Editing Openai/6946e998-79f0-8007-9a65-19828450838d (section)

=== Assistant: Astrophysicists usually phrase this as “η⊕” “eta-Earth”: the average number of roughly Earth-size, roughly Earth-flux p… ===
Astrophysicists usually phrase this as “η⊕” (“eta-Earth”): the average number of roughly Earth-size, roughly Earth-flux planets in the circumstellar habitable zone (HZ) per star. There is not a single exact number, but current Kepler-based estimates give a useful range.

Key points:
# You have to specify what you mean by “habitable” Typically: • star type (e.g., Sun-like FGK dwarfs vs M dwarfs) • planet size (e.g., 0.5–1.5 R⊕, “rocky”) • “conservative” vs “optimistic” HZ boundaries (based on climate models of liquid water on the surface)
# Best current numbers for Sun-like stars (FGK dwarfs) A 2020 Kepler+Gaia analysis (Bryson et al. 2021) defined η⊕ as the number of 0.5–1.5 R⊕ planets in the habitable zone of stars with Teff 4800–6300 K. They found:arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/abs/2010.14812|publisher=arxiv.org|access-date=2025-12-21}}</ref> • Conservative HZ: η⊕ ≈ 0.37–0.60 planets per star (with large error bars, roughly −0.2 to +0.9). • Optimistic HZ: η⊕ ≈ 0.58–0.88 planets per star. Interpreted informally: on the order of one rocky HZ planet for every 2–3 Sun-like stars, possibly more, possibly fewer, depending on systematics. An earlier widely cited Kepler radial-velocity survey (Petigura et al. 2013) suggested ~22% of Sun-like stars have Earth-size planets in their HZ.PNAS<ref>{{cite web|title=PNAS|url=https://www.pnas.org/doi/10.1073/pnas.1319909110|publisher=pnas.org|access-date=2025-12-21}}</ref> NASA’s later public summary of multiple studies described this as “about half of Sun-like stars could host rocky, potentially habitable planets,” once you allow a broader range of sizes and HZ definitions.NASA<ref>{{cite web|title=NASA|url=https://www.nasa.gov/missions/kepler/about-half-of-sun-like-stars-could-host-rocky-potentially-habitable-planets/|publisher=nasa.gov|access-date=2025-12-21}}</ref> So a reasonable “headline” for Sun-like stars is: • Chance that a given Sun-like star has at least one roughly Earth-size HZ planet: probably of order 20–60%, with ~50% often quoted as a plausible mid-range.
# For M dwarfs (red dwarfs) Earlier Kepler work (e.g., Dressing & Charbonneau 2013) found HZ Earth-size occurrence rates of order 0.3 Earth-size HZ planets per M star, with a 95% lower limit ~0.04 per star.ADS<ref>{{cite web|title=ADS|url=https://ui.adsabs.harvard.edu/abs/2013ApJ...767...95D/abstract|publisher=ui.adsabs.harvard.edu|access-date=2025-12-21}}</ref> More recent re-analyses using updated stellar parameters find η⊕ values of order 10%–30% for Earth-size planets in M-dwarf HZs, but with large uncertainties and arguments over completeness.OUP Academic<ref>{{cite web|title=OUP Academic|url=https://academic.oup.com/mnras/article/498/2/2249/5899752|publisher=OUP Academic|access-date=2025-12-21}}</ref>
# “Any” planet in the HZ vs “Earth-like” If you relax the criteria (include super-Earths and mini-Neptunes, say up to 2–3 R⊕), the chance that a given star has at least one planet in its HZ goes up. Many Kepler occurrence studies show that planets of 1–3 R⊕ are common at short periods; extrapolating into the HZ generally yields order-unity numbers (η ≈ 0.5–1 HZ planet per star for that broader size range), but details depend on the radius range and the exact HZ definition.PMC<ref>{{cite web|title=PMC|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7477822/|publisher=pmc.ncbi.nlm.nih.gov|access-date=2025-12-21}}</ref>
# Crude overall answer If you ask, “Given a random main-sequence star with roughly Sun-like metallicity, what is the statistical likelihood that it has at least one terrestrial-size (~0.5–1.5 R⊕) planet in its classical habitable zone?” current Kepler-based estimates suggest: • Rough order of magnitude: tens of percent. • Plausible bracket: ~0.1–0.6 rocky HZ planets per star, depending on star type and assumptions. • That implies something like a 10–60% chance that a random Sun-like star has at least one such planet, with many analyses favoring a “few-tenths per star” regime.
# Important caveats • Systematics are large: completeness corrections, false positives, stellar radii, and HZ boundaries all matter. Recent reviews stress that η⊕ remains quite uncertain, even though it’s clearly not “near zero.”ResearchGate<ref>{{cite web|title=ResearchGate|url=https://www.researchgate.net/publication/397479313_Are_We_There_Yet_Challenges_in_Quantifying_the_Frequency_of_Earth_Analogs_in_the_Habitable_Zone|publisher=researchgate.net|access-date=2025-12-21}}</ref> • This is purely geometric/occurrence “habitability”: it says nothing about atmospheres, water, or actual biology. • Multiplicity: several systems (e.g., TRAPPIST-1) have multiple HZ planets, so the probability that a star hosts at least one will be somewhat higher than η⊕ if η⊕ is interpreted as “planets per star.”

If you want, I can translate this into a Drake-equation-style term and propagate a rough prior on “habitable planets per star” for galaxy-wide estimates.