Dark clouds

Earth is surrounded by a layer of atmosphere that is transparent to solar radiation (i.e. the atmosphere material cannot be heated directly by the Sun), but can retain and radiate infrared heat from the Earth (which can be heated by the Sun).

Suppose an enormous amount of black soot enters the atmosphere (perhaps from volcanos or nuclear weapons) that can be heated by the Sun directly. When the soot surrounded Earth reaches equilibrium, it has the stable temperature $T_{soot}$.

Find $T_{soot}/T_0$, where $T_0$ is the temperature of the Earth with no atmosphere whatsoever.

Assumptions

• The surface of the Earth and the soot layer are perfect blackbodies.
• Approximate the soot as a spherical shell around the Earth.