Problem 4. You’re an astronomer studying the origin of the solar system, and you

Question

Problem 4. You’re an astronomer studying the origin of the solar system, and you’re evaluating the hypothesis that sufficiently small particles were blown out of the solar system by the force of sunlight. The force per area, or radiation pressure, 2 is related to the light intensity by p = S/c. To evaluate this hypothesis, consider the relative size of this outward (away from the sun) radiation force and the inward (toward the sun) gravitational force on a particle of density ? and radius r at a distance R from the sun. Under what conditions (distance from the sun, particle size, etc) does the outward force exceed the inward force? Assume the particles are spherical with a density of 2 g/cm3 , and that they absorb all incoming radiation. The total power output of the sun is P = 3.85 × 1026 W.

Explanation / Answer

consider a particle of radius r, at distance d from the sun
then
volume of particle, V = 4*pi*r^3/3
mass of particle = rho*4*pi*r^3/3 (where rho is density of particel)

hence
frontal area = pi*r^2
force due to radiation pressure on the particle = P*A
A = pi*r^2
P = S/c
S = Power/4*pi*d^2
hence
Fp = Power*r^2/4*d^2*c

from equilibrium
Power*r^2/4d^2*c = GM*rho*4*pi*r^3/3d^2 ( where M is mass of sun)
Power = GM*rho*16*pi*rc/3
hence for outward Force to exceed

GM*rho*16*pi*rc/3 < power
power = 3.85*10^26 W
G = 6.67*10^-11
M = 1.989*10^30 kg
c = 3*10^8 m/s
hence

1732.087162*rho*r < 1
hence
density = rho
radius of particle = r
rho*r < 5.77338*10^-4 kg/m^2
for rho = 2 g/cm^3 = 2000 kg/m^3
r < 2.88669*10^-7 m
r < 0.2886690755 um
hence very fine particle of radius of the order of micro meters get blown off

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