Large quantities of dust should have been left behind after the creation of the
ID: 1547499 • Letter: L
Question
Large quantities of dust should have been left behind after
the creation of the solar system. Larger dust particles, comparable in size to soot and sand grains, are common. They create shooting stars when they collide with the Earth’s atmosphere. However, very small dust particles are conspicuously absent in our solar system. Astronomers believe that these very small dust particles have been “blown out” of the solar system by the Sun.
Compare the forces of gravity and radiation pressure on such dust particles to calculate the diameter of the smallest dust particles that can remain in the solar system over long periods of time.
(Assume that the dust particles are spherical, completely black, and have a density of 2000 kg/m3. Also, assume that the Sun emits electromagnetic radiation with power 3.9 x 1026 W.)
Explanation / Answer
This is basically about solar sailing. We want the radiation force that produces equilibrium with the sun's gravitational force.
With pluto having been fired from the planetary society, the radius of the solar system R can now be defined as that of neptune's orbit, ~4.5E12 m.
However, both gravitational and radiation forces are inversely proportional to R^2, so R is irrelevant. If radiation drives a particle away from the sun when near it, it will drive it out of the solar system.
But we'll include R in the equations so you can see the derivation of the answer.
Power intensity I = power/(4R^2)
Radiation pressure on a nonreflecting (black) body P = I/c (see ref.)
Radiation force Fr = PA = Pr^2, where A is the disk area the particle presents to the radiation and r is the particle radius.
Thus Frad = power*r^2/(4cR^2) = power*r^2/(4cR^2)
Fgrav = GMm/R^2, where M is solar mass and m is particle mass = density*4/3*r^3
Fgrav = Frad = GM*density*4/3*r^3/R^2 = power*r^2/(4cR^2) ==>
GM*density*4/3*r = power/(4c) ==>
r = 3*power/(4c*GM*density*4) = 3*power/(16c*GM*density) (answer)
Using G = 6.67428E-11 N-m^2/kg^2 and M (solar mass) = 1.99E30 kg, I get r = 2.923E-7 m, diameter = 5.846E-7 m
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