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 F_{s} is the solar constant (1358 W/m^{2}), 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).