starship-design: Required microwave-antenna size. 100-light-year trip.

[DotarSojat@aol.com]

Hi all

At 01:43 EDT on 10/21/96, Kevin wrote:

>I really think this is the best design we've come up with yet.
>Aside from the cost and political will issues, none of this
>technology is beyond our capability.  We know how to make solar
>collectors, masers, linear accelerators and closed system
>ecologies.
>
>The question remains, can we build them large enough, precise
>enough, efficient enough and will they last long enough to make
>it to TC and back agin.  But then, these are engineering prob-
>lems, not physics problems.

I think Kevin's note from which the above quote was taken is a
very good wrap-up of the beam-driven-sail concept, except that
the calculated results that he draws from my 9/20 note "Deceler-
ation of sail pushed by constant-power beam" were qualified by
the condition, "ignoring inverse-square effects."  In my 9/11
note, I wrote:

"(Note: This exercise may turn out to be purely academic because
the inverse-square effects...would be much larger [than the
Doppler-shift effects].)"

Kevin acknowledged this issue when he wrote, at 11:13 EDT on
10/21:

>This is one of the problems that's worried Tim.  Tim fears that
>the beam will spread out too much.  I think that the beam can be
>focused into a nearly paralell stream of photons.  thus there
>will be no spreading and diffusion.  _IF_ this can be achieved,
>then range won't be a problem at all.  The ship will be able to
>travel to any star within a good 100 light years on a 3-4 light
>year pulse of maser energy.  Since most of a 100 light-year trip
>will be spent at insanly high fractions of C, I expect the one-
>way trip time to be on the order of ten years for the crew.  of
>course a return mission would arrive a good 250 years after
>departure.

The above quotes indicate that there are some uncertainties that
need to be cleared away with calculations of (1) numbers regard-
ing antenna-size requirements to avoid too much beam attenuation
due to inverse-square effects, and (2) numbers regarding a
100-lt-yr trip.

1. ANTENNA SIZE REQUIREMENT (A tutorial; read at your own risk.)

In the "far field" [range > De^2/(2.44 lambda), to be shown
below] of the emitting aperture, focusing to produce a convergent
beam (or even "a nearly paralell stream of photons") is not poss-
ible, and Fraunhofer diffraction (linear divergence of beam
width, or inverse-square dependence of power per unit area, with
distance) applies.  A diffraction-limited beam ("perfect" antenna
shape) contains about 84 percent of its power within the main
lobe that has an angular radius of 1.22 lambda/De (where lambda
is the wavelength of the emitted radiation and De is the diameter
of the emitting aperture).

In the "near field," Fraunhofer-diffraction considerations are
not available to define the scaling of beam-cross-section
("spot") size with distance.  But focusing to produce a conver-
gent beam is possible, and we can use simple geometrical-optics
considerations to determine the focused-spot size.  For a
reflecting beam-forming "mirror" (antenna) with a focal length,
f, the distance from the mirror to the image spot, di, is related
to the distance from the mirror to the object, do, by the
relation--
     (1/di) + (1/do) = (1/f)   ,

and the magnification, i.e., the ratio of the image size to the
object size, is given simply by the ratio, di/do.  In our case,
the distance from the mirror to the image spot (on the sail) is
many, many focal lengths of the mirror, so do does not need to be
changed much from f to compensate for the change in di as the
sail moves away from the emitter.  Therefore the focused image
spot size in effect increases directly with di (for constant
object size), resulting in an inverse-square dependence of power
per unit area again.

The range boundary between the "near" and "far" fields can be
defined as the distance at which the focused image spot size is
equal to the size of the beam-forming aperture, and at that range
the spot size is that given by Fraunhofer diffraction (with a
factor of 2 to go from radius to diameter), so--
     spot diameter = De = range * 2 * 1.22 * lambda/De   ,

or, solving for the range that gives the boundary between the
near and far fields,
     range = De^2/(2.44 lambda)   .                       Q.E.D.

If the image spot size at the boundary is that given by Fraun-
hofer diffraction, and the spot size has the same dependence on
distance in the near field as the Fraunhofer spot has in the far
field, then we can use the Fraunhofer angular width to determine
the spot size in the near field.  (What we can't do in the near
field is take the same shape of the power distribution within the
spot.  The power distribution within the spot is governed by
"Fresnel" diffraction, which we'll assume gives a gross distrib-
ution not enough different to bother us here.)

If we set the Fraunhofer-diffraction angular width equal to the
angle subtended by the sail diameter (Ds) at the range R, we can
get an expression for the diameter of the emitting antenna (De)
that is required to put "84 percent" of the emitted power within
the area of the sail:
     De = 2.44 * lambda * R/Ds   .

For microwaves with lambda = 1 cm and a sail diameter of 100 km,
say, the required diameter of the emitting antenna grows 2.31
million km for each light-year of distance to the sail, to keep
the main lobe just within the sail diameter.  This is for an
antenna whose figure (shape) is correct to a fraction of a wave-
length.

For an emitting-antenna diameter of 2.31 million km (and wave-
length of 1 cm) for a 1-lt-yr distance to the sail, the boundary
between near and far fields is at a range of about 23 thousand
lt-yr, far enough away to say we're operating in the near field.
[Note: at 1 lt-yr distance, the spot size is
(1 lt-yr/23,000 lt-yr) * 2.31 x 10^6 km = 100 km, the diameter of
the sail, as desired.]

Are we prepared to build an expanding antenna this large?  It
looks as if the M(aser)ARS should be a L(aser)ARS.  For any wave-
length, however, figure control is going to remain a headache.

2. DATA FOR 100-LIGHT-YEAR TRIP (in spite of the above)

A run with the program SAILTRIP (appended to my 9/20 note) with
DSTAR = 100 lt-yr gives the following data for the one-way trip:

Proper (ship) time = 23.8473 yr
Apparent (Earth) time = 104.2934 yr
Total Earth time for emission = 4.2934 yr
Peak proper velocity = 7.2603 lt-yr/yr
Peak apparent velocity = 0.9906 lt-yr/yr
Minimum acceleration (at turnover) = 0.0685 g
Exhaust velocity for deceleration = 0.89346 lt-yr/yr
Mass ratio for deceleration (eta = 1.0) = 33.62

The trip time for the crew is not as short as Kevin estimated
("ten years") because the acceleration is so low for most of the
trip.  His estimates for emission time ("3-4 ... year"), Earth
time for round trip ("a good 250 years") and speed ("insanly
high fractions of C") are pretty good, however.

Rex   <DotarSojat@aol.com>  (Rex Finke)


P.S. To Timothy: You're right in advising Nick not to use aspects
of both particle and wave mechanics at the same time to define
the Doppler effects.  (And welcome back, Nick; even though the
Group became aware of me late, I have continuous copies of the
LIT Engineering Newsletter from its inception through about
April, 1995, to make me aware of previous member contributions.)