At our distance from the Sun, the intensity of solar...

At our distance from the Sun, the intensity of solar radiation is 1370 $\mathrm{W} / \mathrm{m}^{2}$ . The temperature of the Earth is affected by the greenhouse effect of the atmosphere. This phenomenon describes the effect of absorption of infrared light emitted by the surface so as to make the surface temperature of the Earth higher than if it were airless. For comparison, consider a spherical object of radius $r$ with no atmosphere at the same distance from the Sun as the Earth. Assume its emissivity is the same for all kinds of electromagnetic waves and its temperature is uniform over its surface. (a) Explain why the projected area over which it absorbs sunlight is $\pi r^{2}$ and the surface area over which it radiates is $4 \pi r^{2} .$ (b) Compute its steady-state temperature. Is it chilly?

At our distance from the Sun, the intensity of solar radiation is 1370 $\mathrm{W} / \mathrm{m}^{2}$ . The temperature of the Earth is affected by the greenhouse effect of the atmosphere. This phenomenon describes the effect of absorption of infrared light emitted by the surface so as to make the surface
temperature of the Earth higher than if it were airless. For comparison, consider a spherical object of radius $r$ with no atmosphere at the same distance from the Sun as the Earth. Assume its emissivity is the same for all kinds of electromagnetic waves and its temperature is uniform over its surface. (a) Explain why the projected area over which it absorbs sunlight is $\pi r^{2}$ and the surface area over which it radiates is $4 \pi r^{2} .$ (b) Compute its steady-state temperature. Is it chilly?

Key Concepts

Blackbody Radiation and the Stefan-Boltzmann Law

Blackbody radiation refers to the idealized emission of electromagnetic energy by an object purely due to its temperature, as described by the Stefan-Boltzmann law. This law states that the power radiated per unit area is proportional to the fourth power of the object's absolute temperature, which is crucial for computing the equilibrium temperature.

Greenhouse Effect

The greenhouse effect is a process where certain gases in an atmosphere trap infrared radiation, effectively warming the surface beyond what would be expected from direct solar heating alone. While not applied in the calculation for an object without an atmosphere, it is essential for understanding temperature differences in planetary atmospheres.

Projected Area versus Radiating Surface Area

This concept clarifies the difference between the area exposed to incoming radiation and the area over which radiation is emitted. For a spherical object illuminated by a directional source like the Sun, the incoming radiation effectively covers a circular cross-section (?r²), while the object emits radiation uniformly from its entire surface (4?r²).

Steady-State Energy Balance

Steady-state energy balance occurs when a system's energy input equals its energy output, resulting in a constant temperature. In the context of an object in space, the solar radiation absorbed must equal the energy radiated away to maintain thermal equilibrium.

Transcript

Please give Ace some feedback