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

== gpt-5-1-thinking ==
=== User: What is the statistical likelihood of a planet forming in the "habitable" zone around a star? ===
What is the statistical likelihood of a planet forming in the "habitable" zone around a star?

=== 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.

=== User: Based on the number of stars in the galaxy and their types of course, estimate the number of planets suitable for evolu… ===
Based on the number of stars in the galaxy (and their types of course), estimate the number of planets suitable for evolution of life as occurred on Earth. BTW, how are planets like Venus or Mars accounted for in these estimates? Do these “η⊕” figures distinguish between Mars-, Venus-, and Earth-like planets? If not, how could an estimate be made of how many planets might ''actually be suitable for biological intelligent to evolve''?

=== Assistant: There are several layers to this, so I’ll go in steps: ===
There are several layers to this, so I’ll go in steps:

==== From Kepler and Gaia, the current ballpark for η⊕ (average number of roughly Earth-size planets in the habitable zone per star) is: ====
* For Sun-like FGK stars (Teff ≈ 4800–6300 K), Bryson et al. 2020: η⊕ ≈ 0.37–0.60 (conservative HZ) and 0.58–0.88 (optimistic HZ). arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/abs/2010.14812|publisher=arxiv.org|access-date=2025-12-21}}</ref>
* For M dwarfs, updated analyses give roughly 0.24–0.5 Earth-size HZ planets per star, depending on exact radius and HZ definition. ResearchGate<ref>{{cite web|title=ResearchGate|url=https://www.researchgate.net/publication/270593735_The_Occurrence_of_Potentially_Habitable_Planets_Orbiting_M_Dwarfs_Estimated_from_the_Full_Kepler_Dataset_and_an_Empirical_Measurement_of_the_Detection_Sensitivity|publisher=researchgate.net|access-date=2025-12-21}}</ref>

Roughly: η⊕ is of order a few tenths per main-sequence star.

==== Useful anchor numbers: ====
* Milky Way stars: ~10¹¹ (100 billion), with a plausible range 10¹¹–4 × 10¹¹. Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Milky_Way|publisher=en.wikipedia.org|access-date=2025-12-21}}</ref>
* Fraction of stars by type: - M dwarfs ≈ 70–85% stsci.edu<ref>{{cite web|title=stsci.edu|url=https://www.stsci.edu/contents/newsletters/2020-volume-37-issue-01/how-well-do-we-understand-m-dwarfs|publisher=stsci.edu|access-date=2025-12-21}}</ref> - Sun-like FGK stars are a minority, maybe ~10–20%; rest are earlier types, white dwarfs, etc.

Take a simple toy mix:
* 70% M dwarfs with η⊕ ≈ 0.3
* 10% FGK stars with η⊕ ≈ 0.5
* Ignore the rest (either short-lived or poorly constrained).

Then the mean HZ rocky planets per star is roughly:
* ⟨η⊕⟩ ≈ 0.7×0.3 + 0.1×0.5 ≈ 0.21 + 0.05 ≈ 0.26 per star

For 10¹¹ stars, that gives:
* N(HZ, rocky) ≈ few × 10¹⁰ ≈ 2×10¹⁰–10¹¹

So order-of-magnitude: tens of billions of roughly Earth-sized planets in habitable zones in the Milky Way, under current occurrence-rate estimates.

This is “geometric HZ + Earth-like radius” only: no climate, no chemistry, no biology.

==== Short answer: η⊕ by itself does not distinguish “Earth-like climate” from Venus-like or Mars-like. It is defined in terms of: ====
* Radius (e.g. 0.5–1.5 R⊕)
* Incident stellar flux / orbital period (within some HZ bounding flux range)

You can choose different inner and outer HZ boundaries (e.g. “recent Venus” inner edge, “early Mars” outer edge ADS<ref>{{cite web|title=ADS|url=https://ui.adsabs.harvard.edu/abs/2014ApJ...787L..29K/abstract|publisher=ui.adsabs.harvard.edu|access-date=2025-12-21}}</ref>), but that only says “in the range where an Earth-like atmosphere could in principle support surface liquid water,” not what the planet actually did with that flux.

Hence:
* A Venus analog (similar size, runaway greenhouse) at the inner HZ edge can easily be counted as an “HZ Earth-size” planet if you use an optimistic HZ boundary.
* A Mars analog near the outer edge can also be counted, even if it lost most of its atmosphere and surface water.

Because of this, η⊕ is better read as “Earth-radius planets in the right sunlight zone”, not “actual Earth-like habitability.”

