Maxwell's equations of electromagnetic radiation imply that light carries momentum, as does the quantum theory of light. By Newton's Second Law, changing the momentum of light by reflection results in an applied force and, by his Third Law, a reactive force acts upon the reflector, as shown in the figure. The resultant force is perpendicular to a perfect flat reflector. This force is what drives a sailing ship.
The solar intensity W at 1 AU is 1367 W/m2. The pressure P on a flat perfect reflector is given by:
where c is the speed of light, R the distance from the Sun's center in AU, and alpha is the angle between Sun and sail normal.
The idea of using the acceleration at 1 AU with the sail facing the Sun as a comparative reference originated with H.C. Kelly [Kelly,1962]. Wright coined the expression characteristic acceleration and assigned the symbol ac to represent this value. Using an efficiency factor, eta, of typically about 0.90, pressure, characteristic acceleration, and sail loading, sigma, are related by:
Sail loading is total mass divided by sail area, given in g/m2, and is also called areal density.
An aluminum coating is probably the best choice for the reflective surface. It provides a reflectivity of 0.88 to 0.91 across the visible spectrum.
A ship can have multiple sails, each with different orientations. Each sail has some curvature in it. Using finite element analysis, the local pressure on parts of sails can be integrated to determine the total force and moments acting on the ship. A resulting approximation of the force versus its angle theta, valid for a square sail, is:
Sailing motion is determined by theta, the angle between the solar radial and the ship's total force vector. A positive theta generally adds energy and causes outward motion. A negative theta generally reduces energy and causes inward motion. The projection of the acceleration vector onto the velocity vector determines the actual change in energy in the local gravitational field. A sail does not extract energy from or put energy into the reflected light to accomplish its sailing.
If the reflection does not extract energy from or inject energy into the photons, how does the sailing ship gain or lose energy? The reflected photons have the same energy flux they had prior to the interaction, but a different momentum vector. It is this altered momentum vector that gives the ship an accelerating force that allows it to work against the gravitational field to gain or lose energy within the field. The absorbed photons are the energy lost from the impinging flux. The absorbed energy is re-radiated from the sail, with some helping and some hindering the ship’s motion.
The solar wind, charged particles streaming out from the Sun, also exerts a force on a sail (or any other object in space). However, the wind pressure is smaller than solar pressure by a factor of 5000 to 10,000 and is usually ignored when calculating trajectories for sailing ships.
When a solar sailing ship has a component of motion toward or away from the Sun, the wavelength of the incident light is slightly altered by the Doppler effect. This causes a minor change in the power density of the incident light, but is so small that it is ignored.
The abberation of sunlight is the result of a body's lateral motion relative to the source of the light. This means the light does not appear to come precisely from the center of the Sun, but at a very small angle (typically less than 0.01 deg). The angle is so small that it is ignored in trajectory calculations.
Solar pressure calculations for a sail usually assume that the light from the Sun comes from a point source. However, the Sun is a large body with a diameter of 1.4 million km (860 million statute miles). If precision is needed for trajectory calculations close to the Sun, its actual diameter should be taken into account.