Fictional Exoplanet creation

Thorsten

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I've long been interested in working out the how planets elsewhere in space might actually be. We can't know of course, but the more I've been thinking, the more I realized that we still can have a pretty good idea by applying science what can be and what can't.

So over the years I've been delving some in astrophysics, geology, meteorology, chemistry, biology, exobiology as well as studies of the worlds in out solar system (which are fascinating enough I have to say) to get an idea of what the laws governing what happens actually are. As well as developed a simulation code that does at least some orbital dynamics, thermodynamics and such like.

Now I've been talking to a fellow SciFi author who was interested in an Earth-sized world tidally locked into an orbit around an M-class red dwarf star - we came up with questions of whether Earth plants would grow there at all, how weather and climate would be like, how the light would actually look like... So I started a concept study for such a world, which I decided to call 'Stormhold' (mainly for the super-cyclone at the sub-solar point).

Basically it's a nice exercise in applied science - question is - anyone here interested in more details? I wouldn't mind at all documenting the journey, tossing more ideas around how things could work out etc. - I just don't want to bore everyone.

So, as appetizer, here's a first view of Stormhold from space, with the light my best attempt to re-create the 3700 K surface temperature of the star (as you can see, while astronomers call it 'red' we wouldn't actually perceive the light as such...) With the most prominent feature the super-cyclone at the subsolar point.


stormhold_3d_color.jpg
 

N_Molson

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As you say this is a fascinating topic, and you end creating your own worlds, which is quite cool ! Now that Orbiter is Open Source, I guess nothing stands in the way of a more complex, dynamic climate model, other than free time and computing power. (y)
 

MaxBuzz

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unfortunately, the search and study of exoplanets is now likely to cause pessimism 95% of all exoplanets are discovered by indirect methods (Doppler spectroscopy, transit photometry...) all methods just show that there is something , everything else is fantastic

the forum offered to make a generator of planetary systems (with the ability to create your own solar system) would be very cool
 

jedidia

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the forum offered to make a generator of planetary systems (with the ability to create your own solar system) would be very cool
There's enough of those to go around nowadays, I don't think we need to brew our own vintage.
In any case, exoplanet speculation is one of the most interesting parts of science fiction writing, so feel free to go right ahead Thorsten! :)
 

Linguofreak

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I don't think you would get a super-cyclone: the subsolar point would be on the equator, and winds travelling along the equator would not be deflected by Coriolis forces. You would have a weather system generated by the heating at the subsolar point, but it would be more complex complex than just a cyclone centered on that point.
 

Thorsten

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I don't think you would get a super-cyclone: the subsolar point would be on the equator, and winds travelling along the equator would not be deflected by Coriolis forces.

It's a common misconception that you'd need much Coriolis forces to get a cyclone - conservation of angular momentum will easily suffice to spin it up (as you can easily verify by pulling the plug on a bathtub - Coriolis force across a few cm is negligible, and yet the water spins merrily) and the energy to keep it spinning comes from the 'hot core' in terms of hot, water-saturated air that condenses and rains down.

I do agree that the weather system would be - significantly - more complex than a cyclone centered there - but I'd ask to defer the discussion of that after I got to present my reasoning why I believe the weather system would be as I worked it out.

I guess nothing stands in the way of a more complex, dynamic climate model, other than free time and computing power.

I will say at this point that I plan to release my code (just as others I've created) under the GPL license - right now it is not in a shape that I would feel comfortable bothering anyone with it and the documentation is non-existent. But if anyone wants it before I whip it into shape - I don't plan on keeping it, so anyone is welcome to have it.

***

So, let's start with the basic parameters. Said fellow SciFi author was envisioning a story where colonists enroute to somewhere else strand on a planet and have to make do under difficult circumstances - the world should be tidally locked, but it should be marginally possible to grow Earth plants there. What constraints does that actually pose?

As for the tidal locking the rotation period, we have the approximate locking time formula

09666f7b38e7f0d4145398b3caf4118e4fa87a3b


which shows that the semimajor axis a is the main driving force whereas planetary radius R or planet and star masses m_p and m_s are corrections. So the planet needs to be close to the star for a to become small enough to yield tidal locking within the system lifetime - that automatically means a lower luminosity star (since stellar luminosity is tightly linked to stellar mass, it also means a less massive star).