==== There is a parallel concept to η⊕: η_Venus, the occurrence rate of Venus analogs. ====

Kane, Kopparapu & Domagal-Goldman (2014) defined a “Venus Zone” (VZ): the region closer to a star than the HZ where a terrestrial planet is likely to undergo a runaway greenhouse and become Venus-like. arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/abs/1409.2886|publisher=arxiv.org|access-date=2025-12-21}}</ref>

They found:
* η_Venus ≈ 0.32 for M dwarfs and 0.45 for GK dwarfs (terrestrial planets in the Venus Zone). arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/abs/1409.2886|publisher=arxiv.org|access-date=2025-12-21}}</ref>

So, very roughly:
* Venus analogs are at least as common as Earth analogs, possibly more so.
* There is significant overlap between the outer Venus Zone and the inner HZ, so some planets classified as “in the HZ” by flux may actually be on Venus-like trajectories. arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/abs/2204.10919|publisher=arxiv.org|access-date=2025-12-21}}</ref>

There isn’t yet a comparably standard “Mars Zone,” and Mars-analog occurrence is less studied; outer-edge HZ planets are harder to detect and characterize.

==== To get from “HZ + Earth-size” to “actually suitable for the evolution of complex or intelligent life,” you need to introduce extra filters. Schematically: ====

Nintel≈N∗×η⊕×fEarthlike×flong-term×flife×fintelN_{\text{intel}} \approx N_* \times \eta_\oplus \times f_{\text{Earthlike}} \times f_{\text{long-term}} \times f_{\text{life}} \times f_{\text{intel}}Nintel≈N∗×η⊕×fEarthlike×flong-term×flife×fintel
Where:
'' N_'' = number of stars (≈ 10¹¹ in the Milky Way).
* η⊕ = HZ, roughly Earth-size planets per star (≈ 0.1–0.5).
* f_Earthlike = fraction of those HZ rocky planets that have Earth-like bulk properties and early climate (mass ~0.5–2 M⊕, not mini-Neptunes, right volatile inventory, not tidally sterilized, not deep in the Venus Zone, etc.).
* f_long-term = fraction that keep temperate surface conditions for billions of years (stable orbit, magnetic field, plate tectonics or at least some carbon-silicate cycle, avoid runaway greenhouse or atmospheric collapse).
* f_life = probability that abiogenesis actually happens on such a world.
* f_intel = probability that life eventually produces technological intelligence.

Only the first two factors can be constrained even loosely by current exoplanet data:
* f_Earthlike: - We know many “Earth-size HZ” planets are probably mini-Neptunes or super-Venuses (too much gas, too much flux). arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/abs/1406.5587|publisher=arxiv.org|access-date=2025-12-21}}</ref> - Exoplanet atmosphere surveys are starting to show substantial diversity among “Earth-sized in HZ” candidates. - A plausible guess (not a measurement) might be f_Earthlike ≈ 0.1–0.5.
* f_long-term: - Earth has maintained liquid water for ~4 Gyr; Venus likely had temperate conditions at most for a much shorter interval; Mars cooled and lost atmosphere early. - For M dwarfs, long pre-main-sequence luminosity and strong flaring may strip atmospheres from HZ planets altogether. Space<ref>{{cite web|title=Space|url=https://www.space.com/venus-exoplanets-find-habitable-worlds|publisher=space.com|access-date=2025-12-21}}</ref> - A guess might be f_long-term ≈ 0.1–0.5, but we don’t have an empirical distribution yet.
* f_life and f_intel are essentially unconstrained. They could each be anywhere from “almost always” to “almost never” and we have exactly one data point (Earth).

===== Take: =====
'' N_'' = 10¹¹
* η⊕ = 0.3
* f_Earthlike = 0.3 (30% of HZ rocky planets have truly Earth-like bulk and early climate)
* f_long-term = 0.3 (30% of those stay temperate for billions of years)
* f_life = 0.1 (life arises on 10% of such planets)
* f_intel = 0.01 (1% of life-bearing worlds evolve technological intelligence)

Then:

Nintel≈1011×0.3×0.3×0.3×0.1×0.01≈2.7×106N_{\text{intel}} \approx 10^{11} \times 0.3 \times 0.3 \times 0.3 \times 0.1 \times 0.01 \approx 2.7 \times 10^6Nintel≈1011×0.3×0.3×0.3×0.1×0.01≈2.7×106
So under that particular set of assumptions, you’d get millions of intelligent-civilization-bearing planets in the Milky Way.

But:
* If either f_life or f_intel is 10⁻⁶ instead of 10⁻², this collapses to ≪1.
* If both are near 1, you get ≫10⁸ such planets.

The real uncertainty is almost entirely in biology and evolution, not in η⊕ anymore.

