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 Problem A circular solar sail with 100% reflectivity and a radius (R) of 600 m. is at rest in the Earth's orbit. The sail is oriented so that its normal points directly to the Sun. The total mass of the sail and its payload ) is 900 kg. The sail moves toward Mars along a trajectory of increasing r. As it does so, its velocity (v) and acceleration (a) increase. Calculate the radiative pressure on the sail, a(r), v(r), v at the orbit of Mars, and how many days are required to reach Mars. (Other symbols used: c (speed of light), (mass of Sun), L (solar luminosity), r0 (astronomical unit)., rm (distance from Sun to Mars). Hints:   The problem is most easily solved by doing the calculations in the order mentioned in the problem statement. When integrating over the distance travelled, integrate from  to .

Data

 (2.1)

 (2.2)

 (2.3)

 Useful Equations Pressue of Fully Reflective Sail Pointing Directly Toward Light Source. Total Flux on the Sail is 2f.      Luminosity as a Function of Flux       Total Force = Light + Gravity      Velocity as a Function of Distance     Time as a Function of Velocity

Solution

The formula for radiative pressure on a perfectly reflecting surface is:

For a perfectly reflecting sail pointing directly to the Sun and with pressure expressed as force per unit area:

Since

 (4.1)

where R is the radius of the sail, L is the luminosity of the Sun, c is light speed, and r is the distance to the Sun. The force of gravity exerted by the Sun is

Thus the total force (light + gravity) on the sail is:

 (4.2)

Divide the force on the sail by its mass to get the acceleration as a function of distance:

 (4.3)

 (4.4)

Calculate the velocity:

Therefore,

 (4.5)

Substituting the distance to Mars (1.524 A.U.) for the dummy variable ρ:

 (4.6)

The sail would be moving at 23.8 km/sec by the time it reached the orbit of Mars.

To calculate the time required for the voyage:

 (4.7)

 (4.8)

It would take about 67 days for the solar sail to reach the orbit of Mars.

A more accurate formula would take into consideration  the reflectivity of the sail:

where Fs is the solar constant (1358 W/m2), c is the speed of light,  is the area of the sail, ρ is the surface reflectance (1 for perfect reflectance), and θ is the angle of incidence (Sellers, 2004).

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 References Sellers, J. (2004). Understanding Space: An Introduction to Astronautics. Boston: Custom Publishing.