When I was a boy, I read a number of books by Dr. Seuss. One that immediately captured my interest was On Beyond Zebra. I don’t recall much about the book at this point in my life other than the fact that it involved additional letters of the alphabet. Each new letter was introduced and illustrated by the author. The idea there might be unknown letters piqued my youthful interest. Here are the new letters that appeared in that book:
The first book I ever read by Isaac Asimov (1920-1992) was: The Universe, From Flat Earth to Quasar, which I recently re-read. The two books may seem far apart, but I made a connection between them when I came across a section on how the sun generates its energy. The Sun uses nuclear fusion to convert mass to energy. This process is understood using the famous equation E = mc2 developed by Albert Einstein (1879-1955).
The pressures at the center of the sun cause four hydrogen atoms to fuse into a single helium atom. After this process occurs there is a mass imbalance, the four hydrogen atoms have more combined mass than the resulting single helium atom, and the extra mass is converted into energy.
Dr. Asimov states that about 4.2 Tg (Teragrams) of mass is converted to energy every second inside of the sun. He uses pre-metric terms to describe this value as “4 600 000 tons of mass per second.” Unfortunately so does Wikipedia: “the Sun fuses about 620 million metric tons of hydrogen each second.” As I understand it 1 million is 106 and a “metric ton” is a Megagram or 106 grams for 4.2 x 1012 grams per second or 4.2 Tg per second. That’s a lot of grams. Dr. Asimov inquires: “Is it possible for the Sun to support this steady drain of mass at the rate of millions of tons per second? Yes, it certainly is, for the loss is infinitesimally small compared with the total vast mass of the sun.” The currently accepted mass of the sun is, approximately 2 x 1030 kg. This means it’s 2 x 1033 grams, and the proper metric prefix would be?—oh, well, there isn’t exactly a metric prefix for this value. The last magnifying metric prefix is Yotta, which allows the mass to be written as 2 000 000 000 Yg (Yottagrams). Which by current convention it appears there are about three extra metric prefixes needed to express the mass of the sun with a 2, and a minimum of two extra prefixes to use 2000 as a magnitude.
So what does “infinitesimally small” mean? Well the mass lost each second, divided by the total mass of the sun, is 4.2 x 1012 grams/2 x 1033 grams. This value is one divided by 476.19 x 1018 or 0.000 000 000 000 000 000 002 which is quite a tiny ratio. I believe this is indeed a small enough ratio to be “infinitesimally small.” Recall we are talking about 4.2 Tg per second of mass loss. Each gram has 90 TJ (Terajoules) of energy contained within it’s mass. If my computation is correct, then 378 x 1024 joules are released each second. This would be 378 YJ (Yottajoules) per second. We are approaching the limits of the metric prefix Yotta, and in only 1000 seconds we would have 378 000 YJ and see that a new prefix might be useful to describe the power released.
What is notable is that the mass of the sun is not readily expressed with a metric prefix, and it’s not all that massive for a star. It appears that the masses of stars are indeed astronomical. The most massive star is suspected to be R136a1 which is approximately 256 solar masses (a solar mass is the mass of the Sun). This means it has a mass of 512 x 1033 grams or 512 000 000 000 Yg. Clearly we are on beyond Yotta at this point. While I’ve made it clear in the past that astronomical distances are readily expressed with metric prefixes, this is not the case for stellar masses. One can see why R136a1 is described in terms of an equivalent number of solar masses and the metric system is not employed.
Asimov also makes this surprising statement:
Release of energy is always at the expense of disappearance of mass, but in ordinary chemical reactions, energy is released in such low quantities that the mass-loss is insignificant. As I have just said, 670,000 gallons of gasoline must be burned to bring about the loss of 1 gram (1/27 of an ounce). Nuclear reactions produce energies of much greater quantities, and here the loss of mass becomes large enough to be significant.
What I’ve been able find in my research on this subject is both minimal and contentious. It is mostly stated that the amount of mass lost in chemical reactions is “unmeasurable.” The few who venture to put numbers to paper (including a textbook example) end up with magnitudes on the order of 10-33 grams. One example computation has 70 x 10-33 grams as the amount of mass lost in the given chemical reaction. This would be 0.000 000 070 yg (yoctograms) and would indicate a possible need for at least two more metric prefixes. It appears that, at least in theoretical discussions, it might be useful to have two more metric prefixes on the dividing side of the prefixes.
Currently there are 20 metric prefixes from yocto to Yotta. Adding two more prefixes on the magnification side would be useful for some of this astronomical work. It would probably make sense to add a pair to the reducing prefixes also. This would increase the total number to 24 metric prefixes. This is a lot of prefixes, but is far less than the number of magnitudes scientific notation would allow, which would be 60. What I would propose is to consider adding the new prefixes, but at the same time remove the prefix cluster around unity: deca, hecto, deci and centi. They could be separated and relegated into a set of atavistic prefixes which are no longer considered proper modern usage. They would be included as an appendix to the modern prefixes for historical reference, but discouraged for modern use. This simplification would reduce the number of prefixes back to 20 and also provide a larger dynamic range for scientific description.
In early grades it makes sense to me that only the prefixes micro, milli, Kilo and Mega would be taught as the Common Set of Prefixes. These would be the prefixes that students would generally encounter in everyday life (if the US was metric and fully engaged). In Junior High and High School the new set of prefixes I’ve proposed could be taught as the Complete Set of Prefixes. I would argue that all students (and their teachers) should have to memorize and use all these metric prefixes (without the prefix cluster around unity) in their instruction. Textbook authors should not shy away from using Megameters for planetary dimensions, Gigameters for the solar system, and all the other appropriate uses of metric prefixes.
People have objected to my proposal that we teach all students to use all the metric prefixes. They employ the argument that the Common Set of Prefixes is all that is needed for an ordinary person, and the Complete Set of Prefixes is for engineers, scientists and technical people. I reject this view entirely. It produces a scientific apartheid that keeps the public from understanding the important issues of the day, which involve engineering and science more and more everyday. What I have discovered when working with large questions, such as how much the salinity of the ocean would change if we dumped all our fresh water into it, or how much carbon is being belched into our atmosphere over a given period of time, is that these problems are tamed using appropriate metric prefixes. They allow an ordinary citizen to comfortably work with the magnitudes involved. If one talks about hundreds of billions of tons, that is a metaphor, and is not information. If the goal of education in the US is to create the most numerate population on the planet, then a good command of the magnitudes of all the metric prefixes is essential.
I would like to see a song which fixes the order of the metric prefixes in a person’s mind from the smallest to the largest, something similar to Tom Lehrer’s Element Song. Some manner of meaningless acrostic or other method of recalling the order of the 1000 based prefixes should also be developed. With the prefix cluster around unity eliminated, all the magnitudes will be of 1000 and any parsed base unit can be determined. This would allow anyone to look at 1 000 000 000 000 000 grams and immediately relate it to the acrostic or song and “sound out” the size of the number as 1 Pg (Petagram), or conversely be able to take the 1 Pg and work out how many sets of three zeros one would need to express it. This would also be the case for 0.000 000 000 000 001 grams. It could be “sounded out” as 1 fg (femtogram).
When all the metric prefixes no longer apply, that’s when a modern student should viscerally realize they are discussing dimensions that are so large or so small they are mind blowing, and on beyond yocto and Yotta. These values truly exist in an amazing far distant realm.