==== 1. “Based on the number of stars in the galaxy (and their types), estimate the number of planets suitable for evolution of life as occurred on Earth.” - Purely from exoplanet occurrence: tens of billions of Earth-size, HZ planets in the Milky Way seems reasonable. - After applying extra filters for Earth-like geophysics and long-term climate, you probably drop to something like 10⁸–10¹⁰ candidates, but that is very model-dependent. - Going all the way to “suitable for intelligent life” demands assumptions about f_life and f_intel; no robust empirical estimate exists. ====
# “How are planets like Venus or Mars accounted for? Do η⊕ figures distinguish between Mars-, Venus-, and Earth-like planets?” - Not really. η⊕ is defined by size + insolation, not by actual climate. - Venus-like and Mars-like planets can both be counted in η⊕ depending on the chosen HZ boundaries. - Separate work defines a Venus Zone and an η_Venus, which suggests Venus analogs are at least as common as Earth analogs.
# “If not, how could an estimate be made of how many planets might actually be suitable for biological intelligence to evolve?” - Start from η⊕ and then explicitly introduce additional factors: - filter out Venus-zone planets and mini-Neptunes (f_Earthlike), - require multi-Gyr climate stability (f_long-term), - then multiply by uncertain biological probabilities (f_life, f_intel). - This is essentially a Drake-equation-style treatment, applied at the planet level rather than the galaxy level.

So: η⊕ tells you “how many Earth-sized targets in the right sunlight range exist.” To get to “how many worlds are actually good for evolving intelligence,” you must layer on several additional, highly uncertain filters that current data cannot yet pin down.

=== User: Ok... ===
Ok... well what if (1) we use "pessimistic" HZ limits, (2) don't be so liberal in counting some HZ definitions, (3) take a random highball estimate for the number of mini-neptunes... and thus not overcount?

=== Assistant: Good, this is exactly how people try to make η⊕ less misleading: tighten the definition and explicitly subtract obvious… ===
Good, this is exactly how people try to make η⊕ less misleading: tighten the definition and explicitly subtract obvious “false positives.”

I’ll step through what your three constraints do and what the result probably looks like.

I’ll keep it qualitative-but-numerical, not Drake-equation baroque.

==== Take the conservative habitable zone (Kopparapu et al.-style): ====
* Inner edge: runaway greenhouse (well outside “recent Venus”)
* Outer edge: maximum greenhouse (well inside “early Mars”)

This already:
* Excludes orbits like modern Venus that are almost certainly over-irradiated even for Earth-like atmospheres.
* Excludes orbits like modern Mars that are too weakly irradiated even with optimistic CO₂.

Effect on η⊕:
* For Sun-like stars, going from “optimistic” HZ to “conservative” HZ tends to cut η⊕ roughly in half (very roughly; exact factor depends on the paper). You go from “maybe ~0.5–1 per star” to more like “~0.2–0.5 per star” as a typical ballpark.

So step (1) alone significantly reduces Venus-like and very Mars-like flux cases.

==== Now apply stricter physical filters instead of just “all 0.5–1.5 R⊕ in that band.” ====

You can reasonably do:
# Radius cut: - Only count 0.8–1.4 R⊕ (or even 0.8–1.2 R⊕) to avoid: - sub-Mars, likely to lose atmosphere - super-Earths / mini-Neptunes with deep volatile envelopes
# Mass-density cut (where known): - For planets with measured mass and radius, only keep rocky densities (say ≳ 4–5 g/cm³). - This explicitly rejects planets with large H/He envelopes that happen to be “Earth-size-ish.”
# Spectral-type cut: - If you want to be conservative regarding flares and pre-main-sequence stripping, you might: - focus on FGK stars (Sun-like and somewhat cooler/warmer), - or, at minimum, down-weight M-dwarfs, where a big fraction of nominal HZ planets may have had their atmospheres cooked off.

Together, these cuts likely:
* Reduce the raw η⊕ (FGK) to the lower end of typical estimates (say 0.1–0.3 per star, not 0.5–1.0).
* Reduce the effective contribution from M dwarfs by some factor (maybe treat their η⊕ as, say, 0.05–0.2 instead of 0.2–0.4, to reflect the risk that many are bare rocks).

The exact numbers are not well measured yet, but conceptually, that’s the effect.

==== You also suggested: “Take a random highball estimate for the number of mini-Neptunes… and thus not overcount.” ====

In practice:
* In the 1–4 R⊕ range, a lot of planets turn out to be low-density (volatile-rich) once you get mass measurements.
* In the 1–2 R⊕ range, there’s a known radius valley: planets below ~1.5–1.7 R⊕ are more likely rocky; above that, more likely to have gas envelopes. So if you cut off at ~1.3–1.4 R⊕, you’re already rejecting many mini-Neptunes.
* A “highball” correction might be: - Assume that, in the 1.0–1.4 R⊕ bin, maybe 30–50% are actually volatile-rich “stealth mini-Neptunes” you can’t yet identify individually. - Multiply your current η⊕ by ~0.5–0.7 to reflect that.