Inserting some numbers, I could plausibly have a tidally locked earth-sized planet for a distance of up to 0.7 AU and a star of up to 0.8 solar masses (note that the outcome isn't necessarily tidal locking - Mercury would be locked according to the formula, but is in reality in a 2:3 resonance).

The other end of the puzzle was more tricky - what spectral distribution of light do plants actually need to grow? Can they grow under pure red light? It turns out there's actually research from indoor gardening which shows that plants don't grow well under 100% red LED light, but with a 10% admixture of blue light they grow well.

Looking at the Planck formula to get the amount of red vs. blue light from a blackbody radiator at given temperature then establishes that for 3700 K surface temperature the amount of red to blue light is about 9:1, but for a cold M-class star the ratio drops to 16:1.

So from these constraints, we can have anything between a hot M-class star and most of the K-class stars.

One might think that the requirement to have liquid water is another major constraint, but at this point it really is not. Dependent on the albedo of the planet as well as the amount of greenhouse effect, the distance at which one gets reasonable temperatures can really vary a lot (to give an example - it would be quite possible to have instead of a hot hellhouse of Venus in the same orbit a frozen ice planet - all that's required is to assume that the albedo is very high because the planet is basically white, and that it's cold enough that the CO2 in the atmosphere is frozen out - it would then stay there, never create a runaway greenhouse effect that thaws the ice, never bring water vapour in the atmosphere and never create the hot Venus we all know and love).

So, since I wanted to emphasize differences to Earth, I decided to go to the lower limit of star mass and make my star - Mime (after the dwarf smith) - an M-class star with 3700 K surface temperature (which explains the color of the light in the above picture). The resulting planet is well within the tidal locking region, so I decided that it has somewhat more internal heating and volcanism than earth due to the energy it receives that way (note that tidally locked moons such as Io in the solar system tend to be rather active volcanically) and since I don't like perfect alignments, I also decided there are 'seasons' caused by an orbital eccentricity and there's a small axis tilt with respect to the orbit.

I'll close this for the day and then proceed next time with the basic thermodynamics for average surface temperatures and more on the habitability zone
 

Linguofreak

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It's a common misconception that you'd need much Coriolis forces to get a cyclone - conservation of angular momentum will easily suffice to spin it up (as you can easily verify by pulling the plug on a bathtub - Coriolis force across a few cm is negligible, and yet the water spins merrily) and the energy to keep it spinning comes from the 'hot core' in terms of hot, water-saturated air that condenses and rains down.

I suppose you might get a cyclone in the long-period orbit / low Coriolis case, but I guess my point is that, particularly in the short period case, general wind patterns are likely to have reflection symmetry across the equator, with straight line winds along the equator. I'm not sure where the constraint for growing Earth plants puts you as far as long-period / short period, it will certainly impose a minimum period, but how long that will be I'm not certain of.

I do agree that the weather system would be - significantly - more complex than a cyclone centered there - but I'd ask to defer the discussion of that after I got to present my reasoning why I believe the weather system would be as I worked it out.

I'd certainly like to hear that.
 

Thorsten

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I suppose you might get a cyclone in the long-period orbit / low Coriolis case, but I guess my point is that, particularly in the short period case, general wind patterns are likely to have reflection symmetry across the equator, with straight line winds along the equator.

Even in the shortest period cases, Coriolis forces are at best 1/20 of what they are on Earth (under Stormhold assumtions ~1/50 of Earth) - that's not a significant deflection of any airmass, so to good approximation the symmetry is rotational around the subsolar point, and the equator is just one line of many which cross it.
 

N_Molson

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In technical terms, translated in Orbiter language, the most difficult thing is to get a texture and heightmap generator, in which you can input some parameters. As said above, it has been done before and isn't that crazy. @Artlav even managed to get his own procedural generator working.

I'd say that other things such as atmospheric parameters are the fun part. Well, it would probably be a good thing to have one or several layers of "climate maps" too, that would tell Orbiter where clouds or winds are likely to be, and in which shape/direction. It is the future, but a foreseeable future.
 

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In technical terms, translated in Orbiter language, the most difficult thing is to get a texture and heightmap generator,

Well, that I haven't really been interested in doing automatically. Geology is complicated, one needs to work out what processes form the land - was it formed by ice or volcanism? Vegetation is complicated, one needs to work out how the weather and climate is in different places - do plants grow or not? A texture and a heightmap follows from having an idea of all these things.

***

Okay, let's proceed with the science.

