From: roberts@cmr.ncsl.nist.gov (John Roberts)
Subject: Re: Solar sailing
X-Added: Forwarded by Space Digest
Organization: National Institute of Standards and Technology formerly National Bureau of Standards
Original-Sender: isu@VACATION.VENARI.CS.CMU.EDU
Distribution: sci
Date: Sat, 21 Nov 1992 20:42:12 GMT


-From: n4hy@wahoo.UUCP (Bob McGwier)
-Subject: Solar Sailing
-Date: 20 Nov 92 16:15:35 GMT
-Organization: IDA Center for Communications Research

-I have a local high school student asking me for information on Solar
-sailing.  I have programs that will allow him to manipulate the sail
-if I knew how to calculate `thrusts' etc. from the photon pressure.
-Any details you care to send, primarily references, that will allow me
-to help this very bright student, I would appreciate it.

Let P be the power per unit area of an incident beam of directed light
(in other words, all the photons are moving in the same direction - a
reasonable approximation for a solar sail at a considerable distance from
the sun).

The pressure per unit area (call it p) of such a beam of light is P/c,
where c is the speed of light. If your solar sail is black and perpendicular
to the beam of light, then that's the outward force your sail will
experience. If your sail is black and is tilted at an angle of theta with
respect to the beam (where theta=0 means that the beam is perpendicular
to the surface), then you get an outward force of P/c * cos(theta), and a
"sideways" force of P/c * sin(theta) (valid for absolute value of theta 
between 0 and 90 degrees). However, it should be noted that this formula is
valid only with respect to the power that is actually intercepted by the 
sail - a tilted sail has less "effective" area, varying with cos(theta).
So the amount of thrust you'd actually get with your black sail (per unit
area of sail) is P/c * (cos(theta))^2 outward, and
P/c * cos(theta) * sin(theta) sideways.

Most proposed solar sails are reflective, not black. That's because the
principles of conservation of momentum show that reflecting the beam of
light can greatly increase the thrust of the sail. The best results
are with a 100% reflective sail perpendicular to the beam of light (theta = 0);
that doubles the thrust. If theta is nonzero, then the reflected beam will
have an angle of 2 * theta, so the formulas for a completely reflective
sail are modified to

p(outward)  = P/c * cos(theta) * (cos(theta) + cos(2 * theta))

p(sideways) = P/c * cos(theta) * (sin(theta) + sin(2 * theta))

(Note that if abs(theta) is between 45 and 90 degrees (pi/4 and pi/2 radians),
the reflection term actually decreases the outward thrust - under those
conditions, a black sail would actually provide more outward thrust than
a reflective sail. Also, if your sail is less than 100% reflective, then the
reflection term in the above equations is directly proportional to the
reflectivity.)

Now for something more difficult, and not directly relevant to traditional
solar sail designs: suppose the incoming beam is diffuse rather than
directed, and suppose the sail is a diffuse reflector rather than a
specular reflector? If the light is reaching the surface with equal
intensity from all directions, then a spherical integration shows that the
*net* effect is an outward thrust perpendicular to the surface, with an
intensity equal to *half* of what would be expected from a directed beam
of equal power. If the incoming light is a directed beam and the surface
of the sail is a diffuse reflector (in other words, if it reflects the light
equally in all directions), then the reflective component of the thrust
is again perpendicular to the surface, and equal to half of that of a
specular reflection.

If the incoming light is diffuse, and the solar sail is reflective, then 
I *think* that the reflective component is the same whether the sail is a
specular or a diffuse reflector. I don't know of any easy way to work out
the math, but it seems to me that the reflected light would be equally
distributed in either case.

(Could somebody please verify that? I need it for my Dyson sphere 
calculations.)

John Roberts
roberts@cmr.ncsl.nist.gov

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