How Large Should a Solar Sail Be to Counteract the Sun's Gravitational Pull?

[ft92]

Homework Statement

t has been proposed that a spaceship might be propelled in the solar system by radiation pressure, using a large sail made of foil.

Assume that the mass of the ship + sail is 1780.4 kg, that the sail is perfectly reflecting, and that the sail is oriented perpendicular to the Sun's rays. How large must the surface area of the sail be if the radiation force is to be equal in magnitude to the Sun's gravitational attraction?

The rate at which the Sun emits energy is 3.90 × 10²⁶ W. The Sun's mass is 1.99 × 10³⁰ kg. The gravitational constant is 6.67 × 10-11 N m²/kg².

Homework Equations

P_reflect= 2E_f cos²ϑ /c

The Attempt at a Solution

E_f= (3.9×10²⁶ W)/A = (3.9 ×10²⁶)/ 4πR² = 3.1×10²⁵/R²

let surface area be A_f :

force on ship= P_reflect A_f= (2×3.1×10²⁵×cos²0)(A_f)/ 3×10⁸×R²

F= 2.07×10¹⁷ (A_f)/R²

F_{gravitational at R}= GMm/R² = (6.67×10^-11×1.99×10³⁰×1780.4)/R²

let F=F_grav =(6.67×10^-11×1.99×10³⁰×1780.4)/(2.07×10¹⁷)= 1.14 ×10⁶m² do you agree with my result?

 

[gneill]

The result looks reasonable. Just over a kilometer square.