Ken Tapping, August 17, 2011

One of the stories often told about Albert Einstein is that he could measure the universe on the back of an envelope, while the rest of us need large telescopes. Like most stories of this kind, it is a bit misleading. However, it is likely that Einstein did employ one of the most useful tools in science, one that we all use at one time or another. It often happens when we are talking with other scientists over coffee and wish to illustrate what we are suggesting. We grab the only piece of paper close by, which could be an old envelope, a paper napkin or some other impromptu writing surface. We do a “back of the envelope calculation.”

In one of these calculations, the problem is stripped down to its barest essentials. With no textbooks around we have to work with whatever physics is in our heads, and we use only information that is well established and widely accepted, so that we are likely to remember it. Calculations like this usually do not produce a precise answer, but they at least put you in the ballpark, sometimes better, and are usually intended to indicate whether the problem is worth examining in full detail. Quite often, these simple approaches produce very intriguing answers. The following is an example where a basic measurement, combined with some high-school mathematics and physics, and some well-established information, will reveal something quite surprising.

If you take a piece of black cardboard and hold it facing the Sun on a really clear day, that cardboard will rapidly get hot, because it is capturing almost 1400 watts of solar energy. Now imagine that piece of cardboard is only one square metre of a huge sphere of black cardboard surrounding the Sun, with a diameter equal to the diameter of the Earth’s orbit. That cardboard sphere will capture all the Sun’s energy. We don’t have to make this sphere because it is easy to calculate the area of that sphere and then multiply by the power captured per square metre, giving the Sun’s total energy output. The result is a huge number, four followed by 26 zeros. That sounds large, but there are stars producing over 100,000 times that amount!

Let’s keep going. The Sun’s diameter is about 1.4 million kilometres. Using basic physics in combination with the science of helioseismology, we have a rough idea of what the Sun is like inside, just as seismology has revealed the internal structure of our world. We have found that the energy is produced by nuclear fusion in the Sun’s core, which has a diameter of about 350,000 km. So its volume is about 2 followed by 25 zeroes. If we divide this number into the total energy output we get the average amount of energy produced in each cubic metre of the Sun’s core. The result is a bit of a surprise – less than 20 watts. A light bulb puts out more than that. A typical human body produces about ten times that amount of heat. So does a good compost heap!

The big difference is that each cubic metre of the Sun’s core can keep this up for millions or billions of years, because there is plenty of hydrogen fuel in that cubic metre. Moreover, from a gardening point of view, we like our compost heaps to complete their work in a much shorter time than that.