The starshot project is a project to use lasers that would hit a huge solar sail

Question

The starshot project is a project to use lasers that would hit a huge solar sail (StarChip=Nanocraft that is a gram-scale spacecraft) in hopes to accelerate the spacecraft to something near the speed of light. The project has a spacecraft of mass 35 grams with each square of the sail being 5 m long. If the sail has a 99.9995% reflectivity and the propulsion power comes from the 155 gigawatt laser. The laser is one the operates at 12.4 Micrometers and the laster is a CO2 one.

a) For this size of sail, what is the force that is applied to it?
b) Lets say this is non-relativistic, how long does it take to accelerate it to 1/10 the speed of light?
c) If the light fills the sail and the sail only with noting being lost and having the reflectivity above, how much light pressure is produced given these cases?
d) The sail then absorbs power, how much is that power being absorbed?
e) what will its momentum be when it is moving at 20.0% of the speed of light? Neglect relativistic effects.

Explanation / Answer

given

m = 35 g

reflectivity, e = 0.9999995

laser power, Po = 155*10^9 W

lambda = 12.4 um

a. area = 5*5 = 25 m^2

radiation pressure = 2*E*e/c

E = hc/lambda

P = 2*h*e/lambda

force = 25P = 2.689*10^-27 N

but this is force due to one photon

if all the phtons from the power source strike the target

then

force = 25*2*E*e/c

hence

F = 25833.323 N per second

b. acceleration = a = F/m= 738094.942 m/s/s

hence

c/10 = a*t

t = 40.6451 s

c. pressure = F/25 = 1033.33292 Pa

d. if the sail absorbs power, then it absorbs all of the power Po = 155*10^9 W and uses that for propulsion

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