An Introduction to Modern Astrophysics · Problem 21.16

The Milky Way Galaxy — Carroll & Ostlie

Physics — Astrophysics · poynting-robertson-spiral-time-saturn

Problem

(a) Beginning with Eq. (21.6), show that the time required for a spherical particle of radius

R

and density

\rho

to spiral into Saturn from an initial orbital radius

R_0

is given by

t_{\rm Saturn} = \frac{8\pi\rho c^2}{3L_\odot}\,R\,r_S^2\,\ln\left(\frac{R_0}{R_S}\right),

where

R_S

is the radius of the planet and

r_S

is its distance from the Sun. Assume that the orbit of the particle is approximately circular at all times and that it is always a constant distance from the Sun. (b) The E ring is known to contain dust particles having average radii of 1

\mu

m. If the density of the particles is 3000 kg m

^{-3}

, how long would it take for a typical particle to spiral into the planet from an initial distance of 5

R_S

? (c) Compare your answer in part (b) to the estimated age of the Solar System. Could the E ring be a permanent feature of the Saturnian system without a source to replenish the ring? Note that the small moon Enceladus orbits Saturn in the E ring.

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