# Energy from the Sun

## ACS Climate Science Toolkit | Energy Balance

Although much hotter on the inside, we can closely approximate the surface of the sun, from which its emission occurs, as a black body at a temperature of about 5800 K. The Stefan-Boltzmann equation then gives the energy flux emitted at the sun’s surface.

S_{S} = (5.67 × 10^{–8} W·m^{–2}·K^{–4})(5800 K)^{4} = 63 × 10^{6} W·m^{–2}

The surface area of a sphere with a radius r is 4πr^{2}. If r_{S} is the radius of the Sun, the total energy it emits is S_{S}4πr_{s}^{2}. As the radiation is emitted from this spherical surface, it is spread over larger and larger spherical surfaces, so the energy per square meter decreases, as illustrated schematically in the diagram below.

The figure at the right compares the experimental solar emission curve observed outside the Earth’s atmosphere to the emission curve for a 5800 K black body located at the sun’s distance from the Earth. The structure in the experimental curve is a result of absorption of some wavelengths by atoms and ions in the cooler layers outside the sun’s emitting surface.

When the energy emitted by the sun reaches the orbit of a planet, the large spherical surface over which the energy is spread has a radius, d_{P}, equal to the distance from the sun to the planet. The energy flux at any place on this surface, S_{P}, is less than what it was at the Sun’s surface. But the total energy spread over this large surface is the same as the total energy that left the sun, so we can equate them:

S_{S}4πr_{s}^{2} = S_{P}4πd_{P}^{2}

S_{P} = S_{S}(r_{s}/d_{P})^{2}

Values for the average planetary distances, d_{P}, and the corresponding S_{P}, calculated using r_{s} = 700,000 km, are given in the table below.

When radiation from the sun reaches a planet, it does not strike all areas of the planet at the same angle. It strikes directly near the equator, but more obliquely near the poles. To find the amount of energy entering the planetary atmosphere (if any) averaged over the entire planet, consider the diagram. The total amount of radiation incident on the planet (and atmosphere) is equal to the amount the planet intercepts to cast the imaginary shadow shown in the diagram. That is, S_{P}πr_{P}^{2}. If the average energy flux over the area of the planet is S_{ave}, the total energy for the planet is S_{ave}4πr_{P}^{2}. These two total energies must be equal, so: S_{ave} = S_{P}/4. These average fluxes are also included in the table. To find out how this incoming energy is connected to the temperature of the planets see __Predicted Planetary Temperatures__.