Astrophysics for a Ten-Year-Old Mind

A joule expended continuously for one second is a watt, which means a liter of water, converted to energy could power a 1 watt light bulb for 9 • 1016 seconds, or 2.5 • 1013 hours. Of course if it were a 100 watt bulb it would only burn for 2.5 x 1011 hours, and you might find it more interesting to burn 1011 light bulbs for two and a half hours instead. It is then reasonable to ask your partner how large an area that many light bulbs would cover. She'll find, after a little work (especially if she hasn't seen square roots yet), that if she allows an area of about 10 cm by 10 cm for each bulb, she can tile a square roughly thirty-one kilometers on a side with 1011 light bulbs.

What you've just done is arithmetic. Now for the science.

What would happen if you turned all those light bulbs on for two and a half hours?

Would it explode? The amount of energy released would be a little less than a tenth of the energy released in the enormous hydrogen bomb tests of the early 1960s, but the area over which you are releasing it is larger, and you are releasing it over a longer time.

Or would it generate tornadoes? You are releasing less than one one-hundredth of the energy in a typical hurricane, but hurricanes form over vast stretches of the ocean and typically take a week or so to blow themselves out. You are releasing the energy over a comparatively tiny area in less than three hours.

Or would most of the energy escape through the clear sky as a brilliant beacon? Wondering about such possibilities are the roots of science and very appropriate for ten-year olds and curious adults.

Incidentally, the average ten-year old uses about the same amount of energy as a 100 watt bulb, which is why classrooms can warm up so readily. In fact, there is a direct equivalence to food calories which makes a broad range of interesting and silly calculations possible. One joule is equal to about four food calories.

Of course, the calculations can also be serious and scientific. A little bit of algebra that is very appropriate to introduce to fifth graders will allow you to find the mass you need to meet any particular energy requirement. The equation now looks like m = e/c2. For example, given that the Sun emits 4 X 10²⁶ joules per second, how much mass does it convert each second in order to burn? If the Sun is 5 X 10⁹ years (or 1.6 X 10¹⁷ seconds) old, how much mass has it used so far?

It doesn't matter very much what the calculations are. It doesn't matter if the distances or quantities are accurate. What does matter is that students will be exercising the math appropriate to their grade level on one of the most remarkable equations known. Without a refresher from time to time it is unlikely that they will remember the details into adulthood. They don't need to. What is important is that having played with the equation once, they will know that it is tractable, accessible, and so much more than a trademark of science.

MICHAEL CHABIN develops web-based animations that teach math to kids. Some of his animations can be seen at http://www.hypatia.org. He was once operator of the 2.3 meter telescope at Steward Observatory for the University of Arizona, and he is married to an astronomer. His email address is chabin@hypatia.org.

Minding Your Joules

The problem is, of course, understanding what your answer means. Energy is given in joules and not many kids know what a joule is. But they should; it is an extremely intuitive unit. A joule is the effort required to push one kilogram over a meter at an acceleration of one meter per second squared. To get a feel for it, pick up a liter bottle of water and toss it ten centimeters into the air. You've just expended about a joule. Here are some others:

on (Very) Tiny scales...

• breaking a single bond in human DNA requires just 10-20 joules
• a firing neuron needs only 10-9 joules
• for a hop, a flea requires about 10-7 joules

on human scales...

• shooting a basketball jump shot from the free-throw line takes about 15 joules
• throwing a fastball requires 120 joules
• rolling a strike with a bowling ball will cost you 230 joules

on planet scales...

• the largest hydrogen bomb ever tested produced 2.4 X 10¹⁷ joules
• a typical hurricane amounts to about 3.8 X 10¹⁹ joules

on astronomical scales...

• a supernova releases around 10⁴⁴ joules
• one estimate gives the energy yield of the Big Bang as 10⁶⁸ joules

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