# Solar Sailing 201

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## Introduction

Solar Sailing 101 is a qualitative introduction to solar sailing. That is, there was almost no math involved. This document will introduce the basic mathematics behind solar sails. Upon completion, the reader should be able to:

• Explain the inverse square law.
• Calculate the acceleration of a solar sail in orbit around a star.
• Plot the orbit of a solar sail that is pointed straight at the sun.
• Using a computer, calculate the trajectory of a sail held at a constant angle with respect to the sun.

## Inverse Square Law

Solar pressure and gravity both decrease as you travel farther from the sun which approximates an inverse square law, which will be explained here. The photons leaving the surface of the sun expand in a continually growing sphere. These larger spheres have the same original amount of sunlight, and hence momentum, but it is spread out over a larger area, so the solar pressure is lower. Gravity works similarly, but with different principles <TBD>.

The area of the sun, with radius $r_s = 6.960\times 10^8 m$, is $4 \pi r_s^2$. Farther from the sun, the area is $4 \pi r^2$. The ratio of the two is ${r_s^2 \over r^2}$. Thus, the area of the sphere has decreased by a factor of ${1 \over r^2}$, the inverse square of the distance r from the sun.

## Gravity

Solar sails, like other objects, are subject to gravity when travelling through space. This discussion will be limited to gravity from the sun and not the planets or other bodies. This is simpler, because both sunlight (which pushes outward) and gravity (which pulls inward) come from the same source.

The acceleration $a_g$ of a solar sail due to the sun's gravity is ${\mu \over r^2}$, where $\mu$ is the gravitational parameter of the sun.