# Thrust for solar sail equation derivation

In the new SMAD (Chapter 18 Space Mission Engineering and Analysis) they have the equation for thrust of a solar sail and a reduced version. Variables are described in this question.

I would like to reproduce this simplification. Can someone explain how they got from the first one to the simplified version? What numerical values (the actual values too, not only their descriptions) were likely to have been used to get the numerical value of $9.113 \times 10^{-6}$?

• Could you elaborate on what each of the used variables mean? Commented Dec 7, 2017 at 23:57
• Here's a clue for you: Get the value of speed of light in m/s, the value of the solar flux in W/(m*m) at 1 A.U., and the factor of 2, and see if you can come up with that magic number. Then I bet you can figure the rest out. Commented Dec 8, 2017 at 0:08

# Unit

When dealing with a magic number, it's important to know what its unit should be.

$$F=\frac{2RSA}{c}\sin^2\theta=9.113\times10^{-6}\frac{RA}{D^2}\sin^2\theta$$

• $F$ is a thrust (=Force) in Newton.
• $R$, $2$ and $\sin^2$ are unitless.
• $A$ is an area in $m^2$.
• It means that $\frac{S}{c}$ is a pressure in Pascal.

By simplifying $R$, $A$ and $\sin$, we get:

$$\frac{2S}{c}=9.113\times10^{-6}\frac{1}{D^2}$$

• $D$ is a distance in $\mathrm{AU}$
• It means that $9.113\times10^{-6}$ is a force in a non-conventional unit : $\mathrm{Pascal} * \mathrm{AU}^2$

# Value

This equation should be valid anywhere in the solar system. In particular, it should work on Earth ($D = 1 \mathrm{AU}$), where $S$ is the solar constant : $1361 \mathrm{W/m²}$.

Using qalculate with $2*S/c$:

> 2 * 1361W/m² / c to Pa

((2 * (1361 * watt)) / (meter^2)) / speed_of_light = approx. 9.0796147 uPa


Close enough! It seems that $1366 \mathrm{W/m²}$ has been used to get $9.113\times10^{-6} \mathrm{Pa}$.

Using inverse-square law, you get the desired equation.