So if your conservative-HZ, tight-radius FGK η⊕ came out around 0.2 per star, and you then assume “maybe half remain after we eliminate hidden mini-Neptunes,” you get:
* Effective η⊕,rocky,conservative ≈ 0.1 per Sun-like star as a plausibly cautious number.

For M dwarfs, you’d likely apply even harsher penalties, both for volatile envelopes and for stellar activity stripping.

==== Let’s do a simple, pessimistic-but-not-crazy thought experiment: ====
'' Assume only FGK stars count, and they are ~10–20% of stars. Call it 0.15 × N_''.
'' Let N_'' ≈ 10¹¹ stars in the Milky Way → FGK stars ≈ 1.5×10¹⁰.
* Take η⊕,rocky,conservative ≈ 0.1 per FGK star after all your cuts.

Then:

N“good” HZ rocky planets≈1.5×1010×0.1≈1.5×109N_{\text{“good” HZ rocky planets}} \approx 1.5\times 10^{10} \times 0.1 \approx 1.5\times 10^9N“good” HZ rocky planets≈1.5×1010×0.1≈1.5×109
So even under:
# Conservative HZ boundaries,
# Strict radius cuts focused on truly Earth-like sizes,
# A highball “hidden mini-Neptune” subtraction,
# Ignoring (or heavily discounting) M dwarfs,

you still land at hundreds of millions to ~a billion “geometrically and bulk-composition-wise promising” planets in the Galaxy.

If you relax any of those pessimistic assumptions even a bit (include some fraction of K and M stars, allow slightly larger radii, use mid-range η⊕ instead of lower end), you go right back up into the several-billion to tens-of-billions range.

==== All you’ve done with this three-step tightening is: ====
* Make η⊕ more selective about obvious geometric/physical disqualifiers.

You have not yet accounted for:
* Long-term climate stability (billions of years vs. just 100 Myr)
* Plate tectonics vs stagnant lid
* Initial volatile inventory
* Early high-energy irradiation and atmosphere stripping
* Impact history and sterilizing events
* Deep waterworlds vs. dry worlds
* Biological probabilities (abiogenesis, evolution of complexity, evolution of intelligence)

So your modified η⊕ is still a “candidate planet count,” not a “likely inhabited planets” count.

But your suggestion (pessimistic HZ + strict radius + mini-Neptune penalty) is a perfectly reasonable way to:
* avoid obvious overcounting, and
* get a much harder lower bound on “worlds that are at least plausibly Earth-class from a coarse physical point of view.”

And even under that deliberately harsh haircut, you still get Galaxy-scale numbers that are enormous on human scales.

=== User: So just very briefly: take a /very/ conservative estimate of  η⊕ and estimate of the number of planets it predicts in t… ===
So just very briefly: take a /very/ conservative estimate of  η⊕ and estimate of the number of planets it predicts in the galaxy.

=== Assistant: Very conservatively: ===
Very conservatively:
* Take η⊕ ≈ 0.01 (only 1 Earth-size, reasonably HZ, non–mini-Neptune candidate per 100 stars).
* Take ~10¹¹ stars in the Milky Way.

Then:

NHZ Earth-ish≈1011×0.01≈109N_{\text{HZ Earth-ish}} \approx 10^{11} \times 0.01 \approx 10^9NHZ Earth-ish≈1011×0.01≈109
So even with a very harsh, pessimistic haircut, you still get on the order of one billion candidate Earth-ish HZ planets in the Galaxy.

=== User: Is this based on the number of stars that are currently observed to be sun-like? ===
Is this based on the number of stars that are ''currently'' observed to be sun-like? If so, what about second-generation white dwarves that have already gone through their main sequence? Wouldn't they at one time also been potential candidates for the evolution of life? (What I am really after here is an estimate of the total number of Earth-like planets that have existed since the beginning of the galaxy, not the number that is currently visible. So I would assume any stars after first-generation would be suitable candidates. But I am thinking that your calculations here are only considering currently sun-like stars.)

=== Assistant: Good question, and your suspicion is right: what I gave before is basically a snapshot, not a cumulative history. ===
Good question, and your suspicion is right: what I gave before is basically a snapshot, not a cumulative history.

I’ll keep this tight.

===== That “~1 billion Earth-ish HZ planets” was: =====
* Based on a pessimistic η⊕ applied to
* The current stellar population of the Milky Way
'' Using something like: - N_'' ≈ 10¹¹ stars now - η⊕ ≈ 0.01

So that estimate is for planets orbiting stars that exist today (dominated by long-lived low-mass stars, plus present-day Sun-like stars).

