At Earth's distance from the sun, the solar flux, S_{s}, in space is about
1.4 kilowatts per square meter. This is enough power to run a hair dryer continuously, but not enough to power a car. If one
assumes perfect reflectivity from the mirror, the force due to this flux is given in the following table.

Force due to the solar photon flux |

Symbol |
Definition |
Units |

S_{s} |
Solar flux at the Earth's radius from the sun = 1400 |
Watts/m^{2} |

A |
Area of the sail mass |
metersSpeed of light = 3 x 1010 |

c |
Speed of light = 3 x 10^{8} |
m/s |

**Solar Force vs. Earth's Winds**

For a 40.2 square meter sail, the light force is only about 3.8x10-4 Newtons. Consequently,
the pressure from the sun is roughly nine orders of magnitude weaker than what we can harness from the wind on the surface
of the Earth. This force is so gentle that it would be completely swamped by atmospheric friction and is hence unnoticeable
in our day to day activities.

**IN SPACE PHOTONS WIN!!**
**Solar Sailing** - Acceleration due to gravity

Before we can explore the motion of a spacecraft with a light sail attached to it,
we must first understand its natural motion in the presence of the sun's gravitational field. If one places a spacecraft in
orbit around the sun, it moves on a trajectory that is defined by the sum of all the forces acting on it. The dominant force
acting on an orbiting space craft is the centripetal force due to the gravitational pull of the sun as described in figure
and table below. This force is balanced by the outward centrifugal force, mv^{2}/r, of the spacecraft's motion.

Centripetal force due to gravity |

Symbol |
Definition |
Units |

G |
Gravitational constant = 6.672 x 10^{-11} |
Newton-meters^{2}/kg^{2} |

M |
Mass of the sun |
kilograms |

m |
Mass of the spacecraft |
kilograms |

r |
Distance between the centers of gravity for M and m |
Meters/sec^{2} |

a |
Acceleration due to gravity |
Meters/sec^{2} |

v |
Orbital velocity of the satellite of radius r |
Meters/sec |

For example, unfurling a solar sail on a spacecraft which is in a stable centripetal
orbit and orienting the sail 90 degrees to the direction of the solar flux reduces the net radial force due to the pull of
gravity to

where this time we have modified the force due to the photon flux to reflect that
the sail reflectivity includes a reflectivity factor, k (0 < k < 1). R_{e} is the radius of the earth's orbit
around the sun and R is the distance of the spacecraft from the sun. The ratio R_{e}/R, rescales the solar flux from
that given for Earth's orbit (see the table on Force due to the solar photon flux) to accommodate any other radial range from the sun. Because the photon pressure occurs
with no change in the satellite velocity, v, the outward centrifugal acceleration is no longer balanced against the combined
radial pull of solar gravity and the outward solar photon pressure. The imbalanced forces change the orbital trajectory from
a circle into an ellipse as shown in the figure above.

**Newton's third law**

Newton's third law states that for every action, there is an equal and opposite reaction.

One can observe this effect by tilting the solar sail so that its surface is no longer
normal to the incident photons. Then the momentum component of the photon flux, which is parallel to the direction of motion,
translates into a change in the tangential velocity. This is illustrated in the figure below.

If the photons are reflected along the direction of the spacecraft's motion, the
imparted momentum pushes the spacecraft both outward from the sun as well as applying a decelerating force along its tangential
direction of motion. It could be said that this is the light sail version of tacking because the spacecraft moves opposite
the direction of the applied outward photon pressure.

The other maneuver of interest occurs when the mirror is tilted so that photons are
reflected behind the trajectory. This imparts a forward momentum to the spacecraft, causing it to spiral outward as shown
in the figure above.

**Laser assisted light sailing**

Light sailing works well for inner planet missions and for activities extending out
to the Mars orbit. However, the solar flux falls off as the inverse square of the distance from the sun. Thus for missions
beyond the Jupiter orbit, an alternative to solar propulsion is to use directed light from a high power laser. As a pioneer
inventor in the field of interstellar propulsion, Robert Forward has an avid interest in developing methods for boosting the intensity of light that
can be delivered to a light sail. His goal is to reduce the cruise duration of a trip from our solar system to the nearest
star from 6500 years to a time frame on the order of 40 years.

(http://solarsail.jpl.nasa.gov/introduction/how-sails-work.html)