There’s an interesting question in the comments of the last solar sail post:
I have a question that’s been bugging me about solar sails for ages: what about the fact that light pressure falls off over distance? Every time I see the idea discussed, this is never mentioned…
He’s right. As the sail gets farther from the sun, the intensity of the light reaching the sail diminishes. By the same token, the sun’s gravitational force is diminishing as well. To keep going, the radiation pressure has to be greater than the force from the sun’s gravity.
Where I0 is the power output of the sun (3.07 x 1025 W, by my rough calculation), and k is some proportionality factor depending on the sail, used to write that intensity as a force. And the distances d cancel. So if the sail is ever good enough to resist the sun’s gravity, it will never completely stop working due to diminished light because gravity is falling off at the same rate.
The total force (and thus the acceleration) is the difference of those two quantities, and will be decreasing since d is getting larger, reducing the forces in both directions. So the sail will become less and less effective.
You can find the final kinetic energy by integrating that force from the starting point at the Earth’s orbit to infinity. You can use that to find the final velocity.
Reader Challenge: Find that speed! Don’t worry about orbits, just assume the sail is going in a straight line path directly out of the solar system. Bonus points for also plugging in some reasonable values for k and finding an order-of-magnitude estimate for the actual speed.
Update: Here’s the equation I get for final velocity:
R is the sail’s starting distance (earth’s orbital distance), and sigma is the mass per area of the sail. Assuming no payload other than the mass of the sail itself, the size of the sail cancels out of the final equation.