The intensity of light with a mean wavelength of 589 nm falling on a solar sail is 780 watts/m^2. The sail is oriented perpendicular to the direction of the incoming light, in order to intercept as much light as possible.
The momentum of a photon is p = E / c.
The total momentum falling on the sail in one second is therefore equal to the total energy of all the photons falling in one second divided by c.
The energy in one second, per square meter, at 780 watts / m^2 = 780 (J / s) / m^2 is
The momentum of all the photons required to deliver 780 J of energy is
We thus have a total momentum of ( 2.6 * 10^-6 kg m / s ) / m^2 * 35000 m^2 = 9.1 * 10^-2 kg m/s.
This will be the momentum of all the incoming photons in any 1-second time interval. If the surface of the sail is a perfect mirror, the reflected photons will carry away an equal momentum and the magnitude of the momentum change will be double that of the incoming momentum:
By the impulse-momentum theorem the average force on the sail will be
The acceleration of the 1930 kg mass will therefore be
At this rate the time required for a velocity change of .1 c = 3 * 10^7 m/s will be
We note that 10^12 sec is a bit in excess of 30,000 years.
General SolutionIn general if we have light intensity I, in watts / m^2, striking an area A at a perpendicular, then in time interval `dt we have total energy
Since the momentum of a photon with energy E is E / c, the total momentum must equal the total energy of all the photons, divided by c:
If the photons are perfectly reflected by the area then the magnitude of the total momentum change in time interval `dt is
The rate of momentum change, which is the force exerted, is therefore
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