4. [25 points] A tiny spherical dust particle of radius r is in the gravitational field of the Sun. It has a mass density r = 1.00 gm/cm³ and is a distance d from center of the Sun.

Useful information: Mass of Sun = M_sun = 1.99 ´ 10³⁰ kg, light output of Sun = P_sun = 3.90 ´ 10²⁶ watts, Newton’s gravitational constant = G = 6.67 ´ 10^-11 N m²/kg², and Newton’s Law of Gravitation (similar to Coulomb’s Law) is F₁₂ = G m₁ m₂/r₁₂². Express your answers for parts (a) through (e) below in terms of r, d, r, M_sun, P_sun, and G. Assume that the particle is completely black and absorbs all solar radiation that strikes it.

(a) [4 points] Write algebraic expressions for the mass m of the dust particle and for the area A of sunlight that it intercepts and absorbs. (Symbols only!)

The mass is m = r (4/3)pr³.

The cross sectional area is A = pr² (Note: NOT 4pr² , which is the surface area.)

(b) [4 points] Write an algebraic equation for the force of gravity F_g acting on the dust particle. (Symbols only!)

The gravitational force is F_g = G m M_sun/d² = G r (4/3)pr³ M_sun/d²

(c) [4 points] Write an algebraic equation for the intensity I of sunlight illuminating the particle. (Hint: intensity is power per unit area) (Symbols only!)

The light intensity is I = P_sun/4pd²

(d) [4 points] Write an algebraic equation for the force F_lp on the dust particle due to light pressure. (Symbols only!)

Since the dust particle is completely absorbing, F_lp = dp/dt = (1/c)dU/dt = (1/c)IA = (1/c)(P_sun/4pd²)(pr²)
F_lp = r² P_sun/(4c d²)

(e) [4 points] Write the equation for the condition under which F_g = F_lp. (Symbols only!)

The forces are equal when G r (4/3)pr³ M_sun/d² = r² P_sun/(4c d²) or
G r (4/3)pr M_sun= P_sun/(4c)

(f) [5 points] What is the smallest radius that such a dust particle can have and still remain in the solar system (i.e., such that F_g F_lp)? (Calculate the numerical value).

Solving the above equation for r, we have: r = 3P_sun/(16p r c G M_sun)
= 3(3.90 ´ 10²⁶ watts)/[(16p)(1000 kg/m³)(3.00 ´ 10⁸ m/s)(6.67 ´ 10^-11 N m²/kg²)(1.99 ´ 10³⁰ kg )
= 5.75 ´ 10^-7 m.