Soon after the solar system was formed radiation pressure from the sun blew smal

Question

Soon after the solar system was formed radiation pressure from the sun blew small particles of dust out of the system. What radius must a spherical dust particle with a density of 6000 kg/m3 have in order for the gravitational force on the particle to be equal to the force of radiation pressure?

Possibly useful information: gravitational constant G = 6.67x10-11 Nm2/kg2, Power output of the sun Psun= 3.9x1026 Watts, Mass of sun Msun=2.0x1030 kg.

You may assume that the dust particles absorb all incoming light.

A) 3.2x10-9 m

B) 8.1x10-5 m

C) 1.2x10-6 m

D) 5.6x10-6 m

E) 9.7x10-8 m

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How do you complete a problem like this?

Explanation / Answer

E) 9.7*10^-8 m

we know, radius of sun, R = 6.96*10^8 m

Intensity at the surface of the sun, I = Power/Area

= 3.9*10^26/(4*pi*(6.96*10^8)^2)

= 6.407*10^7 W/m^2

Pressure exerted by radiation, P = I/c

= 6.407*10^7/(3*10^8)

= 0.21357 Pa

given, rho = 6000 kg/m^3

let r is the radius of the particle.

volume, V = (4/3)*pi*r^3

mass of the particle, m = rho*V

in the equilibrium

G*M*m/R^2 = P*A

G*M*rho*V/R^2 = P*pi*r^2

G*M*rho*(4/3)*pi*r^3/R^2 = P*pi*r^2

G*M*rho*(4/3)*r/R^2 = P*2

r = (3/4)*P*R^2/(G*M*rho)

= (3/4)*0.21357*(6.96*10^8)^2/(6.67*10^-11*2*10^30*6000)

= 9.7*10^-8 m <<<<<<<<<------------Answer

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