Radiation Pressure — Question 6

Question 6

Radiation and gravity compete for a small absorbing grain

A star has luminosity L and mass M. A spherical black grain has radius a and density rho at distance r much larger than a. Assume isotropic stellar radiation, complete absorption, and isotropic reradiation in the grain rest frame. Neglect motion-dependent effects.

1. Derive the ratio of radial radiation force to gravitational attraction.

2. Find the grain radius for equal forces and state how the ratio varies with distance and size.

Question 6 - Solution

Radiation and gravity compete for a small absorbing grain

(a) Method and calculation

$$F_{\mathrm{rad}}=\frac{L}{4\pi r^2c}\pi a^2,\quad F_g=\frac{GM}{r^2}\frac{4\pi\rho a^3}{3}.$$$$\boxed{\beta=F_{\mathrm{rad}}/F_g=\frac{3L}{16\pi cGM\rho a}}.$$

(b) Interpretation and further calculation

$$\beta=1\Rightarrow\boxed{a_{\mathrm{crit}}=\frac{3L}{16\pi cGM\rho}}.$$$$\boxed{\beta\propto a^{-1},\quad\partial\beta/\partial r=0}.$$

Physical check

The absorbing sphere intercepts its projected area pi a squared, while its mass scales as a cubed.