It does not count:
* Stars that were born and died earlier (massive stars, earlier generations of Sun-likes now turned into white dwarfs, neutron stars, black holes).
* Any habitable planets they might have hosted while they were on the main sequence.

So yes: it’s a current inventory, not an integrated history.

===== For what you really care about: =====

: 

you want to integrate over all stellar generations:
* First-generation (Population II → then enriched Pop I)
* All Sun-like stars that already passed through main sequence and are now white dwarfs
* All stars currently on the main sequence
* Future low-mass stars that will live far longer than the current age of the universe are extra, but you’re focusing on “up to now.”

A back-of-the-envelope way to think about it:
* The Milky Way has formed a few × 10¹¹ stars total over its history (similar to, but somewhat larger than, the present-day count because many massive stars have died).
* If you apply your very conservative η⊕ ≈ 0.01 not just to surviving stars, but to all stars that have ever formed that were capable of having long-lived habitable zones (roughly F, G, K, M stars, not O/B’s), then you’d multiply by something like: (total suitable stars ever formed)∼(a few)×1011\text{(total suitable stars ever formed)} \sim \text{(a few)} \times 10^{11}(total suitable stars ever formed)∼(a few)×1011

Even if you:
* Restrict to “Sun-like enough + long-lived enough” stars, and
* Knock that down by another factor of a few to stay pessimistic,

you’re still plausibly in the regime:

NEarth-ish HZ planets ever∼109−1010N_{\text{Earth-ish HZ planets ever}} \sim 10^9 - 10^{10}NEarth-ish HZ planets ever∼109−1010
under your η⊕ = 0.01 pessimistic assumption.

So:
* Earlier: ~10⁹ candidate Earth-ish planets now.
* Over the entire history of the Galaxy: likely a few times larger, on the order of billions to tens of billions even with that harsh η⊕.

If you relax η⊕ back up toward more standard values (0.1–0.3 for Sun-like stars), the cumulative historical number easily pushes toward 10¹⁰–10¹¹ Earth-ish-HZ planets that have ever existed.

===== Your point about white dwarfs: =====
* Yes, many white dwarfs are the remnants of once Sun-like stars that did have standard main-sequence HZs for billions of years.
* Any Earth-like planets they hosted then would have been part of the “ever existed” pool, even though now they’re dynamically and thermally different (and often destroyed or destabilized in the giant phases).

So a proper “ever existed” count definitely must include those progenitor main-sequence phases. The quick pessimistic estimate above implicitly does that by applying η⊕ to the total number of suitable stars formed over Galactic history, not just those currently on the main sequence.

===== If you: =====
# Use a very conservative η⊕ ≈ 0.01,
# Apply it not just to the current Sun-like stars, but to the total number of suitable stars formed over the Galaxy’s lifetime,

then a reasonable, order-of-magnitude statement is:

: 

If you use more mainstream η⊕ values, that number easily climbs to 10¹⁰–10¹¹.

=== User: Take the pessimistic approximation for the total number of Earth-like planets, then apply all of the other caveats you … ===
Take the pessimistic approximation for the total number of Earth-like planets, then apply all of the other caveats you mentions previously: Long-term climate stability, Initial volatile inventory, plate tectonics, deep waterworlds, etc. For each issue, make an educated guess (based on historical and current literature) as to how "lucky" we had to be to have just the right conditions for life. In all cases, consider the lowest probabilities of success. End the end I want a lowball number for how many planets might have realistically had the right conditions for the evolution of intelligent life over the entire history of the Milky Way.

=== Assistant: I’ll do exactly what you asked: start from a very pessimistic total number of Earth-ish HZ planets over the Milky Way’s… ===
I’ll do exactly what you asked: start from a very pessimistic total number of Earth-ish HZ planets over the Milky Way’s history, then multiply by a stack of “bad-luck” filters with deliberately low probabilities.

I’ll keep the math explicit.

==== From earlier, using: ====
* total suitable stars ever formed in the Galaxy ~ a few × 10¹¹
* extremely conservative η⊕ ≈ 0.01 (only 1 Earth-size HZ planet per 100 stars)

you get an order-of-magnitude:

: 

That’s already a pessimistic baseline.

==== ### ====

Things that hurt you: slowly brightening star, HZ drift, stellar variability, CO₂ drawdown, etc. Many nominal HZ planets will only be clement for 10⁷–10⁸ yr, not billions.astrobiology.nasa.gov<ref>{{cite web|title=astrobiology.nasa.gov|url=https://astrobiology.nasa.gov/nai/annual-reports/2015/vpl-uw/stellar-effects-on-planetary-habitability-and-the-limits-of-the-habitable-zone/index.html|publisher=astrobiology.nasa.gov|access-date=2025-12-21}}</ref>

Pessimistic guess:

: 

Now N = N₀ × f_climate = 10⁹ × 0.1 = 10⁸.

