# what are the formulae to calculate the momentum and pressure of sunlight?

A spacecraft is powered through a solar sail, which reflects sunlight. The area of the sail is 1 km^2, oriented perpendicular to the direction to the Sun. The spacecraft is at 1 AU from the Sun. The sunlight falling on the sail has a power of 1380W/m^2.

(a) Calculate the momentum in this sunlight, and from this derive the pressure exerted on the solar sail by the sunlight;

(b) What is the maximum mass of the sail for which the force from light exceeds the gravitational force from the Sun?

For (a), which forumulae should be used here? I didn't find anything useful online and have no idea of a systematic textbook that I can read.

For a single photon, the momentum p=E/c which depend on frequency or wavelength. But here only enegry flux and surface area are given.

For (b), I presume I should equate the gravitational force and the light pressure to solve for the mass. Or do I have to consider the escape speed of the solar system?

PS: This question is at the level of first year uni physics.

Thank you.

• I am not completely familiar with this, but you know that the sunlight falling on the sale has a power of 1380W/m^2 (so not what the solar panel actually generates). I think this means that 1380J worth of photon energy is falling on the panel every second. From this you can just fill in p = E/c where E=1380J – Thomas Wagenaar Oct 8 '17 at 18:14
• @ThomasW Although the dimensions are correct, I'm still a bit afraid to use it. Actually I'm not even sure how p=E/c for a photon is derived. – lkjhgtf Oct 8 '17 at 18:24
• For the last part of your comment see this question physics.stackexchange.com/questions/2229/… – Thomas Wagenaar Oct 8 '17 at 18:25
• Given the enegry flux and surface area, I can get the power. And as P=Fv=Fc, I can get the F, namely the light pressure. It is correct? But the question asks me to derive it from the momentum. Confused now. – lkjhgtf Oct 8 '17 at 18:27
• For part (a) of your question, you might like to read parts of this Feynman lecture, particularly below Equation (27.15) and around Equation (27.21). – diracula Oct 8 '17 at 18:28

The force exerted by radiation on a surface has a magnitude of $$F = \frac{1+R}{c}\int \vec{N}\cdot d\vec{A},$$ where $\vec{N}$ is the time-averaged Poynting vector (power per unit area) of the incoming radiation in Wm$^{-2}$, with a direction given by the direction of the electromagnetic waves (or photons). This is integrated over the area $A$ of the solar sail, but the scalar product takes account of the angle of the sail to the radiation. The term $R$ is the reflectivity of the sail, where $R=1$ doubles the force applied to the sail, because the momentum change of the light is doubled.