Solution

$12.1.24$ Estimate the maximum size of aluminum dust particles that would move away
from the Sun in outer space under the pressure of solar radiation.

The power radiated by the Sun is $P$, so the intensity of the radiation at a distance $r$ from the Sun is:

\begin{equation} I = \frac{P}{4 \pi r^2} \end{equation}

Thus, the pressure exerted by the light on the dust particle is:

\begin{equation} \mathcal{P} = 2 \frac{I}{c} = \frac{P}{4 \pi r^2 c} \end{equation}

The corresponding force is:

\begin{equation} F = 2 \pi r_1^2 \mathcal{P} = \frac{P r_1^2}{2 r^2 c} \end{equation}

This force must be at least equal to the gravitational force:

\begin{equation} F = \frac{G M_s m_d}{r^2} \quad \text{where} \quad m_d = \rho \frac{4 \pi r_1^3}{3} \end{equation}

Finally:

\begin{equation} \frac{G M_s m_d}{r^2} = \frac{P r_1^2}{4 r^2 c} \rightarrow r_1 = \frac{3 P}{8 \pi M_s G c \rho} \approx 1 \, \mu\text{m} \end{equation}