The next question is working out where a habitable orbit is around the chosen star. For that, one starts with the radiative flux of the star (which we can get from the Stefan-Boltzmann Law given the surface temperature and the stellar radius) and computes how much of that is caught by the planet. Now, part of that is reflected, so the result depends on the albedo.

But Stormhold is a bit more tricky, because the irradiation is strongly asymmetric because of the tidal locking, so an average temperature isn't going to help much, we need to see lines of equal temperature to see where a habitable ring might be.

For that purpose, there is numerics. :cool:The code divides up the surface in a collection of elements, then puts the planet into the chosen orbit around the planet, tilts it with the axial tilt, moves it and rotates it in discrete timesteps and computes for each position how much radiation reaches each of the surface elements. By doing this along the whole orbit, we can then see the energy distribution on the ground position-differentially.


On Earth, this comes out pretty much independent on longitude (the small variations actually are albedo variations between landmasses and ocean) and one nicely sees the variation with latitude - high latitude receives little energy:

earth_energy_deposition.png

Mercury has the funny 3:2 rotation resonance as well as a sizeable eccentricity, so doing the exercise for Mercury shows two bumps in the energy deposition - some longitudes receive more energy than others

mercury_energy_deposition.png

Finally, for Stormhold one sees that really only one hemisphere receives any energy, the rest remains pretty cold - as well as a reduction at high latitudes:

energy_deposition.png

So from this one has to go to temperatures.

This is actually rather tricky. One can compute instantaneous radiative equilibrium temperatures relatively easily by postulating that the incoming radiation is equal to the outgoing radiation flux, so each surface element heats to the temperature to fulfill that condition. However, that's not a particularly good model to track daily variations, since it implies that temperatures drop to zero Kelvin as soon as the sun is down and there is no radiation flux incoming.

To make it more real, one needs to introduce a measure of thermal inertia - the ground that receives the radiation has a certain heat capacity, so it stores heat while it is warmed and releases it when the sun is down.

At this point the calculation becomes more dependent on assumptions, because - what exactly is the thermally active part? Clearly we can't heat hundreds of meters of rock strata during an Earth day - but we can expect to heat more during a Mercury orbit - which has a much longer periodicity. It also means that the simulation now takes some time to reach equilibrium, so it has to run 'dry' for an orbit or two before equilibrium establishes itself and values can be read off.

But at least that's the general idea how to get temperatures - let the incoming radiation heat elements which have a heat capacity, use that reservoir to determine temperatures when there is no incoming radiation.

My idea to calibrate the whole operation was to use Mercury and Earth as test cases, where Mercury basically fixes the thermal inertia model and Earth needs to fix thermal inertia and heat transport through atmosphere and water.

I'll describe some of that next.:coffee:
 

Thorsten

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As mentioned before, the thermal inertia is a somewhat important ingredient. Now, it isn't hard to find out the heat capacity of rock - but the main question is how deep into the ground rock (water,...) participates in the thermal evolution.

Since fundamentally heat is transported inside the ground with a diffusion equation, it is at least clear that the size scale has to go like the square root of the timescale (because that is a general characteristic of diffusion equations). So the problem boils down to fixing a size scale for one Earth day, and then using the scaling with time to apply the value to other timescales.

Of course, for the Stormhold project, there is yet another complication, because the 'day' (the sidereal rotation) isn't even the relevant timescale for thermal evolution - because the day merely compensates the orbital rotation. Instead, the 'year'ly cariation, i.e. the motion from apoapsis to periapsis, is the relevant scale.

Anyway, using a basic size scale of 30 cm rock influenced by thermal variations during a day, scaling this with a Mercury day and appliying it to Mercury in its orbit gives these temperature fields at periapsis

mercury_T_periapsis.jpg


and at apoapsis

mercury_T_apoapsis.jpg


which is halfway realistic.

Applying the same values to Earth on the other hand gives too harsh contrasts between high and low latitudes. The reason is of course the atmosphere - it not only does Greenhouse effect, but also transport.

Generally, atmospheres are complicated beasts, because while they influence the thermal balance, the thermal balance also influences the atmosphere. For instance, at Earth gravity and temperature one can't have a hydrogen atmosphere component - it would boil off into space, thermal speeds of molecules are frequently above escape velocity. So the atmosphere must be heavy enough to be bound at the given temperature.

