The eclipse argument for God’s existence

Can you calculate the approximate probability that a satellite would form so as to yield perfect eclipses? Or specifically our moon, in the fomation of the solar system.

ChatGPT said:

Thought for 20 seconds

Great question — fun to make this concrete. I ran a straightforward Monte-Carlo experiment (and reached the same qualitative conclusion you’d expect): a moon that just happens to subtend essentially the same angle as the Sun (the geometry that makes long, dramatic total eclipses possible) is not impossible but quite uncommon if you treat moon sizes and orbital radii as plausibly variable. Below I show (A) what I assumed, (B) the numerical results, (C) how to read them into a probability for “a moon like ours”, and (D) important caveats.

A — Assumptions used (explicit)

I built a simple, transparent model conditioned on an Earth-like planet orbiting a Sun-like star at 1 AU. These are the main choices:

• Star radius = Sun (696,340 km), planet at 1 AU (149,597,871 km).

• Planet radius = Earth (6,371 km) only to set sensible orbital ranges.

• Moon physical radius sampled log-uniformly between 100 km and 4,000 km (this covers tiny moons up to large satellites like Earth’s).

• Moon orbital semi-major axis sampled log-uniformly between 3 × R⊕ (rough Roche limit proxy) and 60 × R⊕ (Earth–Moon ≈ 60 R⊕).

• Computed instantaneous apparent angular diameters from geometry and compared the moon’s angular diameter to the Sun’s.

• Ran 200,000 trials and counted the fraction that match to various tolerances (±0.5%, ±1%, ±2%, ±5% of the Sun’s angular diameter).

This setup estimates P(match | a moon exists with radius & orbit drawn from the above ranges). I did not try to model the astrophysical probability that a giant-impact moon forms — that’s a separate multiplicative factor discussed below.

B — Numerical results (probability conditional on a moon with those parameter ranges)

Fraction of random moon radius/orbit pairs whose angular size matches the Sun’s from the planet:

• within ±0.5% → ≈ 0.25% (≈ 1 in 400)

• within ±1% → ≈ 0.51% (≈ 1 in 196)

• within ±2% → ≈ 1.03% (≈ 1 in 97)

• within ±5% → ≈ 2.6% (≈ 1 in 38)

If I condition only on large moons (radius ≥ 1,000 km — roughly the “giant impact / big moon” regime), the probabilities are a little smaller:

• within ±1% → ≈ 0.28%.

• within ±5% → ≈ 1.4%.

So: given a moon of plausible sizes and orbital radii, a moon whose apparent diameter matches the Sun to ±1% is on the order of a few tenths of a percent to a few percent (depending on the precise parameter restrictions).

C — From the conditional probability to the overall probability for “a system like ours”

The numbers above are conditional on a moon existing with the sampled parameter ranges. To get the overall probability that a terrestrial planet ends up with a moon that produces near-perfect eclipses you must multiply by:

1.  P(planet gets a sizable moon at all) — the astrophysical probability of forming a large moon (giant-impact formation, capture, co-accretion, etc.). Estimates for Earth-class giant-impact moons are uncertain; many dynamical studies suggest they are rare — ballpark figures used in the literature range from a few percent down to well under 1% per terrestrial planet.

2.  P(match | moon) — the conditional probability from the simulation ( ~0.3%–0.5% within ±1% for large-moon parameter regimes).

Example scenarios:

• If P(large moon forms) = 10%, and P(match|moon) = 0.5%, overall P ≈ 0.10 × 0.005 = 0.0005 (0.05%) → ~1 in 2,000 Earth-like planets.

• If P(large moon forms) = 1%, overall P ≈ 0.01 × 0.005 = 0.00005 (0.005%) → ~1 in 20,000.

• If you accept a looser tolerance (±5%), overall probability could be tens of times larger — but large tolerances weaken the “perfect eclipse” claim.

So the dominant rarity usually comes from the low rate of forming a large moon (giant-impact scenario) rather than the geometric matching itself — the geometry is unlikely but not vanishingly so; having a very large moon in the right orbital zone is the rarer event.

D — Important caveats & physical complications (why this is approximate)

1.  Tidal evolution. Many moons (including ours) migrate outward over time due to tidal exchange of angular momentum. A moon born closer may have matched the Sun’s angular size at one epoch and drifted away later (or vice versa). So the match probability depends on the time window you care about.

2.  Orbital eccentricity and inclination. I sampled orbital radius as a single semimajor axis. Instantaneous apparent size also depends on eccentricity (perigee/apogee) and phasing; that will increase the chance of occasional good “matches” during an orbit.

3.  Formation channels. Different moon-formation processes produce different size & distance distributions. Giant-impact moons (Earth–Moon) are rare but produce large satellites; co-formation tends to make many small moons. My simple model did not attempt to model those distributions from first principles.

4.  Host star and orbital distance. I fixed a Sun-like star and 1 AU. For other star sizes or planet orbital radii, the required size/distance ratio changes (e.g., larger stars or closer planets change the needed moon geometry). So the chance differs across stellar types and habitable zones.

5.  Definition of “perfect eclipse.” Exact equality of angular diameters is a continuum event (probability zero in a continuous model). I therefore used tolerances (±1%, etc.). Your sense of “perfect” may allow somewhat larger or smaller tolerances.

E — Bottom line / takeaway

• Given a moon exists with broadly plausible sizes and orbital radii, the chance that its instantaneous angular diameter matches the Sun’s to within about ±1% is on the order of 0.3%–0.6% (roughly a few tenths of one percent).

• But the overall probability a terrestrial planet ends up with such a moon is the product of that geometric chance and the (probably small) probability of forming a large moon — producing overall odds that are quite small (likely well below 1% per terrestrial planet, possibly orders of magnitude smaller depending on how rare giant-impact moons are).

• So, it’s uncommon but not astronomically impossible for a planet to host a moon that produces “perfect” solar eclipses; the real rarity likely comes from the moon’s origin (giant impact) rather than the geometric match itself.