What is the largest achievable oxygen-atmosphere world?

Question

I am trying to calculate the largest world that will meet the following criteria:

• Exists within the Goldilocks zone of a G2V star similar to our sun.

• Is survivable long-term by anatomically modern humans with a minimum of technology.

• Has an atmosphere that is roughly 21% oxygen, with no significant quantities of toxic gases. The atmospheric composition of the inert elements is irrelevant, however no significant quantities of free hydrogen may exist.

• There must be sufficient water for a somewhat earthlike water cycle.

• There need not be any solid surface. I am aware that a planet with these characteristics will likely be similar to a gas giant.

• Humans will exist upon artificial platforms suspended at an appropriate level by handwavium.

• The Oxygen/Carbon dioxide cycle will be driven by plants growing upon the artificial platforms, as well as self-buoyant photosynthetic organisms in the atmosphere as necessary.

• Elements other than those strictly necessary to support the existence of humans will be present in a reasonable, justifiable abundance.

• Any radiation must be survivable by humans long-term without protective gear.

Assuming a hypothetical surface at an altitude with an atmospheric pressure of 1 ATM and gravity of 1g, with air temperatures in the range of those that occur naturally on Earth, what would the largest practical diameter of this world be?

How thick would the human-survivable and human-habitable atmosphere be, i.e. how high above and how far below the hypothetical 1-ATM, 1g 'surface' could humans survive short-term and indefinitely?

Would there necessarily be a discernable solid, rocky surface or would the atmosphere transition to some solid state at a sufficient depth at some distance above the rocky core?

Edit:

Since there are currently no answers that calculate the size of the world in question, there are some additional criteria:

• With a pure air world, the radius may be in the millions of kilometres, but the stars would not be visible, and indeed sunlight itself may not be visible. The maximum distance that may be seen through clear air is on the order of 240km, so I'd like the air thickness above the 1 ATM level to be around 200km so that the inhabitants can see stars.

• The minimum molecular weight retained must be such that no more than about 5% of the atmospheric composition is Helium.

• Unprotected humans can survive in normal air from about 0.5 ATM (due to hypoxia at lower pressures) to about 10 ATM (due to Nitrogen narcosis starting at about 2.5 ATM). As I don't want my human inhabitants to be able to reach any surface unprotected, I would like the maximum pressure to be at least 15 ATM.

• There needs to be enough water and carbon for their respective cycles, and enough iron and other elements to support life and human industry.

Answer 1

The main factor in answering this question turned out to be the requirement for having hydrogen and helium being largely absent from the atmosphere. According to the principle of Jeans Escape, at a temperature of 288K/15°C, Hydrogen and Helium are not retained in significant quantities until the planetary escape velocity is in excess of around 15km/s.

Since we want the largest possible world with an escape velocity at or below 15km/s and a surface gravity of 9.80665 m/s, this imposes a lower limit on the world's density, as escape velocity is proportional to the world's mass and inversely proportional to radius, while gravity is proportional to mass and inversely proportional to radius squared:

Volume: $$v=\frac{4}{3}\pi r^3$$

Mass: $$M=v\rho$$

Escape Velocity: $$E_v=\sqrt{\frac{2GM}{r}}$$

Gravity: $$g=\frac{GM}{r^2}$$

where: $E_v$: Escape velocity, $G$: Gravitational Constant ($6.67\times10^{-11} m^3kg^{-1}s^{-2}$), $g$: surface gravity, $M$: mass, $v$: Volume, $r$: radius, $\rho$: Density.

So, by plugging the numbers into my spreadsheet, the largest $r$ (and the smallest $\rho$) where $E_v \leq 15000 ms^{-1}$ and $g=9.80665ms^{-2}$ occurs at approximately $\rho = 3058kgm^{-3}$ and $r = 11471km$.

Lower density results in a higher escape velocity if gravity remains equal to $9.80665 ms^{-2}$, and a higher density results in a lower radius.

From a density of $3058kgm^{-3}$, we can suppose that the planet would be largely water and a variety of high-pressure ices, with a small amount of other, heavier substances.

So, to add an atmosphere... I don't want unprotected humans reaching the surface (alive), so with Nitrogen narcosis becoming fatal at around 10 ATM, I decided to go with a surface pressure of 20 ATM. With standard air (about 78% N, 21% O plus trace elements), the scale height is:

$$H=\frac{GT}{mg}=8,428.733048m$$

Where $G$: Gas Constant $8.31446261815324 JK^{−1}mol^{−1}$, $T$: temperature (288K), $m$: Air molar mass ($0.02896968 kgMol^{-1}$), and $g$: gravity.

From this, we can solve for the various altitudes of significance where the Temperature Lapse Rate is naively assumed to be zero for the sake of convenience:

$$H=LN\left(\frac{P}{P_0}\right)\cdot\left(\frac{GT}{-gm}\right)+H_0$$

Where $P$: Pressure, $P_0$: Reference Pressure and $H_0$: Reference Altitude.

The 10 ATM altitude (At or below which Nitrogen narcosis becomes potentially fatal) is about 5,842m.

The 2.5 ATM altitude (where Nitrogen Narcosis begins to be noticeable) is about 17,527m.

The 1 ATM altitude (where the majority of the human inhabitants will live on suspended platforms) is 25,250m

The 0.5 ATM altitude, above which the partial pressure of oxygen is no longer sufficient to support human life is 31,093m

The Armstrong Limit (0.0617 ATM) above which water boils at human body temperature and consciousness even with 100% oxygen is no longer possible for humans without a pressure suit or similar protection is 48,728m

The Karman Line, above which space officially begins (defined as an arbitrary altitude at which aerofoils can no longer generate significant lift), is at about 120,000m altitude.

Answer 2

This is not an exact answer, but so called "water worlds" may be the best candidates.

Gliese 436 b is an exoplanet believed to be composed primarily of water. It's more than 4 times bigger than Earth and more than 21 times heavier, but its surface gravity is only 1.18 g Temperature, though, is too hot - about 439 °C, but this is a question of placing it in the goldilock zone.

Water has benefit of having relatively low density and high ability to resist compression, which makes it an ideal candidate to build large habitable worlds.

Unfortunately I can't provide the math to estimate how exactly big the water world can get, or what should be the size of water world for the surface gravity to be exactly 1 g.

See also Does gravity effect the density of water on an ocean planet?