Then there's the Greenhouse effect (we'll cover that in more detail at some point) - heavily driven by water vapour in the atmosphere. But if the atmosphere gets warmer, it can hold more water, so the Greenhose effect increases, so it can hold yet more water... There's equilibrium points that are earth-like, where the overall role of the Greenhouse effect on thermal balance is limited - but there's also the Venus-like equilibrium where the process continues until all water is in the atmosphere which is heated to unbearable temperatures.

There's snow and albedo - the colder the air is, the more snow stays on the ground. Snow has a high albedo, so it cools the planet, so more snow remains, so the albedo increases yet further. Again this has an Earth-like equilibrium where the seasonal variations are enough to undo it, and a runaway scenario called 'snowball earth' - that seems to have happened more than once in the past (it really depends on which latitudes the continents drift...)

The code generally doesn't check whether the solution is self-consistent, that is something the user has to do, so usually with an exoplanet it's down to iterating the conditions a few times (for Earth, we of course know orbital parameters and the amount of Greenhouse effect and so on...)

Heat transport on Earth is pretty complicated, it involves convection cells (Hadley cells) of limited extent because Coriolis force prohibits large-distance migration of airmasses, sea currents dependent both on underwater topology and Coriolis forces... It's... pretty hard to even halfway simulate that in any meaningful detail without a supercomputing facility.

So the transport code does something rather simple - it does transport on average. If there's a temperature differential between two cells, it permits heat transport between them proportional to the difference and a selectable transport coefficient. So we don't have detailed transport, but we do have transport 'of the magnitude of Earth' or 'a tenth of Earth' to get the right ballpark.

By selecting both thermal inertia and atmosphere transport, one can achieve plausible average daily temperature profiles on Earth at different locations and seasons.

So... using an earth-sized albedo, a somewhat smaller than earth-size Greenhouse effect, earth-sized transport with air and sea currents and then iterating a bit with the details of the orbital parameters gives these solutions for the star and the planet:



Mime
------------
Mass [m_sun]: 0.4
Surface T [K]: 3700
Luminosity [L_sun]: 0.0279554
Radius [Mkm]: 0.283751
Mean density [g/cm^3]: 8.3118
Spectral fraction IR 0.818121
Spectral fraction vis 0.170524
Spectral fraction UV 0.0113552

Stormhold
------
Mass [m_earth]: 1.05
Radius [R_earth]: 1.01
Mean density [g/cm^3]: 5.61902
Surface gravity [g]: 1.03034
Semimajor axis [Mkm]: 28.4236
Eccentricity 0.1
Periapsis [Mkm]: 25.5812
Apoapsis [Mkm]: 31.266
Period [days]: 47.8297
Rotation period [d]: 234648
Sid. rot. period [d]: 47.82
Inclination [deg] 3

Thermal properties
------------------
Max. irrad. [W/m^2]: 1307.44
Min. irrad. [W/m^2]: 875.229
Albedo: 0.3
T_rad periapsis [K]: 280.776
T_rad apoapsis [K]: 254.005


and these temperature profiles across the surface at periapsis

temp_summer.png

and apoapsis

temp_winter,gif.png

so there is quite some temperature difference left between day and night side, but there clearly is a ring of habitable temperatures that is much larger than in the atmosphere-less case.
 

Linguofreak

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One thing to be aware of: a 1:1 spin-orbit resonance is likely to damp out eccentricity, and significant eccentricities are likely to cause capture into resonances like Mercury's 3:2 instead of tidal locking. There's also the issue of significant tidal heating for non-circular orbits, though I'm not sure where that becomes an issue as you go towards closer habitable zones and higher tidal forces.
 

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One thing to be aware of: a 1:1 spin-orbit resonance is likely to damp out eccentricity,
But at what timescale? There's much more energy stored in orbital motion than in rotation, so it must take longer to dampen that.

There's also the issue of significant tidal heating for non-circular orbits, though I'm not sure where that becomes an issue as you go towards closer habitable zones and higher tidal forces.

The calculation at least assumes a significantly higher internal heat flux (combined with more volcanism) than Earth, so yes, I agree there'd be tidal heating - though not on the level of Io.
 

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So, how does one go about and estimate the magnitude of the Greenhouse effect from an atmosphere composition, how do the two tie together?

