cosmology

Avogadro’s Number and the Stars: An Anthropic Near-Miss

Avogadro’s number — the count of atoms in a mole — is about 6\times 10^{23}. The number of stars in the observable universe is, depending on whose galaxy survey you trust, somewhere between 10^{22} and 10^{24}. These are, to within the squint of an order of magnitude, the same number. Sean Carroll asked if this is a coincidence.

There are other big number coincidences that actually do have a deep reason.
[examples]

These deep reasons are often due to anthropic selection: the existence of observers like us requires things like stable stars, solid planets, and creatures that don’t immediately die due to fall damage if they trip, and that requires relationships between the dimensionless numbers describing physics to be in relatively narrow ranges (that is, usually a few orders of magnitude around “just right”). Is there a good anthropic selection effect for the mole-star coincidence?

Both numbers in the dimensionless currency

The trick to making any large-number coincidence say something is to write both quantities in terms of the same fundamental dimensionless constants. Dimensionless, because they cannot depend on our own parochial Earth-measures.

The most relevant one here is the gravitational fine-structure constant, \alpha_G \equiv \frac{G m_p^2}{\hbar c} \approx 6\times10^{-39}, the gravitational analogue of the ordinary fine-structure constant \alpha \approx 1/137. It measures the gravitational attraction between two protons against the natural quantum-electromagnetic scale, and its smallness — gravity is roughly 10^{38} times feebler than electromagnetism — generates almost every large number in physics. The whole game below is figuring out *which power* of \alpha_G each of our two numbers is.

The stars

The number of stars in the observable universe can be estimated as:
N_\star \sim \frac{\text{baryons within the horizon}}{\text{baryons per star}}.

Both pieces are classic results from the anthropic-cosmology literature of the late 1970s, principally Carr and Rees (1979) building on Carter (1974):

Baryons per star.:A star’s mass is essentially the Chandrasekhar mass, fixed by balancing gravity against quantum-mechanical (electron-degeneracy and EM) pressure. It contains
N_{\star,\text{atoms}} \sim \alpha_G^{-3/2} \sim 10^{57} \text{ nucleons}, about 2\times10^{30} kg or one solar mass.

Baryons in the horizon: The number of nucleons within the observable universe is the Eddington–Dirac large number, N_{\text{obs}} \sim \alpha_G^{-2} \sim 10^{78}\text{-}10^{80}.

Carter’s insight was that this isn’t an arbitrary coincidence: requiring the universe to last long enough for stars to form and synthesise the elements ties the horizon’s baryon content to \alpha_G^{-2}.

Dividing, N_\star \sim \frac{\alpha_G^{-2}}{\alpha_G^{-3/2}} = \alpha_G^{-1/2} \sim 10^{19}, and then structure-formation and horizon prefactors — the baryon fraction \Omega_b, the efficiency of turning gas into stars, the choice of particle horizon versus Hubble radius — drag this up by a few orders to the observed 10^{22}–$10^{24}$. So, modulo the usual pile of order-unity factors raised to powers, \boxed{N_\star \sim \alpha_G^{-1/2} \times (\text{structure factors}).}

Avogadro

Now the harder and more disreputable half. Avogadro’s number is not a constant of nature in the way the Chandrasekhar mass is — since 2019 it is a defined integer, fixed by our choice of the kilogram, and originally it was “the number of atoms in 12 grams of carbon-12”. So what we are really asking is: why is a human-chosen unit of mass about 10^{23} nucleons? And that reduces to: why is a human about 10^{27}–10^{28} atoms?

Here the relevant scholarship is the maximum-organism-size argument of Press (1980) and Barrow and Tipler’s The Anthropic Cosmological Principle (1986). The cleanest version asks how large a complex, solid, land-dwelling creature can be and still survive toppling over under its own weight — bond energy versus gravitational potential energy released in a fall. Grinding through the algebra (the surface gravity of the largest possible rocky planet itself scales as \alpha_G^{1/2}, which is where gravity enters), the number of atoms in the largest fall-survivable organism comes out as N_{\max} \sim \left(\frac{\alpha}{\alpha_G}\right)^{3/4}\beta^{3/4} \sim \alpha_G^{-3/4}\times(\text{atomic factors}),
where \beta = m_e/m_p \approx 1/1836. Numerically (\alpha/\alpha_G)^{3/4} \sim 10^{27}, and the \beta^{3/4}\sim 10^{-2.4} correction brings a robustly-built creature down to \sim 10^{24}–10^{25} atoms, i.e. tens of grams to a few kilograms. Thinking beings sit an order or two above this “unbreakable” ceiling — which is why we *do* break bones falling our own height as adults.

To turn an organism mass into a unit of mass, posit that any creature will define its everyday mass unit as some perceptually convenient fraction \phi of its own body — a handful, a mouthful, \phi \sim 10^{-2} to 10^{-3}. Then its “mole” — unit mass divided by nucleon mass — is
\boxed{N_A \sim \phi\,\beta^{3/4}\left(\frac{\alpha}{\alpha_G}\right)^{3/4} \sim \alpha_G^{-3/4}\times(\text{chemistry + convention factors}) \sim 10^{24}.}

The observed 6\times10^{23} sits comfortably inside. As a first-principles estimate of Avogadro’s number from the strength of gravity, that is a little startling.

The swindle

Stars and Avogadro are both enormous because gravity is \sim10^{38} times weaker than electromagnetism. Very cool.

But… N_\star \sim \alpha_G^{-1/2}, N_A \sim \alpha_G^{-3/4}.

These are different powers of \alpha_G. Against the base \alpha_G^{-1}\sim10^{38.2}, the bare exponents predict \alpha_G^{-1/2}\sim10^{19}, \alpha_G^{-3/4}\sim10^{29}, which are ten orders of magnitude apart. The coincidence we set out to explain — that N_\star and N_A are both \sim10^{23} — is therefore not an identity of exponents. It is the result of the two prefactor piles pushing from opposite directions: the stellar count is dragged up by \sim10^{3}–10^{5} of structure-formation factors, and the Avogadro count is dragged down by \sim10^{-5}–10^{-6} of \beta^{3/4}\phi chemistry-and-convention factors. They meet in the middle by a kind of accident.

The argument explains the scale compellingly and the coincidence barely at all.