**Answer: The Sun transmits its energy to Earth across the emptiness of space via radiation. Each square meter of surface at a temperature, T, emits radiation at a rate of σT4, where T is expressed in Kelvin (important!) and σ = 5.67×10−8 W/m²/K4. This constant is easy to remember via the sequence 5-6-7-8. Ignoring for now the subtleties of greenhouse gases, the surface of Earth—typically at 288 K—emits 390 W/m². The Sun, on the other hand, at 5800 K, emits 64 MW per square meter!**

**Summing over the area of the spherical Sun, at 109 times the radius of Earth, we find the total radiant power of the Sun to be a whopping 3.9×1026 W. Now that’s a light bulb! The Sun’s radiant energy spreads into all directions, creating a sphere of light. At the distance of the Earth, that sphere has an area of 4πr² ≈ 2.8×1023 m², where r is the mean Earth-Sun distance. Dividing these huge figures, we find that the radiant intensity at Earth is 1370 W/m²—which I hope will be a familiar number by now for Do the Math readers.**

**We can also turn the σT4 relation on its head and say that a patch of full sun (at the ground) receiving 1000 W/m² corresponds to a radiant temperature of 364 K, or a blistering 91°C. This means that a black panel in full sun could get this hot if no paths other than radiation were available for cooling the panel. We would then say that the panel is in radiative equilibrium with the Sun. But air can carry away heat by convection. The self-convection of a hot, flat plate will be about 10 W/m² per degree of difference between the panel and the surrounding air. Requiring the sum of radiative and convective losses to add up to the input power of 1000 W/m² yields a solution of about 55°C (328 K; 131°F) if the surrounding air is at 20°C. This assumes that the plate has no heat paths available through the (insulated) back side. If, on the other hand, it is a thin panel allowing convection on both sides, it will be cooler—although the “heat rises” phenomenon will suppress heat flow on the back side relative to the front, if the plate is indeed level. Just for fun, if we get an additional 5 W/m²/K of convective loss off the back, the equilibrium temperature drops to 47°C (117°F). It all seems reasonable.**

**Passive Solar: Putting Heat to Use**

**The simplest way to replace fossil fuel energy with solar energy is called a window. A single uncoated piece of glass will transmit 92% of visible light (the rest reflected) when light comes straight in (down to 75% at a 20° grazing incidence, 60% at 10° grazing). The glass is opaque to ultraviolet light and mid- to far-infrared (IR) light, but lets over 95% of the unreflected incident solar spectrum pass.**

**Considering that windows in houses/buildings tend to be vertical, we can evaluate the energy input through windows, taking transmission loss, reflection loss, and angular foreshortening into effect. Because the Sun is higher in the sky in the summer, the window appears foreshortened to the direct sunlight, and also reflects more. So a south-facing window automatically admits more heat in the winter than in the summer, with no adjustment. Putting an overhang over the window—ideally with some vertical space between the window and overhang—can eliminate the summer noon-time contribution entirely. The figure below illustrates the fraction of incident direct-sun energy (think 1000 W/m²) admitted by the window. Vertical reference lines indicate the noon-time elevation of the sun at a latitude of 40° for the winter and summer solstices. The noon-day sun will be somewhere between these values all year. Adjustment to other latitudes involves a simple shift of the dashed lines by the latitude difference.**

**Explanation:**