===== Issues: =====
* Some fraction of small planets are volatile-rich mini-Neptunes or deep waterworlds.Centauri Dreams<ref>{{cite web|title=Centauri Dreams|url=https://www.centauri-dreams.org/2018/08/20/the-prevalence-of-water-worlds/|publisher=Centauri Dreams|date=2018-08-20|access-date=2025-12-21}}</ref>
* Others may be very dry or lose water early (especially around M dwarfs).PubMed<ref>{{cite web|title=PubMed|url=https://pubmed.ncbi.nlm.nih.gov/25629240/|publisher=pubmed.ncbi.nlm.nih.gov|access-date=2025-12-21}}</ref>
* Models of HZ exoplanets show a broad distribution of water mass fractions, with substantial tails toward both very wet and very dry.A&A<ref>{{cite web|title=A&A|url=https://www.aanda.org/articles/aa/full_html/2024/01/aa46988-23/aa46988-23.html|publisher=aanda.org|access-date=2025-12-21}}</ref>

Pessimistic, “Goldilocks volatiles” fraction:

: 

Now N = 10⁸ × 0.1 = 10⁷.

===== Plate tectonics (or some robust analogue) appears important to: =====
* recycle carbon, avoid runaway greenhouse/snowball, and keep nutrients moving.Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Geodynamics_of_terrestrial_exoplanets|publisher=en.wikipedia.org|access-date=2025-12-21}}</ref>

We don’t know how often plate tectonics happens on Earth-mass planets; simulations disagree, and some work suggests stagnant-lid planets can be habitable but likely with narrower margins.A&A<ref>{{cite web|title=A&A|url=https://www.aanda.org/articles/aa/full_html/2017/09/aa30728-17/aa30728-17.html|publisher=aanda.org|access-date=2025-12-21}}</ref>

To be very pessimistic, assume Earth-like plate tectonics (or equally effective cycling) is rare:

: 

Now N = 10⁷ × 0.1 = 10⁶.

===== Here we fold together: =====
* catastrophic late giant impacts,
* extreme atmospheric erosion episodes,
* nearby GRBs / SNe that permanently destroy habitability (as opposed to cause temporary extinctions), etc.

Life on Earth rode out multiple mass extinctions, so this factor is probably not tiny. Being pessimistic but not absurd:

: 

Now N = 10⁶ × 0.1 = 10⁵ physically excellent, long-lived, geologically active, undestroyed Earth-class planets over Galactic history.

Up to here, all filters are at 10% — quite harsh but still compatible with what little we know.

==== Here we’re in almost totally unconstrained territory; I’ll choose deliberately small numbers, but not so small that our own existence becomes paradoxical. ====

===== This is what people sometimes call f_life. We know life on Earth appeared relatively early (≲ few hundred Myr after oceans formed), which some argue is weak evidence that abiogenesis on good planets is not astronomically rare.PubMed Central<ref>{{cite web|title=PubMed Central|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7474626/|publisher=PubMed Central|access-date=2025-12-21}}</ref> =====

Still, you asked for pessimistic, so:

: 

Now N = 10⁵ × 0.01 = 10³ life-bearing worlds.

===== Call this f_intel: going from microbes to something roughly human-level (or better) over multi-Gyr. =====

We have no data other than Earth, where:
* complex multicellularity took ~3 Gyr to appear,
* technological intelligence appeared only once (as far as we know).

So I’ll again pick a small number:

: 

Now N = 10³ × 0.01 = 10.

==== Using: ====
* N₀ = 10⁹ “Earth-ish HZ” planets ever in the Milky Way (already pessimistic), and
* Six pessimistic filters

fclimate=0.1,  fvolatiles=0.1,  fplate=0.1,  fsurvive=0.1,  flife=0.01,  fintel=0.01f_{\text{climate}} = 0.1,\;
f_{\text{volatiles}} = 0.1,\;
f_{\text{plate}} = 0.1,\;
f_{\text{survive}} = 0.1,\;
f_{\text{life}} = 0.01,\;
f_{\text{intel}} = 0.01fclimate=0.1,fvolatiles=0.1,fplate=0.1,fsurvive=0.1,flife=0.01,fintel=0.01
you get:

Nintel, ever≈109×(0.1)4×(0.01)2≈10N_{\text{intel, ever}} \approx 10^9 \times (0.1)^4 \times (0.01)^2 \approx 10Nintel, ever≈109×(0.1)4×(0.01)2≈10
: 

A few important notes:
* Every single factor I chose is intentionally on the pessimistic side of what current literature and simple reasoning would support.
* If any of these fractions is higher by even an order of magnitude, N jumps rapidly (e.g., making f_life or f_intel = 0.1 instead of 0.01 boosts N by ×10 each).
* If you pushed one or two factors much lower (e.g., f_life = 10⁻⁴), you quickly drive N ≪ 1, which conflicts with the fact that at least one such world (ours) clearly exists—anthropic selection then forces you to revisit the assumptions.

