NYTimes.com: About a year from now, if all goes well, a box called LightSail-1, about the size of a loaf of bread, will pop out of a rocket some 500 miles above the Earth.
There in the vacuum it will unfurl four triangular sails (see left image for an artist's rendition of LightSail-1 by Rick Sternbach. Credit: Planetary Society) as shiny as moonlight and only barely more substantial. Then it will slowly rise on a sunbeam and move across the stars.
LightSail-1 will sail only a few hours and gain a few miles in altitude.
But those hours will mark a milestone in the quest to navigate the cosmos on winds of starlight the way sailors for thousands of years have navigated the ocean on the winds of the Earth.
Regarding extreme forms of solar or stellar sails, the following calculations might be of help to those interested in just how extreme stellar sail first pass terminal velocities could be.
A former physics professor of mine from George Mason University in Fairfax Virginia, USA, provided me with the following succinct mathematical derivations.
Assuming fraction f of the starlight is reflected straight back and the sail moves radially outward,
the equation of motion is
B[(1 + (B EXP 2)]dB/[(1 − B)EXP 2] {[1 − (B EXP2)] EXP 3/2} = p (R0/x) EXP 2 where B = v/c, v is the speed of the sail, x is the distance from the star, R0 is the initial distance from the star,
P = 2fA(u0)R0/[Mo(C EXP 2)]
Where;
A is the area of the sail, m0 is its rest mass, and u0 is the energy density of starlight at x = R0; thus, u(x) = (u0)[(R0/x) EXP 2].
Adopting f = 1, a value of M0/A = (10 EXP −8) kg/(meter EXP 2) = the effective mass specific reflecting area of the sail craft, and u0 ~ L/[4(pi)(Ro EXP 2)C] with L the Sun’s luminosity and R0 = 0.03AU, I find p ~ 5 × (10 EXP −2).
Note that sails with such a high mass specific area might consist of carbon nanofiber materials that are highly conducting if not virtually superconducting, and which are of a grid like nature wherein the sail is a weave consisting of mostly empty space. For the above value of M0/A, a metalized carbon nanotube weave comprised of cross woven nanofibers wherein parallel fibers seperated by about 100 nm might in theory work.
Note also that the equation of motion can be integrated analytically to find the terminal speed.
Just integrate from zero to its terminal value and x from R0 to infinity.
This yields for the terminal velocity:
{[(1 − (B EXP 2)] EXP (1/2)} [7 − 14B + 11 (B EXP 2) + 2(B EXP 3)]/(1 − B ) EXP3 = 7 + 15p
With p = 5 × (10 EXP −2), the terminal velocity = 0.251 C.