The intensity of solar radiation reaching the top of the...

The intensity of solar radiation reaching the top of the Earth's atmosphere is $1340 \mathrm{W} / \mathrm{m}^{2} .$ The temperature of the Earth is affected by the so-called greenhouse effect of the atmosphere. That effect makes our planet's emissivity for visible light higher than its emissivity for infrared light. For comparison, consider a spherical object with no atmosphere, at the same distance from the Sun as the Earth. Assume that its emissivity is the same for all kinds of electromagnetic waves and that its temperature is uniform over its surface. Identify the projected area over which it absorbs sunlight and the surface area over which it radiates. Compute its equilibrium temperature. Chilly, isn't it? Your calculation applies to (a) the average temperature of

Key Concepts

Solar Constant

The solar constant defines the intensity of solar radiation incident on an area perpendicular to the Sun's rays at the top of a planet's atmosphere. It is a fundamental parameter in calculating the energy inputs for bodies in space and sets the baseline for assessing their thermal behavior and energy balance.

Geometry of Absorption and Radiation

For a spherical object, the absorption of solar radiation is determined by its projected (cross-sectional) area, which is a circle, while the radiation of energy occurs over its entire, much larger, surface area (the sphere's surface). This geometric discrepancy is critical in understanding why a body in space has a lower equilibrium temperature than might be assumed from its absorption area alone.

Energy Balance and Equilibrium Temperature

The concept of energy balance states that, in thermal equilibrium, the power absorbed by an object equals the power it emits as radiation. The Stefan-Boltzmann law links the radiation power to temperature, allowing the calculation of the equilibrium temperature of a spherical body by equating incoming solar flux (adjusted for projected area) to outgoing thermal radiation (from the full surface area).

Emissivity

Emissivity is a measure of how effectively a surface emits thermal radiation compared to an ideal blackbody. It plays a crucial role in a body's energy balance, as differences in emissivity between various wavelengths (such as visible and infrared) influence the temperature an object reaches under incident solar radiation.

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