Avogadro’s Number and the Stars: An Anthropic Near-Miss – Andart II
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Avogadro’s number — the count of atoms in a mole — is about . The number of stars in the observable universe is, depending on whose galaxy survey you trust, somewhere between and . 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 heartbeats per lifetime (due to allometric scaling). Stars live for roughly a light-crossing time of their radius multiplied by (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, the gravitational analogue of the ordinary fine-structure constant . It measures the gravitational attraction between two protons against the natural quantum-electromagnetic scale, and its smallness — gravity is roughly times feebler than electromagnetism — generates almost every large number in physics. The whole game below is figuring out *which power* of each of our two numbers is.
The stars
The number of stars in the observable universe can be estimated as:
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
about 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, . Additional microphysical and cosmological factors bringing it toward
Carter’s insight was that this isn’t an arbitrary coincidence: requiring the universe to last long enough for...