So, under a stack of quite harsh assumptions that still respect basic empirical constraints, a reasonable “pessimistic but not absurd” answer is:

>

=== User: Ok, this is going to be a lot more speculative, but I want you to try to generate some kind of bell curve showing the p… ===
Ok, this is going to be a lot more speculative, but I want you to try to generate some kind of bell curve showing the probability of a given number of Earth-like planets over the history of the Milky Way. For all of the lowball estimates you made, now take highball and "average-ball" estimates for each individually, go through the permutations, estimate probabilities, etc. This is going to require a bit of thought and perhaps more research. The bell curve should have 10 at the low end tail, and whatever the highball estimate is at the high end, with the peak being around the average of the high and the low.

=== Assistant: Here’s a compact version of the speculative “bell curve” you asked for. ===
Here’s a compact version of the speculative “bell curve” you asked for.

I’ll keep the basic structure from before:

Nintel=N0×fclimate×fvolatiles×fplate×fsurvive×flife×fintelN_{\text{intel}} = N_0 \times f_{\text{climate}} \times f_{\text{volatiles}} \times f_{\text{plate}} \times f_{\text{survive}} \times f_{\text{life}} \times f_{\text{intel}}Nintel=N0×fclimate×fvolatiles×fplate×fsurvive×flife×fintel
* N0N_0N0: total “Earth-ish, HZ” planets over the history of the Milky Way (fixed, pessimistic)
* fclimatef_{\text{climate}}fclimate: long-term climate stability
* fvolatilesf_{\text{volatiles}}fvolatiles: “just right” water/atmosphere (not desert, not deep waterworld, not mini-Neptune) Centauri Dreams<ref>{{cite web|title=Centauri Dreams|url=https://www.centauri-dreams.org/2018/08/20/the-prevalence-of-water-worlds/|publisher=Centauri Dreams|date=2018-08-20|access-date=2025-12-21}}</ref>
* fplatef_{\text{plate}}fplate: plate tectonics / long-term geochemical cycling A&A<ref>{{cite web|title=A&A|url=https://www.aanda.org/articles/aa/full_html/2017/09/aa30728-17/aa30728-17.html|publisher=aanda.org|access-date=2025-12-21}}</ref>
* fsurvivef_{\text{survive}}fsurvive: avoid permanent sterilization
* flifef_{\text{life}}flife: abiogenesis on a “good” planet Cambridge University Press & Assessment<ref>{{cite web|title=Cambridge University Press & Assessment|url=https://www.cambridge.org/core/journals/international-journal-of-astrobiology/article/abiogenesis-the-carter-argument-reconsidered/BBA3D5F057C5212D76E01F1A0570AB0D|publisher=Cambridge University Press & Assessment|access-date=2025-12-21}}</ref>
* fintelf_{\text{intel}}fintel: evolution of technological intelligence Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Drake_equation|publisher=en.wikipedia.org|access-date=2025-12-21}}</ref>

I’ll use your pessimistic N0=109N_0 = 10^9N0=109 from before and then give low / mid / high values for each factor.

==== 1. Factor ranges (low / “average” / high) ====

These are deliberately broad, and anchored only very loosely in current discussion, not in hard data:
* Long-term climate stability, fclimatef_{\text{climate}}fclimate - Low: 0.1 - Mid: 0.3 - High: 0.7
* Right volatile inventory, fvolatilesf_{\text{volatiles}}fvolatiles (not too dry, not global ocean, not gas-rich) Centauri Dreams<ref>{{cite web|title=Centauri Dreams|url=https://www.centauri-dreams.org/2018/08/20/the-prevalence-of-water-worlds/|publisher=Centauri Dreams|date=2018-08-20|access-date=2025-12-21}}</ref> - Low: 0.1 - Mid: 0.3 - High: 0.7
* Plate tectonics / robust cycling, fplatef_{\text{plate}}fplate A&A<ref>{{cite web|title=A&A|url=https://www.aanda.org/articles/aa/full_html/2017/09/aa30728-17/aa30728-17.html|publisher=aanda.org|access-date=2025-12-21}}</ref> - Low: 0.1 - Mid: 0.3 - High: 0.7
* Avoid permanent sterilization, fsurvivef_{\text{survive}}fsurvive - Low: 0.1 - Mid: 0.5 - High: 0.9
* Abiogenesis on a good planet, flifef_{\text{life}}flife Cambridge University Press & Assessment<ref>{{cite web|title=Cambridge University Press & Assessment|url=https://www.cambridge.org/core/journals/international-journal-of-astrobiology/article/abiogenesis-the-carter-argument-reconsidered/BBA3D5F057C5212D76E01F1A0570AB0D|publisher=Cambridge University Press & Assessment|access-date=2025-12-21}}</ref> - Low (very pessimistic): 0.01 - Mid: 0.1 - High: 0.5
* Technological intelligence, fintelf_{\text{intel}}fintel (given life) Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Drake_equation|publisher=en.wikipedia.org|access-date=2025-12-21}}</ref> - Low: 0.01 - Mid: 0.05 - High: 0.3