The basics of the Greenhouse effect are

  • energy is radiated to the ground primarily in optical wavelength to which the atmosphere is transparent
  • the ground heats - but since it is cooler than the sun, it radiates primarily in IR wavelengths
  • to IR however the atmosphere is not transparent, so part of the IR radiation gets captured and heats the atmosphere
  • the heated atmosphere again radiates IR - partially back to the ground, partially into space
  • the part that is radiated back to the ground ends up heating the planet

To do this in detail is... pretty complicated, and even with a few simplifications it remains non-trivial. If the Greenhouse effect is a small correction, i.e. mostly the radiation goes through the atmosphere (as on Earth), one can for instance get by computing for 'the atmosphere' (and not do transport between different layers in the atmosphere as one would have to do for Venus).

The interaction of IR radiation with a gas requires that the molecule has excited states in the right energy range. That rules out diatomic gases (O2, N2, H2,...) - they don't really interact with IR photons.

A gas with the right states has a complicated lineshape that determines the interaction strength as a function of wavelength - here's CO2 (note this is a log-plot):

co2hitran2004micronslog.jpg


Strength varies over six orders of magnitude, and the detailed structure is... again pretty darn complicated, so we use a simplified parametrization of the data to keep it halfway tractable (the original HITRAN data has half a million individual lines for CO2 alone).

So basically interaction strength times number of molecules along the path gives the percentage of absorbed photons, when that gets large one has to exponentiate to account for the fact that a photon which already has been absorbed can't be absorbed again later, and out comes a pattern of wavelength-dependent transmission through the atmosphere.

This is CO2 absorption (1 means complete absorption) for the CO2 density in Earth atmosphere:

absorption_atmosphere_co2.jpg


This is what it would be if all of Earth's atmosphere would be pure CO2:

absorption_atmosphere_purec02.jpg


(one can see in the latter case how increasing the number of molecules dramatically makes minor lines in the spectrum suddenly more important while the major lines broaden)

The idea is then to take a blackbody radiation spectrum of warm Earth, pass it through this filter function (needs to be done for other Greenhouse gases - such as water vapour or methane just as well) and compare the energy in the spectrum before and after applying the filter function - the difference is what is absorbed by the atmosphere, and within the simple approximation half of that goes back to Earth.

That of course changes the temperature at which Earth radiates, so one re-computes the blackbody spectrum and iterates the calculation to get a self-consistent result in the end.

The exercise is rather illuminating, in that it explains for instance why methane is important as Greenhouse gas on Earth - it's not that methane would be a huge absorber - far from it - the reason is that methane has lines just in a window where both CO2 and H2O allow full transmission - and having methane closes that window. It's also not that CO2 is the main Greenhouse gas - water is. The role of CO2 is to warm a bit so that the atmosphere can hold more water - and then water keeps warming quite a bit more. In a way, CO2 and methane really regulate the water vapour content of the atmosphere, and that does the real absorption.

Anyway - using this (simplified) computation allows to play with Stormhold's atmosphere composition and see what gas combination would give what amount of Greenhouse effect - which of course feeds back into the temperature evolution simulation.

Lots of this really is iterating scenarios, running and re-running the code to identify self-consistent parts which fit together well enough, one can't simply start with a set of parameters, run it and expect a reasonable outcome, usually that doesn't happen.

Anyway, in the end here's what I came up with - on average the atmosphere is pretty dry (half of the planet is at permanent 200 K) and due to the frequent rains the volcanic CO2 gets washed out fairly quickly, so the content is rather low.

Code:
Atmospheric composition
-----------------------
     by volume     by mass
O2  : 24.99   %     27.50   %
CO2 : 0.009999 %     0.01513 %
N2  : 73.94   %     71.20   %
Ar  : 0.8299  %     1.142   %
H2O : 0.2294  %     0.1420  %
CH4 : 0.0001700 %     9.353e-05 %
Atmospheric IR retention: 0.3331

Atmosphere
----------
Surface pressure [mbar]: 1215.9
Scale height [m]:        6788.79
Column mass [kg]:        12029.5
Dry lapse rate [K/km]:   9.97911
Surface temperature [K]: 267.391
Emissive temp.      [K]: 240.09
 

Linguofreak

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I've heard the greenhouse effect explained in terms of the "photosphere" for outgoing radiation being at a certain altitude. Insolation (adjusted for albedo) determines the power that has to be radiated at the "photosphere", which fixes the temperature of the photosphere. Then adiabatic heating determines the temperature at all other altitudes (and thus at the surface).

"Photosphere" is the first term that came to mind for me. I guess you could also say "cloud tops", if you define clouds in terms of opacity at thermal, rather than visual, wavelengths.
 

