Reference: Carroll, Bradley W. & Ostlie, Dale A. (2007), An Introduction to Modern Astrophysics, 2nd Edition; Pearson Education – Chapter 3, Problem 3.2.

The luminosity of the Sun is

$\displaystyle L_{S}=3.839\times10^{26}\mbox{ watts} \ \ \ \ \ (1)$

On a sphere of radius ${r}$ centred at the Sun, the power received is therefore

$\displaystyle F=\frac{L_{S}}{4\pi r^{2}} \ \ \ \ \ (2)$

which is the inverse square law for radiation from a spherically symmetric source. At the distance of the Earth, ${r=1\mbox{ AU}=1.496\times10^{11}\mbox{ m}}$ so the solar flux, known as the solar irradiance, is

$\displaystyle F_{S}=1365\mbox{ W m}^{-2} \ \ \ \ \ (3)$

This is the power received when the Sun is directly overhead, which can occur only between the tropics of Cancer and Capricorn. At my latitude of around ${56^{\circ}\mbox{ N}}$, the highest the Sun can get in the sky occurs at noon on the summer solstice, when the Sun’s declination is around ${+23^{\circ}}$, putting it at an altitude of ${\left(90-56+23\right)=57^{\circ}}$. The flux here is therefore

$\displaystyle F_{summer}=F_{S}\sin57^{\circ}=1145\mbox{ W m}^{-2} \ \ \ \ \ (4)$

At the winter solstice, the Sun’s maximum altitude is ${90-56-23=11^{\circ}}$, giving a flux of

$\displaystyle F_{winter}=F_{S}\sin11^{\circ}=260\mbox{ W m}^{-2} \ \ \ \ \ (5)$

I’ve actually measured this in my own backyard with my weather station (see the High Solar Rad graph about halfway down the page) and the numbers do check out; in June the highest recorded radiation is between 1100 and 1200 and in December it peaks at around 200.

A somewhat dated comparison is to look at the radiation emitted by an old-fashioned 100
W light bulb. In order for us to receive the solar irradiance from such a bulb, we’d need to be at a distance

$\displaystyle r=\sqrt{\frac{100}{4\pi\times1365}}=7.6\mbox{ cm} \ \ \ \ \ (6)$

More modern light bulbs such as LEDs emit just as much light as the old high-power bulbs but consume much less power (a typical LED consumes under 5 watts). This is because most of the energy in an LED goes into emitting light while in an old 100W bulb, a large portion of the energy is radiated as heat (or to be more precise, infrared radiation).

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