Consider a small, spherical particle of radius r located in space a distance R from the Sun, of mass MS. Assume the particle has a perfectly absorbing surface and a mass density rho. The value of the solar intensity at the particle's location is S. Calculate the value of r for which the particle is in equilibrium between the gravitational force and the force exerted by solar radiation. Your answer should be in terms of S, R, rho, and other constants. (Use the following as necessary: MS, R, rho, c, S, and G.)

r = √ G c M ρ R/S

Explanation:

For this exercise we will use the equilibrium equation

F₁ - F₂ = 0

F₁ = F₂

Where F₁ is the gravitation force and F₂ the force of the radiation pressure.

The equation for gravitational force

F₁ = G m M / r²

Density is defined by

ρ = m / V

m = ρ V

F₁ = G M ρ V / r²

The radiation pressure has as an expression

P = S / c

Where S is the vector of Poynting (intensity) and c the speed of light

The pressure is defined by

P = F₂ / A

F₂ = P A

F₂ = A S / c

Let's match the two equations

G M ρ V / r² = A S / c

The volume of a body is the product of its area by the other dimension

V = A R

G M ρ (A R) / r² = A S / c

r = √ G M ρ R c / S

r = √ G c M ρ R/S