# The Math

```Hi Brian,

In regards to your request for the math on using a solar sail for more than
just accelerating outward from the sun, I am sorry to say no. I was going to
attempt it, but after typing in just a few of the basic equations I realized
that it was an enormous task which I can't spare the time for.

Basically, you can do anything with a solar sail that you can with a normal
watercraft sail. You can run, reach, tack, back up, and even perform
"station keeping." In fact, a solar sail almost makes a better in system
drive than it does an interstellar drive. Its best performance regime is
when it is close to a star, not far away.

Sorry, it's not that I don't want to go into the math, it is more a matter
of not wanting to type three pages on the basic physics of solar sailing
just to get to the point where I can present deceleration (1 page) radial
acceleration (1 page) and tangential acceleration (1 1/2 pages) adequately.
Instead I am going to recommend you to an excellent primer that covers all
of these topics:

"The Starflight Handbook"
by Eugene Mallove and Gregory Matloff
ISBN 0-471-61912-4

It also addresses many other topics which are frequently covered in this
discussion group. When you have finished that, you might also enjoy a
somewhat more detailed:

"High Performance Solar Sails and Related Reflecting Devices"
by K. E. Drexler
AIAA Paper 79-1418

and

"The Alpha Centauri Probe"
by J. H. Bloomer
in "Proceedings of the 17th International Astronautical Congress"

The deceleration equations were developed by J. T. Early and presented (I
think) in JBIS 37 (1984)

You are quite right in your deduction that you are actually decelerating
against orbital motion and thereby falling in toward the star on a parabolic
orbit. A more efficient manuever would be to accelerate out from the sun (or
other star) toward a gas giant, furl your sail and then perform a gravity
assist manuever back towards the star on a HYPERBOLIC orbit. This combined
with a powered perihelion manuever using a ramjet or equal) around the star
will allow you to achieve the greatest possible amount of boost from the
solar sail design.

Utilizing a maximum acceleration of 14 G this will produce a final velocity
after boost phase of 0.003 c. for a 10,000 ton Explorer Class vehicle. This
would put it at Alpha Centauri in about 1200 years. Higher accelerations (as
much as 400 G) are possible but not for manned vehicles. The limiting factor
is of course the tensile strength of the sail and cables. A Pathfinder type
vehicle could make the same trip in about 350 years. This all assumes no
further acceleration from the sail beyond boost phase. (Its practically
useless beyond a certain distance unless you are going to use a LOT of
lasers or something.)

There are additional considerations such as skin temperature at perihelion
to consider, but most of these values were worked out in the late 80's and
pronounced feasible.

Rather than the wire mesh design I keep hearing here, I would suggest a thin
film metalloid connected in a "parachute" arrangement for a manned mission.
Such a sail can be made in space using vacuum deposition techniques and
theoretically provides the highest possible thrust to weight ratio as well
as being easy to manufacture. The materials for 100 sq km of such a sail
would fit in one space shuttle payload bay. You would probably want to be
able to manufacture several such sails during the course of the mission in
case of damage.

I hope you can find these references, I think you will find them interesting.

Lee Parker
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+  Weave a circle 'round him thrice, and close your eyes with holy dread...   +
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```