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What amount of stellar flux that a planet receives is required to boil away all oceans on said planet? We're thinking of Q🜨 (Earth = 1.00 Q🜨). For comparison: Venus ~ 1.93 Q🜨, Mercury ~ 6.57 Q🜨, Mars ~ 0.43 Q🜨.

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    $\begingroup$ It would depend on the atmosphere. $\endgroup$
    – Kilisi
    Oct 26, 2023 at 20:45
  • $\begingroup$ Atomic Rockets Boom Chart (projectrho.com/public_html/rocket/usefultables.php) says about 4.5E27 joules, or slightly over an exaton of TNT. $\endgroup$
    – KEY_ABRADE
    Oct 26, 2023 at 20:47
  • $\begingroup$ I'm talking about a planet similar in mass, size, density, albedo, and atmospheric composition to Earth (0.7 - 1.5 M🜨, 0.9 - 1.2 R🜨, 0.2 - 0.4 bond albedo). $\endgroup$ Oct 26, 2023 at 20:52

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What you're looking for is probably the Komabayashi-Ingersoll limit for the establishment of a runaway greenhouse effect (as happened on Venus).

If you read Komabayashi's original paper, it states that the effect (after the initial forcing required to saturate the atmosphere and make it greenhouse-like) "operates in a way of positive feedback to boil off the ocean unless precipitation, albedo of cloud, and general circulation stabilize the equilibrium."

Given the uncertainty of several factors (from actual cloud albedo across a suitable range of frequencies, to climate, precipitations and altered circulation, which might establish long-lived "clear windows" allowing radiant cooling), I believe it's not currently possible to calculate a specific flux value that could trigger the effect. Actually, Komabayashi's paper shows an ocean average temperature limit of 6 °C, which obviously isn't realistic, and Ingersoll]2 places this limit at 0.57 cal / cm^2 / min, or almost exactly 400 W/m^2, which also isn't realistic since the real solar flux is about three times that. Both their "grey" models seem therefore too pessimistic.

Considering unforced radiant cooling (i.e. assuming a long-wave transparent atmosphere, a "clear" model which is therefore too optimistic), an estimate is easier (it only involves Stefan-Boltzmann's Law) about 115% of current flux value (which should happen in five or six billion years).

But it is at least conceivable that a complete boiling-off of the oceans could be started now: Kombayashi and Ingersoll put the limit at 0.3, the clear model puts it at 1.15, we don't know where the true Earth atmospheric model (if we had it) would put it.

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  • $\begingroup$ When calculating the habitable zone, I use an inner limit of 1.35 and an outer limit of 0.45, which would extend the Sun's habitable zone from 0.86 - 1.49 AU, which to me sounds pretty reasonable. Is that a realistic assumption or would a planet similar to Earth orbiting the Sun at 0.86 AU already be way too close? $\endgroup$ Oct 26, 2023 at 21:30
  • $\begingroup$ 35% more flux with the current Earth atmosphere would definitely cause a runaway effect with the current Earth atmosphere. The Goldilocks estimate doesn't completely take into account atmospheric effects (actually, based on distance, Earth temperature ought to be about -13 °C using the basic Stefan-Boltzmann formula 0.5*(SUN(1-albedo)/(emittivitySBConstPI*OrbitalRadius^2))^-4 ). $\endgroup$
    – LSerni
    Oct 26, 2023 at 23:10
  • $\begingroup$ From the following values, which one is the best to use as a hard limit for oceans evaporation and runaway greenhouse effects: 1.40, 1.35, 1.30, 1.25, 1.20, 1.15, 1.10, 1.05, 1.00? $\endgroup$ Oct 27, 2023 at 22:12
  • $\begingroup$ I'm sorry, for that you'd need a climatologist - and actually there isn't agreement on details like those. Some are worried that even 1.0 might be enough in the right conditions (and long timescales, of course). For worldbuilding purposes, I believe all those values are defensible (the greater the value, the shorter the timescale). $\endgroup$
    – LSerni
    Oct 28, 2023 at 8:10
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There are almost certainly holes in the following analysis, but I invite comments and corrections.

The Earth's oceans are very roughly 100× the mass of Earth's atmosphere. If they all turned to gas the atmosphere would weight almost 100× as much, so to keep the surface of the earth dry, it would have to be kept at around 300°C, and the upper atmosphere would be correspondingly hotter.

This would result in Boltzmann radiation increasing about 20×, implying that insolation would need a similar increase to maintain the temperature.

But...

The ideal gas equation PV=nRT suggests that the volume of the atmosphere would increase ~340× (n increases by 160× and T increases by 2.1×), making it thousands of kilometres deep.

