Poynting–Robertson effect

Remarks

Idea: in the frame of the moving grain, the starlight arrives slightly from the front because of the aberration of light. So the radiation force has a small component against the motion. Equivalently, in the frame of the star, the light re-emitted by the grain is Doppler shifted forward and carries away momentum along the motion. The drag gives a torque $-F_{\text{PR}} \, r$. It equals the rate of change of the orbital angular momentum $m \sqrt{G M_\star r}$, with $m = \frac{4}{3}\pi s^3 \rho$, and this gives $dr/dt$. Here $M_\star$ is the mass of the star and $G$ is the gravitational constant. Integrating $r \, dr$ from $r_0$ to $0$ gives $t_{\text{PR}}$.

Example: for a grain with $s = 1$ mm and $\rho = 2000$ kg/m$^3$ starting at $1$ astronomical unit from the Sun, the formula gives about $4 \times 10^{13}$ s, roughly $1.4$ million years. Because $t_{\text{PR}}$ grows with $s$, small grains are removed first, which clears dust from the inner Solar System and feeds the zodiacal cloud.

For very small grains the hypothesis on radiation pressure fails: the outward pressure is comparable to gravity and the grain is blown out of the system instead of spiralling in.

John Henry Poynting described the effect in 1903, and Howard P. Robertson gave its relativistic treatment in 1937.

Going deeper

For a student

The effect is a first-order relativistic correction to radiation pressure, of relative size $v/c$, but it acts steadily over many orbits and so changes the orbit secularly. The spiral time is proportional to the size of the grain and to the square of the initial distance. It sets the lifetime of dust in debris disks and in the Solar System, together with collisions and the outward push of radiation pressure.