## Solar Sailing Fundamentals

As is true for aerocapture and the electrodynamic tether, the solar photon sail is an example ofpropellantless field propulsion. Thrust is produced not by the expelling of onboard fuel, but by an external force field—in this case the pressure of solar photons striking the sail. As such, application of the sail can only produce acceleration within the inner solar system or in the presence of another copious source of unidirectional photons.

If we assume a highly reflective, non-transmissive sail with a reflectivity to sunlight of -Rsaii, and if we discount the small contribution to acceleration by photons absorbed by the sail, spacecraft acceleration due to solar radiation pressure can be approximately calculated (in meters per square second) as:

1 + ^sail ACCsail ^c^-sail cMs where c is the speed of light (300,000 kilometers per second), Ms/C is the total spacecraft mass in kilograms, Sc is the solar flux (about 1,400 watts per square meter at the Earth's distance from the Sun), and ^sail is the sail's area normal to the Sun in square meters. Solar photon flux decreases with the square of solar distance. At twice the Earth's solar separation (2 Astronomical Units), Sc has fallen to about 350 watts per square meter, or one-quarter of its previous value.

The phenomenon which describes this decrease in sunlight intensity is known as the "inverse square law,'' and is common in many physical processes involving a point source of radiation (like the Sun). Figure 13.1

FIGURE 13.1 The inverse square law demonstrated.

Intensity = i/tf

### FIGURE 13.1 The inverse square law demonstrated.

illustrates how the inverse square law works. As the distance from the Sun to the solar sail increases, the amount of light spreads out over a larger and larger area, and the light falling on the sail becomes correspondingly less intense. If you were to double the distance between the solar sail and the Sun, the amount of light falling on the sail would only be one-quarter of its previous value—4 is 2 squared and is in the denominator of the fraction, hence the term "inverse square." Similarly, if you were to move the sail to a distance 4 times further from the sun, the amount of sunlight falling on the sail would be down to one-sixteenth of its original value.

To maximize sail acceleration, it is necessary to minimize spacecraft mass, maximize sail reflectivity and work with the largest possible sail areas. Sail engineers often characterize sail performance using a parameter called the "lightness factor." The lightness factor is the ratio of solar radiation-pressure force on the sail to solar gravitational force.

Since solar gravitational force also decreases with the square of the solar distance, lightness factor is constant for a sailcraft anywhere within the solar system. One interesting result can be obtained by assuming a lightness factor of exactly 1, for a sail unfurled normal to the Sun near the Earth at a solar distance of 1 Astronomical Unit (150 million kilometers). Application of Newton's Laws of Motion reveals that this craft moves off into space in a straight line at a constant velocity of 30 kilometers per second. Such a craft could reach Mars in about a month. But astronauts would then have to address the "small" question of how to slow down!

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