A dust grain orbiting the Sun (or in a planetary ring system) absorbs and then re-emits solar radiation. Since the light is radiated from the Sun isotropically and re-emitted by the grain preferentially in the direction of motion, the particle is decelerated (it loses angular momentum) and spirals in toward the object it is orbiting. This process (known as the Poynting–Robertson effect) is just a consequence of the headlight effect discussed in Example 4.3.3.
(a) If a dust grain orbiting the Sun absorbs 100\% of the energy that strikes it and all of the energy is then re-radiated so that thermal equilibrium is maintained, what is the luminosity of the grain? Assume that the particle's cross-sectional area is σg and its distance from the Sun is r.
(b) Show that the rate at which angular momentum is lost from a grain is given by
dtdL=−4πr2σgmc2L⊙L,(21.6)
where m and L=mvr are the mass and angular momentum of the grain, respectively, and L⊙ is the luminosity of the Sun. Hint: Think of radiated photons as carrying an effective mass away from the grain; the effective mass of a photon is just mγ=Eγ/c2.