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starship-design: Motion of sail driven by constant-power beam -- Excellent work Rex!
DotarSojat@aol.com wrote:
>
> Hi all
[Snip]
> 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
[snipped for brevity, but a first-class job]
> 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) .
> __________
So, let me see if I understand this. For a trip to Tau Ceti (12 Lt-yrs
away),
(interpolating from above 6 is almost halfway in between 5.3 and 6.6)
The ship accelerates for 4.25 years crew time to turnaround. It
achieves
a final velocity of .933 C and the acceleration felt by the crew is .186
Earth's transmitters have been on for 1.63 years (as measured by earth)
and turnaround would be achieved at 7.618 years earth time.
Doubling these numbers (where appropriate) gives for a one-way trip:
15 earth years for travel.
8.5 crew years for travel.
3.5 earth years of transmitter time.
With 10 years for in-system study (and return maser construction),
This gives a crew time of 27 years for the entire mission. and an earth
time of 40 years. This is not too bad. The crew would be able to see
earth again, the time differential upon return is not so horrendous that
all your family would be dead. your twin would be "only" 13 years
older than you. I think this is even favorable in terms of "political
return" A politician who voted for this would expect to see results
in about 30 years (travel time plus transmission time back to earth)
Long enough to be forgetable if it goes bad, short enough to hope to
be alive to see it if it works.
>
> These results seem to confirm Kevin's intuitive estimates regard-
Can I say "I told you so"? Aw, come on be a sport. ;-)
> 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.
I thought Tim's treatsy on the subject of deceleration with a beamed
poser source showed that it will work. That is to say that the ship
receives ennough power via the antenna to power an ion engine with
a fairly decent mass ratio.
=============begin included text =============================
Calculations for the deceleration phase of the MARS design.
by Timothy van der Linden (T.L.G.vanderLinden@student.utwente.nl)
Last modified April 14th, 1996
[derivation snipped]
Vstart Vexh optimal Fuel:ship-ratio Energy per kg of ship (in Joules)
0.1 0.062 5.36 7.45E14 7.73E14
0.2 0.121 5.84 3.25E15 3.50E15
0.3 0.180 6.40 8.11E15 9.08E15
0.4 0.240 7.06 1.64E16 1.91E16
0.5 0.300 7.87 2.99E16 3.64E16
0.6 0.364 8.91 5.23E16 6.69E16
0.7 0.433 10.38 9.21E16 1.25E17
0.8 0.512 12.72 1.73E17 2.53E17
0.9 0.615 17.75 4.04E17 6.62E17
0.99 0.803 52.00 3.12E18 7.02E17
0.9996 0.906 238.81 2.91E19 9.26E19
Note that the power of the maser-beam is NOT constant during the
decelerating
phase, it is supposed that it decreases while the ship gets lighter
(because it repulses mass).
======================End included text==============================
By repluses mass, I think Tim means the beam tends to accelerate the
ship more than the engine can compensate. this was supposed to be a
minimum energy solution, perhaps a constant energy beam will raise the
costs. But turning down a power beam is very easy. (think dimmer
switch)
Also Fuel:ship ratio is Reaction Mass:Dry Mass ratio I think.
For a top speed of .9333 C, we are looking at a mass ratio of "only"
30.4 (interpolated). this seems to me to much better than any other
mission discussed so far.
>
> (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.)
I think we can make some advances here using :
1) Coherent energy (masers instead of microwaves)
2) Large focusing elements in outer solar system (ala Robert Foreward)
3) Phased array transmitters to simulate large arperature ~ 400km
With proper microwave optics (moptics? :) yeah, Moptics!) I think we
can make the beam nearly non-divergent. Not totally, but enough to
make this mission possible. Remember, Tim's derivation requires a
diminished beam during deceleration. Inverse square is too much
diminished certainly, but at least it's the right direction for a
change.
I think this mission profile is looking more and more doable. the high
energy costs were (mostly) due to:
1) The extremely large ship 1 E 09 Kg when I calculated that number.
2) The desire to accelerate at 1G the entire time.
3) The foolish notion of increasing the exhaust velocity to save mass.
#1 can be changed any time we wish. We just have to settle on a mass
that seems reasonable (although in this group, the very word is open to
debate)
#2 has been shown by Rex and others for the folly it is. A gradual
decline
into lunar gravity, and a gradual climb back out will certainly be
doable
for the astronauts. no clumsy swiveling sections, no artificial gravity
generators (although if we had those, we could do this mission easily ;)
#3 Was shown by Tim to be foolish. According to the above chart, the
optimum Vexh is .685 (interpolated)
A beamed mission is safer than anti-matter (and easier); it is faster
and
less costly than a fusion fuel mission (excepting perhaps the hybrid
mission Kelly has proposed). It only requires one major advance (self
replicating machines to make the solar collectors.) Other than that,
the technology is all present day.
--
Kevin "Tex" Houston http://umn.edu/~hous0042/index.html