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I've heard the greenhouse effect explained in terms of the "photosphere" for outgoing radiation being at a certain altitude. Insolation (adjusted for albedo) determines the power that has to be radiated at the "photosphere", which fixes the temperature of the photosphere. Then adiabatic heating determines the temperature at all other altitudes (and thus at the surface).

Yes - that's what I'm trying to say here

one can for instance get by computing for 'the atmosphere' (and not do transport between different layers in the atmosphere as one would have to do for Venus).

- it's more complicated than that if you have high absorption, then you don't have a single effective photosphere but many - each with its own window - and for that you need a radiation transport code.

But... you still need to go through lineshapes, because the photosphere picture doesn't talk about what percentage of radiation is absorbed in the first place.
 

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Yes - that's what I'm trying to say here



- it's more complicated than that if you have high absorption, then you don't have a single effective photosphere but many - each with its own window - and for that you need a radiation transport code.

But your description sounds like you're calculating multiple "bounces" for wavelengths where the atmosphere is optically thick. Would it be an appropriate simplification to do the following?

1) Calculate, for each wavelength, the altitude at which the optical depth equals a certain value (I believe 2/3 is used for determining the position of the photosphere in stellar astrophysics).
2) Assume that that wavelength is emitted only from that altitude, because emission at that wavelength from lower altitudes doesn't escape, so we keep it as part of atmospheric heat and don't bother calculating it, and emission at higher altitudes is suppressed by the fact that, to my understanding, materials don't emit efficiently at wavelengths at which they are transparent, so we don't calculate emission at higher altitudes than our chosen optical depth.
3) Assume that all energy emitted downwards at a given wavelength from its characteristic altitude remains part of the thermal energy of the atmosphere (i.e, we don't even bother calculating downward emission), while all energy emitted upwards escapes.
4) For wavelengths with a non-zero optical depth at the surface that is less than our cutoff, calculate blackbody emission at the surface, and then simply calculate the fraction emitted that escapes. That which doesn't simply remains part of atmospheric heat.
 

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Would it be an appropriate simplification to do the following?

I don't know, and therefore I can't really recommend it.

I have to say in advance that I don't have a full transport code myself (although I know how it should be written and could do it, given sufficient time and motivation). But my current code can't do Venus.

I guess in essence you're saying that once the atmosphere is optically thick, the problem simplifies (and becomes a diffusion equation). This is in a sense true, but it doesn't work wavelength by wavelength as your argument seems to assume.

Once you absorb a at a given wavelength, the energy ends up heating the local patch of atmosphere and contributes to a broad distribution of radiation across many wavelengths - part of which is absorbed, part of which escapes through the non-absorbing windows.

So it doesn't ever boil down to a diffusion equation, it boils down to a diffusion equation with loss terms at given windows where there is no absorption - but the radiation in the window is re-populated wherever there is any absorption at different wavelength, There is constant mixing of wavelengths.

Which conceptually makes the radiation from a 'photosphere' questionable, because you do get radiation from higher layers if they absorb at different wavelength - because again the atmosphere there radiates in a broad distribution.

In addition, the distinction between whether an atmosphere does diffusion or convection might be marginal - so a little 'better than diffusion' transport might mean no convection.

So what you're saying hinges on approximations which are not obvious up-front - it might work in some cases, it might not in others, but I see no compelling argument that it must 'usually' work - which is why I would not really try to solve the problem that way. You may get lucky and end up close to reality, but then again you may not. That wavelength mixing might really be a bummer. I don't know either way.
 

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Which conceptually makes the radiation from a 'photosphere' questionable, because you do get radiation from higher layers if they absorb at different wavelength - because again the atmosphere there radiates in a broad distribution.

My understanding is that emissivity equals absorbtivity, so layers at a low optical depth will have low emissivity for the wavelength concerned. Higher layers will only have high emissivity for wavelengths at which the optical depth is already high at that layer, so radiation downwards from those layers will tend to be absorbed quickly, except for exactly those wavelengths in which they emit minimal radiation.

In other words, at an altitude where the mean free path of a photon of a given wavelength is long, the radiation flux in that wavelength can be expected to be heavily biased in the upward direction.
 

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My understanding is that emissivity equals absorbtivity,

In the total energy balance yes, but not per wavelength. What you say is true for line emission, but that's not the mechanism we're dealing with here - we have line absorption but broad blackbody emission.
 
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