My guess is that such a deep atmosphere would be unstable on a planet as small as Earth, prone to evaporating off into space; I'm hoping other readers might be able to cite an effective limit to the mass of atmosphere that Earth could realistically retain.

edits:

  1. A discussion of the Kárman Line in SpaceStackExchange is informative. It has a chart showing that atmospheric pressure vs altitude is very close to logarithmic (below 80 km): roughly 10× pressure increase for every 16~18 km descended. This suggests that increasing the total atmospheric mass a hundred-fold would only increase the depth by 30~300 km, not the "thousands" I've suggested above. (That's still rather vague because I haven't yet accounted for the temperature ramp.)

  2. It's fairly well established citation-needed that a strong geomagnetic field is necessary to keep an atmosphere over millions of years. Mercury, the Moon, and Mars have weak or no magnetic fields, and have lost their atmospheres (or not acquired them). Conversely Venus is smaller than earth, but it also has a magnetic field and retains an atmosphere; indeed, it's closer to the conditions I've outlined above than to the current day conditions on earth.

If you're designing a world the important factor would be whether the solar wind can overwhelm the planet's magnetosphere. If this happens, the atmosphere could be stripped away not by insolation, but rather "swept" away. Then the oceans will start to boil once the pressure drops to about 4kPa at sea level, stopping only when the sea surface freezes. Sublimation will continue after that, but much more slowly.

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  • $\begingroup$ The requirement for temperature to be 311 °C or above is a very good point. However, I don't think that the radiation increase would necessarily be linear: even today, the temperature of the upper atmosphere hasn't a clear relation to the temperatures below the inversion point, and there is a significant increase above the Karman limit [ scied.ucar.edu/learning-zone/atmosphere/… ]. Increase in the atmospheric radius is another very good point. At those heights, (ionized) air molecules would much more easily escape into space. $\endgroup$
    – LSerni
    Oct 27, 2023 at 19:50
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TL;DR: you either need implausibly high stellar fluxes, or geologically long periods of time, if you want to dessicate an Earthlike world.


As a very quick-and-dirty first approximation, you can pull out the simple planetary equilibrium temperature equation: $$T_{eq} = \left[ {F_s(1-A)} \over {4\sigma} \right]^{1 \over 4}$$ where $T_{eq}$ is the temperature of the planet's surface in Kelvin, $F_s$ is the solar flux at the orbital radius of your planet, in watts per square meter, $A$ is the planetary albedo, which is about .3 for Earth. Finally, $\sigma$ is the Stefan-Boltzmann constant.

If $T_{eq}$ is at least 374K (the boiling point of water), then it is going to start being awkward for liquid water to exist on the planet. You can re-arrange the equation to get $4\sigma(\pu{374K})^4\over.7$, or ~6340 W/m2, which is about 4.7x the solar irradiance that Earth currently receives at the top of the atmosphere.

But wait: there's a lot of water in the ocean... on Earth, there's about 2-and-a-bit orders of magnitude more water than there is air. As water is vaporised, surface pressure goes up, which raises the boiling point:

Phase diagram of water

Working out the actual equilibrium surface pressure seems too hard right now, but in order to ensure everything stays steamy you might need an equilibrium temperature of maybe 647K which requires an irradiance of more like 57000 W/m2... more than forty times what Earth gets. The resulting temperatures and pressures could result in supercritical water at the surface, but whether that counts as an ocean is entirely up to you.

That flux is an over estimate of course, on account of water vapor being a very potent greenhouse gas... you probably wouldn't need to increase irradiance that much before you end up liberating huge quantities of water vapor that would then drive a runaway greenhouse effect producing a pressure-cooker world with a hot, dense, thick atmosphere of water vapor, but it does at least give you a reasonable upper-bound of required stellar flux.

As you can see it is rather higher than you might plausibly get from the sort of star that's capable of forming planets. You might have to be content with either hot, liquid oceans, or with waiting for the UV flux at the top of the atmosphere to split water via photolysis causing loss of hydrogen to space and eventually causing the world to lose all its water.

This process will eventually dessicate the Earth, with only a 10-20% increase in luminosity of the Sun, and the passing of 1-3 billion years. The elevated solar fluxes I've suggested above would greatly increase temperatures and UV flux, which would significantly accelerate the process, but it is unlikely to ever be prompt.

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  • $\begingroup$ This addresses much the same logic as my answer, but with a lot more detail, thankyou! $\endgroup$ Nov 3, 2023 at 12:55
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The oceans don’t have to boil to be lost. Once the average surface temperature is hot enough — around 50°C for the Earth, well short of boiling point — evaporation leads to increased levels of water vapour in the upper atmosphere where it’s broken down to hydrogen and oxygen by solar UV radiation and the hydrogen escapes. Over sufficient time — and we’re talking hundreds of millions of years — this process will dry out the oceans. It may well have happened to Venus. https://en.wikipedia.org/wiki/Future_of_Earth#Loss_of_oceans

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  • $\begingroup$ I was under the impression that the earth is gaining hydrogen from the solar wind faster than it's losing it from ionic evaporation? $\endgroup$ Nov 3, 2023 at 12:52

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