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Sailing with Solar Propulsion


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The Physics of Solar Sailing

Both NASA and the European Space Agency are developing solar sails and, although never tested, the concept is quite simple. A solar sail is essentially a giant mirror that reflects photons of sunlight back in the direction they came from.

Although photons do not have mass, they are considered to have momentum, so according to the law of conservation of momentum, the photon loses some of its energy to the sail as it bounces off, giving the sail a shove in the opposite direction.  (http://www.primidi.com/2003/07/03.html)

At Earth's distance from the sun, the solar flux, Ss, 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.

formula for force due to the solar photon flux    Force due to the solar photon flux
Symbol Definition Units
Ss Solar flux at the Earth's radius from the sun = 1400 Watts/m2
A Area of the sail mass metersSpeed of light = 3 x 1010
c Speed of light = 3 x 108 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, mv2/r, of the spacecraft's motion.

centripetal force

formula for centripetal force due to gravity     Centripetal force due to gravity
Symbol Definition Units
G Gravitational constant = 6.672 x 10-11 Newton-meters2/kg2
M Mass of the sun kilograms
m Mass of the spacecraft kilograms
r Distance between the centers of gravity for M and m Meters/sec2
a Acceleration due to gravity Meters/sec2
v Orbital velocity of the satellite of radius r Meters/sec

Elliptical motion induced  by the solar flux acting on a light sail

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

formula

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). Re is the radius of the earth's orbit around the sun and R is the distance of the spacecraft from the sun. The ratio Re/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.

Figure 5: Reflecting the photons forward along the direction of motion slows the spacecraft down.

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.

Figure 6: Reflecting the photons backward away from the direction of motion speeds the spacecraft up.

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)

Three main mission scenarios have been proposed for the use of light sails:

  • Small solar kites (several meters in diameter) to replace the attitude control thrusters. *Mariner 10 spacecrafts.  Attitude control thrusters are tiny thrusters that determine the way that the ship is positioned in 3-dimensional space.
  • Large light sails (100x100 m2) for propulsion and maneuvering on inner planet missions
  • Larger light sails (200x200 m2) for outer planet missions.
  • Super light sails (1000x1000 m2) for interstellar missions. Instead of using solar power, the light sail would be propelled by a 65 GW ( One billion (109) watts) laser system.
Sample road map showing more complex light sail configurations.
Mission name Objective Areal density Sail size
GeoStorm monitoring
  • Detect flux of solar radiation via light sail motion to give early warning of solar storms
  • Demonstrate deployment and handling of moderate sail sizes
    10 gm/m2 70m x 70m
    Inner planet missions Faster commute to Mercury
    • Demonstrate navigation using deceleration, and acceleration via solar radiation pressure
    10 gm/m2 100m x 100m
    Outer planet missions Faster commute to Saturn
    • Combining light sail maneuvers with gravity assists to achieve more complex trajectories
    5 g/m2 200m x 200m
    Inter stellar missions Visit neighboring stars
    • Replace solar source with laser source
    • Achieve v~0.01c terminal velocity.
    1 g/m2 103m x 103m

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

    Next page: more physics.