In Problem 21.16, the Poynting–Robertson effect was shown to be important in understanding the dynamics of ring systems. The Poynting–Robertson effect, together with radiation pressure, is also important in clearing the Solar System of dust left behind by comets and colliding asteroids (the dust that is responsible for the zodiacal light).
(a) Beginning with Eq. (21.6), found in Problem 21.15, show that the time required for a spherical particle of radius R and density ρ to spiral into the Sun from an initial orbital radius of r≫R⊙ is given by Eq. (22.4). Assume that the orbit of the dust grain is approximately circular at all times.
(b) Find the radius of the largest spherical particle that could have spiraled into the Sun from the orbit of Mars during the Solar System's 4.57-billion-year history. Take the density of the dust grain to be 3000 kg m−3.