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. Mammals get roughly 10^9 heartbeats per lifetime (due to allometric scaling). Stars live for roughly a light-crossing time of their radius multiplied by 1/\alpha_G (see below; this is due to Carter). Carbon-12 has a 7.65 MeV excited state at almost exactly the energy needed for triple-alpha fusion to proceed (Hoyle).

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?

This is a derivation of why both numbers live in the same large-number ecosystem, followed by a demonstration that the near equality itself is mostly accidental.

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. (However, real star masses are set by the initial mass function and fragmentation physics supply substantial astrophysical pre-factors bounded by this.)

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}. Additional microphysical and cosmological factors bringing it toward \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.

The criterion-sensitivity makes it worse. The exponent 3/4 in the Avogadro estimate is not robust: depending on whether you bound the organism by fall-survival, static crushing, or Euler buckling, it wanders over the range [3/4, 1]. Against a base of 10^{38}, each \Delta p = 0.1 in that exponent moves the prediction by 3.8 orders of magnitude! A coincidence whose quality depends on which engineering failure mode you assume for hypothetical aliens is not a coincidence that is trying to tell you something deep.

If I wanted to get a particular coincidence, I had more than enough hidden knobs to twist to ensure I got it.

Is all anthropic stuff bogus?

Clearly not. But not everything can be derived tightly, and when the estimates get a lot of prefactors or choices for exponents, then there is too much room for fudging. It is like Fermi estimation: estimates need to be within the right order of magnitude and the errors roughly evenly distributed between too large and too small.

Sometimes coincidences tell interesting stories. As noted in Tegmark, Aguirre, Rees & Wilczek (2006) three epochs in the early universe — matter-radiation equality, recombination, and the onset of structure growth — all happen at suspiciously similar redshifts (z \sim 1100-3400), spanning a tiny time window when they apparently do not have a reason to. As they point out these timescales are tied together by \alpha, \beta and the baryon-to-photon ratio that also strongly affect the existence of observers like us. We couldn’t find ourselves in a universe where these epochs were very different.

Carter and Rees pointed out that \alpha_G \sim \alpha^{20}. The reason is that our kind of life requires both stars where energy transport is by radiation and by convection. Radiative stars are the high-mass ones that go supernova and scatter the heavy elements life is built from. Convection-driven stars are red-yellow stars with early winds and gentler temperatures that are implicated in forming rocky planets, and whose surface temperatures are low enough that their light can drive molecular chemistry rather than breaking bonds. The mass dividing the two regimes works out to roughly \alpha_G^{-2}\alpha^{10} m_p​, while actual stellar masses cluster a few decades around \alpha_G^{-3/2} m_p. Those two mass scales have to overlap. Setting them comparable gives the requirement \alpha_G \sim \alpha^{20}. Carter’s own, more careful, 1974 inequality was \alpha_G \lesssim \alpha^{12}\left(\frac{m_e}{m_p}\right)^{4} which turns into the power 20 if one uses the empirical near-equality m_e/m_p \approx \alpha^2 (good to about an order of magnitude) – itself bound by anthropic constraints (see Barrow & Tipler).

Anthropic arguments can give interesting bounds. In 1987 Weinberg made an anthropic bound: \Lambda must be small enough that galaxies can form before vacuum energy halts collapse, which predicted a value within a couple orders of magnitude of what was measured a decade later. Similarly, we know proton halflife must be more than 10^{17} years, or we would die of radiation poisoning.

A large-number relation is only worth taking seriously when it is an identity of exponents in some fundamental way, not a collision of prefactors.

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