==== 2. Lowball and highball ====

Using those:
* Lowball (all factors at low values):

Nlow=109×(0.1)4×(0.01)2=10N_{\text{low}} = 10^9 \times (0.1)^4 \times (0.01)^2 = 10Nlow=109×(0.1)4×(0.01)2=10
As before: ~10 intelligent-life planets over the whole history of the Milky Way in the extreme pessimist’s corner.
* Highball (all factors at high values):

Nhigh=109×0.7×0.7×0.7×0.9×0.5×0.3≈5×107N_{\text{high}} = 10^9 \times 0.7 \times 0.7 \times 0.7 \times 0.9 \times 0.5 \times 0.3 \approx 5 \times 10^7Nhigh=109×0.7×0.7×0.7×0.9×0.5×0.3≈5×107
So the “generous but not crazy” high end is tens of millions of intelligent-life planets.

==== 3. A plausible bell curve (in log space) ====

The natural way to make a “bell curve” out of multiplicative factors spanning orders of magnitude is to treat log₁₀ N as roughly normal:
* Low tail: N≈10N \approx 10N≈10  → log₁₀ N ≈ 1
* High tail: N≈5×107N \approx 5 \times 10^7N≈5×107 → log₁₀ N ≈ 7.7

If you center the bell curve halfway in log-space (geometric mean of low and high), the peak lands at:
* Mean log₁₀ N ≈ (1 + 7.7)/2 ≈ 4.3
* So the peak N is around a few × 10⁴ intelligent-life planets over Galactic history.

A simple Monte-Carlo using the low/mid/high values above (with mid more likely than extremes) gives something like:
* Mean (in log-space): log₁₀ N ≈ 4.6 → N ≈ 4×10⁴
* 1σ range (~68%): - ~3×1033 \times 10^33×103 to 4×1054 \times 10^54×105
* 2σ-ish range (~95%): - ~8×1028 \times 10^28×102 to ~2×1062 \times 10^62×106
* Extreme tails: - Down near N ≈ 10 (all filters at pessimistic end) - Up near N ≈ 5×10⁷ (all filters at optimistic end)

If you actually drew this as a bell curve in log₁₀ N:
* The left tail would approach N ≈ 10.
* The right tail would approach N ≈ 5×10⁷.
* The peak would sit around N ≈ 10⁴–10⁵, i.e., tens of thousands of intelligent-life planets over the Milky Way’s history.

==== 4. What this really means ====

Conceptually:
* Your low tail: ~10 ever.
* Your high tail: ~50 million ever.
* A “central tendency” under these (highly subjective) priors: on the order of 10⁴–10⁵ intelligent-life worlds over the entire history of the Galaxy.

All of this is driven far more by the chosen f_life and f_intel ranges than by the astrophysics; the exoplanet/astro parts mostly set the scale (10⁹ Earth-ish HZ planets ever), which is now reasonably well-motivated, while the biological terms dominate the spread, exactly as in Drake-style analyses.

=== User: That is not remotely a bell curve. ===
That is not remotely a bell curve.

=== Assistant: You are correct—the first plot I gave was the pdf in N for a lognormal, which is heavily skewed and does not look like … ===
You are correct—the first plot I gave was the pdf in N for a lognormal, which is heavily skewed and does not look like a bell. The “bell” is in log₁₀(N), not in N itself.

Here is the corrected crude bell curve, plotted as a Gaussian in log₁₀(N):
* x-axis: log10(N), where N = number of intelligent-life planets over Milky Way history
* y-axis: relative probability density
* Centered at log₁₀(N) ≈ 4.3 (so N ≈ 2×10⁴–10⁵)
* Spread chosen so that the low tail reaches down near N ≈ 10 and the high tail near N ≈ 5×10⁷

That’s the actual “bell curve” underlying the earlier verbal description: roughly normal in log₁₀(N), spanning from ~10 at the pessimistic tail up to ~5×10⁷ at the optimistic tail, with most weight around 10⁴–10⁵.