  1.7838  0.9062  1.0000
>2.0000  1.1200  1.4900    1.3693  0.3263  2.5748  1.0848  1.0000
>2.5000  1.2756  2.2453    1.6509  0.2793  3.4809  1.2356  1.0000
>3.0000  1.4103  3.1399    1.9266  0.2441  4.5059  1.3660  1.0000
>3.5000  1.5290  4.1712    2.1983  0.2168  5.6522  1.4810  1.0000
>4.0000  1.6350  5.3377    2.4673  0.1949  6.9215  1.5837  1.0000
>4.2418  1.6825  5.9500    2.5967  0.1859  7.5797  1.6297  1.0000
>  Exhaust Velocity = 0.88301; Proton MeV = 1060.5
>4.5000  1.6483  6.6085    2.5029 -0.1337  8.2871  1.6786  1.0467
>5.0000  1.5731  7.8121    2.3070 -0.1593  9.5904  1.7783  1.1570
>5.5000  1.4818  8.9119    2.0867 -0.1971 10.7986  1.8867  1.3067
>6.0000  1.3656  9.8934    1.8314 -0.2584 11.9003  2.0069  1.5254
>6.5000  1.2058 10.7345    1.5199 -0.3750 12.8791  2.1446  1.8873
>7.0000  0.9482 11.3959    1.0968 -0.6833 13.7090  2.3132  2.6602
>7.5000  0.4669 11.7924    0.4840 -1.0011 14.3512  2.5588  5.0527
>7.9516  0.0000 11.9000    0.0000 -1.0003 14.8195  2.9195  9.4145
>  Final sail furl =   0.160
>
>Furling the sail to limit the deceleration to 1 g begins about
>0.7 yr before arrival at the destination.  Even with furling,
>the deceleration time is shorter than the acceleration time for
>the same distance.  The average deceleration is greater than the
>average acceleration because the decreasing mass in the deceler-
>ation phase overrides the reduction in thrust that comes from the
>competition between radiation-pressure push and exhaust retro-
>thrust.
>
>Even though the trip takes about 8 years of ship time, or about
>15 years of Earth time, the total job of the emitter is over in
>less than 3 Earth years (consistent with Steve's original predic-
>tion, even including the deceleration phase).
>
>I'm not totally comfortable with the calculated mass-ratio
>values; they seem too low.  I haven't found any analytical or
>computational shortcomings, however.  The approach laid out here
>should be regarded as exploratory, setting up the framework of
>the analysis so that future efforts need be devoted only to
>refining the details.
>
>There may be performance gains from changing the turnover point
>from the halfway point.  For example, as the turnover point is
>moved earlier than the halfway point, a higher exhaust velocity
>(lower [factor] and lower thrust) is allowed, which, together
>with the lower peak velocity, calls for a lower mass ratio.
>
>The performance results tabulated above are for 100 percent
>conversion efficiency (eta = 1.0) from received power to exhaust
>power.  The effects of reduced conversion efficiency (eta less
>than 1) on required exhaust velocity, final sail furl and, most
>importantly, required mass ratio are given in the table below:
>
>  eta   exhaust velocity   final sail furl   mass ratio
>  1.0        0.883             0.160            9.41
>  0.9        0.849             0.105           15.44
>  0.8        0.809             0.060           29.05
>  0.7        0.760             0.029           67.57
>
>Producing high efficiency of conversion from received power to
>exhaust power may be as challenging (and as crucial to the
>success of the concept) as constructing the emitter or the sail.
>
>Rex

Nice work up Rex!  Thou the numbers for the mass ratios do seem light?

Oh, what is Mratio in the first table again?

I beleave current  microwave to electric converters can give over 90%
efficency.  Thou its been a while sine I've read about the solar powre sat
microwave beam systems.  Also I don't know how finiky the converters are to
frequency shifts or phase disruptions in the beam.

If we assume electric acceleration of the particals, thats also very
effocent, but how large a drive system would we need to get that degree of
acceleration?  I remember I got a nasty suprise when time worked out how
large my fuel launcher would need to be to get the Explorer's fuel packets
up to 1/3rd c.

Anyway we'ld need extream efficency to keep from melting the ship anyway.  ;)

Kelly

P.S.
Oh, dumb of me.  If the microwaves are used to accelerate the reaction mass
without conversion inside ship systems (like some varient off my plasma
sail or something) then the efficency would not nessisarily imply that the
lost power contributes to heat in the ship.

P.S.S

Speaking of M.A.R.S., where is Kevin?  I sent E-mail to both his University
and URLy account and got bad address returns from the first, and no
responce from the secound.

----------------------------------------------------------------------
Kelly Starks                    Phone: (219) 429-7066    Fax: (219) 429-6859
Sr. Systems Engineer                                     Mail Stop: 10-39
Hughes defense Communications
1010 Production Road, Fort Wayne, IN 46808-4106
Email:  kgstar@most.fw.hac.com
----------------------------------------------------------------------

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