Why are there so many important numbers?

kw: book reviews, nonfiction, science, numbers, short biographies

A subject that is exercising many physicists and cosmologists is why so many peculiar numbers are needed to define the physics of the Universe, and why they are so seemingly unrelated. Even more, some of them, according to the current theories, need to take rather precise values or the Universe cannot exist, or if it can, it cannot support carbon-based life.

For example, the efficiency of conversion of hydrogen to helium in stars like the sun is very nearly 0.007. (A proton weighs 1.00739 AMU, where the C¹² nucleus is defined to weigh 12.0, and a helium nucleus weighs 4.0015 AMU; 4×1.00739 = 4.02956; subtracting 4.0015 gives 0.02806; dividing by 4.02956 yields 0.00696). Were the efficiency as low as 0.006, to quote James D. Stein, "The neutron and proton would not bond to each other, deuterium would not form, and the Universe would consist of nothing but hydrogen" (We'll get back to the error in this statement later). And were it a little higher, at 0.008, "…it would be far to easy for protons to bond together," and the "big bang" would seemingly have gone on to bang away all the Universe to helium and heavier elements in short order: no hydrogen means no water, and any life that forms would need a different fluid.

Given that nobody has yet determined some tiny (five or fewer) set of really fundamental constants, from which everything else can be derived, we have quite a number of them. The recent discovery of the Higgs boson was supposed to pave the way for a more fundamental physical theory, but that seems about as far off as it did before. My most recent printout of the CODATA list of "important" constants runs to several pages.

The book is Cosmic Numbers: The Numbers That Define Our Universe by James D. Stein, a mathematics professor at CSU Long Beach. Out of the zoo of CODATA constants, he has chosen 13 to explain to us, and even better, he presents short biographies of the scientists whose work led to an understanding of each of them.

Some numbers have dimensions, meaning that their numerical value depends on the system of measurement. Such is Avogadro's Number, 6.0221413×10²³, the number of atoms in 12 grams of the carbon-12 isotope. It is the ratio of the gram to the AMU. By extension, it is the definition for a mole of any substance, where a mole is the weight in grams equal to the atomic or molecular weight of the atoms or molecules. Thus, one mole of pure isotopic iron as Fe⁵⁶ is 56 grams (or, strictly speaking, 55.9349393 grams, because the atomic weight of that isotope of iron is 55.9349393 AMU). Now, suppose instead of grams, we had in history defined a unit mass to be something else, call it a marg, with a mass about 1.66 times as large. Then Avogadro's Number would be, nearly exactly, 10²⁴, and it is likely that scientists would lobby hard to get the marg redefined to make that number exact. Something similar happened fifty or so years ago, when the inch was redefined to be exactly 25.4mm.

Other numbers are dimensionless, such as absolute zero. This is an extrapolated temperature, defined according to the ideal gas law, at which no more heat can be extracted from a substance, and the atomic or molecular motions that define what we mean by "temperature" would cease completely, except for the tiny gyrations needed to avoid violating Heisenberg's uncertainty principle. The "temperature" 0K (K for "kelvins", which have the same size as Celsius degrees), AKA 0R (in which a Réamur is equal in size to a Fahrenheit degree, but the scale begins at absolute zero), needs no units. Zero is zero.

Another dimensionless number is 1/137, the Fine Structure Constant, initially derived from spectroscopy in a magnetic field. Its actual value is 1/137.036 and about six more digits. Though it can be derived from more fundamental constants such as the unit charge and the speed of light, all the units cancel out, so it is the same numerically in all possible systems of units. This isn't one of Dr. Stein's examples. He presents only two dimensionless constants, Avogadro's Number and the efficiency of hydrogen fusion, discussed above. In the latter chapter (Chapter 10), I was surprised at a number of errors that the physicists among his reviewers ought to have caught.

One was the fusion of proton with neutron, mentioned above. Highly energetic P-P collisions are required for the protons to physically approach close enough for one to emit a positron and become a neutron. Then the strong force can take over and fuse the two. The value of actual interest here is the efficiency of P+P→D+e⁺ conversion. A deuteron weighs 2.01355 AMU, so the conversion efficiency is 0.00061. I suspect it is this number, not the 4P→He⁺⁺ efficiency, that matters most. Another error was quite a long discussion of the mechanics of the P-P chain, in which the text uses "electron" instead of "proton" throughout. Electron collisions don't matter in the core of a star. The substance is a plasma. In essence, it is a mass of colliding protons (and deuterons and other nuclei) in a thin soup of unbound electrons, where there is nearly (or entirely) no impediment to P-P collisions except their own positive electric charge. At much lower energies (temperatures up to a few hundred degrees rather than tens of millions), H-H collisions that occur are primarily mediated by interactions between the electron clouds of the H atoms.

OK, gripe over. I confess to being rather staggered by that, but the rest of the book is a delight. We learn, not just the scientific endeavors of Boltzmann or Newton or Boyle, but their lives and something of their personalities. Science is a human activity, and one could say it is the most human of activities: figuring out how things work is our stock in trade (even if we devote our adolescence to figuring out how the opposite sex works!). The beauty of the numbers Dr. Stein has chosen lies in the sheer brilliance needed to first see the requirement for such a quantity, then to ferret out a way to determine what it is. We may see them as obvious in hindsight, but, for example, prior to Newton's insight, a law of common gravitational attraction just didn't fit in anybody's head.

A story is told of Napoleon, challenging his generals to make an egg stand on end. After they'd all given up, he held the egg and rapped it gently on the tabletop, enough to crush the end just a little. Then it stood. One general protested, "Well, that is obvious!", to which Napoleon replied, "It wasn't obvious before you saw me do it."