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. But very small dust particles are conspicuously absent. Astronomers believe that the very small dust particles have been blown out of the solar system by the sun. By comparing the forces on dust particles, determine 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, black, and have a density of $2000\mathrm{kg}/{\mathrm{m}}^3$. The sun emits electromagnetic radiation with power $3.9\times {10}^{26}\mathrm{W}$.

Step by step solution

01

Data given

$\begin{array}{r}\rho =2000\mathrm{kg}/{\mathrm{m}}^3\\ P=3.9\times {10}^{26}\mathrm{W}\end{array}$

$c=3\times {10}^8\mathrm{m}/\mathrm{s}$

$A=4\pi r$

02

Compare the dust particle's forces

Equation that we must always use:

$\begin{array}{c}{F}_g=\frac{4\pi }{3}\frac{G{M}_{sun}\rho R^3}{r^2}\\ \end{array}$

${\mathrm{F}}_{\text{rad}}=\frac{{\mathrm{P}}_{\mathrm{sun}}}{4\mathrm{c}}\frac{{\mathrm{R}}_{\mathrm{dust}}}{{\mathrm{r}}^2}$

To remain dust inside the planetary system $F_{\mathrm{rad}}\le F_g$,

$F_{\mathrm{rad}}\le F_g$

$\frac{4\pi }{3}\frac{G{M}_{sun}\rho R^3}{r^2}\le \frac{P_{sun}}{4c}\frac{R_{\text{dust}}}{r^2}$

03

Find Rdust 

At last calculate$R_{\text{dust}}$

$R_{\text{dust}}\ge \frac{3P_{sun}}{16\pi G{M}_{sun}c\rho }$

$R_{dust}=\frac{3\left(3.9\times {10}^{26}W\right)}{16\pi \left(6.67\times {10}^{-11}\right)\left(2\times {10}^{30}m\right)\left(3\times {10}^{8}m/s\right)(2000kg/m^3)}$

$R_{\text{dust}}=2.93\times {10}^{-7}\mathrm{m}\to 0.293\mu \mathrm{m}$

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